Accelerated Numerical Simulation of Bloodflow in Aneurysms Using Lattice Boltzmann Methods and Multigrid Sarntal 2005 18.09.2005 Jan Götz
Accelerated Numerical Simulation of Bloodflow in Aneurysms Using Lattice
Boltzmann Methods and Multigrid
Sarntal 2005
18092005
Jan Goumltz
Outline
1 What are aneurysms
2 Numerical Basics
3 Simulation
What are aneurysms
Definition amp Description Symptoms Causes amp Prevention Diagnostics Treatment
1 What are aneurysms
2 Numerical Basics
3 Simulation
Definition amp Description 1
Greek bdquoDilatationldquo
An aneurysm is a local dilatation (balooning) of a blood vessel
Localisation larger arteries in soft tissuendash brainndash aorta
1 near heart
2 abdominal
1 What are aneurysms
2 Numerical Basics
3 Simulation
Bernoulli-Principle
bdquoDecrease in velocity occurs simultaneously with increase in pressureldquo
Definition amp Description 2
High velocity low pressure
Low velocity high pressure
1 What are aneurysms
2 Numerical Basics
3 Simulation
Symptoms
often no symptoms felt by the patient pulsing sensation pain if aneurysm is pressing on
internal organs or nerves rupture causes sudden pain and
severe internal bleeding objective diagnosis from X-ray-
angiography or computer tomography
1 What are aneurysms
2 Numerical Basics
3 Simulation
Causes amp Prevention
80 are arteriosclerotic diseases rest vessel infection injuries or borne
in (eg Marfan syndrome)
healthy lifestyle can prevent most aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Diagnostics 1
X-Ray-Angiography (exact size 2D shape is used also during surgery)
MRI (for exact size 3D shape) CT (for exact size 3D shape) Ultrasound (low cost imprecise) physical examination
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example Angiography
Diagnostics 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example physical examination
Diagnostics 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Case study 12 year old girl with headaches and vision problems
Diagnostics 4
Angliogram Rotational angliogram in 3D
1 What are aneurysms
2 Numerical Basics
3 Simulation
Treatment 1
invasive intervention clipping bypass non-invasive intervention
ndash coils (GDC) ndash stents
conservative treatment with medication1 What are
aneurysms
2 Numerical Basics
3 Simulation mortality rates
preventive surgery 2-5
surgery after rupture 50
Example stents
Treatment 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Lattice Boltzmann Multigrid Simplifications
Numerical Basics
1 What are aneurysms
2 Numerical Basics
3 Simulation
bdquoYou only need to know where you are everything else is not importantldquo
1 microscopic Hamiltonlsquos-equations
2 mesoscopic Lattice-Boltzmann
3 macroscopic Navier-Stokes
Lattice Boltzmann 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
size
What is the Lattice Boltzmann method
1 can be imaged as a type of cellular automaton
2 divide simulation region into a Cartesian grid of squarecubic cells
3 each cell only interacts with its direct neighbourhood
4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities
Lattice Boltzmann 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Outline
1 What are aneurysms
2 Numerical Basics
3 Simulation
What are aneurysms
Definition amp Description Symptoms Causes amp Prevention Diagnostics Treatment
1 What are aneurysms
2 Numerical Basics
3 Simulation
Definition amp Description 1
Greek bdquoDilatationldquo
An aneurysm is a local dilatation (balooning) of a blood vessel
Localisation larger arteries in soft tissuendash brainndash aorta
1 near heart
2 abdominal
1 What are aneurysms
2 Numerical Basics
3 Simulation
Bernoulli-Principle
bdquoDecrease in velocity occurs simultaneously with increase in pressureldquo
Definition amp Description 2
High velocity low pressure
Low velocity high pressure
1 What are aneurysms
2 Numerical Basics
3 Simulation
Symptoms
often no symptoms felt by the patient pulsing sensation pain if aneurysm is pressing on
internal organs or nerves rupture causes sudden pain and
severe internal bleeding objective diagnosis from X-ray-
angiography or computer tomography
1 What are aneurysms
2 Numerical Basics
3 Simulation
Causes amp Prevention
80 are arteriosclerotic diseases rest vessel infection injuries or borne
in (eg Marfan syndrome)
healthy lifestyle can prevent most aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Diagnostics 1
X-Ray-Angiography (exact size 2D shape is used also during surgery)
MRI (for exact size 3D shape) CT (for exact size 3D shape) Ultrasound (low cost imprecise) physical examination
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example Angiography
Diagnostics 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example physical examination
Diagnostics 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Case study 12 year old girl with headaches and vision problems
Diagnostics 4
Angliogram Rotational angliogram in 3D
1 What are aneurysms
2 Numerical Basics
3 Simulation
Treatment 1
invasive intervention clipping bypass non-invasive intervention
ndash coils (GDC) ndash stents
conservative treatment with medication1 What are
aneurysms
2 Numerical Basics
3 Simulation mortality rates
preventive surgery 2-5
surgery after rupture 50
Example stents
Treatment 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Lattice Boltzmann Multigrid Simplifications
Numerical Basics
1 What are aneurysms
2 Numerical Basics
3 Simulation
bdquoYou only need to know where you are everything else is not importantldquo
1 microscopic Hamiltonlsquos-equations
2 mesoscopic Lattice-Boltzmann
3 macroscopic Navier-Stokes
Lattice Boltzmann 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
size
What is the Lattice Boltzmann method
1 can be imaged as a type of cellular automaton
2 divide simulation region into a Cartesian grid of squarecubic cells
3 each cell only interacts with its direct neighbourhood
4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities
Lattice Boltzmann 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
What are aneurysms
Definition amp Description Symptoms Causes amp Prevention Diagnostics Treatment
1 What are aneurysms
2 Numerical Basics
3 Simulation
Definition amp Description 1
Greek bdquoDilatationldquo
An aneurysm is a local dilatation (balooning) of a blood vessel
Localisation larger arteries in soft tissuendash brainndash aorta
1 near heart
2 abdominal
1 What are aneurysms
2 Numerical Basics
3 Simulation
Bernoulli-Principle
bdquoDecrease in velocity occurs simultaneously with increase in pressureldquo
Definition amp Description 2
High velocity low pressure
Low velocity high pressure
1 What are aneurysms
2 Numerical Basics
3 Simulation
Symptoms
often no symptoms felt by the patient pulsing sensation pain if aneurysm is pressing on
internal organs or nerves rupture causes sudden pain and
severe internal bleeding objective diagnosis from X-ray-
angiography or computer tomography
1 What are aneurysms
2 Numerical Basics
3 Simulation
Causes amp Prevention
80 are arteriosclerotic diseases rest vessel infection injuries or borne
in (eg Marfan syndrome)
healthy lifestyle can prevent most aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Diagnostics 1
X-Ray-Angiography (exact size 2D shape is used also during surgery)
MRI (for exact size 3D shape) CT (for exact size 3D shape) Ultrasound (low cost imprecise) physical examination
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example Angiography
Diagnostics 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example physical examination
Diagnostics 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Case study 12 year old girl with headaches and vision problems
Diagnostics 4
Angliogram Rotational angliogram in 3D
1 What are aneurysms
2 Numerical Basics
3 Simulation
Treatment 1
invasive intervention clipping bypass non-invasive intervention
ndash coils (GDC) ndash stents
conservative treatment with medication1 What are
aneurysms
2 Numerical Basics
3 Simulation mortality rates
preventive surgery 2-5
surgery after rupture 50
Example stents
Treatment 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Lattice Boltzmann Multigrid Simplifications
Numerical Basics
1 What are aneurysms
2 Numerical Basics
3 Simulation
bdquoYou only need to know where you are everything else is not importantldquo
1 microscopic Hamiltonlsquos-equations
2 mesoscopic Lattice-Boltzmann
3 macroscopic Navier-Stokes
Lattice Boltzmann 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
size
What is the Lattice Boltzmann method
1 can be imaged as a type of cellular automaton
2 divide simulation region into a Cartesian grid of squarecubic cells
3 each cell only interacts with its direct neighbourhood
4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities
Lattice Boltzmann 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Definition amp Description 1
Greek bdquoDilatationldquo
An aneurysm is a local dilatation (balooning) of a blood vessel
Localisation larger arteries in soft tissuendash brainndash aorta
1 near heart
2 abdominal
1 What are aneurysms
2 Numerical Basics
3 Simulation
Bernoulli-Principle
bdquoDecrease in velocity occurs simultaneously with increase in pressureldquo
Definition amp Description 2
High velocity low pressure
Low velocity high pressure
1 What are aneurysms
2 Numerical Basics
3 Simulation
Symptoms
often no symptoms felt by the patient pulsing sensation pain if aneurysm is pressing on
internal organs or nerves rupture causes sudden pain and
severe internal bleeding objective diagnosis from X-ray-
angiography or computer tomography
1 What are aneurysms
2 Numerical Basics
3 Simulation
Causes amp Prevention
80 are arteriosclerotic diseases rest vessel infection injuries or borne
in (eg Marfan syndrome)
healthy lifestyle can prevent most aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Diagnostics 1
X-Ray-Angiography (exact size 2D shape is used also during surgery)
MRI (for exact size 3D shape) CT (for exact size 3D shape) Ultrasound (low cost imprecise) physical examination
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example Angiography
Diagnostics 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example physical examination
Diagnostics 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Case study 12 year old girl with headaches and vision problems
Diagnostics 4
Angliogram Rotational angliogram in 3D
1 What are aneurysms
2 Numerical Basics
3 Simulation
Treatment 1
invasive intervention clipping bypass non-invasive intervention
ndash coils (GDC) ndash stents
conservative treatment with medication1 What are
aneurysms
2 Numerical Basics
3 Simulation mortality rates
preventive surgery 2-5
surgery after rupture 50
Example stents
Treatment 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Lattice Boltzmann Multigrid Simplifications
Numerical Basics
1 What are aneurysms
2 Numerical Basics
3 Simulation
bdquoYou only need to know where you are everything else is not importantldquo
1 microscopic Hamiltonlsquos-equations
2 mesoscopic Lattice-Boltzmann
3 macroscopic Navier-Stokes
Lattice Boltzmann 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
size
What is the Lattice Boltzmann method
1 can be imaged as a type of cellular automaton
2 divide simulation region into a Cartesian grid of squarecubic cells
3 each cell only interacts with its direct neighbourhood
4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities
Lattice Boltzmann 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Bernoulli-Principle
bdquoDecrease in velocity occurs simultaneously with increase in pressureldquo
Definition amp Description 2
High velocity low pressure
Low velocity high pressure
1 What are aneurysms
2 Numerical Basics
3 Simulation
Symptoms
often no symptoms felt by the patient pulsing sensation pain if aneurysm is pressing on
internal organs or nerves rupture causes sudden pain and
severe internal bleeding objective diagnosis from X-ray-
angiography or computer tomography
1 What are aneurysms
2 Numerical Basics
3 Simulation
Causes amp Prevention
80 are arteriosclerotic diseases rest vessel infection injuries or borne
in (eg Marfan syndrome)
healthy lifestyle can prevent most aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Diagnostics 1
X-Ray-Angiography (exact size 2D shape is used also during surgery)
MRI (for exact size 3D shape) CT (for exact size 3D shape) Ultrasound (low cost imprecise) physical examination
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example Angiography
Diagnostics 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example physical examination
Diagnostics 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Case study 12 year old girl with headaches and vision problems
Diagnostics 4
Angliogram Rotational angliogram in 3D
1 What are aneurysms
2 Numerical Basics
3 Simulation
Treatment 1
invasive intervention clipping bypass non-invasive intervention
ndash coils (GDC) ndash stents
conservative treatment with medication1 What are
aneurysms
2 Numerical Basics
3 Simulation mortality rates
preventive surgery 2-5
surgery after rupture 50
Example stents
Treatment 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Lattice Boltzmann Multigrid Simplifications
Numerical Basics
1 What are aneurysms
2 Numerical Basics
3 Simulation
bdquoYou only need to know where you are everything else is not importantldquo
1 microscopic Hamiltonlsquos-equations
2 mesoscopic Lattice-Boltzmann
3 macroscopic Navier-Stokes
Lattice Boltzmann 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
size
What is the Lattice Boltzmann method
1 can be imaged as a type of cellular automaton
2 divide simulation region into a Cartesian grid of squarecubic cells
3 each cell only interacts with its direct neighbourhood
4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities
Lattice Boltzmann 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Symptoms
often no symptoms felt by the patient pulsing sensation pain if aneurysm is pressing on
internal organs or nerves rupture causes sudden pain and
severe internal bleeding objective diagnosis from X-ray-
angiography or computer tomography
1 What are aneurysms
2 Numerical Basics
3 Simulation
Causes amp Prevention
80 are arteriosclerotic diseases rest vessel infection injuries or borne
in (eg Marfan syndrome)
healthy lifestyle can prevent most aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Diagnostics 1
X-Ray-Angiography (exact size 2D shape is used also during surgery)
MRI (for exact size 3D shape) CT (for exact size 3D shape) Ultrasound (low cost imprecise) physical examination
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example Angiography
Diagnostics 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example physical examination
Diagnostics 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Case study 12 year old girl with headaches and vision problems
Diagnostics 4
Angliogram Rotational angliogram in 3D
1 What are aneurysms
2 Numerical Basics
3 Simulation
Treatment 1
invasive intervention clipping bypass non-invasive intervention
ndash coils (GDC) ndash stents
conservative treatment with medication1 What are
aneurysms
2 Numerical Basics
3 Simulation mortality rates
preventive surgery 2-5
surgery after rupture 50
Example stents
Treatment 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Lattice Boltzmann Multigrid Simplifications
Numerical Basics
1 What are aneurysms
2 Numerical Basics
3 Simulation
bdquoYou only need to know where you are everything else is not importantldquo
1 microscopic Hamiltonlsquos-equations
2 mesoscopic Lattice-Boltzmann
3 macroscopic Navier-Stokes
Lattice Boltzmann 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
size
What is the Lattice Boltzmann method
1 can be imaged as a type of cellular automaton
2 divide simulation region into a Cartesian grid of squarecubic cells
3 each cell only interacts with its direct neighbourhood
4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities
Lattice Boltzmann 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Causes amp Prevention
80 are arteriosclerotic diseases rest vessel infection injuries or borne
in (eg Marfan syndrome)
healthy lifestyle can prevent most aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Diagnostics 1
X-Ray-Angiography (exact size 2D shape is used also during surgery)
MRI (for exact size 3D shape) CT (for exact size 3D shape) Ultrasound (low cost imprecise) physical examination
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example Angiography
Diagnostics 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example physical examination
Diagnostics 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Case study 12 year old girl with headaches and vision problems
Diagnostics 4
Angliogram Rotational angliogram in 3D
1 What are aneurysms
2 Numerical Basics
3 Simulation
Treatment 1
invasive intervention clipping bypass non-invasive intervention
ndash coils (GDC) ndash stents
conservative treatment with medication1 What are
aneurysms
2 Numerical Basics
3 Simulation mortality rates
preventive surgery 2-5
surgery after rupture 50
Example stents
Treatment 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Lattice Boltzmann Multigrid Simplifications
Numerical Basics
1 What are aneurysms
2 Numerical Basics
3 Simulation
bdquoYou only need to know where you are everything else is not importantldquo
1 microscopic Hamiltonlsquos-equations
2 mesoscopic Lattice-Boltzmann
3 macroscopic Navier-Stokes
Lattice Boltzmann 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
size
What is the Lattice Boltzmann method
1 can be imaged as a type of cellular automaton
2 divide simulation region into a Cartesian grid of squarecubic cells
3 each cell only interacts with its direct neighbourhood
4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities
Lattice Boltzmann 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Diagnostics 1
X-Ray-Angiography (exact size 2D shape is used also during surgery)
MRI (for exact size 3D shape) CT (for exact size 3D shape) Ultrasound (low cost imprecise) physical examination
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example Angiography
Diagnostics 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example physical examination
Diagnostics 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Case study 12 year old girl with headaches and vision problems
Diagnostics 4
Angliogram Rotational angliogram in 3D
1 What are aneurysms
2 Numerical Basics
3 Simulation
Treatment 1
invasive intervention clipping bypass non-invasive intervention
ndash coils (GDC) ndash stents
conservative treatment with medication1 What are
aneurysms
2 Numerical Basics
3 Simulation mortality rates
preventive surgery 2-5
surgery after rupture 50
Example stents
Treatment 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Lattice Boltzmann Multigrid Simplifications
Numerical Basics
1 What are aneurysms
2 Numerical Basics
3 Simulation
bdquoYou only need to know where you are everything else is not importantldquo
1 microscopic Hamiltonlsquos-equations
2 mesoscopic Lattice-Boltzmann
3 macroscopic Navier-Stokes
Lattice Boltzmann 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
size
What is the Lattice Boltzmann method
1 can be imaged as a type of cellular automaton
2 divide simulation region into a Cartesian grid of squarecubic cells
3 each cell only interacts with its direct neighbourhood
4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities
Lattice Boltzmann 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Example Angiography
Diagnostics 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Example physical examination
Diagnostics 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Case study 12 year old girl with headaches and vision problems
Diagnostics 4
Angliogram Rotational angliogram in 3D
1 What are aneurysms
2 Numerical Basics
3 Simulation
Treatment 1
invasive intervention clipping bypass non-invasive intervention
ndash coils (GDC) ndash stents
conservative treatment with medication1 What are
aneurysms
2 Numerical Basics
3 Simulation mortality rates
preventive surgery 2-5
surgery after rupture 50
Example stents
Treatment 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Lattice Boltzmann Multigrid Simplifications
Numerical Basics
1 What are aneurysms
2 Numerical Basics
3 Simulation
bdquoYou only need to know where you are everything else is not importantldquo
1 microscopic Hamiltonlsquos-equations
2 mesoscopic Lattice-Boltzmann
3 macroscopic Navier-Stokes
Lattice Boltzmann 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
size
What is the Lattice Boltzmann method
1 can be imaged as a type of cellular automaton
2 divide simulation region into a Cartesian grid of squarecubic cells
3 each cell only interacts with its direct neighbourhood
4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities
Lattice Boltzmann 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Example physical examination
Diagnostics 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Case study 12 year old girl with headaches and vision problems
Diagnostics 4
Angliogram Rotational angliogram in 3D
1 What are aneurysms
2 Numerical Basics
3 Simulation
Treatment 1
invasive intervention clipping bypass non-invasive intervention
ndash coils (GDC) ndash stents
conservative treatment with medication1 What are
aneurysms
2 Numerical Basics
3 Simulation mortality rates
preventive surgery 2-5
surgery after rupture 50
Example stents
Treatment 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Lattice Boltzmann Multigrid Simplifications
Numerical Basics
1 What are aneurysms
2 Numerical Basics
3 Simulation
bdquoYou only need to know where you are everything else is not importantldquo
1 microscopic Hamiltonlsquos-equations
2 mesoscopic Lattice-Boltzmann
3 macroscopic Navier-Stokes
Lattice Boltzmann 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
size
What is the Lattice Boltzmann method
1 can be imaged as a type of cellular automaton
2 divide simulation region into a Cartesian grid of squarecubic cells
3 each cell only interacts with its direct neighbourhood
4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities
Lattice Boltzmann 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Case study 12 year old girl with headaches and vision problems
Diagnostics 4
Angliogram Rotational angliogram in 3D
1 What are aneurysms
2 Numerical Basics
3 Simulation
Treatment 1
invasive intervention clipping bypass non-invasive intervention
ndash coils (GDC) ndash stents
conservative treatment with medication1 What are
aneurysms
2 Numerical Basics
3 Simulation mortality rates
preventive surgery 2-5
surgery after rupture 50
Example stents
Treatment 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Lattice Boltzmann Multigrid Simplifications
Numerical Basics
1 What are aneurysms
2 Numerical Basics
3 Simulation
bdquoYou only need to know where you are everything else is not importantldquo
1 microscopic Hamiltonlsquos-equations
2 mesoscopic Lattice-Boltzmann
3 macroscopic Navier-Stokes
Lattice Boltzmann 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
size
What is the Lattice Boltzmann method
1 can be imaged as a type of cellular automaton
2 divide simulation region into a Cartesian grid of squarecubic cells
3 each cell only interacts with its direct neighbourhood
4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities
Lattice Boltzmann 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Treatment 1
invasive intervention clipping bypass non-invasive intervention
ndash coils (GDC) ndash stents
conservative treatment with medication1 What are
aneurysms
2 Numerical Basics
3 Simulation mortality rates
preventive surgery 2-5
surgery after rupture 50
Example stents
Treatment 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Lattice Boltzmann Multigrid Simplifications
Numerical Basics
1 What are aneurysms
2 Numerical Basics
3 Simulation
bdquoYou only need to know where you are everything else is not importantldquo
1 microscopic Hamiltonlsquos-equations
2 mesoscopic Lattice-Boltzmann
3 macroscopic Navier-Stokes
Lattice Boltzmann 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
size
What is the Lattice Boltzmann method
1 can be imaged as a type of cellular automaton
2 divide simulation region into a Cartesian grid of squarecubic cells
3 each cell only interacts with its direct neighbourhood
4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities
Lattice Boltzmann 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Example stents
Treatment 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
Lattice Boltzmann Multigrid Simplifications
Numerical Basics
1 What are aneurysms
2 Numerical Basics
3 Simulation
bdquoYou only need to know where you are everything else is not importantldquo
1 microscopic Hamiltonlsquos-equations
2 mesoscopic Lattice-Boltzmann
3 macroscopic Navier-Stokes
Lattice Boltzmann 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
size
What is the Lattice Boltzmann method
1 can be imaged as a type of cellular automaton
2 divide simulation region into a Cartesian grid of squarecubic cells
3 each cell only interacts with its direct neighbourhood
4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities
Lattice Boltzmann 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Lattice Boltzmann Multigrid Simplifications
Numerical Basics
1 What are aneurysms
2 Numerical Basics
3 Simulation
bdquoYou only need to know where you are everything else is not importantldquo
1 microscopic Hamiltonlsquos-equations
2 mesoscopic Lattice-Boltzmann
3 macroscopic Navier-Stokes
Lattice Boltzmann 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
size
What is the Lattice Boltzmann method
1 can be imaged as a type of cellular automaton
2 divide simulation region into a Cartesian grid of squarecubic cells
3 each cell only interacts with its direct neighbourhood
4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities
Lattice Boltzmann 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
bdquoYou only need to know where you are everything else is not importantldquo
1 microscopic Hamiltonlsquos-equations
2 mesoscopic Lattice-Boltzmann
3 macroscopic Navier-Stokes
Lattice Boltzmann 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
size
What is the Lattice Boltzmann method
1 can be imaged as a type of cellular automaton
2 divide simulation region into a Cartesian grid of squarecubic cells
3 each cell only interacts with its direct neighbourhood
4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities
Lattice Boltzmann 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
What is the Lattice Boltzmann method
1 can be imaged as a type of cellular automaton
2 divide simulation region into a Cartesian grid of squarecubic cells
3 each cell only interacts with its direct neighbourhood
4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities
Lattice Boltzmann 2
1 What are aneurysms
2 Numerical Basics
3 Simulation
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
D3Q19 model for 3 dimensions with 19 discrete velocity-directions
Lattice Boltzmann 3
1 What are aneurysms
2 Numerical Basics
3 Simulation
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Evolution equation to be computed
)(t
fff
)()(1
txftxftxf eqaaaa
ttxftxf
tttxf
aa
aa
)(
)(
The Boltzmann equation
Lattice Boltzmann 4
BGK
Discretisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
We can do this in two steps
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()( 1
21 iiaaiaina yxfheyhexf
collision
advectionstreaming
Lattice Boltzmann 5
1 What are aneurysms
2 Numerical Basics
3 Simulation
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
collision and streaming
Lattice Boltzmann 6
1 What are aneurysms
2 Numerical Basics
3 Simulation
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)
Lattice Boltzmann 7
1 What are aneurysms
2 Numerical Basics
3 Simulation
22
2
9
2
33 ueuuewf aaa
eqa
131 iforwi72181 iforwi198361 iforwi
a
af a
aa feu
density velocity
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Why no bdquonormalldquo multigrid
hhh fuL
hhhhhhh uLfuLuL ~~
hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case
rarr We need another approach for our nonlinear problem
Multigrid 1
1 What are aneurysms
2 Numerical Basics
3 Simulation
exact solution approx solution hu~
hu
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Options for nonlinear MG
1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised
2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method
Multigrid 2
for LB-application FAS is better
1 What are aneurysms
2 Numerical Basics
3 Simulation
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
equations
Multigrid 3
term on RHS is called defect correction
)(ˆhh
Hhh
HhHHH uRIuILuL
)(ˆhh
Hhh
HhHHH uRIuILuL
nhHhH
hH
nh
nh uIuIuu ˆ1
This operator is the direct injection
1 What are aneurysms
2 Numerical Basics
3 Simulation
correction of uh
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Blood is a suspension of
bull formed blood cells (red white etc)
bull some liquid particles
bull an aqueous solution (plasma)
Simplifications 1
At high shear rate (γlt100 sec-1) blood can be treated as Newtonian
We focus on large vessels rarr high shear rates
1 What are aneurysms
2 Numerical Basics
3 Simulation
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Fluid-structure interaction
Simplifications 2
To first approximation we neglect the effect of elastic walls
This is reasonable because in large arteries the effect is quite minor
Additionally we assume blood as homogenous and incompressible
1 What are aneurysms
2 Numerical Basics
3 Simulation
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Simulation
bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example
1 What are aneurysms
2 Numerical Basics
3 Simulation
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Goal of the Simulation 1
Recall
Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50
And
The number one cause of death in a developed nation is a heart- or vascular disease
1 What are aneurysms
2 Numerical Basics
3 Simulation
rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important
clinical applications need fast simulations
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Goal of the Simulation 2
use as initial condition
rarr faster convergence stability
1 What are aneurysms
2 Numerical Basics
3 Simulation
1 preliminary time-independent incompressible velocity-field
2 periodically forced time-dependant (pulsating) velocity-field
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Why Lattice Boltzmann
1 LBM results in an accurate reproduction of the Navier-Stokes-equations so why NOT
2 very complex geometries are readily handled
3 LBM is simple to implement and modify
4 changing the geometry during simulation is possible
5 calculate pressure and other stresses locally in time and space
6 very good parallelization vectorisation and cache-optimisation
1 What are aneurysms
2 Numerical Basics
3 Simulation
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
performance FAS vs linear
1 What are aneurysms
2 Numerical Basics
3 Simulation
Non-Linear LBE Time-Step
Linear LBE Jacobi Relaxation
computation of micro variables 18
9x9 matrix vector multiplications 146
collision 75 other terms36
advection 0 under-relaxation27
total 93 total 209
rarr We already have an existing non-linear LBE method so lets use it
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
The algorithm 1
)()(1
)()(ii
eqaii
naii
naiia yxfyxfyxfyxf
n
)()(
21 iiaaiaia yxfheyhexf
naHa
na fDff 11
collision
advection
relaxation
This equation is
new
DH is called the defect correction
1 What are aneurysms
2 Numerical Basics
3 Simulation
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
The algorithm 2
)(1
)(1
1)(
21
21
aiaieqa
aiaiaiiah
heyhexf
heyhexfyxffR
How to get the defect correction
hhHhh
HhHH fRIfIRD 2ˆ
1
2
1 What are aneurysms
2 Numerical Basics
3 Simulation
hhHhh
HhHH fRIfIRD ˆ
In a standard FAS the defect correction would be
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
The algorithm 3
Why do we have no bdquostandardldquo
1 What are aneurysms
2 Numerical Basics
3 Simulation
2
13
ch
In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05
And For smaller the convergence rate of the LBM-scheme degrades
rarr Trick Use a constant but rescale the correction generated by the coarse grid
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Example
Flowlines in a saccular aneurysm
1 What are aneurysms
2 Numerical Basics
3 Simulation
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde
Links amp useful stuff
bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick
bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis
ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner
httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI
Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical
company) httpwwwprosper-hospitalde
Download
This presentation can be downloaded from
wwwsarntalerlangen-rocktde