-
COMPUTATIONAL INTELLIGENCE
IN
MULTISCALE AND BIOMEDICAL
ENGINEERING
TADEUSZ BURCZYŃSKI Institute of Fundamental Technological
Research
Polish Academy of Sciences (IPPT PAN)
and
Cracow University of Technology
JUBILEE SCIENTIFIC CONFERENCE „PRACTICAL APPLICATIONS OF
INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”
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Intelligence and Interdependence between macro and micro
http://hunch.net/~yan/solid.mechanics.html
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Contents
• Intelligent computing
• Multiscale inverse problems
• Computational Intelligence Systems (CIS)
• Optimal Design on the micro-macro levels
• Identification problems on the micro-macro levels
• Smart design materials in nano-scale
• Concluding remarks
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Three important areas of intelligent computing methods,
namely:
• Evolutionary Computing based on Evolutionary Algorithms
(EA)
• Immune Computing based on Artificial Immune Systems (AIS)
• Swarm Computing based on Particle Swarm Optimizers (PSO)
are presented as intelligent computing (Artificial Intelligence
- AI) methods.
Criteria of AI: • Turing test,
• Intelligent actions:
- heuristics,
- learning,
• Rational perpetration.
COMPUTATIONAL INTELLIGENCE
INTELIGENT COMPUTING METHODS
-
Common features of intelligent
bio-inspired methods
• Formulation based on population (set of problems in each
iteration).
• Operators simulate some biological or natural processes.
• Stochastic approach.
• The great probability of finding global solutions (possibility
of closing to
the global optimum also when the starting population is in local
optimas
basins).
• Impact of the best solutions on next iterations, even the
worst solution can
have impact.
• Time consuming but there is possibility to speed up by
parallel computing
and grid environment.
-
Intelligent optimization methods inspired by biological/natural
mechanisms – soft computing
Obje
ctive
fun
ctio
n v
alu
e pathogens
Obje
ctive
fun
ctio
n v
alu
e Individuals
Evolutionary
algorithms (EA) Artificial immune
systems (AIS)
The goal of AIS
find the most dangerous pathogen
i.e. the global optimum
of objective function
The goal of EA
find the fittest chromosom
i.e. the global optimum
of objective function
Obje
ctive
fun
ctio
n v
alu
e Locations
The goal of PSO
find the best location
i.e. the global optimum
of objective function
Particle swarm
optimizers (PSO)
-
Evolutionary algorithm (EA)
Distributed EA
Sequential EA
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Artificial Immune System (AIS)
Parameters of AIS:
• the number of memory cells
• the number of the clones
• crowding factor
• Gaussian mutation
B-cell with antibodies
T-cell (non self protein recognition)
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Particle Swarm Optimization (PSO)
Parameters of PSO:
• number of the particles,
• number of design variables,
• inertia weight,
• two acceleration coefficients,
• two random numbers with uniform distribution,
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Parallel Bioinspired Algorithm
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Hybrid Bioinspired Algorithm
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The number of subpopulations
The number of chrom.
Simple crossover
Gaussian mutation
1 20 100% 100%
2 10 100% 100%
Comparison for he mathematical function
46
The number of memory
cells
The number of the clones
Crowding factor
Gaussian mutation
2 4 0.45 40%
The Rastrigin function
21
( ) 10 10cos 2n
i i
i
F x n x x
5.12 5.12ix
for n=2
min 0,0, ,0 0.0F x F
The stop condition: F(x)
< 0.1
The optimal parameters of AIS
The optimal parameters of EA
The optimal parameters of PSO
Number of particles
Interia weight w
Acceleration coefficient c1
Acceleration coefficient c2
74 1 1.9 1.9
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Multiscale approach in engineering problems
Nano
-
Multiscale Modelling
10 -9 10 -6 10 -3 10 0
Length, m
10 -15
10 -12
10 -9
10 -6
10 -3
10 0
10 3 T
ime,
s
Atomistic
Dislocations
Substructures
Grain/Phase
Macro-Interface
FEM/BEM
Celular
Automata
Dislocation
Dynamics
Molecular Dynamics
Tight
Binding
Ab-Initio Physical
Chemical
Biological Mechanical
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Inverse Problems in
Multiscale Modelling
B. Inverse problems: Optimization
Identification
Optimization: minimization of a given objective function in
macro
scale with respect to design variables in micro scale of the
structure
Identification: evaluation of some geometrical or material
parameters
of the structures in micro scale having measured information
in
macro scale.
-
CIS Computational Intelligent System
Soft computing Hard computing
FEM (Finite Element Method)
BEM (Boundary Element Method)
MM (Meshless Methods)
MD (Molecular Dynamics)
Bio-inspired
Methods
AI
Ansys
Nastran
Marc
Mentat
Lammps
In-house software
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Computational Intelligent System - interfaces
EA
AIS
PSO
Evolutionary Computing
Immune Computing
Swarm Computing
Multiobjective Computing
Computational Intelligent System
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Optimization
Problems of
Multiscale
Modelling
Macro-Micro
Nano
-
Numerical homogenization
Numerical homogenization by using RVE
(Representative Volume Element)
-
Numerical homogenization - requirements
• Separation of scales
• Averaging theorem
• Hill’s condition (the equality of the averaged micro-scale
energy density and the macro-scale energy density at the selected
point of macro-structure corresponding to the RVE)
l and L are characteristic lengths of body in
macro/micro scales.
average macroscopic value
volume of RVE element
stress nad strain tensors
temperature gradient and heat fluxes
periodic boundary conditions
1l
L
1
RVE
RVE
RVE
d
RVE
ij ij ij ij
, ,i i i iT q T q
ij ij
,i iT q
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Numerical homogenization
• Hook’s law
• Fourier’s law
'
ij ijkl ijc
'
,i ij iq k T
• Tensor of effective elastic constants
• Tensor of effective thermal constants
11 12 13
21 22 23
31 32 33'
44
55
66
0 0 0
0 0 0
0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
ij
c c c
c c c
c c cc
c
c
c
11
'
22
33
0 0
0 0
0 0
ij
k
k k
k
-
Numerical homogenization
avg. - average
Macro-stresses
Homogenization
Macro-strains
Localization
BVP
BVP – Boundary Value Problem
Macro
Micro
RVE
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Optimal design on macro-micro scales
min oDV
J
1 2, ,..., ,...i n
Ch
DV B cell x x x x
P
Constraints: min max
0, 1,2,..
, ,1,2,..
( , , ), 0,1,2,...
i i i
J m
x x x i n
J J u m
xi – design variables, play the role of geometrical,
material
or topologcal parameters in the micro scale
where
DV=design vector
J0 – objective function described
in the macro scale
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Meso scale:
Grains Micro scale: Single grain
Nano scale:
Molecular/atomic
level
Macro scale:
Structure
Illustration of optimization in multiscale approach
0 0( , , )J J u 1 2, ,..., ,...i nDV x x x x
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Design variables
RVE
Material parameters
Shape parameters
Topology parameters
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Evolutionary/immune/swarm optimization in multiscale
in macro scale in micro scale
RVE
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DEA parameters: 2 subpopulations
20 chromosomes in each
Rank selection
Gasuss mutation
Simple crossover
g7,
g8
The best solution
in the 1st generation
The best solution
in the last generation
0 0 maxmin ,Ch
J where J u 1 2 3 4 5 6 7 8, , , , , , ,Ch g g g g g g g g
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Optimization of Functionally Graded Materials in Multsicale
Modelling
The function or composition changes gradually in the
material
http://www.unl.edu/emhome/faculty/bobaru/project_shape_optim.htm
FGM in nature – clam shell
Bamboo
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Functionally Graded Materials
The function or composition changes gradually in the
material
Metal-ceramic FGMs
http://sbir.nasa.gov/SBIR/successes/ss/3-079text.html
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Optimization of FGM parameters
macromodel Micromodel - RVE
Minimization of inclusions total volume
-
z
1
dA
Z
n
z A
f h
maxiu u
6 design parameters - diameters di
Minimization of inclusions total volume
Constraint on maximum displacement value
-
Displacements map for the best solution (umax=4)
Minimization of inclusions total volume
the resuts 1 - 0.187501 2 - 0.137236 3 - 0.123124 4 - 0.104760 5
- 0.143142 6 - 0.101725
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FGM material for tooth implant
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The simplified model of implant-bone systen with FGM material –
optimization of porosity
Minimization of porosity p1 (mat1) and p2 (mat2)
Constraints on max eqivalent stress value in the bone area are
imposed
Box constraints on prosity [0.0; 0.4]
igl
voidsi
ch
V
Vp
ppch
ppf
],[
min
21
21F
F
-
Macromodel
FEM MSC.Nastran
Micromodel
FMBEM model
RVE
Optimization of functionally graded materials in multiscale
modelling
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Distribution of equivalent stresses in the optimal design
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Multicriteria Optimal Design of porous microstructures
1min d
u
def
ux
f u
Optimization functionals for termomechanical problems
• minimization of displacement on selected part of the
boundary
2min d
q
def
qx
f q
• minimization of heat flux on selected part of the boundary
3max d
q
def
qx
f q
• maximization of heat flux on selected part of the boundary
4
d
maxd
por
RVE
pordef
xRVE
f
• maximization of the porosity of the microstructure
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Numerical example
Boundary conditions
P0(total)=100N
T0=0°C
T1=100°C
Constraints (5 design variables)
Z1 – [0.53 – 0.92] Z2 – [0.09 – 0.45] Z3 – [0.09 – 0.45] Z4 –
[0.08 – 0.47] Z5 – [0.09 – 0.45]
Macromodel
of aluminium plate 50x50x1 under thermomechanical loadings
RVE model of microstructure with void modeled using NURBS
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Variants of multicriteria optimal design of materials
Variant 1
Variant 2
1min d
u
def
ux
f u
minimization of displacement on selected part of the
boundary
2min d
q
def
qx
f q
minimization of heat flux on selected part of the boundary
3max d
q
def
qx
f q
• maximization of heat flux on selected part of the boundary
4
d
maxd
por
RVE
pordef
xRVE
f
• maximization of the porosity of the microstructure
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Results of multicriteria optimization (variant 1)
1 d
u
def
uf u
2 d
q
def
qf q
-
Results of multicriteria optimization (variant 2)
3 d
q
def
qf q
4
d
d
por
RVE
pordef
RVE
f
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Identification: macro-micro
Goal – find the FEM microscale model parameters on the base of
experimental measurements in macro
-
Real structure
FEM model
What material properties for FEM gives the same results in
sensor points as in real structure ?
-
0
0
1 1
min
ˆˆ
DV
m m
i i i i
j j
J
where
J a u u b
IDENTIFICATION
1 2, ,..., ,...i nDV x x x x
xi – design variables, play the role of material or
geometrical
parameters in the micro scale
min max , ,1,2,..i i ix x x i n
ˆ ,
ˆ
i i
i i
u and u computed and measured displacements
and computed and measured strains
-
http://www.ucc.ie/bluehist/CorePages/Bone/Bone.htm
Identification of material parameters of a bone tissue
The femur bone is build from trabecular and compact bone. The
identification of material properties of single trabeculae.
K.Tsubota, T. Adachi, S. Nishiumi , Y. Tomita, ATEM'03,
JSME-MMD, 2003
G. M. Kurtzman, 2006
femur trabecular bone RVE
single trabeculae
-
Identification can be performed in two stages: I) Identification
of anisotropic homogenized material properties of RVE on the basis
of measurements for femur II) Identification of isotropic material
properties of trabeculae on the basis of homogenized RVE
anisotropic material properties
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Problem formulation for II stage (RVE)
Design parameters:
chi=[Young modulus E, Poisson ratio ] material properties of
single trabeculae
chi=[g1, g2]
The objective function: where: - RVE homogenized material
properties from macromodel - computed homogenized RVE material
properites n - number of coefficients (9) The homogenized
anisotropic material properties for RVE: a[i] ={E11 E22 E33 E12 E13
E23 E44 E55 E66}
The constraints on design parameters values:
1
ˆmin ( )
n
i i
i
F a a
chˆia
ia
min maxi i ig g g
11 12 13
22 23
33
44
55
66
0 0 0
0 0 0
0 0 0
0 0
. 0
x x
x x
z z
xy xy
yz yz
zx zx
E E E
E E
E
E
sym E
E
-
The FEM model for RVE created on the basis of microCT. The bone
sample was taken form the femur.
~70,000 DOF
-
The fitness function value for the best chromosome in
subpopulations
iteration
fitness
Actual Found
E [MPa] 3300.0 3305.5
0.330 0.329
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Creation of new graphene-like
materials by means of the hybrid
parallel evolutionary algorithm
Nano level optimization of graphene allotropes
by means of a hybrid parallel evolutionary algorithm
Journal Computational Materials Science (in press)
by A.Mrozek, W. Kuś, T. Burczyński
-
Carbon allotropes
-diamond
- graphite/graphene
-nanotubes/nanorings
etc.
- fullerenes
- amorphous state
-
Graphene-like 2D materials / hybridization of carbon atoms
acetylenic linkages,
nanowires benzene rings, base of the graphite/
graphene honeycomb lattice
-
Optimal searching for the new atomic structures:
• proper interaction model
• optimization’s algorithm
• stable configurations of atoms correspond to the minima
on the potential energy surface
-
Bond Order (BO) potentials for molecular dynamics simulations of
carbon/hydrocarbons
• LCBOP (I + II) Long range Carbon Bond Order Potential • REBO
(Reactive Empirical Bond Order)
• AIREBO (Adaptive Intermolecular REBO) a variant of the REBO
with additional torsion and (Lennard-Jones-like) long-range terms*
• ReaxFF (Reactive Force Fields) with equilibration of atomic
charge All of them can handle various hybridization states of
carbon atoms
*used in this work S.J. Stuart, A.B. Tutein, J.A. Harrison, A
reactive potential for hydrocarbons with intermolecular
interactions, The Journal of Chemical Physics, 112(14), 2000, pp.
6472–6486
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Evolutionary optimization of atomic structure - minimization of
the potential energy
• Fitness function - the total potential energy of the
considered atomic cluster (sum
over all atomic interactions)
• Design variables/genes: the real-valued Cartesian
Coordinates of each atom in the considered cluster
• Constrains: all atoms can move freely in the triclinic
or rectangular unit cell with imposed periodic boundary
conditions
• Neighborhood-dependent behavior of carbon atoms (i.e.
hybridization’s states,
bond’s lengths and angles) is handled by AIREBO potential and
conjugated
gradient-based molecular statics solver
• Periodicity of the lattice is guaranteed by the molecular
static solver
, , ,
REBO LJ TORSION
ij ij kijl
i j i k i j l i j k
FF E E E
-
START - generation of initial
population
(creation of randomly-generated
positions of atoms)
minimization of potential energy using
molecular statics/gradient method
fitness function evaluation
selection
modification of genes
using evolutionary operators
Stop condition
?
Proposed hybrid evolutionary-molecular computational system
END
N Y
- LAMMPS
- E.A.
-
Parallel hybrid gradient/evolutionary algorithm (small, 2D
problems)
Multiple instances of LAMMPS
- LAMMPS
- E.A.
-
Parallel hybrid gradient/evolutionary algorithm (large, 3D
problems)
-
Proposed hybrid evolutionary/gradient algorithm
• modular structure (each part can be replaced with appropriate
equivalent,
e.g.
- AIREBO -> ReaxFF (Reactive Force Fields)
- EA -> AIS etc.
• ready for 3D optimization (and not only carbon atoms…)
-
Validation & Results obtained using prototype version of the
algorithm
-
Dimensions:12x10 Å triclinic unit cell, 25 atoms
4 threads
100 individuals
124800 FF evaluations
10% mutation & crossover
0 200 400 600 800 1000 1200 1400 1600-156
-154
-152
-150
-148
-146
-144
-142
-140
-138
Example of the progress of optimization
-
Supergraphene, as presented in: Enyashin A.N., Ivanovskii A.L.,
Graphene allotropes, Physica Status Solidi, 248, 8, 2011, pp.
1879-1883
bond’s lengths computed using classical MD and AIREBO potential:
Mrozek A., Burczynski T.,
Examination of mechanical properties of graphene allotropes by
means of computer simulation,
CAMES, 20, 4, 2013, pp. 309-323.
-
Supergraphene found by g-optim algorithm:
(in 34th generation)
Triclinic unit cell: 10x6Å, 8 atoms
(finally relaxed to 10.65x6.08Å)
0 10 20 30 40 50 60 70 80 90 100-50.5
-50
-49.5
-49
-48.5
-48
-47.5
-47
-46.5
-46
-45.5
1.32Å
1.38Å
potential energy (eV) vs. generation
-
Graphyne, as presented in: Enyashin A.N., Ivanovskii A.L.,
Graphene allotropes, Physica Status Solidi, 248, 8, 2011, pp.
1879-1883
bond’s lengths computed using classical MD and AIREBO potential:
Mrozek A., Burczynski T.,
Examination of mechanical properties of graphene allotropes by
means of computer simulation,
CAMES, 20, 4, 2013, pp. 309-323
-
Graphyne found by g-optim algorithm:
(in 23th generation)
Triclinic unit cell: c.a. 10x6Å, 12 atoms
(finally relaxed to 10.2x5.9Å)
0 10 20 30 40 50 60-81
-80
-79
-78
-77
-76
-75
-74
-73
-72
potential energy (eV) vs. generation
-
Structure found by g-optim algorithm:
(in 223 generation)
Orthogonal unit cell: 4x7Å, 8 atoms
2 threads
100 individuals
22300 FF evaluations
10% mutation & crossover
0 50 100 150 200 250 300 350 400 450 500-52.2
-52.1
-52
-51.9
-51.8
-51.7
potential energy (eV) vs. generation
1. Example of the „new” graphene-like materials X
-
-5.94eV
-6.31eV
-8.01eV
-6.31eV
-5.94eV
-8.01eV
-5.94eV
a) b)
Unit cell close-up X
-
2. Example of the „new” graphene-like materials Y
Structure found by g-optim algorithm:
(in 55th generation)
Orthogonal unit cell: 6x4Å, 8 atoms
2 threads
100 individuals
5500 FF evaluations
10% mutation & crossover
0 20 40 60 80 100 120 140 160 180 200-54.5
-54
-53.5
-53
-52.5
-52
-51.5
-51
-50.5
-50
potential energy (eV) vs. generation
-
-6.81eV
-6.87eV
-6.81eV
-6.59eV
-6.59eV
a) b)
Unit cell close-up Y
-
equilibrate the investigated „nanospecimen”
at the desired temperature
apply a certain, finite deformation the structure (dε, dγ)
equilibrate the structure
compute all the necessary
time-averaged values (displacements/deformations,
components of microstress tensor)
Examination of the mechanical properties (tensile / shear tests
at the finite temperature)
1 1
2
N N
i i i ij ij
i j i
m
σ v v r f
-
Tensile test – strain(%) - stress(N/m) curve
Young moduli (0-5%) vertical axis: 176 N/m horizontal axis:
183,4 N/m
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
5
10
15
20
25
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
5
10
15
20
25
30
Mechanical Properties of X
-
Tensile test – strain(%) - stress(N/m) curve
Young moduli (0-5%) horizontal axis: 226 N/m vertical axis: 280
N/m
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
5
10
15
20
25
0 0.05 0.1 0.15 0.2 0.250
5
10
15
20
25
30
35
Mechanical Properties of Y
-
Concluding remarks
• Two-scale macro-micro materials design needs special
analytical and
computational techniques and tools.
• Coupled soft and hard computing techniques based on
Computational
Intelligent System (CIS) ensure the great probability of finding
global
solutions. CIS has the great flexibility.
• Effective CIS is based on the parallel computing and grid
environment.
• Optimal material and geometrical parameters on the micro-scale
ensure the
extremum for an objective function in the macro-scale.
• Using CSI it is possible to create material on the nano-scale
……
-
Concluding remarks cont.
• Proposed algorithm gives possibility of finding new flat
carbon networks with unique properties
• Newly created structures are „physically” correct: form proper
basic elements (benzene rings,
triads, acetylenic groups etc.), without alone atoms or
unconnected branches etc.
• The AIREBO potential seems to be reasonable choice for
modeling presented flat carbon
structures (except long-range interactions), where
time-consuming ab-inito methods are not
suitable
• Proposed optimization algorithm is easy to parallelize, since
the most time-consuming step is
molecular statics and FF evaluation
• It is possible to create a new (even 3D) material with
predefined density and
properties using this methodology.
-
^Institute of Computer Science, Cracow University of Technology,
Poland
Thank you for your attention!