ASME V&V 40.4 Code Verification Example Problem Marc Horner Ismail Guler ANSYS, Inc. Boston Scientific Corporation (On behalf of ASME V&V 40.4 Verification Working Group) May 21, 2020 ASME Verification & Validation 2020 Virtual Symposium ASME V&V 40 1
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ASME V&V 40.4 Code VerificationExample Problem
Marc Horner Ismail GulerANSYS, Inc. Boston Scientific Corporation
(On behalf of ASME V&V 40.4 Verification Working Group)
May 21, 2020ASME Verification & Validation 2020 Virtual Symposium
ASME V&V 40
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ASME V&V 40.4 Verification Best Practices Working Group
Verification Best Practices
Code Verification Best Practices Calculation Verification Best Practices
1) Poiseuille flow example problem (complete)2) Womersley flow example problem (active)
1) Nitinol stent FEA example problem (active)2) Hip stem FEA example problem (active)3) Knee tibial tray FEA example problem (active)
Goal: The fundamentals of code and calculation verification are thoroughly reviewed in the ASME V&V 10, 10.1, and 20 standards. However, the computational models used in the evaluation of medical devices can be quite complex and have their own subtleties. The objective of the ASME V&V 40.4 working group is to explore, learn, and employ code and calculation verification best practices on representative examples from the medical device space.
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“Four ups” of VVUQ (adopted from “integral theory” of Ken Wilber)
Waking up ASME V&V 40 (How much V&V?)Being conscious of COU of model, model risk(model influence, decision consequence), etc.
Growing up ASME V&V 10 & 20 (How to do V&V?)Being well versed in VVUQ methodologies(not just textbook knowledge, but experiential knowledge)
Cleaning up Shadow work: allergies and addictionsallergies → dislike/resistance to rigorous VVUQ because it
puts burden on us (convenience bias), etc.addictions→ urge to follow old way of doing things, etc.
(following legacy standard procedures in VVUQ)
Showing up Progress made in “three ups” listed above allows one to become a better citizen of the modeling world and engagefully in that world.
VVUQ is a process for realizing the reality (credibility) of computational models.
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Christopher Basciano1, Jeff Bodner2, Constantine Butakoff3, Carlos Corrales4, Chris Delametter5, Swapnil Dindorkar6, Jun Ding7, Beatriz Eguzkitza8, Alex Francois-Saint-Cyr5, Mark Goodin9, Sharath Gopal10,
• Time-dependent fully coupled solver• Implicit “generalized-alpha” method for time discretization
• Steps taken by solver → fixed (manually specified)• Newton’s method for nonlinear iterations• Relative tolerance of 1E-9 (default = 1E-3) for iterative convergence• Direct solver (PARDISO) for linear system of equations
15
Convergence tolerance for nonlinear iterations in COMSOL
default relative tolerance = 10-3
relative tolerance used in this study = 10-9
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Expected order of accuracy for spatial discretization (FEM) in COMSOL
• First order (linear) elements were used for both the velocity components and pressure (Q1+Q1).
• Ciarlet P. G., The Finite Element Method for Elliptic Problems, SIAM 2002 (first published in 1978)• Second order (p = k + 1 = 2) accuracy is expected for the error in L2 norm for the second order
elliptic boundary value problems when linear (k = 1) finite elements are used.• COMSOL Multiphysics 5.1 Reference Manual
• “The accuracy of SUPG can be shown to be at least O(h^(k+1/2)) where k >= 1 is the order of the basis functions.”o Johnson C., Numerical Solution of Partial Differential Equations by the Finite Element
Method, Dover 2009 (first published in 1987)
expected order of accuracy for spatial discretization 1.5
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Expected order of accuracy for temporal discretization (generalized-alpha) in COMSOL
Implicit “generalized-alpha” method implementation in COMSOL → second order accurate
Jansen K. E., Whiting C. H., and Hulbert G. M., A generalized-alpha method for integrating the filtered Navier-Stokes equations with a stabilized finite element method, Computer Methods in Applied Mechanics and Engineering 2000; 190: 305-319.
expected order of accuracy for temporal discretization → second order
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Discretization levels and Courant number
Courant number based on max. centerline velocity:
CL,max
max
w tC
x
=
Spatial Temporal CourantDiscretization Discretization Number
(mm) (ms)0.25 25 1.251
0.125 25 2.5020.0625 25 5.003
0.5 25 0.6250.5 12.5 0.3130.5 6.25 0.156
Spatial Temporal CourantDiscretization Discretization Number
(mm) (ms)0.25 6.25 0.313
0.125 6.25 0.6250.0625 6.25 1.251
0.25 3.125 0.1560.25 1.5625 0.078
Study # 1 Study # 2
CL,max 0.012508 m/sw =
Spatial Temporal CourantDiscretization Discretization Number
Time history of discretization error (spatial refinement)
Discretization error for axial component of velocity (cm/s) at centerline (x=2cm & r=0cm)
( , )t
x
h exact
h x tf h h f = −
6.25 msth t= =6.25 msth t= =
(Study # 2) (Study # 2)
24
Time history of discretization error (spatial refinement)
Discretization error for axial component of velocity (cm/s) at centerline (x=2cm & r=0cm)
25 msth t= = 6.25 msth t= =
( , )t
x
h exact
h x tf h h f = −
(Study # 1) (Study # 2)
25
Time history of discretization error (temporal refinement)
Discretization error for axial component of velocity (cm/s) at centerline (x=2cm & r=0cm)
62.5 mxh x = =0.25 mmxh x= =
( , )t
x
h exact
h x tf h h f = −
(Study # 2) (Study # 3)
26
Time history of discretization error (temporal refinement)
Discretization error for axial component of velocity (cm/s) at centerline (x=2cm & r=0cm)
( , )t
x
h exact
h x tf h h f = −
62.5 mxh x = =0.25 mmxh x= =
(Study # 2) (Study # 3)
27
Time history of discretization error (temporal refinement) – zoom-in
Discretization error for axial component of velocity (cm/s) at centerline (x=2cm & r=0cm)
0.25 mmxh x= =0.25 mmxh x= =
error decreases with temporal refinement (“apparent convergence”)
error decreases with temporal refinement (“apparent convergence”)
( , )t
x
h exact
h x tf h h f = −
(Study # 2) (Study # 2)
28
DE for axial component of velocity (cm/s) at centerline (t = 10.00 s)
21.8743 & 6.04 10xp g −= = 71.9071 & 7.99 10tq g −= =
spatial refinement temporal refinement
t
x
h p
h x xg h − = t
x
h q
h t tg h − =
0.25 mmxh x= =
6.25 msth t= =
t
x
h p q
h x x t tg h g h
= +(Study # 2)
29
DE for axial component of velocity (cm/s) at centerline (t = 10.25 s)
22.4824 & 1.44 10xp g −= = 72.0124 & 5.93 10tq g −= =
spatial refinement temporal refinement
0.25 mmxh x= =
6.25 msth t= =
t
x
h p
h x xg h − = t
x
h q
h t tg h − =
t
x
h p q
h x x t tg h g h
= +(Study # 2)
30
DE for axial component of velocity (cm/s) at centerline (t = 10.50 s)
21.8743 & 6.04 10xp g −= = 71.9088 & 7.95 10tq g −= =
spatial refinement temporal refinement
0.25 mmxh x= =
6.25 msth t= =
t
x
h p
h x xg h − = t
x
h q
h t tg h − =
t
x
h p q
h x x t tg h g h
= +(Study # 2)
31
DE for axial component of velocity (cm/s) at centerline (t = 10.75 s)
22.4824 & 1.44 10xp g −= = 72.0108 & 5.95 10tq g −= =
spatial refinement temporal refinement
0.25 mmxh x= =
6.25 msth t= =
t
x
h p
h x xg h − = t
x
h q
h t tg h − =
t
x
h p q
h x x t tg h g h
= +(Study # 2)
32
Time history of discretization error (spatial refinement)
L2-norm of discretization error for axial component of velocity (cm/s) at x=2cm1/2
2
2
1( ) where ( , )t t t
x x x
h h h exact
h h h x td f h h fA
= = −
6.25 msth t= =6.25 msth t= =
(Study # 2) (Study # 2)
33
Time history of discretization error (temporal refinement)
L2-norm of discretization error for axial component of velocity (cm/s) at x=2cm1/2
2
2
1( ) where ( , )t t t
x x x
h h h exact
h h h x td f h h fA
= = −
0.25 mmxh x= = 62.5 mxh x = =
(Study # 2) (Study # 3)
34
Time history of discretization error (temporal refinement)
L2-norm of discretization error for axial component of velocity (cm/s) at x=2cm1/2
2
2
1( ) where ( , )t t t
x x x
h h h exact
h h h x td f h h fA
= = −
0.25 mmxh x= = 62.5 mxh x = =
(Study # 2) (Study # 3)
35
Time history of discretization error (temporal refinement) – zoom-in
L2-norm of discretization error for axial component of velocity (cm/s) at x=2cm1/2
2
2
1( ) where ( , )t t t
x x x
h h h exact
h h h x td f h h fA
= = −
error increases with temporal refinement (“apparent divergence”)
0.25 mmxh x= =0.25 mmxh x= =
The term “apparent divergence” was coined by Luis Eca during a discussion of the results on November 20, 2019 at the ASME V&V committee meeting in San Antonio, TX.
error increases with temporal refinement (“apparent divergence”)
(Study # 2) (Study # 2)
36
Time history of discretization error (temporal refinement)
L2-norm of discretization error for axial component of velocity (cm/s) at x=2cm1/2
2
2
1( ) where ( , )t t t
x x x
h h h exact
h h h x td f h h fA
= = −
error increases with temporal refinement (“apparent divergence”)
2
t
x
h p q
h x x t tg h g h
= +
2if 0 ( 0), but 0 ( 0) is approaching from belowt
x
h
x t hg g →2
as t
x
h
h th
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L2-norm of DE for axial component of velocity (cm/s) at x=2cm (t = 10.00 s)
21.9941 & 6.46 10xp g −= =
spatial refinement temporal refinement
2
t
x
h p
h x xg h − =2
t
x
h q
h t tg h − =
0.25 mmxh x= =
6.25 msth t= =
72.2.0415 & 1 16 0tq g −= − =
2
t
x
h p q
h x x t tg h g h
= +(Study # 2)
38
77.1.9542 & 1 73 0tq g −= − =
L2-norm of DE for axial component of velocity (cm/s) at x=2cm (t = 10.25 s)
22.0384 & 3.54 10xp g −= =
spatial refinement temporal refinement
2
t
x
h p
h x xg h − =2
t
x
h q
h t tg h − =
0.25 mmxh x= =
6.25 msth t= =
2
t
x
h p q
h x x t tg h g h
= +(Study # 2)
39
L2-norm of DE for axial component of velocity (cm/s) at x=2cm (t = 10.50 s)
21.9941 & 6.45 10xp g −= =
spatial refinement temporal refinement
2
t
x
h p
h x xg h − =2
t
x
h q
h t tg h − =
0.25 mmxh x= =
6.25 msth t= =
72.2.0415 & 1 16 0tq g −= − =
2
t
x
h p q
h x x t tg h g h
= +(Study # 2)
40
77.1.9542 & 1 73 0tq g −= − =
L2-norm of DE for axial component of velocity (cm/s) at x=2cm (t = 10.75 s)
22.0384 & 3.54 10xp g −= =
spatial refinement temporal refinement
2
t
x
h p
h x xg h − =2
t
x
h q
h t tg h − =
0.25 mmxh x= =
6.25 msth t= =
2
t
x
h p q
h x x t tg h g h
= +(Study # 2)
41
Time history of discretization error (spatial refinement)
Discretization error for flow rate (mL/s) at x=2cm
6.25 msth t= =6.25 msth t= =
( , )t
x
h exact
h x tf h h f = −
(Study # 2) (Study # 2)
42
Time history of discretization error (spatial refinement)
Discretization error for flow rate (mL/s) at x=2cm
25 msth t= = 6.25 msth t= =
( , )t
x
h exact
h x tf h h f = −
(Study # 1) (Study # 2)
43
Time history of discretization error (temporal refinement)
Discretization error for flow rate (mL/s) at x=2cm
0.25 mmxh x= =
( , )t
x
h exact
h x tf h h f = −
62.5 mxh x = =
(Study # 2) (Study # 3)
44
Time history of discretization error (temporal refinement)
Discretization error for flow rate (mL/s) at x=2cm
0.25 mmxh x= =
( , )t
x
h exact
h x tf h h f = −
62.5 mxh x = =
(Study # 2) (Study # 3)
45
Discretization error for flow rate (mL/s) at x=2cm (t = 10.00 s)
22.0017 & 1.12 10xp g −= =
spatial refinement temporal refinement
t
x
h p
h x xg h − = t
x
h q
h t tg h − =
0.25 mmxh x= =
6.25 msth t= =
71.1308 & 1.63 10tq g −= =
t
x
h p q
h x x t tg h g h
= +(Study # 2)
46
Discretization error for flow rate (mL/s) at x=2cm (t = 10.25 s)
32.4025 & 9.91 10xp g −= =
spatial refinement temporal refinement
0.25 mmxh x= =
6.25 msth t= =
78.2.0021 & 1 14 0tq g −= − =
t
x
h p q
h x x t tg h g h
= +
t
x
h p
h x xg h − = t
x
h q
h t tg h − =
(Study # 2)
47
Discretization error for flow rate (mL/s) at x=2cm (t = 10.50 s)
22.0017 & 1.12 10xp g −= =
spatial refinement temporal refinement
0.25 mmxh x= =
6.25 msth t= =
71.1328 & 1.61 10tq g −= =
t
x
h p q
h x x t tg h g h
= +
t
x
h p
h x xg h − = t
x
h q
h t tg h − =
(Study # 2)
48
Discretization error for flow rate (mL/s) at x=2cm (t = 10.75 s)