A preliminary study on some a posteriori error estimates for soft-tissue biomechanics
S. P. A. Bordas1, M. Bucki2, F. Chouly3,M. Duprez4, V. Lleras5, C. Lobos6, A. Lozinski3, P.-Y. Rohan7, S. Tomar1
1Université du Luxembourg, Luxembourg.
2Texisense, Grenoble, France.
3Université Bourgogne Franche-Comté, France.
4Université Aix-Marseille, France.
5Université de Montpelier, France.
6Universidad Técnica Federico Santa Marià, Chile.
7Arts & Métiers ParisTech, France.
Euromech Colloquium 595
29-31 August 2017, Lille
Introduction
Patient-specific finite element models of soft-tissues
Design of personalized medical devices Computer-assisted planning or guidance
Bucki, Lobos, Payan & Hitschfeld 2011 www.Texisense.com 2017c
Michel Duprez A posteriorierror estimates for soft-tissue biomechanics 1
Introduction
Meshing is still a big challenge
Complex geometries
Geometric quality of the elements ? Impact on the accuracy of the results ?
ådiscretization error www.Texisense.com 2017c
Ito Y, Shih A & Soni 2009 Zhang, Hughes
& Bajaj 2010
Ebeida, Patney, Owens & Mestreau 2011 Lobos & Gonzalez 2015 Frey & Georges 2008
Michel Duprez A posteriorierror estimates for soft-tissue biomechanics 2
Introduction
A posteriori error estimate : Optimal / Economical mesh Finite Elements
Solution
New mesh Error Estimate
in each element Solve
Adapt
Hansbo & Johnson 1992 Ainsworth & Oden 1997 Verfürth 1999
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Method
Simplified setting for a preliminary study
2D linear elasticity with (possibly) linearized fiber activation :
minu∈V
R
Ωσ(u) :ε(u)
where σ(u) = σP(u)
| {z } passive part
+ σA
|{z}
fiber activation
with
σA=βT eA⊗eA
eA: fiber direction T : tension
β: activation parameter
Cowin & Humphrey 2001 Payan & Ohayon 2017
Biomechanics of Living Organs : Hyperelastic Constitutive Laws for Finite Element Modeling.
Michel Duprez A posteriorierror estimates for soft-tissue biomechanics 4
Method
Goal oriented error estimates
Variational setting (for linear problems) :u∈Vs.t.
a(u,v) =l(v) ∀v∈V
Here
a(u,v) = Z
Ω
σP(u) :ε(u) and l(v) = Z
Ω
σA:ε(u)
Linearquantity of interest:
J:V3u7→J(u)∈R.
Aim :|J(u)−J(uh)|?
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Method
Dual Weighted Residuals (DWR) error estimates
Fundamental observation :
J(u)−J(uh) =a(u,z)−a(uh,z) =l(z)−a(uh,z) =:ηh(z).
withz∈V s.t. a(v,z) =J(v), ∀v∈V.
åglobal estimator:ηh
Moreover :
J(u)−J(uh) = P
K∈Kh
hRK,z−πhziK+hR∂K,z−πhzi∂K
=: P
K∈Kh
ηK(z−πhz) Error by element
whereπhzis an interpolant andRKandR∂Kare the residual contribution over the cellKand his boundary∂K.
ålocal estimators:ηK
Becker & Rannacher 1997, 2001
Rognes & Logg 2013(Generic implementation in FENICS)
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1st Example : Artery with fibre activation
Le Floc’h et al 2008
Region of interestω(green)
Displacement Activation
Quantity of interest :J(u) =1 2 Z
ω
trε(u)dx∈R
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1st Example : Artery with fibre activation
103 103.5 104
10−6 10−5 10−4
N
J2(u)−J2(uh)
Uniform refinement Adaptive refinement 1/N2 1/N
Initial error estimatorηK
Initial mesh Adapted mesh (5th iteration)
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2nd Example : Tongue with fibre activation
Bijar, Rohan, Perrier & Payan 2015 Region of interestω(green)
Displacement
Activation of Genioglossus
Quantity of interest :J(u) =1 2 Z
ω
trε(u)dx∈R
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2nd Example : Tongue with fibre activation
102 103 104
10−6 10−5 10−4 10−3
N
J(u)−J(uh)
Uniform refinement Adaptive refinement 1/N2 1/N
Initial error estimatorηK
Initial mesh Adapted mesh (5th iteration)
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Conclusion
Motivations
A posteriorierror estimators :
Rarely used in practice for biomedical simulations Allow to quantify the discretization error
Discretization error may have serious implications for guidance/design
Results
DWR estimate : discretization error for a goal-oriented quantity Mesh adaptivity : optimal accuracy with low computational cost Promising preliminary results in a simplified setting
Perspectives
Errors coming from the parameters and the model Hyperelasticity & 3D
Other error estimator
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Acknowledgement
For the discussions
R. Becker F. Hubert J. Ohayon ...
Y. Payan P. Perrier J. Hale
For their financing
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Conclusion
A posteriorierror estimators : Error per cell
Optimal mesh
Improvement of the simulations
Conclusion
A posteriorierror estimators : Error per cell
Optimal mesh
Improvement of the simulations
