Stochastic modeling of a class of stored energy functions for incompressible hyperelastic materials with uncertainties

Staber, Brian; Guilleminot, Johann

doi:10.1016/j.crme.2015.07.008

Cited by 34 publications

(72 citation statements)

References 30 publications

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“…This motivates the use of the stochastic version where uncertainty in the parameters can be taken rigorously into account. Taking into account uncertainties in biomechanics models is a relatively young field and little work has been done in particular for soft tissue with large deformation [37]. In a clinical environment, where safety-critical decisions must be made based on the output of such simulations, being able to propagate uncertainty efficiently through such models is of importance.…”

Section: Hyperelasticity Equation With Stochastic Materials Parametersmentioning

confidence: 99%

Accelerating Monte Carlo estimation with derivatives of high-level finite element models

Hauseux

Hale

Bordas

2017

Computer Methods in Applied Mechanics and Engineering

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In this paper we demonstrate the ability of a derivative-driven Monte Carlo estimator to accelerate the propagation of uncertainty through two high-level non-linear finite element models. The use of derivative information amounts to a correction to the standard Monte Carlo estimation procedure that reduces the variance under certain conditions. We express the finite element models in variational form using the high-level Unified Form Language (UFL). We derive the tangent linear model automatically from this high-level description and use it to efficiently calculate the required derivative information. To study the effectiveness of the derivative-driven method we consider two stochastic PDEs; a onedimensional Burgers equation with stochastic viscosity and a three-dimensional geometrically non-linear Mooney-Rivlin hyperelastic equation with stochastic density and volumetric material parameter. Our results show that for these problems the first-order derivative-driven Monte Carlo method is around one order of magnitude faster than the standard Monte Carlo method and at the cost of only one extra tangent linear solution per estimation problem. We find similar trends when comparing with a modern non-intrusive multi-level polynomial chaos expansion method. We parallelise the task of the repeated forward model evaluations across a cluster using the ipyparallel and mpi4py software tools. A complete working example showing the solution of the stochastic viscous Burgers equation is included as supplementary material.

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Section: Hyperelasticity Equation With Stochastic Materials Parametersmentioning

confidence: 99%

Accelerating Monte Carlo estimation with derivatives of high-level finite element models

Hauseux

Hale

Bordas

2017

Computer Methods in Applied Mechanics and Engineering

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“…Next, following [23][24][25][26][27], for the random nonlinear shear modulus µ(a 0 ), defined by (3.7), we set the mathematical expectations: 12) where, by the constraint (3.11), the mean value µ(a 0 ) is fixed and greater than zero, and the logarithmic constraint (3.12) implies that both µ(a 0 ) and µ(a 0 ) −1 are second-order random variables (i.e. they have finite mean and finite variance).…”

Section: (A) Calibration Of Random Field Parametersmentioning

confidence: 99%

“…Critically, equations (3.11) and (3.12) imply that µ(a 0 ) follows a Gamma distribution (the maximum entropy distribution) [72,73], 13) where the mean value µ(a 0 ) is obtained at step 1. For the random vector (R 1 (a 0 ), · · · , Rn(a 0 )), applying the constraints [23,25] 15) this vector follows a Dirichlet distribution [41,74], D (ξ 1 (a 0 ), · · · , ξn(a 0 )). Then, every random variable Rp(a 0 ), p = 1, · · · , n, follows a standard Beta distribution [40,41], B(ξp(a 0 ), ψp(a 0 )), with ξp(a 0 ) > 0 and ψp(a 0 ) = n q=1,q =p ξq(a 0 ) > 0 satisfying 17) and is calculated from the mean values obtained at step 1.…”

Section: (A) Calibration Of Random Field Parametersmentioning

confidence: 99%

“…Recently, stochastic strategies based on information theory, which aids with the calibration of hyperelastic models for isotropic elastic solids, were proposed in [23][24][25]. Specifically, stochastichyperelastic models were identified from experimental data consisting of the mean values and standard deviations of elastic stresses under finite strain deformations.…”

Section: Introductionmentioning

confidence: 99%

“…In [25], the stochastic approach was employed for the calibration of Ogden-type models to brain, liver, and spinal cord data representing mean values and standard deviation of the first Piola-Kirchhoff stress under finite compression tests. Ogden-type strain-energy functions and their extension to compressible materials in a stochastic framework were formulated originally in [23,24]. Within this framework, strain-energy functions with random field parameters were obtained under a combination of physically realistic and theoretical restrictions, namely:…”

Section: Introductionmentioning

confidence: 99%

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Stochastic isotropic hyperelastic materials: constitutive calibration and model selection

2018

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Biological and synthetic materials often exhibit intrinsic variability in their elastic responses under large strains, owing to microstructural inhomogeneity or when elastic data are extracted from viscoelastic mechanical tests. For these materials, although hyperelastic models calibrated to mean data are useful, stochastic representations accounting also for data dispersion carry extra information about the variability of material properties found in practical applications. We combine finite elasticity and information theories to construct homogeneous isotropic hyperelastic models with random field parameters calibrated to discrete mean values and standard deviations of either the stress-strain function or the nonlinear shear modulus, which is a function of the deformation, estimated from experimental tests. These quantities can take on different values, corresponding to possible outcomes of the experiments. As multiple models can be derived that adequately represent the observed phenomena, we apply Occam's razor by providing an explicit criterion for model selection based on Bayesian statistics. We then employ this criterion to select a model among competing models calibrated to experimental data for rubber and brain tissue under single or multiaxial loads.

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Stochastic modeling of the Ogden class of stored energy functions for hyperelastic materials: the compressible case

Staber

Guilleminot

2016

Z Angew Math Mech

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This paper is devoted to the modeling of compressible hyperelastic materials whose response functions exhibit uncertainties at some scale of interest. The construction of parametric probabilistic representations for the Ogden class of stored energy functions is specifically considered and formulated within the framework of Information Theory. The overall methodology relies on the principle of maximum entropy, which is invoked under constraints arising from existence theorems and consistency with linearized elasticity. As for the incompressible case discussed elsewhere, the derivation essentially involves the conditioning of some variables on the stochastic bulk and shear moduli, which are shown to be statistically dependent random variables in the present case. The explicit construction of the probability measures is first addressed in the most general setting. Subsequently, particular results for classical Neo‐Hookean and Mooney‐Rivlin materials are provided. Salient features of the probabilistic representations are finally highlighted through forward Monte‐Carlo simulations. In particular, it is seen that the models allow for the reproduction of typical experimental trends, such as a variance increase at large stretches. A stochastic multiscale analysis, where uncertainties on the constitutive law of the matrix phase are taken into account through the proposed approach, is also presented.

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Stochastic modeling of a class of stored energy functions for incompressible hyperelastic materials with uncertainties

Cited by 34 publications

References 30 publications

Accelerating Monte Carlo estimation with derivatives of high-level finite element models

Accelerating Monte Carlo estimation with derivatives of high-level finite element models

Stochastic isotropic hyperelastic materials: constitutive calibration and model selection

Stochastic modeling of the Ogden class of stored energy functions for hyperelastic materials: the compressible case

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