An a posteriori error estimator for the spectral fractional power of the Laplacian

E-print/Working paper (2022)

We develop a novel a posteriori error estimator for the L2 error committed by the finite ele- ment discretization of the solution of the fractional Laplacian. Our a posteriori error estimator takes ... [more ▼]

We develop a novel a posteriori error estimator for the L2 error committed by the finite ele- ment discretization of the solution of the fractional Laplacian. Our a posteriori error estimator takes advantage of the semi–discretization scheme using a rational approximation which allows to reformulate the fractional problem into a family of non–fractional parametric problems. The estimator involves applying the implicit Bank–Weiser error estimation strategy to each parametric non–fractional problem and reconstructing the fractional error through the same rational approximation used to compute the solution to the original fractional problem. We provide several numerical examples in both two and three-dimensions demonstrating the effectivity of our estimator for varying fractional powers and its ability to drive an adaptive mesh refinement strategy. [less ▲]

Removing the saturation assumption in Bank-Weiser error estimator analysis in dimension three

in Applied Mathematics Letters (2020), 107

We provide a new argument proving the reliability of the Bank-Weiser estimator for Lagrange piecewise linear finite elements in both dimension two and three. The extension to dimension three constitutes ... [more ▼]

We provide a new argument proving the reliability of the Bank-Weiser estimator for Lagrange piecewise linear finite elements in both dimension two and three. The extension to dimension three constitutes the main novelty of our study. In addition, we present a numerical comparison of the Bank-Weiser and residual estimators for a three-dimensional test case. [less ▲]

Parameter identification problem in bimaterial human skin and sensitivity analysis : Uncertainties in biomechanics of skin

Scientific Conference (2019, December 09)

The proposed paper concerns the prediction of the numerical response of a biomechanical structure submitted to an unknown external loading state. The methodology is based on homogeneous and then ... [more ▼]

The proposed paper concerns the prediction of the numerical response of a biomechanical structure submitted to an unknown external loading state. The methodology is based on homogeneous and then heterogeneous structures such as healthy or pathological cutaneous tissues that can be mechanically tested in vivo under a patchy knowledge of boundary conditions. Experimental data corresponding to the extension of a piece of skin located between two pads with displacement enslavement, represent input data to the numerical model. Data are reaction force on one pad and displacement field between the two pads and all around. The numerical model consists of a representation of the bi-material domain geometry with neo-hookean behaviors. The boundary conditions and loadings of the experimental extension test are imposed. The materials parameters have been identified by inverse method starting from a constrained cost function minimizing the difference between the calculated displacements field and experimental displacements field obtained by digital image correlation and taking into account the reaction force as a constraint. An analysis of the model sensitivity to material parameters is presented. [less ▲]

Uncertainty Quantification in Finite Element Models:Application to SoftTissue Biomechanics

Scientific Conference (2018, July 23)

We present probabilistic approaches aiming at the selection of the best constitutive model and to identify their parameters from experimental data. These parameters are always associated with some degree ... [more ▼]

We present probabilistic approaches aiming at the selection of the best constitutive model and to identify their parameters from experimental data. These parameters are always associated with some degree of uncertainty. It is therefore important to study how this statistical uncertainty in parameters propagates to a safety-critical quantity of interest in the output of a model. Efficient Monte Carlo methods based on variance reduction techniques (Sensitivity Derivatives Monte Carlo methods [Hauseux et al. 2017] and MultiLevel Monte Carlo [Giles 2015] methods) are employed to propagate this uncertainty for both random variables and random fields. Inverse and forward problems are strongly connected. In a bayesian setting [Matthies et al. 2017], developing methods that reduce the number of evaluations of the forward model to an absolute minimum to achieve convergence is crucial for tractable computations. Numerical results in the context of soft tissue biomechanics are presented and discussed. [less ▲]

Experimental and numerical assessment of the mechanics of keloid-skin composites undergoing large deformations

Scientific Conference (2017, August)