Regularization of the noisy Cauchy problem solution approximated by an energy‐like method

Rischette, Romain; Baranger, Thouraya; Andrieux, Stéphane

doi:10.1002/nme.4501

Cited by 13 publications

(7 citation statements)

References 27 publications

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“…Further work will address, first, examples and applications in three-dimensions of space and the derivation of the gradient of the error functional for non-twice differentiable potentials, secondly the issues of regularization: up to what noise level the regularization-free method developed here is valid [1,16,23]? What kind of regularization has to be added above this noise level [47][48][49][50]?…”

Section: Discussionmentioning

confidence: 99%

Solution of nonlinear Cauchy problem for hyperelastic solids

Andrieux

Baranger

2015

Inverse Problems

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The problem of expanding given (measured) fields at the surface of a solid within the solid and up to inaccessible parts of its boundary is addressed for a nonlinear hyperelastic medium. The problem is formulated as a nonlinear Cauchy problem and is solved thanks to a technique consisting of splitting the unknown field into two solutions of well posed problems and minimizing a specially designed error in constitutive equation between the two fields, taking advantage of the convexity of the hyperelastic potential. The minimization involves as unknowns the boundary conditions fields on the inaccessible part of the boundary of the solid. Two illustrations are given, the first one with a twice-differentiable hyperelastic potential describing a material with nonlinear compressibility, the second one deals with a geomaterial with asymmetric elasticity in the tension and compression ranges, and involves an only one-differentiable potential.

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Section: Discussionmentioning

confidence: 99%

Solution of nonlinear Cauchy problem for hyperelastic solids

Andrieux

Baranger

2015

Inverse Problems

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“…Nevertheless, it has been further combined with a Tikhonov technique to overcome instability towards noisy data. 14 Optimal control techniques 15 and quasi-reversibility methods 16,17 are also examples of methods used to solve elliptic inverse Cauchy problems.…”

Section: Introductionmentioning

confidence: 99%

“…It consists in separating the ill‐posed Cauchy problem into two well‐posed problems and then minimizing the gap between these two separate fields. Nevertheless, it has been further combined with a Tikhonov technique to overcome instability towards noisy data 14 . Optimal control techniques 15 and quasi‐reversibility methods 16,17 are also examples of methods used to solve elliptic inverse Cauchy problems.…”

Section: Introductionmentioning

confidence: 99%

Fading regularization FEM algorithms for the Cauchy problem associated with the two‐dimensional biharmonic equation

Boukraa

Amdouni

Delvare

2022

Math Methods in App Sciences

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In this paper, we use the fading regularization method to solve a biharmonic inverse problem, represented by the Cauchy problem. Two formulations are studied and implemented numerically using a finite element method (FEM).We present numerical reconstructions of the missing data on the inaccessible part of the boundary from the knowledge of over-prescribed noisy data for both smooth and piecewise smooth two-dimensional geometries. Numerical examples validate the convergence, stability and efficiency of the proposed numerical algorithm, as well as its capability to deblur the noisy data.

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“…The authors developed in a series of papers solution algorithms for the resolution of the Cauchy problem in linear and nonlinear mechanics [25][26][27][28][29][30][31][32], including nonlinear elasticity and elastoplasticity, stationary and heat equation [33][34][35][36][37][38][39][40][41]. The method relies on a gap functional which is minimized in the proposed approach, provided it is positive, and zero when the gap vanishes.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Cauchy problem and identification in contact mechanics: a solving method based on Bregman-gap

Andrieux¹,

Baranger²

2020

Inverse Problems

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This paper proposes a solution method for identification problems in the context of contact mechanics when overabundant data are available on a part Γm of the domain boundary while data are missing from another part of this boundary. The first step is then to find a solution to a Cauchy problem. The method used by the authors for solving Cauchy problems consists of expanding the displacement field known on Γm towards the inside of the solid via the minimization of a function that measures the gap between solutions of two well-posed problems, each one exploiting only one of the superabundant data. The key question is then to build an appropriate gap functional in strongly nonlinear contexts. The proposed approach exploits a generalization of the Bregman divergence, using the thermodynamic potentials as generating functions within the framework of Generalized Standard Materials, but also Implicit Generalized Standard Materials in order to address Coulomb friction. The robustness and efficiency of the proposed method are demonstrated by a numerical bi-dimensional application dealing with a cracked elastic solid with unilateral contact and friction effects on the crack's lips.

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Regularization of the noisy Cauchy problem solution approximated by an energy‐like method

Cited by 13 publications

References 27 publications

Solution of nonlinear Cauchy problem for hyperelastic solids

Solution of nonlinear Cauchy problem for hyperelastic solids

Fading regularization FEM algorithms for the Cauchy problem associated with the two‐dimensional biharmonic equation

Nonlinear Cauchy problem and identification in contact mechanics: a solving method based on Bregman-gap

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