Mathematical analysis and simulation of fixed point formulation of Cauchy problem in linear elasticity

Ellabib, Abdellatif; Nachaoui, Abdeljalil; Ousaadane, Abdessamad

doi:10.1016/j.matcom.2021.02.020

Cited by 8 publications

(5 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, where η k 1 is the sequence defined from (17). We also have for θ ∈ (0, 1), B θ satisfies the properties of lemma 4.3, B θ is linear, symmetric and positive definite in L 2 (Γ 1 ) and corollary 4.5 leads that B θ is a contraction and lemma 4.6 gives 1 / ∈ sp(B θ ).…”

Section: Remarkmentioning

confidence: 84%

“…The stability of method was also investigated by perturbing the given data with various levels of noise. A similar work has been done for a developed fixed point algorithm in [17] using boundary element method. • The application of the method of fundamental solutions to the Cauchy problem in twodimensional isotropic linear elasticity is investigated in [33].…”

Section: Convergence With Respect To the Grid Size H And Accuracymentioning

confidence: 98%

“…In [44], an invariant method of fundamental solutions for the two-dimensional isotropic linear elasticity. We recently used a fixed point formulation [17], it consists in reformulating the inverse problem into a fixed point one. The solution existence is shown using the topological degree of Leray-Schauder then a fixed point algorithm is constructed to solve the Cauchy problem for linear elasticity equation.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Convergence study and regularizing property of a modified Robin–Robin method for the Cauchy problem in linear elasticity

Ellabib¹,

Nachaoui²,

Ousaadane³

2022

Inverse Problems

Self Cite

View full text Add to dashboard Cite

In this paper, we are interested in solving a Cauchy inverse problem in linear elasticity. For this, we propose a new method based on Robin conditions on the inaccessible boundary, then we study the convergence and regularizing property of the proposed algorithm. We use the finite element method for the discretization of our problem. Further, we treat the spectrum analysis of the discrete problem in order to study the convergence behavior of the proposed method in the discrete case. Finally, we present numerical results which show the efficiency and stability of the proposed approach in the presence of perturbed data. The robustness of the proposed algorithm is also performed for the anisotropic and heterogeneous cases.

show abstract

Section: Remarkmentioning

confidence: 84%

Section: Convergence With Respect To the Grid Size H And Accuracymentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Convergence study and regularizing property of a modified Robin–Robin method for the Cauchy problem in linear elasticity

Ellabib¹,

Nachaoui²,

Ousaadane³

2022

Inverse Problems

Self Cite

View full text Add to dashboard Cite

show abstract

“…It involves finding a solution to the Poisson equation inside a domain, given values on a subset of the boundary. Various numerical methods have been developed to tackle this problem, such as the boundary element method [33,34,[40][41][42], finite element method [18,34,[43][44][45], and finite difference method [28,46]. These methods typically rely on mesh-based discretization techniques and have proven to be effective in solving the linear Cauchy problem.…”

Section: Introductionmentioning

confidence: 99%

Polynomial Approximation of a Nonlinear Inverse Cauchy Problem

2023

acad. sci. j.

View full text Add to dashboard Cite

In this paper, a class of nonlinear inverse boundary problem in the context of heat transfer is considered. We consider a class of nonlinear inverse boundary problems in the context of heat transfer. The problem involves determining the temperature distribution within a domain subject to a Cauchy boundary condition on a part of its boundary. We introduce a transformed variable, which allows us to reformulate the problem as a linear Cauchy problem followed by a series of nonlinear equations. We propose a polynomial expansion method to solve the linear Cauchy problem for the Laplace equation, and we employ the Newton method to solve the resulting nonlinear equations. Importantly, our approach does not rely on mesh-based discretization, allowing for parallel computation and preserving the mesh-free nature of the problem. We present numerical results obtained using our methodology and discuss the effectiveness of the proposed approach. The results show that the method provides a robust and efficient framework for solving nonlinear inverse boundary problems in heat transfer, with potential applications in various engineering and scientific fields.

show abstract

“…Consequently, determining and handling 3 the missing solution and its normal derivative on this remaining part of the boundary is required. These kind of the problems are called Cauchy problems and it's a classical type of inverse problems [21,21,49,50]. In the sense of Hadamard [26] these problems are severe ill posed problems.…”

Section: Introductionmentioning

confidence: 99%

An accelerated alternating iterative algorithm for data completion problems connected with Helmholtz equation

Berdawood

Nachaoui

2023

Stat., optim. inf. comput.

View full text Add to dashboard Cite

This paper deals with an inverse problem governed by the Helmholtz equation. It consists in recovering lackingdata on a part of the boundary based on the Cauchy data on the other part. We propose an optimal choice of the relaxationparameter calculated dynamically at each iteration. This choice of relaxation parameter ensures convergence without priordetermination of the interval of the relaxation factor required in our previous work. The numerous numerical example showsthat the number of iterations is drastically reduced and thus, our new relaxed algorithm guarantees the convergence for allwavenumber k and gives an automatic acceleration without any intervention of the user.

show abstract

Mathematical analysis and simulation of fixed point formulation of Cauchy problem in linear elasticity

Cited by 8 publications

References 19 publications

Convergence study and regularizing property of a modified Robin–Robin method for the Cauchy problem in linear elasticity

Convergence study and regularizing property of a modified Robin–Robin method for the Cauchy problem in linear elasticity

Polynomial Approximation of a Nonlinear Inverse Cauchy Problem

An accelerated alternating iterative algorithm for data completion problems connected with Helmholtz equation

Contact Info

Product

Resources

About