Cauchy Problem In Linear Elasticity Research Articles

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

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.

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

In this work, an inverse problem in linear elasticity is considered, it is about reconstructing the unknown boundary conditions on a part of the boundary based on the other boundaries. A methodology based on the domain decomposition operating mode is opted by constructing a Steklov–Poincaré kind’s operator. This allows us to reformulate our inverse problem into a fixed point one involving a Steklov kind’s operator, the existence of the fixed point problem is shown using the topological degree of Leray–Schauder. The proposed approach offers the opportunity to exploit domain decomposition methods for solving this inverse problem. Finally, a numerical study of this problem using the boundary element method is presented. The obtained numerical results show the efficiency of the proposed approach.

The generalized finite difference method for the inverse Cauchy problem in two-dimensional isotropic linear elasticity

In this study, the generalized finite difference method (GFDM) was used to stably and accurately solve two-dimensional (2D) inverse Cauchy problems in linear elasticity by using the Navier equations. In Cauchy problems, overdetermined boundary conditions are imposed on parts of the boundary, whereas there are missing boundary conditions on some parts of the boundary. In Cauchy problems, conventional numerical methods generally generate highly ill-conditioned matrices and thus provide unstable numerical solutions. Moreover, even if a slight noise is added in the boundary conditions, numerical errors are evidently magnified. The GFDM, one of the most promising meshless methods and an extension of the classical finite difference method, can avoid time-consuming tasks of mesh generation and numerical quadrature. The GFDM was applied in this study to stably solve the 2D Cauchy problems in linear elasticity and four numerical examples are provided to illustrate the consistency and accuracy of the presented meshless numerical scheme. Moreover, the stability of the presented scheme for inverse Cauchy problems was proved by adding noise into the boundary conditions.

Identification of boundary conditions by solving Cauchy problem in linear elasticity with material uncertainties

This work is a contribution to non-destructive testing in the context of uncertainties. It consists in identifying boundary conditions on an inaccessible part of a solid body boundary, from the knowledge of over-specified data given on an accessible part of this boundary. This problem is well known as Cauchy problem. The material properties are considered uncertain. Polynomial chaos expansion is used in order to solve this Cauchy problem in the random context. A stochastic constrained optimization problem is formulated and solved. Numerical experiments are presented.

To solve the inverse Cauchy problem in linear elasticity by a novel Lie-group integrator

In this paper, we propose a simple, iteration free and easy-to-implement numerical algorithm for the solution of inverse Cauchy problem in linear or nonlinear elasticity. The bottom of a finite rectangular plate is imposed by overspecified boundary data, and we seek unknown data on the top side. A spring-damping transform method (SDTM) is introduced to the Navier equations, such that after a discretization by the differential quadrature method, we can apply a novel Lie-group integrator, namely the mixed group-preserving scheme (MGPS), to solve them as an initial value problem. Several numerical examples including nonlinear ones are examined to show that the MGPS can overcome the ill-posed behaviour of the inverse Cauchy problem in elasticity, which has good efficiency and stability against the noisy disturbance, even with an intensity large up to and .

Relaxation procedures for an iterative MFS algorithm for the stable reconstruction of elastic fields from Cauchy data in two-dimensional isotropic linear elasticity

We investigate two numerical procedures for the Cauchy problem in linear elasticity, involving the relaxation of either the given boundary displacements (Dirichlet data) or the prescribed boundary tractions (Neumann data) on the over-specified boundary, in the alternating iterative algorithm of Kozlov et al. (1991). The two mixed direct (well-posed) problems associated with each iteration are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method, while the optimal value of the regularization parameter is chosen via the generalized cross-validation (GCV) criterion. An efficient regularizing stopping criterion which ceases the iterative procedure at the point where the accumulation of noise becomes dominant and the errors in predicting the exact solutions increase, is also presented. The MFS-based iterative algorithms with relaxation are tested for Cauchy problems for isotropic linear elastic materials in various geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the proposed method.

An iterative method for the Cauchy problem in linear elasticity with fading regularization effect

In this paper, an iterative method for solving the Cauchy problem in linear elasticity is introduced. This problem consists in recovering missing data (displacements and forces) on some parts of a domain boundary from the knowledge of overspecified data (displacements and forces) on the remaining parts. The algorithm reads as a least square fitting of the given data, with a regularization term whose effect fades as the iterations go on. So the algorithm converges to the solution of the Cauchy problem. Numerical simulations using the finite element method highlight the algorithm’s efficiency, accuracy, robustness to noisy data as well as its ability to deblur noisy data.

A relaxation method of an alternating iterative algorithm for the Cauchy problem in linear isotropic elasticity

We propose two algorithms involving the relaxation of either the given Dirichlet data (boundary displacements) or the prescribed Neumann data (boundary tractions) on the over-specified boundary in the case of the alternating iterative algorithm of Kozlov et al. [16] applied to Cauchy problems in linear elasticity. A convergence proof of these relaxation methods is given, along with a stopping criterion. The numerical results obtained using these procedures, in conjunction with the boundary element method (BEM), show the numerical stability, convergence, consistency and computational efficiency of the proposed method.

The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity

In this paper, yet another iterative procedure, namely the minimal error method (MEM), for solving stably the Cauchy problem in linear elasticity is introduced and investigated. Furthermore, this method is compared with another two iterative algorithms, i.e. the conjugate gradient (CGM) and Landweber–Fridman methods (LFM), previously proposed by Marin et al. [Marin, L., Háo, D.N., Lesnic, D., 2002b. Conjugate gradient-boundary element method for the Cauchy problem in elasticity. Quarterly Journal of Mechanics and Applied Mathematics 55, 227–247] and Marin and Lesnic [Marin, L., Lesnic, D., 2005. Boundary element-Landweber method for the Cauchy problem in linear elasticity. IMA Journal of Applied Mathematics 18, 817–825], respectively, in the case of two-dimensional homogeneous isotropic linear elasticity. The inverse problem analysed in this paper is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions. The numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method (BEM) for two-dimensional homogeneous isotropic linear elastic materials.

An optimization approach for the Cauchy problem in linear elasticity

An energy error functional is introduced in the context of the ill-posed problem of boundary data recovery in linear elasticity, which is well known as the Cauchy problem. The problem is converted into one of optimization; the computation of the gradients of the energy functional is given for both the continuous and the discrete problems. Links with existing methods for data completion are described and numerical experiments highlight the efficiency of the proposed method as well as its robustness in the case of singular data.

Boundary element-Landweber method for the Cauchy problem in linear elasticity

In this paper, an iterative algorithm based on the Landweber method in combination with the boundary element method is developed for solving the Cauchy problem in isotropic linear elasticity. An efficient regularizing stopping criterion is also employed. The numerical results obtained confirm that the iterative method produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data, respectively.

An iterative boundary element algorithm for a singular Cauchy problem in linear elasticity

In this paper a new iterative boundary element method (BEM) is developed to solve singular Cauchy problems in isotropic linear elasticity. In order to deal with the difficulties associated with the instability caused by the prescription of the boundary data and the low rate of convergence manifested by standard numerical techniques, the iterative algorithm proposed by Kozlov et al. [1] is combined with the singularity subtraction technique, see Portela et al. [2]. A regularising stopping criterion is developed and, moreover, for a given physical geometry and loading of a cracked elastic medium, the convergence and the stability of the algorithm proposed are numerically investigated for both exact and noisy data. The conclusion of this paper is that if a singular problem is encountered then a singularity based technique, such as that described in Sect. 2, should be employed. In addition, if this singular problem is such that inverse analysis in necessary, such as in crack problems, then a suitable regularisation method should be applied to the regular problem obtained by taking into account the singularity, and not to the singular problem with the singularity ignored.

Conjugate Gradient-Boundary Element Method for the Cauchy Problem in Elasticity

In this paper, an iterative algorithm based on the conjugate gradient method (CGM) in combination with the boundary element method (BEM) for obtaining stable approximate solutions to the Cauchy problem in linear elasticity is analysed. An efficient stopping criterion for the CGM proposed by Nemirovskii in 1986 is employed and in addition the accuracy of the iterative algorithm is improved by using a variable relaxation procedure. The numerical results obtained confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data.

Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition

In this paper the singular value decomposition (SVD), truncated at an optimal number, is analysed for obtaining approximate solutions to ill-conditioned linear algebraic systems of equations which arise from the boundary element method (BEM) discretisation of an ill-posed boundary value problem in linear elasticity. The regularisation parameter, namely the optimal truncation number, is chosen according to the discrepancy principle. The numerical results obtained confirm that the SVD+BEM produces a convergent and stable numerical solution with respect to decreasing the mesh size discretisation and the amount of noise added into the input data.

Boundary Element Regularisation Methods for Solving the Cauchy Problem in Linear Elasticity

In this article several boundary element regularisation methods, such as iterative, conjugate gradient, Tikhonov regularisation and singular value decomposition methods, for solving the Cauchy problem in isotropic linear elasticity are developed and compared. Regularising stopping criteria are developed and the convergence, as well as the stability, of the numerical methods proposed are analysed. The Cauchy problem in isotropic linear elasticity can be regularised by various methods, such as the general regularisation methods presented in this article, but more accurate results are obtained by particular methods which take into account the particular structure of the problem, such as the alternating iterative algorithm originally proposed by [V.A. Kozlov, V.G. Maz'ya and A.F. Fomin, (1991). An iterative method for solving the Cauchy problem for elliptic equations. Comput. Maths. Math. Phys ., 31 , 45-52].

Boundary element method for the Cauchy problem in linear elasticity

In this paper, the iterative algorithm proposed by Kozlov et al. [Comput Maths Math Phys 32 (1991) 45] for obtaining approximate solutions to ill-posed boundary value problems in linear elasticity is analysed. The technique is then numerically implemented using the boundary element method (BEM). The numerical results obtained confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. An efficient stopping regularizing criterion is given and in addition, the accuracy of the iterative algorithm is improved by using a variable relaxation procedure. Analytical formulae for the integration constants resulting from the direct application of the BEM for an isotropic linear elastic medium are also presented.

An Inverse Problem in Elasticity With Partially Overprescribed Boundary Conditions, Part I: Theoretical Approach

A Cauchy problem in linear elasticity is considered. This problem is governed by a Fredholm integral equation of the first kind and cannot be solved directly. The regularization method, which has been originally employed by Gao and Mura (1989), is formulated from a different perspective in order to address some of the difficulties experienced in their formulation. The theoretical details are discussed in this paper. Numerical examples are treated to Part II.