A meshless method for solving the cauchy problem in three-dimensional elastostatics

Abstract

An International Joumal Available online at www.sc,encedirect.com computers & .=,=.o= mathematics with applications Computers and Mathematics with Applications 50 (2005) 73-92 www elsevmr com/locate/camwaAbstract--The apphcatlon of the method of fundamental solutions to the Cauchy problem in three-dimensional lsotroplc linear elasticity is mvestlgated The resulting system of linear algebraic equations is ill-conditioned and therefore, its solution is regularized by employing the first-order Tlkhonov functional, while the choice of the regularizatlon parameter Is based on the L-curve method. Numerical results are presented for both under-and equally-determined Cauchy pioblems in a pmcewise smooth geometry. The convergence, accuracy, and stabihty of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solutmn domain, and decreasing the amount of noise added into the Input data, respectively, are analysed. ~)

The MFS for the detection of inner boundariesin linear elasticity

“…Regularization can be achieved either by appropriately limiting the number of functional evaluations, or by introducing penalty terms in the objective cost functional that is minimized. The extension of the proposed technique to inverse inclusion problems in three-dimensional linear elasticity, [15], is deferred to a future work.…”

A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation

“…), have been used increasingly over the last decade for the numerical solution of inverse problems. For example, the Cauchy problem associated with the heat conduction equation [38][39][40][41][42][43][44][45][46][47][48], linear elasticity [49,50], steady-state heat conduction in functionally graded materials [51], Helmholtz-type equations [19][20][21]52], Stokes problems [53], the biharmonic equation [54], etc., have been successfully addressed by using the MFS.…”

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

“…Some iterative methods are based on the use of a sequence of well-posed problems and others on the minimization of an energy-like functional. Numerical algorithms are implemented using different numerical methods, such as the finite element method (FEM) [1,2,3,5,8,9,11,13,27,38], the boundary element method (BEM) [12,14,20,22,26,23,28,29,30,31,32,33,36,37,40,41], the finite difference method [21] or meshless methods [34,35]. Some papers present comparisons between different numerical methods [10,31,37].…”

“…The references [1,5,8,9,11,12,14,20,21,23,25,26] propose different methods of solving the Cauchy problem for the Laplace equation. References [2,3,13,22,23,27,28,29,30,31,32,33,34,35,36,37,38,40,41] deal with the Cauchy problem in linear elasticity. These methods can be classified as Tikhonov type methods [15,22,27,29,33,34,35,38,39,40,41], quasi-reversibility type methods [5,21,25], iterative methods [1,2,3,8,9,…”

“…References [2,3,13,22,23,27,28,29,30,31,32,33,34,35,36,37,38,40,41] deal with the Cauchy problem in linear elasticity. These methods can be classified as Tikhonov type methods [15,22,27,29,33,34,35,38,39,40,41], quasi-reversibility type methods [5,21,25], iterative methods [1,2,3,8,9,11,12,13,14,15,20,23,26,28,32,36,37],... Quasi reversibility methods and Tikhonov regularization methods present the advantage of leading to well posed problems after mod...…”