maximal ergodic theorems for some group actions abstract we prove maximal ergodic inequalities for a sequence of operators and for their averages in the noncommutative l p space we also obtain the corresponding individual ergodic theorems applying these results to actions of a free group on a von neumann algebra we get noncommutative analogues of maximal ergodic inequalities and pointwise ergodic theorems of nevo stein

an operator splitting radial basis function method for the solution of transient nonlinear poisson problems abstract this paper presents an operator splitting radial basis function os rbf method as a generic solution procedure for transient nonlinear poisson problems by combining the concepts of operator splitting radial basis function interpolation particular solutions and the method of fundamental solutions the application of the operator splitting permits the isolation of the nonlinear part of the equation that is solved by explicit adams bashforth time marching for half the time step this leaves a nonhomogeneous modified helmholtz type of differential equation for the elliptic part of the operator to be solved at each time step the resulting equation is solved by an approximate particular solution and by using the method of fundamental solution for the fitting of the boundary conditions radial basis functions are used to construct approximate particular solutions and a grid free dimension independent method with high computational efficiency is obtained this method is demonstrated for some prototypical nonlinear poisson problems in heat and mass transfer and for a problem of transient convection with diffusion the results obtained by the os rbf method compare very well with those obtained by other traditional techniques that are computationally more expensive the new os rbf method is useful for both general irregular two and three dimensional geometry and provides a mesh free technique with many mathematical flexibilities and can be used in a variety of engineering applications

the method of fundamental solutions for the numerical solution of the biharmonic equation abstract the method of fundamental solutions mfs is a relatively new technique for the numerical solution of certain elliptic boundary value problems it falls in the class of methods generally called boundary methods and like the well known boundary integral equation method is applicable when a fundamental solution of the differential equation is known in the mfs the approximate solution is a linear combination of fundamental solutions with singularities placed outside the domain of the problem the locations of the singularities are either preassigned or determined along with the coefficients of the fundamental solutions so that the approximate solution satisfies the boundary conditions as well as possible in many applications these quantities are determined by a least squares fit of the boundary conditions a nonlinear problem which is solved using standard software in this paper the mfs is formulated for biharmonic problems and is applied to a variety of standard test problems as well as to problems arising in elasticity and fluid flow

a numerical study on the solution of the cauchy problem in elasticity this work deals with the cauchy problem in two dimensional linear elasticity the equations of the problem are discretized through a standard fem approach and the resulting ill conditioned discrete problem is solved within the frame of the tikhonov approach the choice of the required regularization parameter is accomplished through the generalized cross validation criterion on this basis a numerical experimentation has been performed and the calculated solutions have been used to highlight the sensitivity to the amount of known data the noise always present in the data the regularity of boundary conditions and the choice of the regularization parameter the aim of the numerical study is to implicitly device some guidelines to be used in the solution of this kind of problems

the collocation points of the fundamental solution method for the potential problem we propose an algorithm for computing harmonic functions in a two dimensional domain with prescribed dirichlet data our algorithm is a variant of what is called a fundamental solution method 1 3 this method requires us to select 2n points in the two dimensional plane n of which are called collocation points and the remaining n are called charge points representing the position of the singularities of the fundamental solution it is known that there exists a set of 2n points by which the error is exponentially small 1 3 6 however these papers are concerned mainly with existence and as far as the authors know few fast and reliable algorithms are known for good position of the points in this paper we propose a new rule for the position of the points and examine its efficiency by numerical experiments the new rule uses fft effectively

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

the minimal error method for the cauchy problem in linear elasticity numerical implementation for two dimensional homogeneous isotropic linear elasticity abstract 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 hao 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

completions of local morphisms and valuations in this note we study injective local morphisms varphi r mathfrak m rightarrow s mathfrak n of local excellent domains in particular we are interested in the problem when the mathfrak n adic topology ons restricts to a topology on r that is linearly equivalent to the mathfrak m adic topology using a valuative criterion we prove this in case r is analytically irreducible and s r is essentially of finite type and we recover and extend a weak version of gabrielov s rank condition

a meshless method for solving the cauchy problem in three dimensional elastostatics the application of the method of fundamental solutions to the cauchy problem in three dimensional isotropic linear elasticity is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularized by employing the first order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for both under and equally determined cauchy problems in a piece wise smooth geometry the convergence accuracy and stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

the method of fundamental solutions and condition number analysis for inverse problems of laplace equation this paper investigates the applications of the method of fundamental solutions together with the conditional number analysis to solve various inverse 2d laplace problems involving under specified and or over specified boundary conditions through the method of fundamental solutions and the condition number analysis it is numerically found that solutions of inverse laplace problems can be obtained without iteration or regularization for small noise levels since the method of fundamental solutions is a boundary type meshless numerical method that can automatically satisfy the governing equation however for larger values of noise levels regularization is still necessary to obtain promising result the present paper mainly focuses on the two types of numerical predictions of inverse 2d laplace problems 1 cauchy problem and 2 shape identification problem good quantitative agreement with the analytical solutions and other numerical methods for small perturbed boundary data is observed by using present meshless numerical scheme

non iterative regularized mfs solution of inverse boundary value problems in linear elasticity numerical reconstruction of the missing boundary data in 2d and 3d linear elasticity noisy over prescribed data are available on the remaining accessible boundary the inverse problem is solved by a meshless method i e the method of fundamental solutions mfs stabilization is achieved via several svd based regularization methods the regularization parameter is selected via the criteria used for inverse problems the numerical reconstruction of the missing dirichlet and neumann data on an inaccessible part of the boundary in the case of two and three dimensional linear isotropic elastic materials from the knowledge of over prescribed noisy measurements taken on the remaining accessible boundary part is investigated this inverse problem is solved using the method of fundamental solutions mfs whilst its stabilization is achieved through several singular value decomposition svd based regularization methods such as the tikhonov regularization method 48 the damped svd and the truncated svd 18 the regularization parameter is selected according to the discrepancy principle 40 generalized cross validation criterion 14 and hansen s l curve method 20

multi parameter tikhonov regularization and model function approach to the damped morozov principle for choosing regularization parameters in this paper we study the multi parameter tikhonov regularization method which adds multiple different penalties to exhibit multi scale features of the solution an optimal error bound of the regularization solution is obtained by a priori choice of multiple regularization parameters some theoretical results of the regularization solution about the dependence on regularization parameters are presented then an a posteriori parameter choice i e the damped morozov discrepancy principle is introduced to determine multiple regularization parameters five model functions i e two hyperbolic model functions a linear model function an exponential model function and a logarithmic model function are proposed to solve the damped morozov discrepancy principle furthermore four efficient model function algorithms are developed for finding reasonable multiple regularization parameters and their convergence properties are also studied numerical results of several examples show that the damped discrepancy principle is competitive with the standard one and the model function algorithms are efficient for choosing regularization parameters

numerical solution of cauchy problems in linear elasticity in axisymmetric situations abstract an iterative method for solving axisymmetric cauchy problems in linear elasticity is presented this kind of problem consists in recovering missing displacements and forces data on one part of a domain boundary from the knowledge of overspecified displacements and forces data on another part of this boundary numerical simulations using the finite element method highlight the algorithm s efficiency accuracy and robustness to noisy data as well as its ability to deblur noisy data an application of the inverse technique to the identification of a friction coefficient is also presented

the mfs for the cauchy problem in two dimensional steady state linear thermoelasticity we study the reconstruction of the missing thermal and mechanical fields on an inaccessible part of the boundary for two dimensional linear isotropic thermoelastic materials from over prescribed noisy cauchy data on the remaining accessible boundary this problem is solved with the method of fundamental solutions mfs together with the method of particular solutions mps via the mfs based particular solution for two dimensional problems in uncoupled thermoelasticity developed in marin and karageorghis 2012a 2013 the stabilisation regularization of this inverse problem is achieved by using the tikhonov regularization method tikhonov and arsenin 1986 whilst the optimal value of the regularization parameter is selected by employing hansen s l curve method hansen 1998

analysis of lavrentiev finite element methods for data completion problems the variational finite element solution of cauchy s problem expressed in the steklov poincare framework and regularized by the lavrentiev method has been introduced and computationally assessed in azaiez et al inverse probl sci eng 18 1063 1086 2011 the present work concentrates on the numerical analysis of the semi discrete problem we perform the mathematical study of the error to rigorously establish the convergence of the global bias variance error

a particular solution trefftz method for non linear poisson problems in heat and mass transfer this paper proposes a computational procedure based on the trefftz method for the solution of non linear poisson problems the problem is solved by finding an approximate particular solution to the poisson equation and using boundary collocation to solve the resulting laplace equation this method results in a global collocation and hence eliminates the need for discretization of the domain in both two and three dimensions the solution procedure proposed earlier in the literature based upon the above method has been reformulated for increased computational efficiency a quasi newton iteration method along with a new heuristic for source point location is used for efficient convergence of the numerical scheme the efficacy of the new formulation has been demonstrated for two classes of problems viz the thermal explosion problem and the diffusion reaction problem in a partially wetted catalyst pellet a brief error analysis of the method as well as future research directions is presented

regularized method of fundamental solutions for boundary identification in two dimensional isotropic linear elasticity we investigate the stable numerical reconstruction of an unknown portion of the boundary of a two dimensional domain occupied by an isotropic linear elastic material from a prescribed boundary condition on this part of the boundary and additional displacement and traction measurements i e cauchy data on the remaining known portion of the boundary this inverse geometric problem is approached by combining the method of fundamental solutions mfs and the tikhonov regularization method whilst the optimal value of the regularization parameter is chosen according to the discrepancy principle various geometries are considered i e convex and non convex domains with a smooth or piecewise smooth boundary in order to show the numerical stability convergence consistency and computational efficiency of the proposed method

the fundamental solution of mindlin plates resting on an elastic foundation in the laplace domain and its applications abstract the aim of this study is to investigate the method of fundamental solution mfs applied to a shear deformable plate reissner mindlin s theories resting on the elastic foundation under either a static or a dynamic load the complete expressions for internal point kernels i e fundamental solutions by the boundary element method for the mindlin plate theory are derived in the laplace transform domain for the first time on employing the mfs the boundary conditions are satisfied at collocation points by applying point forces at source points outside the domain all variables in the time domain can be obtained by durbin s laplace transform inversion method numerical examples are presented to demonstrate the accuracy of the mfs and comparisons are made with other numerical solutions in addition the sensitivity and convergence of the method are discussed for a static problem the proposed mfs is shown to be simple to implement and gives satisfactory results for shear deformable plates under static and dynamic loads

the method of fundamental solutions for inverse boundary value problems associated with the steady state heat conduction in anisotropic media in this paper the method of fundamental solutions is applied to solve some inverse boundary value problems associated with the steady state heat conduction in an anisotropic medium since the resulting matrix equation is severely ill conditioned a regularized solution is obtained by employing the truncated singular value decomposition while the optimal regularization parameter is chosen according to the l curve criterion numerical results are presented for both two and three dimensional problems as well as exact and noisy data the convergence and stability of the proposed numerical scheme with respect to increasing the number of source points and the distance between the fictitious and physical boundaries and decreasing the amount of noise added into the input data respectively are analysed a sensitivity analysis with respect to the measure of the accessible part of the boundary and the distance between the internal measurement points and the boundary is also performed the numerical results obtained show that the proposed numerical method is accurate convergent stable and computationally efficient and hence it could be considered as a competitive alternative to existing methods for solving inverse problems in anisotropic steady state heat conduction copyright 2005 john wiley sons ltd

an invariant method of fundamental solutions for two dimensional isotropic linear elasticity abstract in this paper we propose an invariant method of fundamental solutions for solving boundary value problems in two dimensional isotropic linear elasticity the invariant method of fundamental solutions keeps a very basic natural property the so called invariance property under trivial coordinate changes in the problem description e g dilations and or contractions the problems are solved by the tikhonov regularization method in conjunction with morozov s discrepancy principle finally five examples of boundary value problems are investigated to show the efficiency of this method the numerical convergence accuracy and stability of the proposed method with respect to the number of source points the distance between the sources and the boundary and the amount of noise added into the input data respectively are also analyzed

ergodic theorems for free group actions on von neumann algebras abstract we extend a recent ergodic theorem of a nevo and e stein to the non commutative case let be a faithful normal state on the von neumann algebra a let a i r i 1 generate f r the free group on r generators and let i r i 1 be automorphisms of a which leave invariant define to be the group homomorphism from f r to the automorphisms of a defined on base elements by a i i define w n as the set of all reduced words in f r of length n the identity is a neutral element and w n as the number of elements of w n let n 1 w n a w n a and s n 1 n n 1 k 0 k we then show that if x is in a then s n x converges almost uniformly to an element a to prove the above theorem we prove an ergodic theorem involving completely positive maps of which the free group situation is a special case roughly if p 1 p 2 0 p 1 p 2 1 n positive maps such that 1 n p 1 n 1 p 2 n 1 n 1 with 0 x x and 1 1 1 then with a few technical assumptions we show a convergence result for lim n n x and show that lim n n 1 n 1 k 0 k x converges almost uniformly in the case p 2 0 1 a automorphism our theorems correspond to the non commutative pointwise ergodic theorem of e c lance the results partially generalize a result of kummerer our theorems also include results concerning normal operators on a hilbert space which generalizes work of guivarc h

on ergodic theorems for free group actions on noncommutative spaces we extend in a noncommutative setting the individual ergodic theorem of nevo and stein concerning measure preserving actions of free groups and averages on spheres s 2 n of even radius here we study state preserving actions of free groups on a von neumann algebra a and the behaviour of s 2 n x for x in noncommutative spaces l p a for the cesaro means this problem was solved by walker our approach is based on ideas of bufetov we prove a noncommutative version of rota alternierende verfahren theorem to this end we introduce specific dilations of the powers of some noncommutative markov operators

the method of fundamental solutions for axisymmetric potential problems in this paper we investigate the application of the method of fundamental solutions mfs to two classes of axisymmetric potential problems in the first the boundary conditions as well as the domain of the problem are axisymmetric and in the second the boundary conditions are arbitrary in both cases the fundamental solutions of the governing equations and their normal derivatives which are required in the formulation of the mfs can be expressed in terms of complete elliptic integrals the method is tested on several axisymmetric problems from the literature and is also applied to an axisymmetric free boundary problem copyright 1999 john wiley sons ltd

matrix decomposition mfs algorithms for elasticity and thermo elasticity problems in axisymmetric domains in this work we propose an efficient matrix decomposition algorithm for the method of fundamental solutions when applied to three dimensional boundary value problems governed by elliptic systems of partial differential equations in particular we consider problems arising in linear elasticity in axisymmetric domains the proposed algorithm exploits the block circulant structure of the coefficient matrices and makes use of fast fourier transforms the algorithm is also applied to problems in thermo elasticity several numerical experiments are carried out

an integral equations method combined minimum norm solution for 3d elastostatics cauchy problem abstract in this paper we establish new density results for the equilibrium equations based on the denseness result of the elastic potential functions the cauchy problem for the equilibrium equations is investigated for this ill posed problem we construct a regularizing solution using the single layer potential function the well posedness of the regularizing solution as well as the convergence property is rigorously analyzed the advantage of the proposed scheme is that the regularizing solution is of the explicit analytic solution and therefore is easy to be implemented the method combines minimum norm solution with morozov discrepancy principle to solve an inverse problem convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method the numerical convergence accuracy and stability of the method with respect to the discretisation about the integral equations on pseudo boundary and the distance between the pseudo boundary and the real boundary of the solution domain and decreasing the amount of noise added into the input data respectively are also analysed with some examples

a meshless singular boundary method for three dimensional elasticity problems summary this study documents the first attempt to extend the singular boundary method sbm a novel meshless boundary collocation method for the solution of three dimensional 3d elasticity problems the sbm involves a coupling between the regularized boundary element method bem and the method of fundamental solutions mfs the main idea here is to fully inherit the dimensionality and stability advantages of the former and the meshless and integration free attributes of the later this makes it particularly attractive for problems in complex geometries and three dimensions four benchmark 3d problems in linear elasticity are well studied to demonstrate the feasibility and accuracy of the proposed method the advantages disadvantages and potential applications of the proposed method as compared with the fem bem and mfs are also examined and discussed this article is protected by copyright all rights reserved

the method of fundamental solutions for the cauchy problem in two dimensional linear elasticity in this paper the application of the method of fundamental solutions to the cauchy problem in two dimensional isotropic linear elasticity is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularised by employing the first order tikhonov functional while the choice of the regularisation parameter is based on the l curve method numerical results are presented for both smooth and piecewise smooth geometries as well as for constant and linear stress states the convergence and the stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

inverse cauchy method with conformal mapping application to latent flatness defect detection during rolling process abstract in this paper a semi analytical inverse cauchy problem is presented for a finite cylinder with a cylindrical hole under the surface called the detecting roll applications to latent flatness defect detection during rolling process are considered a steel strip wraps at a known angle around the detecting roll the presented inverse method aims at evaluating the residual stress profile of the strip the inputs are the displacements and free surface traction at the hole surface and the outputs are the resultant forces per unit length in the contact between the strip and the roll thus by equilibrium of the strip the residual stress profile is inferred this paper is based on plane complex elasticity with conformal mapping techniques several 2d computations are performed at several axial positions this work is theoretical the measurement device is not available and technical issues are not broached some numerical examples with synthetic data are provided in order to evaluate accuracy and noise sensitivity results encourage further investigations and technical developments

an energy error based method for the resolution of the cauchy problem in 3d linear elasticity a new method is described for the problem of expanding known displacement fields at the boundary of a solid together with the surface tractions on it towards the solid interior up to the inaccessible part of the boundary the solid is supposedly linearly elastic with known elastic moduli but not necessarily homogeneous nor isotropic the problem is the cauchy problem for the lame operator a new form of this cauchy problem suited for applications associated with surface tangential fields measurements is also stated and studied the method is based on the splitting of the elastic fields into two separate solutions of wellposed problems the gap between these fields is subsequently minimised with respect to the unknown boundary data in order to produce the desired expanded elastic fields the gap used here is an energy error associated with the elastic energy of the system various 3d applications are given including non linear boundary conditions on the unreachable boundary

fading regularization mfs algorithm for inverse boundary value problems in two dimensional linear elasticity abstract we investigate the numerical reconstruction of the missing displacements dirichlet data and tractions neumann data on an inaccessible part of the boundary in the case of two dimensional linear isotropic elastic materials from the knowledge of over prescribed noisy measurements taken on the remaining accessible boundary part this inverse problem is solved using the fading regularization method originally proposed by cimetiere et al 2000 2001 for the laplace equation in conjunction with a meshless method namely the method of fundamental solutions mfs the stabilisation of the numerical method proposed herein is achieved by stopping the iterative procedure according to morozov s discrepancy principle morozov 1966

homomorphisms of local algebras in positive characteristic we study some properties of regularity of homomorphisms of local rings of positive equicharacteristic in particular we extend a theorem of izumi about the linear equivalence of topologies in this case the main tool we use is a monomialization theorem for homomorphisms of power series in positive characteristic

the method of fundamental solutions for inverse source problems associated with the steady state heat conduction this paper presents the use of the method of fundamental solutions mfs for recovering the heat source in steady state heat conduction problems from boundary temperature and heat flux measurements it is well known that boundary data alone do not determine uniquely a general heat source and hence some a priori knowledge is assumed in order to guarantee the uniqueness of the solution in the present study the heat source is assumed to satisfy a second order partial differential equation on a physical basis thereby transforming the problem into a fourth order partial differential equation which can be conveniently solved using the mfs since the matrix arising from the mfs discretization is severely ill conditioned a regularized solution is obtained by employing the truncated singular value decomposition whilst the optimal regularization parameter is determined by the l curve criterion numerical results are presented for several two dimensional problems with both exact and noisy data the sensitivity analysis with respect to two solution parameters i e the number of source points and the distance between the fictitious and physical boundaries and one problem parameter i e the measure of the accessible part of the boundary is also performed the stability of the scheme with respect to the amount of noise added into the data is analysed the numerical results obtained show that the proposed numerical algorithm is accurate convergent stable and computationally efficient for solving inverse source problems in steady state heat conduction copyright 2006 john wiley sons ltd

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

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

the almansi method of fundamental solutions for solving biharmonic problems a new fundamental solutions method for the numerical solution of two dimensional biharmonic problems is described in this method which is based on the almansi representation of a biharmonic function in the plane the approximate solution is expressed in terms of fundamental solutions of laplace s equation and is determined by a least squares fit of the boundary conditions the results of numerical experiments which demonstrate the efficacy of the method are presented

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

singular boundary method for solving plane strain elastostatic problems this study documents the first attempt to apply the singular boundary method sbm a novel boundary only collocation method to two dimensional 2d elasticity problems unlike the method of fundamental solutions mfs the source points coincide with the collocation points on the physical boundary by using an inverse interpolation technique to regularize the singularity of the fundamental solution of the equation governing the problems of interest three benchmark elasticity problems are tested to demonstrate the feasibility and accuracy of the proposed method through detailed comparisons with the mfs boundary element method bem and finite element method fem

boundary element solution for the cauchy problem in linear elasticity using singular value decomposition abstract 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

the generalized finite difference method for the inverse cauchy problem in two dimensional isotropic linear elasticity abstract 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

mathematical foundation of the mfs for certain elliptic systems in linear elasticity the method of fundamental solutions mfs is a trefftz type technique in which the solution of an elliptic boundary value problem is approximated by a linear combination of translates of fundamental solutions with singularities placed on a pseudo boundary i e a surface embracing the domain of the problem under consideration in this work we develop a mathematical framework for the numerical implementation of the mfs in elliptic systems we obtain density results with respect to the c l norms which establish the applicability of the method in certain systems arising from the theory of elastostatics and thermo elastostatics the domains in our density results may possess holes and they satisfy the segment condition

regularized mfs solution of inverse boundary value problems in three dimensional steady state linear thermoelasticity abstract we investigate the numerical reconstruction of the missing thermal and mechanical boundary conditions on an inaccessible part of the boundary in the case of three dimensional linear isotropic thermoelastic materials from the knowledge of over prescribed noisy data on the remaining accessible boundary we employ the method of fundamental solutions mfs and several singular value decomposition svd based regularization methods e g the tikhonov regularization method tikhonov and arsenin 1986 the damped svd and the truncated svd hansen 1998 whilst the regularization parameter is selected according to the discrepancy principle morozov 1966 generalized cross validation criterion golub et al 1979 and hansen s l curve method hansen and o leary 1993