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

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

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

shape constrained partial identification of a population mean under unknown probabilities of sample selection summary estimating a population mean from a sample obtained with unknown selection probabilities is important in the biomedical and social sciences using a ratio estimator aronow lee 2013 proposed a method for partial identification of the mean by allowing the unknown selection probabilities to vary arbitrarily between two fixed values in this paper we show how to use auxiliary shape constraints on the population outcome distribution such as symmetry or log concavity to obtain tighter bounds on the population mean we use this method to estimate the performance of aymara students an ethnic minority in the north of chile in a national educational standardized test we implement this method in the r package scbounds

effective algorithms with circulant block matrices abstract effective numerical algorithms for circulant block matrices a whose blocks are circulant are obtained the eigenvalues of such matrices are determined in terms of the eigenvalues of matrices of reduced dimension and systems of linear equations involving these matrices are solved efficiently using fast fourier transforms

applicability and applications of the method of fundamental solutions in the present work we investigate the applicability of the method of fundamental solutions for the solution of boundary value problems of elliptic partial differential equations and elliptic systems more specifically we study whether linear combinations of fundamental solutions can approximate the solutions of the boundary value problems under consideration in our study the singularities of the fundamental solutions lie on a prescribed pseudo boundary the boundary of a domain which embraces the domain of the problem under consideration we extend previous density results of kupradze and aleksidze and of bogomolny to more general domains and partial differential operators and with respect to more appropriate norms our domains may possess holes and their boundaries are only required to satisfy a rather weak boundary requirement namely the segment condition our density results are with respect to the norms of the spaces c l analogous density results are obtainable with respect to holder norms we have studied approximation by fundamental solutions of the laplacian biharmonic and m harmonic and modified helmholtz and poly helmholtz operators in the case of elliptic systems we obtain analogous density results for the cauchy navier operator as well as for an operator which arises in the linear theory of thermo elasticity we also study alternative formulations of the method of fundamental solutions in cases when linear combinations of fundamental solutions of the equations under consideration are not dense in the solution space finally we show that linear combinations of fundamental solutions of operators of order m 4 with singularities lying on a prescribed pseudo boundary are not in general dense in the corresponding solution space

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

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

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

kansa rbf algorithms for elliptic problems in axisymmetric domains we employ a kansa radial basis function method for the numerical solution of elliptic boundary value problems in three dimensional axisymmetric domains we consider problems governed by the poisson equation the inhomogeneous biharmonic equation and the inhomogeneous cauchy navier equations appropriate discretizations lead to linear systems the coefficient matrices of which possess block circulant structures these systems are then solved efficiently by means of matrix decomposition algorithms and fast fourier transforms methods for choosing appropriate values of the shape parameter are also proposed the effectiveness of the proposed algorithms is demonstrated by considering several numerical examples

orthogonal spline collocation methods for partial differential equations abstract this paper provides an overview of the formulation analysis and implementation of orthogonal spline collocation osc also known as spline collocation at gauss points for the numerical solution of partial differential equations in two space variables advances in the osc theory for elliptic boundary value problems are discussed and direct and iterative methods for the solution of the osc equations examined the use of osc methods in the solution of initial boundary value problems for parabolic hyperbolic and schrodinger type systems is described with emphasis on alternating direction implicit methods the osc solution of parabolic and hyperbolic partial integro differential equations is also mentioned finally recent applications of a second spline collocation method modified spline collocation are outlined

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

deligne s integrality theorem in unequal characteristic and rational points over finite fields if v is a smooth projective variety defined over a local field k with fi nite residue field so that itscohomology over the algebraic closure k is supported in codimension 1 then the mod p reduction of a projective regular model carries a rational point as a consequence if the chow group of 0 cycles of v over a large algebraically closed field is trivial then the mod p reduction of a projective regular model carries a rational point

degenerations of rationally connected varieties we prove that a degeneration of rationally connected varieties over a field of characteristic zero always contains a geometrically irreducible subvariety which is rationally connected

matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions abstract bialecki b and g fairweather matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions journal of computational and applied mathematics 46 1993 369 386 in recent years several matrix decomposition algorithms have been developed for the efficient solution of the linear algebraic systems arising when finite difference finite element galerkin and spectral methods are applied to separable elliptic boundary value problems in a rectangle the success of these methods depends on knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two point boundary value problems in one space dimension the primary purpose of this paper is to provide an overview of matrix decomposition algorithms and show how they can be expressed in terms of a unifying framework particular emphasis is placed on algorithms formulated recently by the authors for solving the linear systems arising in orthogonal spline collocation that is spline collocation at gauss points all of the methods discussed in this paper are modular and possess a great deal of natural parallelism

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

method of fundamental solutions in micromechanics of elastic random structure composites abstract one considers linearly elastic composite media which consist of a homogeneous matrix containing a statistically homogeneous random set of aligned homogeneous heterogeneities of non canonical i e nonellipsoidal shape the new general integral equations connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities the method is based on a recently developed centering procedure where the notion of a perturbator is introduced in terms of boundary interface integrals that makes it possible to reconsider basic concepts of micromechanics such as effective field hypothesis quasi crystalline approximation and the hypothesis of ellipsoidal symmetry the results of this reconsideration are quantitatively estimated for some modeled composite reinforced by aligned homogeneous heterogeneities of non canonical shape some new effects are detected that are impossible in the framework of a classical background of micromechanics

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

interval estimation of population means under unknown but bounded probabilities of sample selection applying concepts from partial identification to the domain of finite population sampling we propose a method for interval estimation of a population mean when the probabilities of sample selection lie within a posited interval the interval estimate is derived from sharp bounds on the hajek 1971 estimator of the population mean we demonstrate the method s utility for sensitivity analysis by applying it to a sample of needles collected as part of a syringe tracking and testing programme in new haven connecticut copyright 2013 oxford university press

fundamental solutions method for elliptic boundary value problems we consider a procedure for solving boundary value problems for elliptic homogeneous equations known as the fundamental solutions method we prove its applicability for some second order operators as well as for fourth order ones the boundary conditions of an elliptic problem are approximated by using fundamental solutions of the corresponding operator with singularities located outside the domain of interest in a specific case of the laplacian an estimate shows that the method discussed possesses convergence properties as good as those of any method using harmonic polynomials as trial functions numerical examples are given among which are simple signorini s problems for harmonic and lame operators

short note the method of fundamental solutions for 2d and 3d stokes problems a numerical scheme based on the method of fundamental solutions mfs is proposed for the solution of 2d and 3d stokes equations the fundamental solutions of the stokes equations stokeslets are adopted as the sources to obtain flow field solutions the present method is validated through other numerical schemes for lid driven flows in a square cavity and a cubic cavity test results obtained for a rectangular cavity with wave shaped bottom indicate that the mfs is computationally efficient than the finite element method fem in dealing with irregular shaped domain the paper also discusses the effects of number of source points and their locations on the numerical accuracy

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

stress intensity factor computation using the method of fundamental solutions mixed mode problems the method of fundamental solutions is applied to the computation of stress intensity factors in linear elastic fracture mechanics the displacements are approximated by linear combinations of the fundamental solutions of the cauchy navier equations of elasticity and the leading terms for the displacement near the crack tip two algorithms are developed one using a single domain and one using domain decomposition numerical results are given copyright 2006 john wiley sons ltd

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

congruences for rational points on varieties over finite fields we prove the existence of rational points on singular varieties over finite fields arising as degenerations of smooth proper varieties with trivial chow group of 0 cycles we also obtain congruences for the number of rational points of singular varieties appearing as fibres of a proper family with smooth total and base space and such that the chow group of 0 cycles of the generic fibre is trivial in particular this leads to a vast generalization of the classical chevalley warning theorem the above results are obtained as special cases of our main theorem which can be viewed as a relative version of a theorem of h esnault on the number of rational points of smooth proper varieties over finite fields with trivial chow group of 0 cycles

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

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