Boundary knot method for some inverse problems associated with the Helmholtz equation

The boundary knot method is an inherently meshless, integration‐free, boundary‐type, radial basis function collocation technique for the solution of partial differential equations. In this paper, the method is applied to the solution of some inverse problems for the Helmholtz equation, including the highly ill‐posed Cauchy problem. Since the resulting matrix equation is badly ill‐conditioned, a regularized solution is obtained by employing truncated singular value decomposition, while the regularization parameter for the regularization method is provided by the L‐curve method. Numerical results are presented for both smooth and piecewise smooth geometry. The stability of the method with respect to the noise in the data is investigated by using simulated noisy data. The results show that the method is highly accurate, computationally efficient and stable, and can be a competitive alternative to existing methods for the numerical solution of the problems. Copyright © 2005 John Wiley & Sons, Ltd.     

A boundary element strategy for elastostatic inverse problems involving uncertain boundary conditions

In this paper, a convenient strategy is developed to find solutions for a class of uncertain‐boundary‐value problems by the Boundary Element Method (BEM). Such problems are ill‐posed, but ill‐conditioning of the associated algebraic systems of equations can be controlled to a large extent, and useful boundary data can be obtained despite ill‐conditioning. Interior data of not only sufficient quantity, but also of good quality at good locations contribute to good solutions. Our strategy permits the condition number of the algebraic systems, as a function of interior‐data locations, to be monitored, such that measured data from displacement sensors and/or strain sensors, at locations found to be good ones for the problem at hand, can be used. The present strategy is based upon the concept of a Green's‐function library through partitioning of the BEM algebraic system. Algebraic systems are solved using least squares via Singular Value Decomposition (SVD). The library idea takes advantage of modern data storage and retrieval technology and permits the process of repeated trials, in order to determine good data sensor locations, to be done quickly and efficiently. Several numerical examples are given to demonstrate the strategy. Some examples examine the consequences of errors in measured data. Copyright © 1999 John Wiley & Sons, Ltd.     

Stochastic inverse heat conduction using a spectral approach

An adjoint‐based functional optimization technique in conjunction with the spectral stochastic finite element method is proposed for the solution of an inverse heat conduction problem in the presence of uncertainties in material data, process conditions and measurement noise. The ill‐posed stochastic inverse problem is restated as a conditionally well‐posed L₂ optimization problem. The gradient of the objective function is obtained in a distributional sense by defining an appropriate stochastic adjoint field. The L₂ optimization problem is solved using a conjugate‐gradient approach. Accuracy and effectiveness of the proposed approach is appraised with the solution of several stochastic inverse heat conduction problems. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Boundary element method applied to three dimensional thermoelastic contact

In this paper we present a boundary element method to analyze and solve three dimensional frictionless thermoelastic contact problems. Although many problems in engineering can be solved with one-dimensional or two-dimensional models, those simplifications there are not possible in many others, such as the design of microelectronics packages. We calculate the stresses, movements, temperatures and thermal gradients on 3D solids. A thermal resistance at the contact zone depends on the local pressure is considered. The problem is solved by a double iterative method, so that in the final solution do not appear tensions in the contact zone or penetrations between the two solids. The solutions are compared with other works, where possible, to validate the method.     

XFEM‐based crack detection scheme using a genetic algorithm

A new computational tool is developed for the accurate detection and identification of cracks in structures, to be used in conjunction with non‐destructive testing of specimens. It is based on the solution of an inverse problem. Based on some measurements, typically along part of the boundary of the structure, that describe the response of the structure to vibration in a chosen frequency or a combination of frequencies, the goal is to estimate whether the structure contains a crack, and if so, to find the parameters (location, size, orientation and shape) of the crack that produces a response closest to the given measurement data in some chosen norm. The inverse problem is solved using a genetic algorithm (GA). The GA optimization process requires the solution of a very large amount of forward problems. The latter are solved via the extended finite element method (XFEM). This enables one to employ the same regular mesh for all the forward problems. Performance of the method is demonstrated via a number of numerical examples involving a cracked flat membrane. Various computational aspects of the method are discussed, including the a priori estimation of the ill‐posedness of the crack identification problem. Copyright © 2007 John Wiley & Sons, Ltd.     

Indirect T‐Trefftz and F‐Trefftz methods for solving boundary value problem of Poisson equation

Abstract

The Poisson equation can be solved by first finding a particular solution and then solving the resulting Laplace equation. In this paper, a computational procedure based on the Trefftz method is developed to solve the Poisson equation for two‐dimensional domains. The radial basis function approach is used to find an approximate particular solution for the Poisson equation. Then, two kinds of Trefftz methods, the T‐Trefftz method and F‐Trefftz method, are adopted to solve the resulting Laplace equation. In order to deal with the possible ill‐posed behaviors existing in the Trefftz methods, the truncated singular value decomposition method and L‐curve concept are both employed. The Poisson equation of the type, ?² u = f(x, u), in which x is the position and u is the dependent variable, is solved by the iterative procedure. Numerical examples are provided to show the validity of the proposed numerical methods and some interesting phenomena are carefully discussed while solving the Helmholtz equation as a Poisson equation. It is concluded that the F‐Trefftz method can deal with a multiply connected domain with genus p(p > 1) while the T‐Trefftz method can only deal with a multiply connected domain with genus 1 if the domain partition technique is not adopted.     

A comparison of different methods to solve inverse biharmonic boundary value problems

The Boundary Element Method (BEM) is applied to solve numerically some inverse boundary value problems associated to the biharmonic equation which involve over‐ and under‐specified boundary portions of the solution domain. The resulting ill‐conditioned system of linear equations is solved using the regularization and the minimal energy methods, followed by a further application of the Singular Value Decomposition Method (SVD). The regularization method incorporates a smoothing effect into the least squares functional, whilst the minimal energy method is based on minimizing the energy functional for the Laplace equation subject to the linear constraints generated by the BEM discretization of the biharmonic equation. The numerical results are compared with known analytical solutions and the stability of the numerical solution is investigated by introducing noise into the input data. Copyright © 1999 John Wiley & Sons, Ltd.     

Global space–time multiquadric method for inverse heat conduction problem

In this paper, a radial basis collocation method (RBCM) based on the global space–time multiquadric (MQ) is proposed to solve the inverse heat conduction problem (IHCP). The global MQ is simply constructed by incorporating time dimension into the MQ function as a new variable in radial coordinate. The method approximates the IHCP as an over‐determined linear system with the use of two sets of collocation points: one is satisfied with the governing equation and another is for the given conditions. The least‐square technique is introduced to find the solution of the over‐determined linear system. The present work investigates two types of the ill‐posed heat conduction problems: the IHCP to recover the surface temperature and heat flux history on a source point from the measurement data at interior locations, and the backward heat conduction problem (BHCP) to retrieve the initial temperature distribution from the known temperature distribution at a given time. Numerical results of four benchmark examples show that the proposed method can provide accurate and stable numerical solutions for one‐dimensional and two‐dimensional IHCP problems. The sensitivity of the method with respect to the measured data, location of measurement, time step, shape parameter and scaling factor is also investigated. Copyright © 2010 John Wiley & Sons, Ltd.     

A fast BE‐FE coupling scheme for partly immersed bodies

Fluid–structure coupled problems are investigated to predict the vibro‐acoustic behavior of submerged bodies. The finite element method is applied for the structural part, whereas the boundary element method is used for the fluid domain. The focus of this paper is on partly immersed bodies. The fluid problem is favorably modeled by a half‐space formulation. This way, the Dirichlet boundary condition on the free fluid surface is incorporated by a half‐space fundamental solution. A fast multipole implementation is presented for the half‐space problem. In case of a high density of the fluid, the forces due to the acoustic pressure, which act on the structure, cannot be neglected. Thus, a strong coupling scheme is applied. An iterative solver is used to handle the coupled system. The efficiency of the proposed approach is discussed using a realistic model problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Local transmitting boundaries for the generalized interpolation material point method

The boundary‐value problems of mechanics can be solved using the material point method with explicit solver formulations. In explicit formulations, even quasi‐static problems are solved as if dynamic, which means that waves are reflected at computational boundaries, generating spurious oscillations in the solution to the boundary‐value problem. Such oscillations can be reduced to a level such that they are barely noticeable with the use of transmitting boundaries. Current implementations of transmitting boundaries in the material point method are limited to the standard viscous boundary. The absence of any stiffness component in the standard viscous boundary may lead to an undesirable finite rigid‐body motion over time. This motion can be minimized through the adoption of the transmitting cone boundary that approximates the stiffness of the unbounded domain. This paper lays out the implementation of the transmitting cone boundary for the generalized interpolation material point method. The cone boundary reflection‐canceling tractions can be applied to either the edges or the centroids of material points; this paper discusses the implications of both approaches.     

Analytic preconditioners for the electric field integral equation

Since the advent of the fast multipole method, large‐scale electromagnetic scattering problems based on the electric field integral equation (EFIE) formulation are generally solved by a Krylov iterative solver. A well‐known fact is that the dense complex non‐hermitian linear system associated to the EFIE becomes ill‐conditioned especially in the high‐frequency regime. As a consequence, this slows down the convergence rate of Krylov subspace iterative solvers. In this work, a new analytic preconditioner based on the combination of a finite element method with a local absorbing boundary condition is proposed to improve the convergence of the iterative solver for an open boundary. Some numerical tests precise the behaviour of the new preconditioner. Moreover, comparisons are performed with the analytic preconditioner based on the Calderòn's relations for integral equations for several kinds of scatterers. Copyright © 2004 John Wiley & Sons, Ltd.     

Three‐dimensional steady thermal stress analysis by triple‐reciprocity boundary element method

The steady thermal stress problems without heat generation can be solved easily by the boundary element method. However, for the case with arbitrary heat generation, the domain integral is necessary. In this paper, it is shown that the problems of three‐dimensional steady thermal stress with heat generation can be approximately solved without the domain integral by the triple‐reciprocity boundary element method. In this method, an arbitrary distribution of heat generation is interpolated by boundary integral equations. In order to solve the problem, the values of heat generation at internal points and on the boundary are used. Copyright © 2005 John Wiley & Sons, Ltd.     

A domain decomposition method for discontinuous Galerkin discretizations of Helmholtz problems with plane waves and Lagrange multipliers

A nonoverlapping domain decomposition (DD) method is proposed for the iterative solution of systems of equations arising from the discretization of Helmholtz problems by the discontinuous enrichment method. This discretization method is a discontinuous Galerkin finite element method with plane wave basis functions for approximating locally the solution and dual Lagrange multipliers for weakly enforcing its continuity over the element interfaces. The primal subdomain degrees of freedom are eliminated by local static condensations to obtain an algebraic system of equations formulated in terms of the interface Lagrange multipliers only. As in the FETI‐H and FETI‐DPH DD methods for continuous Galerkin discretizations, this system of Lagrange multipliers is iteratively solved by a Krylov method equipped with both a local preconditioner based on subdomain data, and a global one using a coarse space. Numerical experiments performed for two‐ and three‐dimensional acoustic scattering problems suggest that the proposed DD‐based iterative solver is scalable with respect to both the size of the global problem and the number of subdomains. Copyright © 2009 John Wiley & Sons, Ltd.     

Investigations on boundary temperature control analysis considering a moving body based on the adjoint variable and the fictitious domain finite element methods

This paper presents an examination of moving‐boundary temperature control problems. With a moving‐boundary problem, a finite‐element mesh is generated at each time step to express the position of the boundary. On the other hand, if an overlapped domain, that is, comprising foreground and background meshes, is prepared, the moving boundary problem can be solved without mesh generation at each time step by using the fictitious domain method. In this study, boundary temperature control problems with a moving boundary are formulated using the finite element, the adjoint variable, and the fictitious domain methods, and several numerical experiments are carried out. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical solution of the variation boundary integral equation for inverse problems

In this paper a procedure to solve the identification inverse problems for two‐dimensional potential fields is presented. The procedure relies on a boundary integral equation (BIE) for the variations of the potential, flux, and geometry. This equation is a linearization of the regular BIE for small changes in the geometry. The aim in the identification inverse problems is to find an unknown part of the boundary of the domain, usually an internal flaw, using experimental measurements as additional information. In this paper this problem is solved without resorting to a minimization of a functional, but by an iterative algorithm which alternately solves the regular BIE and the variation BIE. The variation of the geometry of the flaw is modelled by a virtual strainfield, which allows for greater flexibility in the shape of the assumed flaw. Several numerical examples demonstrate the effectiveness and reliability of the proposed approach. Copyright © 2000 John Wiley & Sons, Ltd.     

Direct method of solution for general boundary value problem of the Laplace equation

A general boundary value problem for two-dimensional Laplace equation in the domain enclosed by a piecewise smooth curve is considered. The Dirichlet and the Neumann data are prescribed on respective parts of the boundary, while there is the second part of the boundary on which no boundary data are given. There is the third part of the boundary on which the Robin condition is prescribed. This problem of finding unknown values along the whole boundary is ill posed. In this sense we call our problem an inverse boundary value problem. In order for a solution to be identified the inverse problem is reformulated in terms of a variational problem, which is then recast into primary and adjoint boundary value problems of the Laplace equation in its conventional form. A direct method for numerical solution of the inverse boundary value problem using the boundary element method is presented. This method proposes a non-iterative and unified treatment of conventional boundary value problem, the Cauchy problem, and under- or over-determined problems.     

二维泊松方程问题Lagrange型有限元p型超收敛算法

该文针对二维泊松方程问题的Lagrange型有限元法提出了一种p型超收敛算法。该法受有限元线法对二维问题降维思想的启发,基于网格结点位移的天然超收敛性,通过从网格中取出一行对边相邻的单元作一子域,将子域内各单元另一对边解答取为原有限元解答,在子域上建立真解近似满足的局部偏微分方程边值问题,对该局部边值问题,沿对边方向单向提高单元阶次进行有限元求解获得单元对边上的超收敛解。单元另一对边上的超收敛解可通过另一方向的单元行类似获得。在单元边超收敛解的基础上,依次取出各个单元,以单元边位移超收敛解为Dirichlet边界条件,双向提高单元阶次对原泊松方程问题进行有限元求解即可获得全域超收敛解。数值算例表明,通过简单的后处理计算本法可显著提高解答的精度和收敛阶。     

The boundary element method applied to a moving free boundary problem

In this paper a boundary problem is considered for which the boundary is to be determined as part of the solution. A time‐dependent problem involving linear diffusion in two spatial dimensions which results in a moving free boundary is posed. The fundamental solution is introduced and Green’s Theorem is used to yield a non‐linear system of integral equations for the unknown solution and the location of the boundary. The boundary element method is used to obtain a numerical solution to this system of integral equations which in turn is used to obtain the solution of the original problem. Graphical results for a two‐dimensional problem are presented. Published in 1999 by John Wiley & Sons, Ltd.     

Analysis of cracked plates using an iterative hybrid technique of boundary element method and distributed dislocation method

An iterative hybrid technique of boundary element method (BEM) and distributed dislocation method (DDM) is introduced for solving two dimensional crack problems. The technique decomposes the problem into (n + 1) subsidiary problems where n is the number of crack branches. The required solution will be the sum of these (n + 1) solutions. The first subsidiary problem is to find the stress distribution induced in the plate in the absence of the crack using BEM. All of the remaining subsidiary problems, are stress disturbance ones that will be solved using DDM. The results will be added and compared with the boundary conditions of the original problem. Iteration will be performed between the plate boundaries and crack faces until all of the boundary conditions are satisfied.