Solving Inverse Conductivity Problems In Doubly Connected Domains By the Homogenization Functions of Two Parameters

In the paper, we make the first attempt to derive a family of two-parameter homogenization functions in the doubly connected domain, which is then applied as the bases of trial solutions for the inverse conductivity problems. The expansion coefficients are obtained by imposing an extra boundary condition on the inner boundary, which results in a linear system for the interpolation of the solution in a weighted Sobolev space. Then, we retrieve the spatial- or temperature-dependent conductivity function by solving a linear system, which is obtained from the collocation method applied to the nonlinear elliptic equation after inserting the solution. Although the required data are quite economical, very accurate solutions of the space-dependent and temperature-dependent conductivity functions, the Robin coefficient function and also the source function are available. It is significant that the nonlinear inverse problems can be solved directly without iterations and solving nonlinear equations. The proposed method can achieve accurate results with high efficiency even for large noise being imposed on the input data

An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation

An incomplete boundary data problem for the biharmonic equation is considered, where the displacement is known throughout the boundary of the solution domain whilst the normal derivative and bending moment are specified on only a portion of the boundary. For this inverse ill‐posed problem an iterative regularizing method is proposed for the stable data reconstruction on the underspecified boundary part. Convergence is proven by showing that the method can be written as a Landweber‐type procedure for an operator formulation of the incomplete data problem. This reformulation renders a stopping rule, the discrepancy principle, for terminating the iterations in the case of noisy data. Uniqueness of a solution to the considered problem is also shown

A boundary integral equation method for numerical solution of parabolic and hyperbolic Cauchy problems

We present a unified boundary integral approach for the stable numerical solution of the ill-posed Cauchy problem for the heat and wave equation. The method is based on a transformation in time (semi-discretisation) using either the method of Rothe or the Laguerre transform, to generate a Cauchy problem for a sequence of inhomogenous elliptic equations; the total entity of sequences is termed an elliptic system. For this stationary system, following a recent integral approach for the Cauchy problem for the Laplace equation, the solution is represented as a sequence of single-layer potentials invoking what is known as a fundamental sequence of the elliptic system thereby avoiding the use of volume potentials and domain discretisation. Matching the given data, a system of boundary integral equations is obtained for finding a sequence of layer densities. Full discretisation is obtained via a Nyström method together with the use of Tikhonov regularization for the obtained linear systems. Numerical results are included both for the heat and wave equation confirming the practical usefulness, in terms of accuracy and resourceful use of computational effort, of the proposed approach

Mathematical and numerical methods for inverse problems using fundamental solutions techniques

Fundação para a Ciência e a Tecnologia - SFRH/BD/27914/200

On the development of an efficient truly meshless discretization procedure in computational mechanics

Supervised by Mandayam A. Srinivasan and Klaus-Jurgen Bathe.Also issued as Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2001.Includes bibliographical references (leaves 157-163).by Suvranu De

On the development of an efficient truly meshless discretization procedure in computational mechanics

Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2001.Includes bibliographical references (leaves 157-163).The objective of this thesis is to present an efficient and reliable meshless computational technique - the method of finite spheres - for the solution of boundary value problems on complex domains. This method is truly meshless in the sense that the approximation spaces are generated and the numerical integration is performed without a mesh. While the theory behind meshless techniques is rather straightforward, the generation of a computationally efficient scheme is quite difficult. Computational efficiency may be achieved by proper choice of the interpolation functions, effective ways of incorporating the essential boundary conditions and efficient and specialized numerical integration rules. The pure displacement formulation is observed to exhibit volumetric "locking" during incompressible (or nearly incompressible) analysis. A displacement/pressure mixed formulation is developed to overcome this problem. The stability and optimality of the mixed formulation are tested using numerical inf-sup tests for a variety of discretization schemes. Solutions to several example problems are presented showing the application of the method of finite spheres to problems in solid and fluid mechanics. A very specialized application of the technique to physically based real time medical simulations in multimodal virtual environments is also presented. In the current form of implementation, the method of finite spheres is about five times slower than the finite element techniques for problems in two-dimensional elastostatics.by Suvranu De.Sc.D

A high-performance boundary element method and its applications in engineering

As a semi-numerical and semi-analytical method, owing to the inherent advantage, of boundary-only discretisation, the boundary element method (BEM) has been widely applied to problems with complicated geometries, stress concentration problems, infinite domain problems, and many others. However, domain integrals and non-symmetrical and dense matrix systems are two obstacles for BEM which have hindered the its further development and application. This thesis is aimed at proposing a high-performance BEM to tackle the above two drawbacks and broaden the application scope of BEM. In this thesis, a detailed introduction to the traditional BEM is given and several popular algorithms are introduced or proposed to enhance the performance of BEM. Numerical examples in heat conduction analysis, thermoelastic analysis and thermoelastic fracture problems are performed to assess the efficiency and correction of the algorithms. In addition, necessary theoretical derivations are embraced for establishing novel boundary integral equations (BIEs) for specific engineering problems. The following three parts are the main content of this thesis. (1) The first part (Part II consisting of two chapters) is aimed at heat conduction analysis by BEM. The coefficient matrix of equations formed by BEM in solving problems is fully-populated which occupy large computer memory. To deal with that, the fast multipole method (FMM) is introduced to energize the line integration boundary element method (LIBEM) to performs better in efficiency. In addition, to compute domain integrals with known or unknown integrand functions which are caused by heat sources or heterogeneity, a novel BEM, the adaptive orthogonal interpolation moving least squares (AOIMLS) method enhanced LIBEM, which also inherits the advantage of boundary-only discretisation, is proposed. Unlike LIBEM, which is an accurate and stable method for computing domain integrals, but only works when the mathematical expression of integral function in domain integrals is known, the AOIMLS enhanced LIBEM can compute domain integrals with known or unknown integral functions, which ensures all the nonlinear and nonhomogeneous problems can be solved without domain discretisation. In addition, the AOIMLS can adaptively avoid singular or ill-conditioned moment matrices, thus ensuring the stability of the calculation results. (2) In the second part (Part III consisting of four chapters), the thermoelastic problems and fracture problems are the main objectives. Due to considering thermal loads, domain integrals appear in the BIEs of the thermoelastic problems, and the expression of integrand functions is known or not depending on the temperature distribution given or not, the AOIMLS enhanced LIBEM is introduced to conduct thermoelasticity analysis thereby. Besides, a series of novel unified boundary integral equations based on BEM and DDM are derived for solving fracture problems and thermoelastic fracture problems in finite and infinite domains. Two sets of unified BIEs are derived for fracture problems in finite and infinite domains based on the direct BEM and DDM respectively, which can provide accurate and stable results. Another two sets of BIEs are addressed by employing indirect BEM and DDM, which cannot ensure a stable result, thereby a modified indirect BEM is proposed which performs much more stable. Moreover, a set of novel BIEs based on the direct BEM and DDM for cracked domains under thermal stress is proposed. (3) In the third part (Part IV consisting of one chapter), a high-efficiency combined BEM and discrete element method (DEM) is proposed to compute the inner stress distribution and particle breakage of particle assemblies based on the solution mapping scheme. For the stress field computation of particles with similar geometry, a template particle is used as the representative particle, so that only the related coefficient matrices of one template particle in the local coordinate system are needed to be calculated, while the coefficient matrices of the other particles, can be obtained by mapping between the local and global coordinate systems. Thus, the combined BEM and DEM is much more effective when modelling a large-scale particle system with a small number of distinct possible particle shapes. Furthermore, with the help of the Hoek-Brown criterion, the possible cracks or breakage paths of a particle can be obtained

Mathematical and Numerical Aspects of Dynamical System Analysis

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”

The Method of Fundamental Solutions and MCMC Methods for Solving Electrical Tomography Problems

Electrical impedance tomography (EIT) is a non-intrusive and portable imaging technique which has been used widely in many medical, geological and industrial applications for imaging the interior electrical conductivity distribution within a region from the knowledge of the injected currents through attached electrodes and resulting voltages, or boundary potential and current flux. If the quantities involved are all real then EIT is called electrical resistance tomography (ERT). The work in this thesis focuses on solving inverse geometric problems in ERT where we seek detecting the size, the shape and the location of inner objects within a given bounded domain. These ERT problems are governed by Laplace’s equation subject either to the most practical and general boundary conditions, forming the socalled complete-electrode model (CEM), in two dimensions or to the more idealised boundary conditions in three-dimensions called the continuous model. Firstly, the method of the fundamental solutions (MFS) is applied to solve the forward problem of the two-dimensional complete-electrode model of ERT in simplyconnected and multiple-connected domains (rigid inclusion, cavity and composite bimaterial), as well as providing the corresponding MFS solutions for the three-dimensional continuous model. Secondly, a Bayesian approach and the Markov Chain Monte Carlo (MCMC) estimation technique are employed in combinations with the numerical MFS direct solver in order to obtain the inverse solution. The MCMC algorithm is not only used for reconstruction, but it also deals with uncertainty assessment issues. The reliability and accuracy of a fitted object is investigated through some meaningful statistical aspects such as the object boundary histogram and object boundary credible intervals