A CQM-based BEM for transient heat conduction problems in homogeneous materials and FGMs

A method for numerical solution of time-domain boundary integral formulations of transient problems governed by the heat equation is presented. The heat conduction problem is analyzed considering homogeneous and non-homogeneous media. In the case of the non-homogeneous media, the conductor material is assumed to be a functionally graded material, i.e., the material properties vary spatially according to known smooth functions. For some specific spatial variations of the material properties, the fundamental solution and the boundary integral equation of the problem are obtained thanks to a change of variables that transforms the original problem to the standard heat conduction problem for homogeneous materials. For the treatment of time-dependent terms, the convolution quadrature method is adopted to approximate numerically the integral equation of the time-domain boundary element method. In the case that the responses are required at a large number of interior points, the convolution performed to calculate them is very time consuming. It is shown that the discrete convolution of the proposed formulation can be computed by means of the fast Fourier transform technique, which considerably reduces the computational complexity. Results for some transient heat conduction examples are presented to validate the numerical techniques studied.     

Boundary element bending analysis of heated elastic plates

Generally, the boundary integral equations for nonlinear and inhomogeneous problems need to be formulated in terms of boundary and domain integrals. The present paper declares the possibility of the transformation of the domain integral term into the boundary integral for the thermal bending analysis of a thin elastic plate,provided that the temperature can be expressed by a linear variation over the thickness.     

Boundary element methods for potential problems with nonlinear boundary conditions

Galerkin boundary element methods for the solution of novel first kind Steklov-Poincaré and hypersingular operator boundary integral equations with nonlinear perturbations are investigated to solve potential type problems in two- and three-dimensional Lipschitz domains with nonlinear boundary conditions. For the numerical solution of the resulting Newton iterate linear boundary integral equations, we propose practical variants of the Galerkin scheme and give corresponding error estimates. We also discuss the actual implementation process with suitable preconditioners and propose an optimal hybrid solution strategy.

     

Stability of a streamline diffusion finite element method for turning point problems

A one-dimensional singularly perturbed problem with a boundary turning point is considered in this paper. Let V^h be the linear finite element space on a suitable grid . A variant of streamline diffusion finite element method is proved to be almost uniform stable in the sense that the numerical approximation u_h satisfies u-u_h∞C|lnε| inf_vhV^hu-v_h∞, where C is independent with the small diffusion coefficient ε and the mesh . Such stability result is applied to layer-adapted grids to obtain almost ε-uniform second order scheme for turning point problems.     

Dynamic analysis of plane elasticity with new complex Fourier radial basis functions in the dual reciprocity boundary element method

In this paper, dual reciprocity (DR) boundary element method (BEM) is reformulated using new radial basis function (RBF) to approximate the inhomogeneous term of Navier’s differential equation (i.e., inertia term). This new RBF, which is in the form of exp(iωr), is called complex Fourier RBF hereafter. The present RBF has simultaneously collected the properties of Gaussian and real Fourier RBF reported in literature together. Consequently, this promising feature has provided more robustness and potency of the proposed method. The required kernels for displacement and traction particular solutions are derived by employing the method of variation of parameters. As some terms of these kernels are singular, a new simple smoothing trick is employed to resolve the singularity problem. Moreover, the limiting values of relevant kernels are evaluated. The validity, accuracy, and strength of the present formulation are illustrated throughout several numerical examples. The numerical results show that the proposed complex Fourier RBF represents more accurate solutions, using less degree of freedom compared to other RBFs available in the literature.     

A sequential algorithm and error sensitivity analysis for the inverse heat conduction problems with multiple heat sources

This paper proposes a sequential approach to determine the unknown parameters for inverse heat conduction problems which have multiple time-dependent heat sources. There are two main aims in this study, one is to derive an inverse algorithm that can estimate the unknown conditions effectively, and the other is to bring up a theoretical sensitivity analysis to discuss what causes the growth of errors. This paper has three major achievements with regard to the literature on IHCPs, as follows: (1) proposing an efficient sequential inverse algorithm that can simultaneously determine several unknown time-dependent parameters; (2) exploring why the sequential function specification method can provide a stable but inaccurate estimation when tackling problems with larger measurement errors; and (3) discussing the sensitivity problem and analyzing what factors cause the growth in error sensitivity. Three examples are applied to demonstrate the performance of the proposed method, and the numerical results show that the accurate estimations can be obtained by alleviating the error sensitivity when the measurement error is considered.     

A meshfree method with dynamic node reconfiguration for analysis of thermo-elastic problems with moving concentrated heat sources

In this work, a novel approach for efficient analysis of transient thermo-elastic problems including a moving point heat source is presented. This approach is based on a meshfree method with dynamic reconfiguration of the nodal points. In order to accurately capture the large temperature gradients at the location of the concentrated heat source, a fine configuration of nodal points at this location is selected. In contrast, a coarser nodal arrangement is used in other parts of the problem domain. During the problem analysis, the fine nodal arrangement moves with the point heat source. Consequently, the meshfree methods are ideally suited to this approach. In the present work, the meshfree radial point interpolation method (RPIM) is adopted for the numerical analyses. Since the density of the nodal points varies in different parts of the domain, the background decomposition method (BDM) is used for efficient computation of the domain integrals. In the BDM, the density of the integration points conform to that of the nodal points and thus the computational effort is minimized. Some numerical examples are provided to assess the accuracy and usefulness of the proposed approach in computation of the temperature, displacement, and stress fields.     

Convergence analysis of an upwind finite volume element method with Crouzeix-Raviart element for non-selfadjoint and indefinite problems

In this paper we construct an upwind finite volume element scheme based on the Crouzeix-Raviart nonconforming element for non-selfadjoint elliptic problems. These problems often appear in dealing with flow in porous media. We establish the optimal order H ¹-norm error estimate. We also give the uniform convergence under minimal elliptic regularity assumption      

The versions of the finite element method for problems with boundary layers

We study the uniform approximation of boundary layer functions for , , by the and versions of the finite element method. For the version (with fixed mesh), we prove super-exponential convergence in the range . We also establish, for this version, an overall convergence rate of in the energy norm error which is uniform in , and show that this rate is sharp (up to the term) when robust estimates uniform in are considered. For the version with variable mesh (i.e., the version), we show that exponential convergence, uniform in , is achieved by taking the first element at the boundary layer to be of size . Numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates, even when few degrees of freedom are used and when is as small as, e.g., . They also illustrate the superiority of the approach over other methods, including a low-order version with optimal ``exponential" mesh refinement. The estimates established in this paper are also applicable in the context of corresponding spectral element methods.

     

A parallel version of the preconditioned conjugate gradient method for boundary element equations

The parallel version of precondition techniques is developed for matrices arising from the Galerkin boundary element method for two-dimensional domains with Dirichlet boundary conditions. Results were obtained for implementations on a transputer network as well as on an nCUBE-2 parallel computer showing that iterative solution methods are very well suited for a MIMD computer. A comparison of numerical results for iterative and direct solution methods is presented and underlines the superiority of iterative methods for large systems.     

Boundary element tearing and interconnecting methods in unbounded domains

Finite element tearing and interconnecting (FETI) methods and boundary element tearing and interconnecting (BETI) methods are special iterative substructuring methods with Lagrange multipliers. For elliptic boundary value problems on bounded domains, the condition number of these methods can be rigorously bounded by C(1+log(H/h))², where H is the subdomain diameter and h the mesh size. The constant C is independent of H, h and possible jumps in the coefficients of the partial differential equation.In certain situations, e.g., in electromagnetic field computations, instead of imposing artificial boundary conditions one may be interested in modelling the real physical behaviour in an exterior domain with a radiation condition. In this work we analyze one-level BETI methods for such unbounded domains and show explicit condition number estimates similar to the one above. Our theoretical results are confirmed in numerical experiments.     

A numerical investigation of blow-up in reaction-diffusion problems with traveling heat sources

This paper studies the numerical solution of a reaction-diffusion differential equation with traveling heat sources. According to the fact that the locations of heat sources are known, we add auxiliary mesh points exactly at heat sources and present a novel moving mesh algorithm for solving the problem. Several examples are provided to demonstrate the efficiency of the new moving mesh method, especially in the case of two or three traveling heat sources. Moreover, numerical results illustrate that the speed of the movement of the heat source is critical for blow-up when there is only one traveling heat source. For the case of two traveling heat sources, blow-up depends not only on the speed but also on the distance between the two traveling heat sources.     

Convergence analysis of an upwind mixed element method for advection diffusion problems

We consider a upwinding mixed element method for a system of first order partial differential equations resulting from the mixed formulation of a general advection diffusion problem. The system can be used to model the transport of a contaminant carried by a flow. We use the lowest order Raviart-Thomas mixed finite element space. We show the first order convergence both for concentration and concentration flux in L²(Ω).     

Finite element approximation with numerical integration for differential eigenvalue problems

Error estimates of the finite element method with numerical integration for differential eigenvalue problems are presented. More specifically, refined results on the eigenvalue dependence for the eigenvalue and eigenfunction errors are proved. The theoretical results are illustrated by numerical experiments for a model problem.     

Finite element and boundary element contact stress analysis with remeshing technique

The fundamental part of the contact stress problem solution using a finite element method is to locate possible contact areas reliably and efficiently. In this research, a remeshing technique is introduced to determine the contact region in a given accuracy. In the proposed iterative method, the meshes near the contact surface are modified so that the edge of the contact region is also an element’s edge. This approach overcomes the problem of surface representation at the transition point from contact to non-contact region. The remeshing technique is efficiently employed to adapt the mesh for more precise representation of the contact region. The method is applied to both finite element and boundary element methods. Overlapping of the meshes in the contact region is prevented by the inclusion of displacement and force constraints using the Lagrange multipliers technique. Since the method modifies the mesh only on the contacting and neighbouring region, the solution to the matrix system is very close to the previous one in each iteration. Both direct and iterative solver performances on BEM and FEM analyses are also investigated for the proposed incremental technique. The biconjugate gradient method and LU with Cholesky decomposition are used for solving the equation systems. Two numerical examples whose analytical solutions exist are used to illustrate the advantages of the proposed method. They show a significant improvement in accuracy compared to the solutions with fixed meshes.     

Global superconvergence of finite element methods for parabolic inverse problems

In this article we transform a large class of parabolic inverse problems into a nonclassical parabolic equation whose coefficients consist of trace type functionals of the solution and its derivatives subject to some initial and boundary conditions. For this nonclassical problem, we study finite element methods and present an immediate analysis for global superconvergence for these problems, on basis of which we obtain a posteriori error estimators. This research was supported in part by the Shahid Beheshti University, the National Basic Research Program of China (2007CB814906), the National Natural Science Foundation of China (10471103 and 10771158), Social Science Foundation of the Ministry of Education of China (Numerical methods for convertible bonds, 06JA630047), Tianjin Natural Science Foundation (07JCYBJC14300).     

Finite element analysis of transient electromagnetic scattering problems

In this paper, Newmark time-stepping scheme and edge elements are used to numerically solve the time-dependent scattering problem in a three-dimensional cavity. Finite element methods based on the variational formulation derived in [23] are considered. Due to the lack of regularity of _r , the existence and uniqueness of the discrete solutions and their convergence are proved by using the concept of collectively compact operators. An optimal convergence rate in the energy norm is also established.     

On the coupling of regularization techniques and the boundary element method for a hemivariational inequality modelling a delamination problem

In this paper, we couple regularization techniques of nondifferentiable optimization with the h‐version of the boundary element method (h‐BEM) to solve nonsmooth variational problems arising in contact mechanics. As a model example, we consider the delamination problem. The variational formulation of this problem leads to a hemivariational inequality with a nonsmooth functional defined on the contact boundary. This problem is first regularized and then discretized by an h‐BEM. We prove convergence of the h‐BEM Galerkin solution of the regularized problem in the energy norm, provide an a priori error estimate and give a numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Identification of heat transfer dynamics for non-modal analysis of thermoacoustic stability

A systematic approach for non-modal stability analysis of thermoacoustic systems with a localized heat source is proposed. The response of the heat source to flow perturbations is obtained from unsteady computational fluid dynamics combined with correlation-based linear system identification. A model for the complete thermoacoustic system is formulated with a Galerkin expansion technique, where the heat source is included as an acoustically compact element. The eigenvalues of the resulting system are obtained from discretization of the solution operator, the maximum growth factor is estimated from the pseudospectra using Kreiss’ theorem.The approach is illustrated with a simple Rijke tube configuration. Results obtained with a simple “baseline” model for the heat source dynamics based on King’s law - widely used in hot wire anemometry - are compared against the more advanced treatment developed here. Analysis of pseudospectra diagrams shows that the choice of the heat source model does influence the sensitivity of eigenvalues to perturbations and hence the non-normal behavior. The maximum growth factor for the system with the heat source model based on King’s law is more sensitive to changes in the heat source location than the CFD-based heat source model.