A boundary element method based on the Laplace-transform technique is developed for transient coupled thermoelasticity problems of two-dimensional finite domain The Laplace-transform method is applied to the time domain and the resulting equations in the transformed field are discretized using the boundary element method The nodal dimensionless temperature and displacements in the transformed domain are inverted to obtain the actual physical quantities, using the numerical inversion of the Laplace-transform method The work is concerned with the thermal and mechanical shocks in a finite domain considering classical coupled theory of thermoelasticity Elastic and thermoelastic wave creation and propagation in a finite domain and their effects on each other are investigated Numerical implementations are presented and compared with the known data

http://www.iust.ac.ir/files/rail/Abstract/hossini/4.pdf

BEM analysis of thermal and mechanical shock in a two-dimensional finite domain considering coupled thermoelasticity

This paper deals with a new boundary element method for analysis of the quasistatic problems in coupled thermoelasticity. Through some mathematical manipulation of the Navier equation in elasticity, the heat conduction equation is transformed into a simpler form, similar to the uncoupled-type equation with the modified thermal conductivity which shows the coupling effects. This procedure enables us to treat the coupled thermoelastic problems as an uncoupled one, A few examples are computed by the proposed BEM, and the results obtained are compared with the analytical ones available in the literature, whereby the accuracy and versatility of the proposed method are demonstrated.

Application of boundary element method to 3-D problems of coupled thermoelasticity

In this work, we introduce multipoint flux (MF) approximation method to the problem of conduction heat transfer in anisotropic media. In such media, the heat flux vector is no longer coincident with the temperature gradient vector. In this case, thermal conductivity is described as a second order tensor that usually requires, at least, six quantities to be fully defined in general three-dimensional problems. The two-point flux finite differences approximation may not handle such anisotropy and essentially more points need to be involved to describe the heat flux vector. In the framework of mixed finite element method (MFE), the MFMFE methods are locally conservative with continuous normal fluxes. We consider the lowest order Brezzi–Douglas–Marini (BDM) mixed finite element method with a special quadrature rule that allows for nodal velocity elimination resulting in a cell-centered system for the temperature. We show comparisons with some analytical solution of the problem of conduction heat transfer in anisotropic long strip. We also consider the problem of heat conduction in a bounded, rectangular domain with different anisotropy scenarios. It is noticed that the temperature field is significantly affected by such anisotropy scenarios. Also, the technique used in this work has shown that it is possible to use the finite difference settings to handle heat transfer in anisotropic media. In this case, heat flux vectors, for the case of rectangular mesh, generally require six points to be described.

A Multipoint Flux Approximation of the Steady-State Heat Conduction Equation in Anisotropic Media

This paper is concerned with a dual reciprocity boundary element method (DRBEM) applied to transient heat conduction problems for time-dependent, anisotropic materials. The integral equation formulation employs the fundamental solution of the Laplace equation for linear isotropic materials. As a result, from the nonlinear and also from the anisotropic parts of the governing differential equation, domain integrals arise in the boundary integral equation. These domain integrals are transformed into boundary integrals by using a new radial basis function. Furthermore, the time derivative is approximated by a time-stepping scheme, which yields additional domain integrals in the formulation. Such a domain integral at each time-step can also be transformed into boundary integrals via the same dual reciprocity method. The details of the proposed DRBEM are presented in what follows. Because of the domain integral arising from material nonlinearity, the solution is inevitably performed in an iterative manner. The usefulness of the present iterative method is demonstrated through comparison of the results obtained by a special computer code developed by the authors with those obtained by the finite element method.

A time-stepping DRBEM for transient heat conduction in anisotropic solids

Analysis of non-stationary temperature fields in laminated strips and plates

This study is concerned with a new boundary element method for two dimensional anisotropic thermoelasticity in steady heat conduction. The governing differential equations of the displacement and temperature fields are transformed into a set of boundary integral equations which include the unknown quantities of displacement, temperature, traction, and heat flux on the boundary. They are discretized by means of the standard boundary element method, and solved under the given boundary conditions. A set of fundamental solutions are derived by the Hormander method. Mathematical formulations are presented in detail for two dimensional thermoelasticity in anisotropic bodies. The proposed solution procedure is applied to some typical examples, and its usefulness is demonstrated through discussion of the results obtained.

Analysis of Two-Dimensional Anisotropic Thermoelasticity by Boundary Element Method.

This paper is concerned with a new boundary element approach based on the time-stepping scheme for two-dimensional thermoelasticity in unsteady heat conduction. Coupling between the temperature and displacement fields is considered in the context of a quasi-static transient state. The time derivative in the differential equations is approximated by the time-stepping scheme. The reduced differential equations are transformed into a set of boundary integral equations using the exact fundamental solutions, which can be derived by the Hormander method. The standard boundary element method is applied to obtain numerical solutions at each time step. Computation begins in the initial state, and advances in turn by a time-marching procedure until the final state is realized. The mathematical formulation is presented in detail. The proposed solution procedure is applied to an example problem to show its usefulness.

Analysis of Two-Dimensional Transient Thermoelasticity by the Time-Stepping Boundary Element Method. In Consideration of Coupling Effect.

This paper is concerned with an application of the boundary element method (BEM) using the dual reciprocity method (DRM) to analyze nonlinear transient heat conduction in anisotropic solids. In this study, because the fundamental solution of this problem has not previously been obtained, the concept (the analog equation method) proposed by Katsikadelis and Nerantzaki is applied. A standard linear partial differential operator, in which the fundamental solution can be obtained, can be extracted from a nonlinear partial differential equation. One can consider the remainder as a body force and solve the equation using DRM. Mathematical formulations of this approach for two-dimensional problems are presented in detail. Two schemes are discussed in this paper: The “isotropic” scheme, in which the operator of steady-state heat conduction in isotropic solids is extracted, and the “anisotropic” scheme, where the operator of the steady-state heat conduction in anisotropic solids is extracted. The proposed solution procedure is applied to a couple of typical examples, and the validity and other numerical properties of the proposed BEM are demonstrated through discussions of the results obtained.

Analysis of Two-Dimensional Nonlinear Transient Heat Conduction in Anisotropic Solids by Boundary Element Method Using Dual Reciprocity Method

This paper is concerned with the application of the boundary element method (BEM) with the analog equation method (AEM), proposed by Katsikadelis and Nerantzaki, and Green's theorem to analyze steady-state heat conduction in anisotropic solids. In this study, the linear differential operator (the Laplacian) of steady-state heat conduction in isotropic solids is extracted from the governing differential equation. The integral equation formulated employs the fundamental solution of the Laplace equation for isotropic solids, and therefore, from the anisotropic part of the governing differential equation, a domain integral appears in the boundary integral equation. This domain integral is transformed into boundary integrals using Green's theorem with a polynomial function. The mathematical formulation of this approach for two-dimensional problems is presented in detail. The proposed solution is applied to two typical examples, and the validity and other numerical properties of the proposed BEM are demonstrated in the discussion of the results obtained.

Analysis of Two-Dimensional Steady-State Heat Conduction in Anisotropic Solids by Boundary Element Method Using Analog Equation Method and Green's Theorem

This paper is concerned with an application of the homotopy boundary element method originally proposed by Liao and Chwang to analysis of nonlinear transient heat conduction in anisotropic solids. Usually, a domain integral arises in the boundary integral equation of this formulation. Some ideas are needed to keep the boundary-only feature of BEM. In this paper, the resulting domain integral is transformed into a boundary integral by the dual reciprocity method using a new set of radial basis functions. Mathematical formulations of this approach for two-dimensional problems are presented in detail. Two schemes are discussed in this paper : The “isotropic” scheme, in which the state before mapping is considered as steady state heat conduction in isotropic solids, and the “anisotropic” scheme, where the state before mapping as steady state heat conduction in anisotropic solids. The proposed solution procedure is applied to a couple of typical examples, and the accuracy and other numerical properties of the proposed BEM are demonstrated through discussions of the results obtained.

Shuji Ishiguro

Papers

Analysis of Two-Dimensional Anisotropic Thermoelasticity by Boundary Element Method.

Analysis of Two-Dimensional Transient Thermoelasticity by the Time-Stepping Boundary Element Method. In Consideration of Coupling Effect.

Analysis of Two-Dimensional Nonlinear Transient Heat Conduction in Anisotropic Solids by Boundary Element Method Using Dual Reciprocity Method

Analysis of Two-Dimensional Steady-State Heat Conduction in Anisotropic Solids by Boundary Element Method Using Analog Equation Method and Green's Theorem

Analysis of Two-Dimensional Nonlinear Transient Heat Conduction in Anisotropic Solids by Boundary Element Method Using Homotopy and Dual Reciprocity