Solution of Coupled Linear Thermoviscoelasticity Problems With Dual Reciprocity Boundary Element Method (Drbem): a Comparison of Direct and Iterative Solutions

Abstract eng:
In this study, both direct and iterative procedures for the dual reciprocity boundary element method solution of isotropic coupled linear thermoviscoelasticity (TVE) problems are considered. The governing equations for coupled linear thermoelasticity is given in Fouriér space as (1) where , , , , and are the material constants, namely, shear modulus, Lame’s modulus, density, thermal conductivity, heat capacity under constant volume and where is , the isotropic heat conduction coefficient. The field variables are the displacement components, and the temperature difference (from reference temperature ), . In the equations, represents the body forces, represents the internal heat generation, represents the imaginary number, e.g., , and is the Fouriér transform parameter, which can also be associated with frequency. Also, in Eq(1), summation convention is in place requiring a summation over a repeated index and a comma in the indices require differentiation with the following indices. In case of TVE, the correspondence principle is applied, changing the material constants with their complex values using a selected viscoelastic model. In this study, viscous dissipation, since it introduces a high nonlinearity to the equations, is not considered. Therefore, Eq(1) would be applicable for the case of TVE replacing and with their complex values in the first of Eq(1). Solution of such problems using BEM is available in literature, yet, the fundamental solutions required in such a procedure is computationally complex, requiring special treatment when singularities arise and also resulting in high computational time. Therefore, to overcome such complexity, dual reciprocity (DR) method [1] can be applied by treating the coupling terms, and as non-homogenous terms. In this way, by adding sufficient number of internal collocation points to the solution domain, DR provides a boundary-only solution. The resulting system of equations can be solved in two ways: (i) The equations in Eq(1) can be solved simultaneously and repeatedly until a certain criteria of convergence is obtained, which is the iterative method [2], or (ii) by combining the matrices, a larger system of equations can be obtained which can be solved in one step to obtain the solution, which is the direct method [3]. In this study, both iterative and direct solution methods are reviewed, and then a comparison between them considering convergence, computational effort and programming issues (such as coding easiness and opportunities as parallelization) is made. In the study, 2D problems are considered and constant elements are used in the solution. To assess the methods, several problems are solved.

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