Generalized convolution quadrature based boundary element method for uncoupled thermoelasticity

Highlights

• Application of the generalised Convolution Quadrature in thermoelastic BEM.

• Variable time step size in a time domain thermoelastic BE formulation.

• Convergence study of a CQ based BEM.

• Realisation of Robin boundary condition for different shape functions of the Dirichlet and Neumann data.

Abstract

Mechanical loads together with changing temperature conditions can be found in a wide variety of fields. Their effects on elastic media are reflected in the theory of thermoelasticity. For typical materials in engineering, very often a simplification of this coupled theory can be used, the so-called uncoupled quasistatic thermoelasticity. Therein, the effects of the deformations onto the temperature distribution is neglected and the mechanical inertia effects as well. The Boundary Element Method is used to solve numerically these equations in three dimensions. Since convolution integrals occur in this boundary element formulation, the Convolution Quadrature Method may be applied. However, very often in thermoelasticity the solution shows rapid changes and later on very small changes. Hence, a time discretisation with a variable time step size is preferable. Therefore, here, the so-called generalised Convolution Quadrature is applied, which allows for non-uniform time steps. Numerical results show that the proposed method works. The convergence behavior is, as expected, governed either by the time stepping method or the spatial discretisation, depending on which rate is smaller. Further, it is shown that for some problems the proposed use of the generalised Convolution Quadrature is the preferable.

Introduction

In many engineering applications not only the deformation is of interest but as well the temperature and their interaction. The presumably most prominent example are thermal stresses, i.e., the stress in an elastic body caused by a heat source or by a change in temperature. An industrial example is hot forming with all its variants. These coupled effects are described in the theory of thermoelasticity.

The theory of thermoelasticity is well known for several decades. In fact, the classical linear approach goes back to Duhamel in 1837 [14] and Neumann in 1885 [31]¹. It integrates the effects of mechanical loads together with those of a temperature field onto an elastic structure. The mathematical description is based on a system of coupled differential equations, consisting of a temperature equation and an equation for the deformations and was established by Biot [5]. The theory can be found in a variety of textbooks, see e.g., Nowacki [32]. This set of two coupled partial differential equations can be simplified by assuming that the thermoelastic dissipation can be neglected. Such an assumption results in a one-sided coupling, i.e., the temperature development is not influenced by the displacement solution but vice versa. The other simplification is whether the inertia terms can be neglected for slow processes. Hence, four different simplifications can be found in literature. For many engineering applications, it can be assumed that the coupling of the temperature field with the displacement solution is negligible and the mechanical inertia effects can be neglected as well. This is the so called Uncoupled Quasistatic theory of Thermoelasticity (UQT), see e.g., [34], which is considered here.

For the numerical solution of this set of governing equations most numerical methods have corresponding formulations. This holds as well for the Boundary Element Method (BEM), which has gained popularity since it requires only the meshing of the body’s surface. The analytical basis of a thermoelastic BEM, i.e., the respective integral representations and fundamental solutions, can be found in a series of papers by Sládek and Sládek [42], [43] and in the book of Kupradze et al. [22].

The analysis of thermoelasticity using the BEM was started by Rizzo and Shippy [35] and Cruse et al. [9]. They treated three-dimensional problems of uncoupled thermoelasticity for steady state heat conduction. BE formulations for transient thermoelasticity in 2D were presented in [45] and for 3D in [8]. First numerical results for an uncoupled formulation can be found in [41]. Further numerical results are presented for planar problems in [10] for coupled and uncoupled quasistatic thermoelasticity. Later on, the same authors showed the general three-dimensional case of UQT together with numerical results [11]. In [12], these authors show the similarities of consolidation in poroelasticity and the quasi-static thermoelasticity with numerical results. A 2D-formulation of the coupled dynamic equations has been presented in [46] using the Laplace transform and a numerical inverse transformation. By rewriting the thermal equation, a special form of the coupling term can be found which allows a formulation of the coupled quasi-static case using the elastostatic fundamental solution, i.e., partly the fundamental solutions of the uncoupled formulation [44]. An approach using particular integrals for the solution of transient thermoelasticity combined with the complementary solution of the steady state problem was presented by Park and Banerjee [33]. Chatterjee et al. [7] presented a simplified re-integration based fast-convolution algorithm for UQT as a memory saving alternative for large scale problems. An extension to anisotropic elastic material properties for fully coupled thermoelasticity is found in the book of Gaul [15] or in [20]. In this formulation, the dual reciprocity method is used to handle the time dependent terms.

The above discussed formulations use either a formulation in the transformed domain with an inverse transformation back to time domain, or they treat the problem in time domain directly. A methodology to compute in time domain but using Laplace domain fundamental solution is the Convolution Quadrature method (CQ), which goes back to Lubich [27], [28]. BEM formulations based on CQ can be found, e.g., for elastodynamics [19], viscoelasticity [38], and poroelasticity [39], [23]. In case of thermoelasticity, the respective formulation has been proposed in [1] with numerical results in 2D. Here, an extension is presented towards 3D, which is for the formulation of the equations straight forward. The main contribution is to allow for a variable time step size. This can be done using the so-called generalized Convolution Quadrature method (gCQ). Lopez-Fernandez and Sauter [24] published this generalisation of the CQ. The numerical realisation can be found in [25]. The extension to use a Runge-Kutta method as underlying time stepping has been presented in [26]. An application of the gCQ in acoustics with absorbing boundary conditions has been published in [36]. This extension to variable time step sizes seems to be well suited for thermoelastic problems as the solution behavior in most applications show in the beginning much larger gradients compared to later times. Hence, an adjustment of the time step size seems to be favourable.

The paper is organised as follows. First, the basic equations and relations of thermoelasticity are recapped. Next, the discretisations in space and time are introduced and the gCQ is explained. The comparison with two 1D solutions shows the performance of the method. Finally, a simplified example from hot forming shows the applicability of the proposed formulation to real world problems.