A new time domain boundary element formulation for generalized dynamic coupled thermoelasticity
Introduction
Dynamic thermoelasticity finds applications in aerospace engineering, pulsed lasers, fast burst nuclear reactors and particle accelerators 1, 2, 3. The classical dynamic coupled thermoelasticity 4, 5 implies an infinite propagation speed for thermal signals, which is physically unacceptable. Generalized thermoelasticity theories, such as the most widely accepted theory of Green and Lindsay [6], eliminate this paradox. Furthermore, classical theory can be easily obtained as a special case of these generalized theories.
Realistic problems of dynamic coupled thermoelasticity can only be solved by numerical methods. The finite element method (FEM) has been used for solving such problems in the framework of both classical 7, 8 and generalized theories [9]. The boundary element method (BEM) in the frequency or Laplace transform domain [10] has also been used for solving problems of classical dynamic coupled thermoelasticity 11, 12, 13, 14, 15, 16 as well as a generalized one 17, 18, 16. However, only Suh and Tosaka 15, 18 and Chen and Dargush [16] have presented numerical results.
The BEM has the advantage of requiring only a surface discretization of the problem in contrast to the FEM which requires a discretization of the surface as well as the interior domain. However, the great complexity of the frequency domain fundamental solution [5] is certainly a problem. Sladek and Sladek [19] presented a dual reciprocity BEM (DR/BEM) formulation in the time domain for solving problems of classical dynamic coupled thermoelasticity without numerical results. They employed as fundamental solutions the elastostatic for the solid and the steady-state one for the temperature and transformed the acceleration and velocity domain integrals into surface ones. This method, even though elegant in form as being a "boundary only" approach, may not be the best for application because of the complexity of this coupled problem in conjunction with the usual question in the DR/BEM of how many interior collocation points are necessary for acceptable accuracy.
In this paper a new time domain BEM formulation and solution procedure is developed for the generalized dynamic coupled thermoelasticity. Velocity and acceleration terms in the governing equations are replaced by finite difference type expressions according to the implicit time integration scheme of Wilson θ and the resulting static-like equations are written in integral form in conjunction with (a) the elastostatic and steady-state heat conduction fundamental solutions and (b) the frequency domain fundamental solution pair of classical coupled thermoelasticity [5]. In both cases, boundary and interior domain discretization are necessary to achieve a matrix formulation of the problem at every time step (domain/boundary element method or D/BEM). Use of the boundary conditions and elimination of the interior unknowns results in a matrix equation system involving only boundary unknowns. This is solved by a stepwise time integration. This method has already been used in dynamic analysis of beams, membranes and flexural plates by Tanaka and Matsumoto [20], Matsumoto et al. [21] and Qin [22], respectively. The time domain D/BEM used here appears to be conceptually and computationally simpler than the DR/BEM [19] and more efficient than the FEM [9] because of the smaller size matrices involved which are nonsymmetric, but so are those in the FEM. However, the present method is usually restricted to problems with finite domains due to its interior discretization exactly as it is the case with the FEM.
Section snippets
Statement of the problem in time domain
The governing equations of motion and energy for generalized dynamic coupled thermoelasticity have the form [16]:where the symbols ∇ and Δ stand for the gradient and Laplacian operators, respectively, cϵ=ρc, b=(3λ+2μm)αϵ, λ and μ are the Lame constants, u, φ0, φ, f and ψ are the displacement vector, reference temperature, temperature difference, body force and heat source, respectively, ρ, c, κ and aϵ are the density,
First integral formulation in space
Rewriting Eq. (5)in the formand taking into account the type of the operator given by Eq. (6), which consists of the uncoupled elastostatic and Laplacian operators, it is easy to see that the thermoelastic problem Eq. (17)admits an integral representation of the formwhere the vector and are the fundamental solutions
Second integral formulation in space
The integral formulation presented in this section is possible only when the two parameters β and γ, appearing in the implicit time integration scheme of Wilson θ, satisfy the relation β=γ2 [2]. In this case Eq. (5)can be written asThe operator is the Laplace domain time-independent Biot [25] operator of coupled thermoelasticity at the frequency s=γ/βT. Thus, the thermoelastic problem described by Eq. (31)admits an
Discretization and solution procedure
This section describes the solution procedure of the problem which is the same for both integral formulations since they are identical in form and their only difference lies in their different integral kernels. The presence of both surface and volume integrals in integral Eq. (22)and Eq. (37)implies that a boundary as well as an interior discretization of the domain is necessary. Assuming quadratic boundary and interior elements (e.g. 3-noded line boundary elements and 8-noded quadrilateral
Conclusions
A new time domain boundary element formulation and solution procedure for the generalized dynamic coupled thermoelasticity has been briefly presented. This approach employs very simple fundamental solutions at the expense of an interior discretization in addition to the boundary one. However, the resulting matrix equations are of smaller size than those obtained by finite elements. The method is also expected to have higher accuracy than the time domain method of ref. [19] which suffers from
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