Keywords: proper generalized decomposition, boundary element method, heat equation.

The boundary element method (BEM) allows efficient solution of partial differential equations whose kernel functions are known. The heat equation is one of these candidates when the thermal parameters are assumed constant (linear model). When the model involves large physical domains and time simulation intervals the amount of information that must be stored increases significantly. This drawback can be circumvented by using advanced strategies, as for example the multi-pole technique.

We propose in this paper an alternative radically different approach that leads to a separated solution of the space and time problems within a non-incremental integration strategy. The technique is based on the use of a space-time separated representation of the unknown field that, introduced in the residual weighting formulation, allows the definition of a separated solution of the resulting weak form. The spatial step can be then treated by invoking the standard BEM for solving the resulting steady state problem defined in the physical space. Then, the time problem that results in an ordinary first order differential equation is solved using any standard appropriate integration technique (e.g. backward finite differences).

In the case of the linear and transient heat equation here considered for the sake of simplicity, the proper generalized decomposition (PGD) leads to the solution of a series of steady state diffusion-reaction problems (accurately solved by using the BEM method) and a series of problems that consist of a simple time dependent ODE.

Separated representations were already applied for solving transient models in the context of finite element discretizations [1,2,3,4], but they never have been used in the BEM framework, and certainly in this context the main advantage is the possibility of defining non-incremental strategies as well as the possibility of avoiding the use of space-time kernels. In principle, this technique seems specially adapted for solving transient problems involving extremely small time steps.

1

A. Ammar, B. Mokdad, F. Chinesta, R. Keunings, "A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids", Journal of Non-Newtonian Fluid Mechanics, 139(3), 153-176, 2006. doi:10.1016/j.jnnfm.2006.07.007

2

A. Ammar, B. Mokdad, F. Chinesta, R. Keunings, "A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids: Part II: Transient simulation using space-time separated representations", Journal of Non-Newtonian Fluid Mechanics, 144(2-3), 98-121, 2007. doi:10.1016/j.jnnfm.2007.03.009

3

F. Chinesta, A. Ammar, A. Falco, M. Laso, "On the reduction of stochastic kinetic theory models of complex fluids", Modeling Simulation in Materials Science Engineering, 15, 639-652, 2007. doi:10.1088/0965-0393/15/6/004

4

A. Ammar, F. Chinesta, P. Joyot, "The nanometric and micrometric scales of the structure and mechanics of materials revisited: An introduction to the challenges of fully deterministic numerical descriptions", International Journal for Multiscale Computational Engineering, 6(3), 191-213, 2008. doi:10.1615/IntJMultCompEng.v6.i3.20

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