Global Finite Element Matrix Construction Based on a CPU-GPU Implementation

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Many physical phenomena in stationary condition such as electrical and magnetic potential, heat conduction, fluid flow and elastic problems in static condition can be described by elliptic partial differential equations (EPDE). The EPDE does not involve a time variable, and so describes the steady state of problems. A linear EPDE has the general form as presented in Eq. (1), where a,b,c are coefficient functions, the term f is a source (excitation) function and u is the unknown variable. All of this function can vary spatially (x,y,z).

∇(c∙∇u)+b∙∇u+au=f (1)

EPDE can be solved exactly by mathematical procedures like Fourier series [1]. However, the classical solution frequently no exists and for those problems where is possible the use of these analytical methods, many simplifications are done [2]. Consequently, several numerical methods have been developed to efficiently solve EPDE such as the finite element method (FEM), finite difference and others.
The FEM have several advantages over other methods. The main advantage is that it is particularly effective for problems with complicated geometry using unstructured meshes [2]. One way to get a suitable framework for solving EPDEs by using FEM is formulate them as variational problems also called weak solution.
The variacional formulation of an EPDE is a mathematical treating for converting the strong formulation to a weak formulation, which permits the approximation in elements or subdomains, and the EPDE
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