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Boundary Element 2

# Boundary Element 2 - Boundary Element Method for Elasticity...

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Boundary Element Method for Elasticity Problems Another general numerical method has recently emerged that provides good computational abilities and has some particular advantages when compared to FEM. The technique known as the boundary element method (BEM) has been widely used by computational mechanics investigators leading to the development of many private and commercial codes. Similar to the finite element method, BEM can analyze many different problems in engineering science including those in thermal sciences and fluid mechanics. Although the method is not limited to elastic stress analysis, our brief presentation will only discuss this particular case. Many texts have been written that provide additional details on this subject, see for example Banerjee and Butterfield (1981) and Brebbia and Dominguez (1992). The formulation of BEM is based on an integral statement of elasticity, and this can be cast into a relation involving unknowns only over the boundary of the domain under study. This originally lead to the boundary integral equation method (BIE), and early work in the field was reported by Rizzo (1967) and Cruse (1969). Subsequent research realized that finite element methods could be used to solve the boundary integral equation by dividing the boundary into elements over which the solution is approximated using appropriate interpolation functions. This process generates an algebraic system of equations to solve for the unknown nodal values that approximate the boundary solution. A procedure to calculate the solution at interior domain points can also be determined from the original boundary integral equation. This scheme also allows variation in element size, shape and approximating scheme to suit the application, thus providing similar advantages as FEM. By discretizing only the boundary of the domain, BEM has particular advantages over FEM. The first issue is that the resulting BEM equation system is generally much smaller than that generated by finite elements. It has been pointed out in the literature, that boundary discretization is somewhat easier to interface with CAD computer codes that create the original problem geometry. Two-dimensional comparisons of equivalent FEM and BEM meshes for a rectangular plate with a central circular hole, hollow cylinder and gear tooth problem and shown in Figures 1 and 2. It is apparent that a significant reduction in the number of elements (by a factor of five) is realized in the BEM mesh. (FEM Discretization: 228 Elements) | | | | | | | | | | | | | | | | | | | | | | | | | | (BEM Discretization: 44 Elements) | | | | | | Figure 1. Comparison of Typical FEM and BEM Meshes

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Figure 2. Comparisons of FEM and BEM meshes.