This paper describes a mesh refinement technique for boundary element method in which the number of elements, the size of elements and the element end location are determined iteratively in order to obtain a user specified accuracy. The method uses¸ norm as a measure of error in the density function and a grading function that ensures that error over each element is the same. The use of grading function along with¸ norm makes the mesh refinement technique applicable to Direct and Indirect boundary element method formulation for a variety of boundary element method applications. Numerical problems in elastostatics, fracture mechanics, and bending of plate solved using Direct and Indirect method in which the density functions are approximated by Linear Lagrange, Quadratic Lagrange or Cubic Hermite polynomials validate the effectiveness of the proposed mesh refinement technique.
This paper describes a mesh refinement technique for boundary element method in which the number of elements, the size of elements and the element end location are determined iteratively in order to obtain a user specified accuracy. The method uses¸ norm as a measure of error in the density function and a grading function that ensures that error over each element is the same. The use of grading function along with¸ norm makes the mesh refinement technique applicable to Direct and Indirect boundary element method formulation for a variety of boundary element method applications. Numerical problems in elastostatics, fracture mechanics, and bending of plate solved using Direct and Indirect method in which the density functions are approximated by Linear Lagrange, Quadratic Lagrange or Cubic Hermite polynomials validate the effectiveness of the proposed mesh refinement technique.
The discretization of the boundary in boundary element method generates integrals over elements that can be evaluated using numerical quadrature that approximate the integrands or semi-analytical schemes that approximate the integration path. In semi-analytical integration schemes, the integration path is usually created using straight-line segments. Corners formed by the straight-line segments do not affect the accuracy in the interior significantly, but as the field point approaches these corners large errors may be introduced in the integration. In this paper, the boundary is described by a cubic spline on which an integration path of straight-line segments is dynamically created when the field point approaches the boundary. The algorithm described improves the accuracy in semi-analytical integration schemes by orders of magnitude at insignificant increase in the total solution time by the boundary element method. Results from two indirect BEM and a direct BEM formulation in which the unknowns are approximated by linear and quadratic Lagrange polynomial and a cubic Hermite polynomial demonstrate the versatility of the described algorithm.
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