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PRACTICAL ASPECTS OF DEVELOPMENT AND USE
OF BOUNDARY ELEMENT METHODS
J O Watson
Mining Engineering
UNSW, Sydney 2052
Australia
By the time the first analyses of stress by boundary elements were carried out, the finite element method was already an accepted method of stress analysis in engineering practice. Early researchers into boundary element methods believed that the reduction of the dimension of the numerical problem to be solved would be decisive in any comparison of the methods. When this was found not to be so, attention turned to the improvement of computational efficiency by the use of improved elements and equation solvers designed to take advantage of special characteristics of systems arising from discretisation of boundary integral equations.
It was found that efficiency is improved by elements with higher order (typically quadratic) variation of displacement, except where the exact solution exhibits singularities. Near crack and notch roots, surface displacements computed using isoparametric quadratic boundary elements vary somewhat erratically compared with those computed using similar finite elements. Shifting of side nodes to quarter points offers a solution, but it leaves stress intensity factors to be determined by postprocessing such as consideration of crack opening displacements, and is not applicable to notches. A more satisfactory approach is the introduction of singular shape functions that multiply stress intensity factors in dominant and subdominant modes of crack or notch opening displacement; nodal stress intensity factors become unknowns to be computed simultaneously with nodal displacements, and uncertainties associated with postprocessing are eliminated.
It remains the case that stresses computed by boundary elements are relatively sensitive to error in surface modelling. This is not the fault of the method: stresses at the surface are indeed sensitive to local variation of surface geometry, but that is not always reflected in results computed by extrapolation from the Gauss points of finite elements. There is a need for data transfer in which, typically, the surface mesh is made coarser but of finite curvature everywhere except at edges and corners of the exact geometry. Whether such coarsening and defeaturing takes place inside or outside the boundary element software is in principle immaterial, but it may prove opportune for developers of boundary elements to include the necessary logic within their software rather than wait for others to provide it.
In the solution of typical problems of practical complexity most of the run time is spent solving simultaneous equations for nodal displacements, and it became evident early in the development of boundary element software that an alternative to Gaussian elimination on a fully populated matrix would have to be found. There now exist several techniques of accelerated solution in which advantage is taken of the decay towards zero of the kernels of the boundary integral equation. In these techniques, far-field approximations are made and solutions are found by successive approximation. Acceptance of far-field approximations in routine engineering analysis must depend upon the availability of error estimates, and solutions by successive approximation are of use only if they converge reliably for convoluted surfaces.
As an adjunct to techniques of accelerated solution, Hermitian boundary elements have been developed. The advantage of Hermitian elements is that all degrees of freedom are associated with nodes at vertices of elements, and nodes at vertices are shared by larger numbers of adjacent elements. This reduces the global number of degrees of freedom. Hermitian cubic elements have been developed for plane strain, and will eventually replace quadratic elements in three dimensional analysis. Splines have also been proposed, but these may impose unacceptable constraints on mesh topology for surfaces in three dimensions.