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<p>We recently laid down the theoretical basis for the consistent formulation of the collocation boundary element method, as it should have been conceived from the beginning. We proposed a convergence theorem for two- and three-dimensional problems of elasticity and potential, which applies to generally curved elements in the frame of an isoparametric analysis. We also showed that the code implementation leads to controllable, highly precise and accurate results for arbitrarily small source-field distances of two-dimensional problems&#xa0;– limited only by the machine’s capacity to represent numbers. On the other hand, there still is the cost-benefit question of how to adequately describe a real problem’s geometry without increasing the number of degrees of freedom (h- and p-mesh refinement). We are proposing that the isoparametric implementation&#xa0;– with the introduced elegance of a convergence theorem – be replaced with a formulation that preserves the problem’s idealized geometry but is not isoparametric, in general. We also introduce a homothetic approach&#xa0;– for nodes and elements adaptively generated according to the same pattern along a boundary patch&#xa0;–, which is highly cost-effective. We present conceptual formulation, code implementation, and numerical illustrations that go from the simple case of an infinite plate with a circular hole to very challenging&#xa0;– physically unrealistic and only mathematically conceivable – topological applications: a multi-connected domain with generally curved boundary patches and presenting cracks, cusp and reentrant angles of virtually zero magnitude, and a strip of material of zero width. This cannot be manufactured in the real world but can be nevertheless simulated provided we have the proper mathematical tools, as presently proposed.</p>

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Consistent Boundary Element Method for Two-Dimensional Problems of Elasticity with Geometry-Preserving, Homothetic Element Generation

  • 期刊论文

  • 发表日期：

    2025年7月4日

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  • Ney Augusto Dumont

摘要

We recently laid down the theoretical basis for the consistent formulation of the collocation boundary element method, as it should have been conceived from the beginning. We proposed a convergence theorem for two- and three-dimensional problems of elasticity and potential, which applies to generally curved elements in the frame of an isoparametric analysis. We also showed that the code implementation leads to controllable, highly precise and accurate results for arbitrarily small source-field distances of two-dimensional problems – limited only by the machine’s capacity to represent numbers. On the other hand, there still is the cost-benefit question of how to adequately describe a real problem’s geometry without increasing the number of degrees of freedom (h- and p-mesh refinement). We are proposing that the isoparametric implementation – with the introduced elegance of a convergence theorem – be replaced with a formulation that preserves the problem’s idealized geometry but is not isoparametric, in general. We also introduce a homothetic approach – for nodes and elements adaptively generated according to the same pattern along a boundary patch –, which is highly cost-effective. We present conceptual formulation, code implementation, and numerical illustrations that go from the simple case of an infinite plate with a circular hole to very challenging – physically unrealistic and only mathematically conceivable – topological applications: a multi-connected domain with generally curved boundary patches and presenting cracks, cusp and reentrant angles of virtually zero magnitude, and a strip of material of zero width. This cannot be manufactured in the real world but can be nevertheless simulated provided we have the proper mathematical tools, as presently proposed.