Browsing by Author "Alves, Carlos J.S."
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- On the application of the method of fundamental solutions to boundary value problems with jump discontinuitiesPublication . Alves, Carlos J.S.; Valtchev, SvilenTwo meshfree methods are proposed for the numerical solution of boundary value problems (BVPs) for the Laplace equation, coupled with boundary conditions with jump discontinuities. In the first case, the BVP is solved in two steps, using a subtraction of singularity approach. Here, the singular subproblem is solved analytically while the classical method of fundamental solutions (MFS) is applied for the solution of the regular subproblem. In the second case, the total BVP is solved using a variant of the MFS where its approximation basis is enriched with a set of harmonic functions with singular traces on the boundary of the domain. The same singularity-capturing functions, motivated by the boundary element method (BEM), are used for the singular part of the solution in the first method and for augmenting the MFS basis in the second method. Comparative numerical results are presented for 2D problems with discontinuous Dirichlet boundary conditions. In particular, the inappropriate oscillatory behavior of the classical MFS solution, due to the Gibbs phenomenon, is shown to vanish.
- Trefftz methods with cracklets and their relation to BEM and MFSPublication . Alves, Carlos J.S.; Martins, Nuno F.M.; Valtchev, Svilen S.In this paper we consider Trefftz methods which are based on functions defined by single layer or double layer potentials, integrals of the fundamental solution, or their normal derivative, on cracks. These functions are called cracklets, and satisfy the partial differential equation, as long as the crack support is not placed inside the domain. A boundary element method (BEM) interpretation is to consider these cracks as elements of the original boundary, in a direct BEM approach, or elements of an artificial boundary, in an indirect BEM approach. In this paper we consider the cracklets just as basis functions in Trefftz methods, as the method of fundamental solutions (MFS). We focus on the 2D Laplace equation, and establish some comparisons and connections between these methods with cracklets and standard approaches like the BEM, indirect BEM, and the MFS. Namely, we propose the enrichment of the MFS basis with the cracklets. Several numerical simulations are presented to test the performance of the methods, in particular comparing the results with the MFS and the BEM.