Browsing by Author "Valtchev, Svilen S."
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- A meshfree method with plane waves for elastic wave propagation problemsPublication . Valtchev, Svilen S.In this paper, we address the meshfree numerical solution of time-harmonic linear elastic wave propagation problems in homogeneous media. In particular, we analyze the asymptotic behavior of the method of fundamental solutions (MFS) with source points located far away from the domain of interest. The asymptotic MFS is shown to be equivalent to a Trefftz method, here referred to as the plane waves method (PWM), based on superposition of shear and compressional elastic plane waves with different directions of propagation. Several numerical examples are included in order to illustrate the equivalence between the asymptotic MFS and the PWM. The convergence and stability of the PWM are also analyzed in smooth settings.
- 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.