Bifurcation indicator based on meshless and asymptotic numerical methods for nonlinear poisson problems

Abstract

We propose in this work new algorithms associating asymptotic numerical method and meshless discretization (MFS‐MPS: Method of fundamental solutions‐Method of particular solutions) to compute branch solutions of nonlinear Poisson problems. To detect singular points on these branches, geometrical indicator, Padé approximants, and analytical bifurcation indicator are proposed. Numerical applications show the robustness and the effectiveness of the proposed algorithms. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 978–993, 2014

Extending the method of fundamental solutions to non-homogeneous elastic wave problems

“…In [8] the method of approximate particular solutions was developed, where the collocation was performed using two sets of RBFs, related in the sense of the DRM. In [36,37] a linear combination of RBFs and fundamental solutions was used for the solution of nonlinear Poisson problems. A set of Helmholtz fundamental solutions with different source points and test frequencies was used in the MFS-K method [6] for the non-homogeneous acoustic wave propagation problem.…”

Fundamental solutions and asymptotic numerical methods for bifurcation analysis of nonlinear bi‐harmonic problems

“…As the series solution is limited by a radius of convergence, a continuation procedure allows one to obtain the whole solution branch in a step by step manner where the step length is computed a posteriori by exploiting the terms of these series. It has been successfully applied in nonlinear solid and fluid mechanics .…”

“…The nearest singularity can also be described by the first real root of the denominator of the Padé approximants that is a rational fraction asymptotically equivalent to the Taylor series . In this paper, the method of Padé approximants will be used . The definition of Taylor series can be defined at the level of the continuous problem and it has to be coupled with a spatial discretization technique.…”

“…The bifurcation indicator has been introduced by Boutyour in solid mechanics and by Tri in fluid mechanics and for nonlinear Poisson problems . Recently, a method based on Taylor meshless method developed for bifurcation analyses within the von Karman plate theory .…”

Combining MFS and PGD methods to solve transient heat equation

“…Meshless methods have been generally proposed to alleviate a part of the difficulties issue of domain meshing and to avoid mesh distortion. Method of fundamental solutions has predominantly been applied to stationary problems governed by elliptic PDEs such as Laplace, Helmholtz, biharmonic, Lamé and Stokes equations [2][3][4][5][6][7][8][9]. However, in recent years some different versions of MFS used for parabolic heat-transfer problems have appeared and have been extensively treated in the literature [10][11][12][13][14][15].…”