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International audience; An efficient numerical model is developed in this work for the bifurcation analysis within Föppl-von Karman plate theory. This procedure is based on the use of the method of fundamental solutions in a high order continuation method that relies on Taylor series expansion with respect to a path parameter. This numerical model has an adaptive step length, which is effective especially for solving nonlinear problems and detecting bifurcation points. Despite of the huge application field of nonlinear elasticity, the method of fundamental solution was very rarely applied in this field and never to a nonlinear elastic plate model. The governing equations are strongly formulated in terms of the two unknowns: the deflection and the stress function. The singular value decomposition regularization is used to overcome the difficulty of the resulting system ill-conditioning. The accuracy and efficiency of the numerical model are illustrated on buckling numerical examples.

International audience; New algorithms, combining asymptotic numerical method (ANM) and method of fundamental solutions, are proposed to compute bifurcation points on branch solutions of a nonlinear bi-harmonic problem. Three methods, mainly based on asymptotic developments framework, are then proposed. The first one consists in exploiting the ANM step accumulation close to the bifurcation points on a solution branch, the second method allows the introduction of an indicator that vanishes at the bifurcation points, and finally the first real root of the Padé approximant denominator represents the third bifurcation indicator. Two numerical examples are considered to analyze the robustness of these algorithms.

In this paper, we propose a dimensionless numerical mesh-free model for the simulation of the compressible isothermal viscous flows. The novelty of this work consists to formulate the Navier-Stokes equations under a dimensionless form and to solve them by a high order mesh-free algorithm to simulate the compressible fluid flows. This algorithm combines a classical implicit Euler scheme, a high order continuation with the Moving Least Squares (MLS) and a homotopy transformation. The MLS approximation and implicit Euler scheme are used respectively for the spatial and temporal discretizations of dimensionless Navier-Stokes equations. The homotopy transformation serves to introduce in dimensionless Navier Stokes equations an arbitrary operator and a parameter without physical dimension. The obtained equations are solved by a high order continuation. The performance of the presented model is tested on the standard benchmark lid-driven cavity problem. Then, the Mach and Reynolds numbers effect is discussed. The obtained results are compared with those of the Finite Difference Method (FDM) coupled with an explicit Runge-Kutta (R-K) scheme and those of literature. This comparison reveals that the results of the dimensionless model are obtained with a less expensive CPU time compared to that of the other algorithms.

International audience; In the present work, we propose a numerical analysis of instability and bifurcations for geometrically nonlinear elasticity problems. These latter are solved by using the Asymptotic Numerical Method (ANM) associated with the Method of Fundamental Solutions (MFS). To compute bifurcation points and to determine the critical loads, we propose three techniques. The first one is based on a geometrical indicator obtained by analyzing the Taylor series. The second one exploits the properties of the Padé approximants, and the last technique uses an analytical bifurcation indicator. Numerical examples are studied to show the efficiency and the reliability of the proposed algorithms

The paper deals with the application of Asymptotic Numerical Method (ANM) for solving non-linear Partial Differential Equations discretized by a meshless technique. In a recent paper [3] , it was proposed to associate ANM and Method of Fundamental Solutions (MFS) in a boundary only framework, which permits one to compute a part of non-linear response curves up to the radius of convergence. In the present paper, a continuation algorithm is presented, that is able to compute any solution branch by using the same basis functions. The discretization technique combines fundamental solutions, method of particular solutions (referred as MPS or MFS–MPS when it is coupled with fundamental solutions) and Analog Equation Method (AEM).

International audience; We propose in this article a numerical algorithm based on the combination of the method of fundamental solutions (MFS) and the proper generalized decomposition technique (PGD) to solve time-dependent heat equation. The MFS is considered as a truly meshless technique well adapted for a wide range of physical problems and the PGD approach can be considered as a reduction technique based on the separated representation of the variable functions. The proposed study relates to a separation between the spatial and temporal coordinates. To show the effectiveness of the proposed algorithm, several examples are presented and compared to the reference results

We show in this work that the Asymptotic Numerical Method (ANM) combined with the Method of Fundamental Solution (MFS) can be a robust algorithm to solve the nonlinear Poisson problem. The ANM transforms the nonlinear problem into a sequence of linear ones which can be solved by MFS. This last method consists of approximating the solution of the linear Poisson problem by a linear combination of fundamental solutions. Some examples are presented to show the efficiency of the proposed method.

International audience; 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, Pade approximants, and analytical bifurcation indicator are proposed. Numerical applications show the robustness and the effectiveness of the proposed algorithms.

Perturbation methods (asymptotic expansions) are usually considered as powerful methods for solving many kinds of non-linear problems. However, these methods are very often applied in a purely anal...