Class Of Meshless Methods Research Articles

Effect of Local Support Configuration on the Precision of Numerical Solutions of Poisson Equation Obtained With Differential Quadrature Method

Differential quadrature methods are devised to numerically solve ordinary and partial differential equations by approximating the derivatives of the unknown function at points of a cloud defined on the domain of interest as weighted sums of the values of such function at other points of the cloud. Local versions of this class of meshless methods restrict the points used in such expansion, by establishing suitable supporting regions. In this paper, we present the local differential quadrature method and we use it to solve a boundary problem in electromagnetism. In order to do this, we evaluate the numerical solutions of the Poisson equation on a 2-D domain. Furthermore, we propose an alternative definition of supporting region that has yielded better solutions than the conventional one. Root-mean-square errors for the approximations with both (alternative and conventional) definitions of local supports are obtained and their dependences with the density of nodes are studied. We find out that the best accuracy obtained with the alternative definition of the local support is due to the smaller condition numbers of the linear systems yielded.

Weak Dual Pairs and Jetlet Methods for Ideal Incompressible Fluid Models in $$n \ge 2$$ n ≥ 2 Dimensions

We review the role of dual pairs in mechanics and use them to derive particle-like solutions to regularized incompressible fluid systems. In our case we have a dual pair resulting from the action of diffeomorphisms on point particles (essentially by moving the points). We then augment our dual pair by considering the action of diffeomorphisms on Taylor series, also known as jets. The augmented weak dual pairs induce a hierarchy of particle-like solutions and conservation laws with particles carrying a copy of a jet group. We call these augmented particles jetlets. The jet groups serve as finite-dimensional models of the diffeomorphism group itself, and so the jetlet particles serve as a finite-dimensional model of the self-similarity exhibited by ideal incompressible fluids. The conservation law associated to jetlet solutions is shown to be a shadow of Kelvin's circulation theorem. Finally, we study the dynamics of infinite time particle mergers. We prove that two merging particles at the zeroth level in the hierarchy yield dynamics which asymptotically approach that of a single particle in the first level in the hierarchy. This merging behavior is then verified numerically as well as the exchange of angular momentum which must occur during a near collision of two particles. The resulting particle-like solutions suggest a new class of meshless methods which work in dimensions $n \geq 2$ and which exhibit a shadow of Kelvin's circulation theorem. More broadly, this provides one of the first finite-dimensional models of self-similarity in ideal fluids.

Review of the modified finite particle method and application to incompressible solids

This paper focuses on the application of the Modified Finite Particle Method (MFPM) on incompressibile elasticity problems. MFPM belongs to the class of meshless methods, nowadays widely investigated due to their characteristics of being totally free of any kind of grid or mesh. This characteristic makes meshless methods potentially useful for the study of large deformations problems and fluid dynamics. In particular, the aim of the work is to compare the results obtained with a simple displacement-based formulation, in the limit of incompressibility, and some formulations proposed in the literature for full incompressibility, where the typical divergence-free constraint is replaced by a different equation, the so-called Pressure Poisson Equation. The obtained results show that the MFPM achieves the expected second-order accuracy on formulation where the equations imposed as constraint satisfies also the original incompressibility equation. Other formulations, differently, do not satisfy the incompressibility constraint, and thus, they are not successfully applicable with the Modified Finite Particle Method.

Use of radial basis functions for solving the second‐order parabolic equation with nonlocal boundary conditions

AbstractNonlocal mathematical models appear in various problems of physics and engineering. In these models the integral term may appear in the boundary conditions. In this paper the problem of solving the one‐dimensional parabolic partial differential equation subject to given initial and nonlocal boundary conditions is considered. These kinds of problems have certainly been one of the fastest growing areas in various application fields. The presence of an integral term in a boundary condition can greatly complicate the application of standard numerical techniques. As a well‐known class of meshless methods, the radial basis functions are used for finding an approximation of the solution of the present problem. Numerical examples are given at the end of the paper to compare the efficiency of the radial basis functions with famous finite‐difference methods. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008