Convergence Analysis of Meshfree Approximation Schemes

This work is concerned with the formulation of a general framework for the analysis of meshfree approximation schemes and with the convergence analysis of the local maximum-entropy (LME) scheme as a particular example. We provide conditions for the convergence in Sobolev spaces of schemes that are n-consistent in the sense of exactly reproducing polynomials of degree less than or equal to n ≥ 1 and whose basis functions are of rapid decay. The convergence of the LME in W^(1,p)_(loc) (Ω) follows as a direct application of the general theory. The analysis shows that the convergence order is linear in h, a measure of the density of the point set. The analysis also shows how to parameterize the LME scheme for optimal convergence. Because of the convex approximation property of LME, its behavior near the boundary is singular and requires additional analysis. For the particular case of polyhedral domains we show that, away from a small singular part of the boundary, any Sobolev function can be approximated by means of the LME scheme. With the aid of a capacity argument, we further obtain approximation results with truncated LME basis functions in H^1(Ω) and for spatial dimension d > 2

Geometrically-exact time-integration mesh-free schemes for advection-diffusion problems derived from optimal transportation theory and their connection with particle methods

We develop an Optimal Transportation Meshfree (OTM) particle method for advection-diffusion in which the concentration or density of the diffusive species is approximated by Dirac measures. We resort to an incremental variational principle for purposes of time discretization of the diffusive step. This principle characterizes the evolution of the density as a competition between the Wasserstein distance between two consecutive densities and entropy. Exploiting the structure of the Euler-Lagrange equations, we approximate the density as a collection of Diracs. The interpolation of the incremental transport map is effected through mesh-free max-ent interpolation. Remarkably, the resulting update is geometrically exact with respect to advection and volume. We present three-dimensional examples of application that illustrate the scope and robustness of the method.Comment: 19 pages, 8 figure

High-order maximum-entropy collocation methods

This paper considers the approximation of partial differential equations with a point collocation framework based on high-order local maximum-entropy schemes (HOLMES). In this approach, smooth basis functions are computed through an optimization procedure and the strong form of the problem is directly imposed at the collocation points, reducing significantly the computational times with respect to the Galerkin formulation. Furthermore, such a method is truly meshless, since no background integration grids are necessary. The validity of the proposed methodology is verified with supportive numerical examples, where the expected convergence rates are obtained. This includes the approximation of PDEs on domains bounded by implicit and explicit (NURBS) curves, illustrating a direct integration between the geometric modeling and the numerical analysis

Second order convex maximum entropy approximants with applications to high order PDE

We present a new approach for second order maximum entropy (max-ent) meshfree approximants that produces positive and smooth basis functions of uniform aspect ratio even for non-uniform node sets, and prescribes robustly feasible constraints for the entropy maximization program defining the approximants. We examine the performance of the proposed approximation scheme in the numerical solution by a direct Galerkin method of a number of partial differential equations (PDEs), including structural vibrations, elliptic second order PDEs, and fourth order PDEs for Kirchhoff-Love thin shells and for a phase field model describing the mechanics of biomembranes. The examples highlight the ability of the method to deal with non-uniform node distributions, and the high accuracy of the solutions. Surprisingly, the first order meshfree max-ent approximants with large supports are competitive when compared to the proposed second order approach in all the tested examples, even in the higher order PDEs

Enhanced local maximum-entropy approximation for stable meshfree simulations

We introduce an improved meshfree approximation scheme which is based on the local maximum-entropy strategy as a compromise between shape function locality and entropy in an information-theoretical sense. The improved version is specifically designed for severe, finite deformation and offers significantly enhanced stability as opposed to the original formulation. This is achieved by (i) formulating the quasistatic mechanical boundary value problem in a suitable updated-Lagrangian setting, (ii) introducing anisotropy in the shape function support to accommodate directional variations in nodal spacing with increasing deformation and eliminate tensile instability, (iii) spatially bounding and evolving shape function support to restrict the domain of influence and increase efficiency, (iv) truncating shape functions at interfaces in order to stably represent multi-component systems like composites or polycrystals. The new scheme is applied to benchmark problems of severe elastic and elastoplastic deformation that demonstrate its performance both in terms of accuracy (as compared to exact solutions and, where applicable, finite element simulations) and efficiency. Importantly, the presented formulation overcomes the classical tensile instability found in most meshfree interpolation schemes, as shown for stable simulations of, e.g., the inhomogeneous extension of a hyperelastic block up to 100% or the torsion of a hyperelastic cube by 200° –both in an updated Lagrangian setting and without the need for remeshing

Second order convex maximum entropy approximants with applications to high order PDE

We present a new approach for second order maximum entropy (max-ent) meshfree approximants that produces positive and smooth basis functions of uniform aspect ratio even for non-uniform node sets, and prescribes robustly feasible constraints for the entropy maximization program defining the approximants. We examine the performance of the proposed approximation scheme in the numerical solution by a direct Galerkin method of a number of partial differential equations (PDEs), including structural vibrations, elliptic second order PDEs, and fourth order PDEs for Kirchhoff-Love thin shells and for a phase field model describing the mechanics of biomembranes. The examples highlight the ability of the method to deal with non-uniform node distributions, and the high accuracy of the solutions. Surprisingly, the first order meshfree max-ent approximants with large supports are competitive when compared to the proposed second order approach in all the tested examples, even in the higher order PDEs.Peer ReviewedPostprint (author's final draft