Convergence Analysis of Meshfree Approximation Schemes

Abstract

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 MaximumEntropy (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 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.

Enhanced local maximum-entropy approximation for stable meshfree simulations

“…Despite this adaptive evolution of shape functions, that scheme is still prone to tensile instability in case of anisotropic deformations (as will be demonstrated in Section 2.4). Further advances in the area of max-ent approximations include the convergence analysis of Bompadre et al (2012), the variational formulation of the optimal support size of max-ent shape functions (Rosolen et al, 2010), max-ent schemes with arbitrary order of consistency (González et al, 2010), as well as tools to evaluate derivatives of max-ent shape functions near the boundary (Greco and Sukumar, 2013).…”

“…Despite this adaptive evolution of shape functions, that scheme is still prone to tensile instability in case of anisotropic deformations (as will be demonstrated in Section 2.4). Further advances in the area of max-ent approximations include the convergence analysis of Bompadre et al (2012), the variational formulation of the optimal support size of max-ent shape functions (Rosolen et al, 2010), max-ent schemes with arbitrary order of consistency (González et al, 2010), as well as tools to evaluate derivatives of max-ent shape functions near the boundary (Greco and Sukumar, 2013).Here, we present an enhanced local max-ent scheme for stable, quasistatic meshfree simulations. In Section 2.1, we present modified local max-ent shape functions that are based on an anisotropic Pareto compromise between maximizing information entropy and minimizing shape function width, which will play a crucial role in eliminating the tensile instability under large deformations.…”

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

“…We specifically resort to the max-ent interpolation scheme proposed by [13]. This interpolation scheme is meshfree and conforming and supplies converging approximations in general W 1,p spaces [14].…”

On the Infinite Particle Limit in Lagrangian Dynamics and Convergence of Optimal Transportation Meshfree Methods

“…See, e.g., [9] and the references therein for a general introduction to the theory of variational integrators and [10] for a convergence analysis on manifolds. As discussed in [8], conforming fields may be interpolated efficiently with max-ent shape functions as developed in [1], for which convergence has been proved in the recent article [6].…”