Cell-based maximum entropy approximants for three-dimensional domains: Application in large strain elastodynamics using the meshless total Lagrangian explicit dynamics method

We present the cell-based maximum entropy (CME) approximants in E3 space by constructing the smooth approximation distance function to polyhedral surfaces. CME is a meshfree approximation method combining the properties of the maximum entropy approximants and the compact support of element-based interpolants. The method is evaluated in problems of large strain elastodynamics for three-dimensional (3D) continua using the well-established meshless total Lagrangian explicit dynamics method. The accuracy and efficiency of the method is assessed in several numerical examples in terms of computational time, accuracy in boundary conditions imposition, and strain energy density error. Due to the smoothness of CME basis functions, the numerical stability in explicit time integration is preserved for large time step. The challenging task of essential boundary condition (EBC) imposition in noninterpolating meshless methods (eg, moving least squares) is eliminated in CME due to the weak Kronecker-delta property. The EBCs are imposed directly, similar to the finite element method. CME is proven a valuable alternative to other meshless and element-based methods for large-scale elastodynamics in 3D. A naive implementation of the CME approximants in E3 is available to download at https://www.mountris.org/software/mlab/cme.Fil: Mountris, Konstantinos A.. Universidad de Zaragoza; EspañaFil: Bourantas, George C.. University of Western Australia; AustraliaFil: Millán, Raúl Daniel. Universidad Nacional de Cuyo. Facultad de Ciencias Aplicadas a la Industria; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mendoza; ArgentinaFil: Joldes, Grand R.. University of Western Australia; AustraliaFil: Miller, Karol. Cardiff University; Reino Unido. University of Western Australia; AustraliaFil: Pueyo, Esther. Centro de Investigacion Biomedica En Red.; España. Universidad de Zaragoza; EspañaFil: Wittek, Adam. University of Western Australia; Australi

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

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

Deriving the continuity of maximum-entropy basis functions via variational analysis

In this paper, we prove the continuity of maximum-entropy basis functions using Variational Analysis techniques. The use of information-theoretic variational principles to derive basis functions is a recent development. In this setting, data approximation is viewed as an inductive inference problem, with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints. For a set of distinct nodes {x i} n i=1 in IRd, the convex approximation of a function u(x) is: u h (x) = � n i=1 pi(x)ui, where {pi} n i=1 are non-negative basis functions, and uh (x) must reproduce affine functions: � n i=1 pi(x) = 1, � n i=1 pi(x)x i = x. Given these constraints, we compute pi(x) by minimizing the relative entropy functional (Kullback-Leibler distance), D(p�m) = � n i=1 pi(x) ln � pi(x)/mi(x) � , where mi(x) is a known prior weight function distribution. To prove the continuity of the basis functions, we appeal to the theory of epi-convergence

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

Analysis and new constructions of generalized barycentric coordinates in 2D

Different coordinate systems allow to uniquely determine the position of a geometric element in space. In this dissertation, we consider a coordinate system that lets us determine the position of a two-dimensional point in the plane with respect to an arbitrary simple polygon. Coordinates of this system are called generalized barycentric coordinates in 2D and are widely used in computer graphics and computational mechanics. There exist many coordinate functions that satisfy all the basic properties of barycentric coordinates, but they differ by a number of other properties. We start by providing an extensive comparison of all existing coordinate functions and pointing out which important properties of generalized barycentric coordinates are not satisfied by these functions. This comparison shows that not all of existing coordinates have fully investigated properties, and we complete such a theoretical analysis for a particular one-parameter family of generalized barycentric coordinates for strictly convex polygons. We also perform numerical analysis of this family and show how to avoid computational instabilities near the polygon’s boundary when computing these coordinates in practice. We conclude this analysis by implementing some members of this family in the Computational Geometry Algorithm Library. In the second half of this dissertation, we present a few novel constructions of non-negative and smooth generalized barycentric coordinates defined over any simple polygon. In this context, we show that new coordinates with improved properties can be obtained by taking convex combinations of already existing coordinate functions and we give two examples of how to use such convex combinations for polygons without and with interior points. These new constructions have many attractive properties and perform better than other coordinates in interpolation and image deformation applications

SIAM J. OPTIM. c ○ XXXX Society for Industrial and Applied Mathematics Vol. 0, No. 0, pp. 000–000 DERIVING THE CONTINUITY OF MAXIMUM-ENTROPY BASIS FUNCTIONS VIA VARIATIONAL ANALYSIS ∗

Abstract. In this paper, we prove the continuity of maximum-entropy basis functions using variational analysis techniques. The use of information-theoretic variational principles to derive basis functions is a recent development. In this setting, data approximation is viewed as an inductive inference problem, with the basis functions being synonymous with a discrete probability distribution, and the polynomial reproducing conditions acting as the linear constraints. For a set of distinct nodes {x i} n i=1 in Rd, the convex approximation of a function u(x) isu h (x) = � n i=1 pi(x)ui, where {pi} n i=1 are nonnegative basis functions, and uh (x) must reproduce affine functions � n i=1 pi(x) =1, � n i=1 pi(x)x i = x. Given these constraints, we compute pi(x) by minimizing the relative entropy functional (Kullback–Leibler distance), D(p�m) = � n i=1 pi(x)ln � pi(x)/mi(x) � , where mi(x) isa known prior weight function distribution. To prove the continuity of the basis functions, we appeal to the theory of epiconvergence