Cell-based maximum-entropy approximants

Abstract

In this paper, we devise cell-based maximum-entropy (max-ent) basis functions that are used in a Galerkin method for the solution of partial differential equations. The motivation behind this work is the construction of smooth approximants with controllable support on unstructured meshes. In the variational scheme to obtain max-ent basis functions, the nodal prior weight function is constructed from an approximate distance function to a polygonal curve in R 2 . More precisely, we take powers of the composition of R-functions via Boolean operations. The basis functions so constructed are nonnegative, smooth, linearly complete, and compactly-supported in a neighborring of segments that enclose each node. The smoothness is controlled by two positive integer parameters: the normalization order of the approximation of the distance function and the power to which it is raised. The properties and mathematical foundations of the new compactly-supported approximants are described, and its use to solve two-dimensional elliptic boundary-value problems (Poisson equation and linear elasticity) is demonstrated. The sound accuracy and the optimal rates of convergence of the method in Sobolev norms are established.

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

“…In the following, we give a brief overview of the minimum relative entropy framework (Section 2.1) along with the nodal prior weight construction using R-functions (Section 2.2). For a detailed description of the CME formulation, we refer the reader to the work of Millán et al 28…”

“…18 Moreover, the basis functions inherit the nodal prior weight functions smoothness. 31,32 Various nodal prior weight functions, such as Gaussian weight function, 20,22,33 quartic polynomial weight function, [34][35][36] level set based nodal weight function, 37,38 exponential nodal weight function, 39,40 and approximate distance function to planar curves 28 have been used to construct maximum entropy approximants with specific desired properties. The expression of the gradient for the MaxEnt basis functions for a node a evaluated on a point x ∈ E d is given by…”

“…A variational approach to adapt in the context of LME has been proposed in the work of Rosolen et al 27 Cell-based maximum entropy (CME) approximants have been proposed as an alternative to LME in the Galerkin framework. 28 CME capitalizes on the background mesh connectivity only to define compact support for basis functions on N-rings (N ≥ 1) of connected cells while the basis functions values are computed without using the connectivity information. Up to date, CME approximants have been implemented to approximate field functions in E 2 .…”

kFEM: Adaptive meshfree finite‐element methods using local kernels on arbitrary subdomains

“…Such a particular case can be considered the lowest‐order member of the kernel ‐FEM (kFEM) family, which we refer to as h ‐kFEM with p =1 in the next sections. It must be noted that our work is not the first attempt to develop cell‐based MaxEnt approximants, or to couple them with standard mesh‐based finite‐element (FE) schemes,() but to our knowledge, it is the first to consider local refinement inside the cells and to not assume the availability of a mesh.…”

Parallel computations in nonlinear solid mechanics using adaptive finite element and meshless methods

“…Recent use of the max-ent method, e.g. in fracture mechanics, can be found in Amiri et al (2014b,a), while other recent examples of the use of max-ent in meshless methods can be found in (Ortiz et al, 2010(Ortiz et al, , 2011Millán et al, 2011;Quaranta et al, 2012;Rosolen et al, 2012;Millán et al, 2015;Peco et al, 2015).…”