descriptionPublicationkeyboard_double_arrow_right Article 01 Feb 2015 English Publisher:Elsevier BVJournal:Computer Methods in Applied Mechanics and Engineering, volume 284, pages 712-731 (issn: 0045-7825,

Authors: Millán, Daniel; Sukumar, N; Arroyo Balaguer, Marino;

doi: 10.1016/j.cma.2014.10.012

handle: 2117/78127 , 2117/84277

Cell-based maximum-entropy approximants

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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 neighbor-ring 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.

Related Organizations

University of California, Davis
United States
Universitat Politècnica de Catalunya
Spain
Universitat Polite`cnica de Catalunya
Spain

Keywords

Relative entropy, boundary value problems, Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs, Classificació AMS::65 Numerical analysis::65N Partial differential equations, boundary value problems, :65 Numerical analysis::65N Partial differential equations, boundary value problems [Classificació AMS], Classificació AMS::65 Numerical analysis::65K Mathematical programming, Mètodes de, :65 Numerical analysis::65K Mathematical programming, optimization and variational techniques [Classificació AMS], CONSTRUCTION, smooth and nonnegative basis functions, Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi numèrica, MESHFREE METHOD, PART I, Engineering, Mechanical, DISTANCE FIELDS, approximate distance function, Numerical analysis, Engineering, Civil, Classificació AMS::65 Numerical analysis::65N Partial differential equations, R-functions, Engineering, Multidisciplinary, CONVOLUTION SURFACES, Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs, :Matemàtiques i estadística::Anàlisi numèrica [Àrees temàtiques de la UPC], Galerkin, Mètodes de, Engineering, Ocean, Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi numèrica::Mètodes numèrics, Approximate distance function, ARBITRARY PLANAR POLYGONS, FORMULATION, Engineering, Aerospace, Engineering, Biomedical, Anàlisi numèrica, :Matemàtiques i estadística::Anàlisi numèrica::Mètodes numèrics [Àrees temàtiques de la UPC], relative entropy, Classificació AMS::65 Numerical analysis::65K Mathematical programming, optimization and variational techniques, ISOGEOMETRIC ANALYSIS, FINITE-ELEMENTS, Galerkin, Computer Science, Software Engineering, Engineering, Marine, Galerkin methods, Smooth and nonnegative basis functions, Engineering, Manufacturing, compact-support, Classical linear elasticity, Engineering, Industrial, MOVING LEAST-SQUARES, Delaunay mesh, Compact-support, optimization and variational techniques

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