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

Abstract

SUMMARYWe 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 nonuniform 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 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 nonuniform node distributions and the high accuracy of the solutions. Surprisingly, the first-order meshfree max-ent approximants with large supports are competitive when compared with the proposed second-order approach in all the tested examples, even in the higher order PDEs.

A stabilized formulation with maximum entropy meshfree approximants for viscoplastic flow simulation in metal forming

“…Moreover, the approximants are multidimensional and lead to well behaved mass matrices. We refer to [60] for a more detailed description of maximum-entropy approximants and their applications. …”

“…In [58] this formulation is extended to Stokes flow in two dimensions and three-dimensional incompressible elasticity and its stability is demonstrated through inf-sup numerical tests. High order max-ent schemes [59][60][61][62] would be another option in order to employ richer approximants for the velocities. However, such approximation schemes are not guaranteed to be LBBcompliant if coupled with constant or linear approximants for the pressure.…”

Blending isogeometric analysis and local maximum entropy meshfree approximants

“…The above program is convex, smooth and feasible for any spa tial dimension d (as long as x 2 convX), and produces C 1 meshfree non negative functions p a ðxÞ [1]. Moreover, the constraints (con sistency conditions) guarantee solutions that reproduce exactly af fine functions (see [6,7,28,29] for higher order approaches). Duality methods provide an efficient route to solving the optimiza tion problem and computing almost explicitly p a ðxÞ at each evalu ation point x.…”

“…The non negativity and first order reproducing conditions endow these approximants with the structure of convex geometry [1], like linear finite ele ment, natural neighbor method [3], subdivision approximants [4], or B Spline and Non Uniform Rational B Splines (NURBS) basis functions [5]. Max ent approximants have been extended to second order [6,7], and to arbitrary order by dropping non negativity [8].…”

“…The resulting approximation scheme automatically retains the non negativity and smoothness of the B Spline and LME parents. Although max ent approximants can be extended to higher order consistency, at the expense of a more involved formulation [6,7], numerical experiments show that first order consistent approximants perform very well, even in high order partial differential equations. In [7], we showed that first order LME approximants attain the same accuracy as 5th or der B Splines for structural vibrations, and are comparable to sec ond order max ent approximation schemes in a fourth order phase field model [23], or in thin shell problems [24,25], where they also compete with subdivision finite elements.…”