Blending isogeometric analysis and local maximum entropy meshfree approximants

Abstract

We present a method to blend local maximum entropy (LME) meshfree approximants and isogeometric analysis. The coupling strategy exploits the optimization program behind LME approximation, treats isogeometric and LME basis functions on an equal footing in the reproducibility constraints, but views the former as data in the constrained minimization. The resulting scheme exploits the best features and overcomes the main drawbacks of each of these approximants. Indeed, it preserves the high fidelity boundary representation (exact CAD geometry) of isogeometric analysis, out of reach for bare meshfree methods, and easily handles volume discretization and unstructured grids with possibly local refinement, while maintaining the smoothness and non-negativity of the basis functions. We implement the method with B-Splines in two dimensions, but the procedure carries over to higher spatial dimensions or to other non-negative approximants such as NURBS or subdivision schemes. The performance of the method is illustrated with the heat equation, and linear and nonlinear elasticity. The ability of the proposed method to impose directly essential boundary conditions in non-convex domains, and to deal with unstructured grids and local refinement in domains of complex geometry and topology is highlighted by the numerical examples.

Introduction

Approximants selected by maximum entropy (max-ent) are non-negative smooth meshfree approximation schemes, optimal from an information theory viewpoint [1], [2]. The non-negativity and first-order reproducing conditions endow these approximants with the structure of convex geometry [1], like linear finite element, 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].

Local maximum entropy (LME) approximants allow us to flexibly control the support of the basis functions on unstructured grids of points [1], [9]. Their non-negativity endow them with variation diminishing properties, as well as with a weak Kronecker-delta property on the boundary of the convex hull of the set of nodes [1], by which interior basis functions vanish at the boundary of the convex hull, and basis functions vanish at any given face unless the corresponding node belongs to that face of the boundary. Thanks to this property, essential boundary conditions can be easily imposed on polygonal convex domains, in contrast with other meshfree methods [10]. Furthermore, the evaluation of the LME basis functions is very efficient using duality methods [1]. The main drawback of these approximants is given by the inherent limitation of meshfree methods to represent complex boundaries with high fidelity. In such methods, the boundaries that can be represented by a mere collection of points are polytopes, either the convex hull or more controllable domains given by alpha shapes [11]. Furthermore, the weak Kronecker-delta property of LME approximants does not hold in non-convex parts of the domain [1].

Motivated by the recent impetus on isogeometric analysis [5], [12], which aims at integrating Computer Aided Design (CAD) technologies, such as B-Splines, NURBS or subdivision surfaces [4], and engineering analysis, we propose here using such high-fidelity description of the boundary of the domain, while approximating the interior with max-ent methods. Remarkably, the limitations of LME approximants and of isogeometric analysis are in some sense complementary, since the main drawback of the latter is precisely the rigidity imposed by the NURBS framework on the volume meshing, which requires special techniques to go beyond tensor product meshes and accommodate trimmed surfaces, local refinement, or incongruent surface descriptions at opposing faces. Some of these issues are partially addressed in 2D with T-Spline technologies [13], [14], [15], [16], [17], hierarchical B-Splines [18] or trimming techniques [19], but largely open in 3D [20], [21]. Three-dimensional subdivision schemes, producing smooth convex approximants from unstructured grids, are still the topic of current research [22].

The goal of the proposed method is to unify in a common framework the geometric fidelity of isogeometric boundary representations with the flexibility of meshfree approximants in the bulk of the domain. Since both B-Splines and LME approximants are convex schemes, we will show that they can be coupled through the constraints in a max-ent program. 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-order B-Splines for structural vibrations, and are comparable to second-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.

In the same spirit of the method presented here, the NURBS enhanced finite element method (NEFEM) [26] adopts a NURBS boundary representation, coupled to standard finite elements in the interior of the domain. This approach exploits the high fidelity geometry representation of isogeometric analysis, but does not insist in preserving the smoothness and positivity of the basis functions, placing more emphasis in the high-order reproducibility conditions. On the other hand, Moving Least Squares (MLS) meshfree basis functions have been coupled with finite elements through the consistency conditions [27].

The paper is organized as follows. Sections 2 Maximum entropy approximation schemes, 3 Isogeometric boundary representation provide the main concepts about max-ent approximations schemes and the isogeometric representation of boundaries. In Section 4, we describe the proposed blending strategy, and in Section 5 we report on illustrative numerical examples. Finally, Section 6 collects the concluding remarks.