Modeling curved interfaces without element‐partitioning in the extended finite element method

Abstract

In this paper, we model holes and material interfaces (weak discontinuities) in two-dimensional linear elastic continua using the extended finite element method on higher-order (spectral) finite element meshes. Arbitrary parametric curves such as rational Bézier curves and cubic Hermite curves are adopted in conjunction with the level set method to represent curved interfaces. Efficient computation of weak form integrals with polynomial integrands is realized via the homogeneous numerical integration scheme-a method that uses Euler's homogeneous function theorem and Stokes' theorem to reduce integration to the boundary of the domain. Numerical integration on cut elements requires the evaluation of a one-dimensional integral over a parametric curve, and hence, the need to partition curved elements is eliminated. To improve stiffness matrix conditioning, ghost penalty stabilization and the Jacobi preconditioner are used.For material interface problems, we develop an enrichment function that captures weak discontinuities on spectral meshes. Taken together, we show through numerical experiments that these advances deliver optimal algebraic rates of convergence with h-refinement (p = 1, 2, … , 5) and exponential rates of convergence with p-refinement (p = 1, 2, … , 7) for elastostatic problems with holes and material inclusions on Cartesian pth-order spectral finite element meshes. KEYWORDSBézier curves, homogeneous numerical integration method, interface stabilization, level set methods, material interface enrichment, spectral X-FEM Int J Numer Methods Eng. 2019;120:607-649.wileyonlinelibrary.com/journal/nme

Moment fitted cut spectral elements for explicit analysis of guided wave propagation

“…To integrate the basis monomials over the physical element domain, local hierarchical meshes in combination with standard Lagrangian element partitions were employed. However, no particular restriction is set on the method used for this computation, and thus a number of alternatives might be used: be it pure quad/octrees [41], high order element partitions based on the blending function method [85], or approaches based on Gauss' divergence theorem [76,77], just to name a few. Although results are promising, the authors feel that further improvements could be achieved by deepening the understanding of the procedure under some key aspects.…”

“…In the XFEM/GFEM and SCM/FCM communities, different strategies have been proposed to tackle these issues (e.g. [75,76,57,77]). In this work, we use quadtree (in 2D) and octree (in 3D) meshes in combination with boundary-conforming element partitions.…”

Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

“…These extensions are discussed and presented in Upreti et al [73]. For some of the considerations and challenges in the representation (implicit and parametric) of curves in enriched computational methods, see Chin and Sukumar [74].…”

“…• x Figure 13: Approximate distance fields on curved domains using φ(x) = 2/W 1 (x), with W 1 (x) given in (16). Surface and contour plots are shown for an (a) elliptical disk, (b) annulus, (c) hypocycloid, and (d) propeller [74].…”

Simple Formula for Integration of Polynomials on a Simplex

“…In particular, because of applications in finite (and extended finite) element methods and also for volume computation in the Natural Element Method (NEM), there has been a recent renewal of interest in developing efficient integration numerical schemes for polynomials on convex and non-convex polytopes For instance the HNI (Homogeneous Numerical Integration) technique developed in Chin et al [4] and based on [11], has been proved to be particular efficient in some finite (and extended finite) element methods; see e.g. Antonietti et al [1], Chin and Sukumar [5], Nagy and Benson [13] for intensive experimentation, Zhang et al [14] for NEM, Frenning [6] for DEM (Discrete Element Method), Leoni and Shokef [12] for volume computation. For exact volume computation of polytopes in computational geometry, the interested reader is also referred to Büeler et al [3] and references therein.…”