Extended virtual element method for the Laplace problem with singularities and discontinuities

Journal Article · Fri Aug 02 00:00:00 EDT 2019 · Computer Methods in Applied Mechanics and Engineering

DOI:https://doi.org/10.1016/j.cma.2019.07.028· OSTI ID:1844125

Benvenuti, E. ^[1]; Chiozzi, A. ^[1]; Manzini, G. ^[2]; Sukumar, N. ^[3]

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1. University of Ferrara (Italy)
2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
3. Univ. of California, Davis, CA (United States)

In this paper, we propose the extended virtual element method (X-VEM) to treat singularities and crack discontinuities that arise in the Laplace problem. The virtual element method (VEM) is a stabilized Galerkin formulation on arbitrary polytopal meshes, wherein the basis functions are implicit (virtual)—they are not known explicitly nor do they need to be computed within the problem domain. In this work, suitable projection operators are used to decompose the bilinear form on each element into two parts: a consistent term that reproduces the first-order polynomial space and a correction term that ensures stability. A similar approach is pursued in the X-VEM with a few notable extensions. To capture singularities and discontinuities in the discrete space, we augment the standard virtual element space with an additional contribution that consists of the product of virtual nodal basis (partition-of-unity) functions with enrichment functions. For discontinuities, basis functions are discontinuous across the crack and for singularities a weakly singular enrichment function that satisfies the Laplace equation is chosen. For the Laplace problem with a singularity, we devise an extended projector that maps functions that lie in the extended virtual element space onto linear polynomials and the enrichment function, whereas for the discontinuous problem, the consistent element stiffness matrix has a block-structure that is readily computed. An adaptive homogeneous numerical integration method is used to accurately and efficiently (no element-partitioning is required) compute integrals with integrands that are weakly singular. Once the element projection matrix is computed, the same steps as in the standard VEM are followed to compute the element stabilization matrix. Numerical experiments are performed on quadrilateral and polygonal (convex and nonconvex elements) meshes for the problem of an $$\textit{L}$$-shaped domain with a corner singularity and the problem of a cracked membrane under mode III loading, and results are presented that affirm the sound accuracy and demonstrate the optimal rates of convergence in the $$\textit{L}^2$$ norm and energy of the proposed method.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE; USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
• Grant/Contract Number: 89233218CNA000001; AC52-06NA25396
• OSTI ID: 1844125
• Alternate ID(s): OSTI ID: 1701931
• Report Number(s): LA-UR-19-20877
• Journal Information: Computer Methods in Applied Mechanics and Engineering, Journal Name: Computer Methods in Applied Mechanics and Engineering Vol. 356; ISSN 0045-7825
• Publisher: Elsevier
• Country of Publication: United States
• Language: English

Related Subjects

42 ENGINEERING
97 MATHEMATICS AND COMPUTING
Crack discontinuity
Enrichment
Mathematics
Partition-of-unity enrichment
Polygonal meshes
VEM
Cracks
Vertex singularity
Virtual element method
X-VEM