Article (Scientific journals)
Efficient optimization-based quadrature for variational discretization of nonlocal problems
Pasetto, Marco; Shen, Zhaoxiang; D'Elia, Marta et al.
2022In Computer Methods in Applied Mechanics and Engineering, 396
Peer reviewed
 

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Keywords :
Finite element discretizations; Generalized moving least squares; Nonlocal models; Numerical quadrature; Peridynamics
Abstract :
[en] Casting nonlocal problems in variational form and discretizing them with the finite element (FE) method facilitates the use of nonlocal vector calculus to prove well-posedness, convergence, and stability of such schemes. Employing an FE method also facilitates meshing of complicated domain geometries and coupling with FE methods for local problems. However, nonlocal weak problems involve the computation of a double-integral, which is computationally expensive and presents several challenges. In particular, the inner integral of the variational form associated with the stiffness matrix is defined over the intersections of FE mesh elements with a ball of radius δ, where δ is the range of nonlocal interaction. Identifying and parameterizing these intersections is a nontrivial computational geometry problem. In this work, we propose a quadrature technique where the inner integration is performed using quadrature points distributed over the full ball, without regard for how it intersects elements, and weights are computed based on the generalized moving least squares method. Thus, as opposed to all previously employed methods, our technique does not require element-by-element integration and fully circumvents the computation of element–ball intersections. This paper considers one- and two-dimensional implementations of piecewise linear continuous FE approximations, focusing on the case where the element size h and the nonlocal radius δ are proportional, as is typical of practical computations. When boundary conditions are treated carefully and the outer integral of the variational form is computed accurately, the proposed method is asymptotically compatible in the limit of h∼δ→0, featuring at least first-order convergence in L2 for all dimensions, using both uniform and nonuniform grids. Moreover, in the case of uniform grids, the proposed method passes a patch test and, according to numerical evidence, exhibits an optimal, second-order convergence rate. Our numerical tests also indicate that, even for nonuniform grids, second-order convergence can be observed over a substantial pre-asymptotic regime. © 2022 Elsevier B.V.
Disciplines :
Engineering, computing & technology: Multidisciplinary, general & others
Author, co-author :
Pasetto, Marco;  University of Califronia, San Diego > Department of Mechanical and Aerospace Engineering
Shen, Zhaoxiang  ;  University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Engineering (DoE)
D'Elia, Marta;  Sandia National Laboratories > Computational Science and Analysis
Tian, Xiaochuan;  University of California, San Diego > Department of Mathematics
Trask, Nathaniel;  Sandia National Laboratories > Center for Computing Research
Kamensky, David;  University of California, San Diego > Department of Mechanical and Aerospace Engineering
External co-authors :
yes
Language :
English
Title :
Efficient optimization-based quadrature for variational discretization of nonlocal problems
Publication date :
2022
Journal title :
Computer Methods in Applied Mechanics and Engineering
ISSN :
0045-7825
Publisher :
Elsevier B.V.
Volume :
396
Peer reviewed :
Peer reviewed
Focus Area :
Computational Sciences
Available on ORBilu :
since 11 July 2022

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