Efficient optimization-based quadrature for variational discretization of nonlocal problems
Efficient optimization-based quadrature for variational discretization of nonlocal problems
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DOI：
10.1016/j.cma.2022.115104
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发表时间：
2022-01
期刊：
影响因子：
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通讯作者：
M. Pasetto;Zhaoxiang Shen;M. D'Elia;Xiaochuan Tian;Nathaniel Trask;D. Kamensky
中科院分区：
文献类型：
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作者：
M. Pasetto;Zhaoxiang Shen;M. D'Elia;Xiaochuan Tian;Nathaniel Trask;D. Kamensky
铸造非局部问题的变分形式和离散它们的有限元（FE）方法便于使用非局部向量微积分证明适定性，收敛性和稳定性的计划。采用有限元方法也有利于复杂区域几何形状的网格划分，并与有限元方法耦合处理局部问题。然而，非局部弱问题涉及双重积分的计算，这在计算上是昂贵的，并提出了一些挑战。特别地，与刚度矩阵相关联的变分形式的内积分被定义在有限元网格单元与半径为δ的球的交点上，其中δ是非局部相互作用的范围。识别和参数化这些交叉点是一个非平凡的计算几何问题。在这项工作中，我们提出了一种正交技术，其中的内部集成是使用分布在整个球的正交点，而不考虑它如何相交的元素，并根据广义移动最小二乘法计算权重。因此，相对于所有以前采用的方法，我们的技术不需要元素的元素集成和完全规避计算的元素球相交。本文考虑了一维和二维分段线性连续有限元近似的实现，重点是单元尺寸h和非局部半径δ成比例的情况，这是典型的实际计算。当边界条件被仔细处理并且变分形式的外积分被精确计算时，所提出的方法在h <$δ→ 0的极限下是渐近相容的，在所有维数下，无论使用均匀网格还是非均匀网格，都具有至少一阶L2收敛性.此外，在均匀网格的情况下，所提出的方法通过补丁测试，并根据数值证据，表现出最佳的，二阶收敛速度。我们的数值试验还表明，即使是非均匀网格，二阶收敛可以观察到大量的前渐近制度。
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 L 2 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.