[论文解读] Efficient optimization-based quadrature for variational discretization of nonlocal problems

Marco Pasetto, Zhaoxiang Shen+4 more|arXiv (Cornell University)|Jan 28, 2022

Numerical methods in engineering参考文献 60被引用 5

一句话总结

本文提出了一种基于优化的积分方法，用于非局部问题的有限元离散化，通过使用广义移动最小二乘法在完整球体上计算积分权重，避免了计算成本高昂的单元-球体交集计算。该方法在均匀网格上实现了至少一阶L2收敛和最优二阶收敛，在非均匀网格上表现稳健，并在h ∼ δ → 0时保持渐近相容性。

ABSTRACT

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-posedeness, 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 $\delta$, where $\delta$ 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 $\delta$ 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 $h \sim \delta o 0$, featuring at least first-order convergence in L^2 for all dimensions, using both uniform and nonuniform grids.

研究动机与目标

• 解决非局部有限元方法中单元-球体交集检测带来的计算瓶颈。
• 开发一种积分格式，实现非局部刚度矩阵的高效且精确组装，而无需逐单元积分。
• 在均匀和非均匀网格上确保渐近相容性和最优收敛速率。
• 通过避免对单元-球体重叠区域的几何重构，最小化实现复杂度。
• 通过数值实验和收敛性分析验证该方法的鲁棒性。

提出的方法

• 该方法在完整影响球体H(x, δ)上分布积分点，内层积分过程中忽略单元边界。
• 通过广义移动最小二乘法（GMLS）计算积分权重，以确保一致性和准确性。
• GMLS框架通过求解最小二乘优化问题来确定权重，使其能够恢复至指定阶次的多项式一致性。
• 该方法避免了显式计算单元-球体交集，后者在几何上复杂且耗时。
• 该方法在标准有限元框架内实现，仅需极少的代码修改。
• 边界条件被仔细处理，以保持渐近相容性和收敛性。

实验结果

研究问题

• RQ1 能否设计一种积分方法，在避免单元-球体交集计算的同时，保持非局部有限元方法中的精度和收敛性？
• RQ2 在均匀和非均匀网格上，该方法在L2和H1范数下可实现的收敛速率是多少？
• RQ3 当h ∼ δ → 0时，该方法是否保持渐近相容性？
• RQ4 该方法能否通过patch测试，并在均匀网格上实现最优收敛？
• RQ5 在非均匀网格的预渐近区域，该方法表现如何？

主要发现

• 该方法在所有维度下，对均匀和非均匀网格，均实现了L2范数下至少一阶收敛。
• 在均匀网格上，该方法在L2范数下表现出最优二阶收敛，并通过了patch测试。
• 在非均匀网格上，该方法在较大预渐近区域内表现出有效的二阶收敛，仅在与patch测试存在微小偏差时才显现出渐近一阶收敛。
• H1范数下的收敛速率始终比L2范数低一个阶次。
• 所有积分点的积分权重均为严格正值，确保了稳定性与鲁棒性。
• 在h ∼ δ → 0的极限下，该方法保持渐近相容性，且实现开销极小。