Efficient Nonlinear Solvers for Nodal High-Order Finite Elements in 3D
Efficient Nonlinear Solvers for Nodal High-Order Finite Elements in 3D
复制标题
3D 节点高阶有限元的高效非线性求解器
DOI：
10.1007/s10915-010-9396-8
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发表时间：
2010
影响因子：
2.5
通讯作者：
Jed Brown
中科院分区：
文献类型：
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作者：
Jed Brown
传统的高阶有限元方法很少用于工业问题，因为随着阶数的增加，雅可比矩阵会迅速失去稀疏性，导致无法承受的求解时间和内存需求。尽管二次和高阶离散化提供了良好的精度和稳定性，但这种效应通常将阶数限制为最多二次。我们提出了一种方法，在该方法中的雅可比矩阵的行动是适用于无矩阵利用六面体元素的张量积的基础上，而更稀疏的矩阵的基础上Q1子元素的节点上的高阶基础组装预处理。有了这个“双阶”方案，存储是独立的频谱阶数和一个自然的录音计划是可用的，以更新一个全精度矩阵自由雅可比矩阵在残差评估。无矩阵雅可比应用程序规避了稀疏矩阵运算的典型内存带宽瓶颈，提供了数倍的浮点性能，并更好地利用共享内存总线的多个内核。计算结果的p-Laplacian和Stokes问题，使用块预条件和AMG，表现出网格独立的收敛速度和弱（有界）的顺序依赖，即使是高度变形的网格和非线性系统的几个数量级的动态范围的系数。对于约5的谱阶，双阶方案需要一半的存储器和类似的时间来组装二次（Q2）元素，使其非常适合一般使用。
Conventional high-order finite element methods are rarely used for industrial problems because the Jacobian rapidly loses sparsity as the order is increased, leading to unaffordable solve times and memory requirements. This effect typically limits order to at most quadratic, despite the favorable accuracy and stability properties offered by quadratic and higher order discretizations. We present a method in which the action of the Jacobian is applied matrix-free exploiting a tensor product basis on hexahedral elements, while much sparser matrices based on Q1 sub-elements on the nodes of the high-order basis are assembled for preconditioning. With this “dual-order” scheme, storage is independent of spectral order and a natural taping scheme is available to update a full-accuracy matrix-free Jacobian during residual evaluation. Matrix-free Jacobian application circumvents the memory bandwidth bottleneck typical of sparse matrix operations, providing several times greater floating point performance and better use of multiple cores with shared memory bus. Computational results for the p-Laplacian and Stokes problem, using block preconditioners and AMG, demonstrate mesh-independent convergence rates and weak (bounded) dependence on order, even for highly deformed meshes and nonlinear systems with several orders of magnitude dynamic range in coefficients. For spectral orders around 5, the dual-order scheme requires half the memory and similar time to assembled quadratic (Q2) elements, making it very affordable for general use.