Fast Barycentric-Based Evaluation Over Spectral/hp Elements
Fast Barycentric-Based Evaluation Over Spectral/hp Elements
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DOI：
10.1007/s10915-021-01750-2
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发表时间：
2021-03
影响因子：
2.5
通讯作者：
Edward Laughton;Vidhi Zala;A. Narayan;R. Kirby;D. Moxey
中科院分区：
文献类型：
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作者：
Edward Laughton;Vidhi Zala;A. Narayan;R. Kirby;D. Moxey
随着谱/hp 元素方法和高阶有限元方法的使用不断普及，社区创建与基本高阶运算相关的高效、优化算法的努力不断增加。正交点解展开评估、刚度和质量矩阵生成以及矩阵组装等核心任务受到了极大的关注。 随着高阶方法适用的问题类型的扩大，以及相应地通过高阶方法完成的数值任务类型的增加，这些核心运算的数量和类型也随之扩大。这项工作的重点是元素内任意点的解扩展评估。此操作是许多后处理应用（例如流线和路径评估）以及现场投影技术（例如迫击炮）的核心。我们将间隔开发的重心插值技术扩展到 2D（三角形和四边形）和 3D（四面体、棱柱、金字塔和六面体）光谱/hp 元素方法。我们为它们的实现提供了高效的算法，并使用 Spectrum/hpelement 库 Nektar++ 通过针对“标准”拉格朗日方法运行一系列基线评估来证明其有效性，其中生成插值矩阵并应用矩阵乘法来评估给定位置的点。我们展示了针对各种元素形状、多项式阶数和维度进行的一系列严格基准测试的结果。我们表明，当要重复评估兴趣点时，与缓存矩阵评估相比，重心方法的执行速度最差。然而，当兴趣点反复变化，使得插值矩阵必须以“标准”方法重新生成时，重心方法会产生更好的性能，最小加速因子为 。此外，当还需要解评估的导数时，重心方法在所有元素和阶数上通常略优于缓存插值矩阵方法，并且加速比最高。最后，我们使用非共形不连续伽辽金模拟研究了标量输运的现实示例，其中我们观察到与基于矩阵的方法相比，重心方法在计算时间上的加速。我们还探讨了两种插值方法的复杂性，并表明与拉格朗日插值矩阵方法的最佳情况空间复杂性相比，重心插值方法需要存储。
As the use of spectral/hpelement methods, and high-order finite element methods in general, continues to spread, community efforts to create efficient, optimized algorithms associated with fundamental high-order operations have grown. Core tasks such as solution expansion evaluation at quadrature points, stiffness and mass matrix generation, and matrix assembly have received tremendous attention. With the expansion of the types of problems to which high-order methods are applied, and correspondingly the growth in types of numerical tasks accomplished through high-order methods, the number and types of these core operations broaden. This work focuses on solution expansion evaluation at arbitrary points within an element. This operation is core to many postprocessing applications such as evaluation of streamlines and pathlines, as well as to field projection techniques such as mortaring. We expand barycentric interpolation techniques developed on an interval to 2D (triangles and quadrilaterals) and 3D (tetrahedra, prisms, pyramids, and hexahedra) spectral/hpelement methods. We provide efficient algorithms for their implementations, and demonstrate their effectiveness using the spectral/hpelement libraryNektar++by running a series of baseline evaluations against the ‘standard’ Lagrangian method, where an interpolation matrix is generated and matrix-multiplication applied to evaluate a point at a given location. We present results from a rigorous series of benchmarking tests for a variety of element shapes, polynomial orders and dimensions. We show that when the point of interest is to be repeatedly evaluated, the barycentric method performs at worstslower, when compared to a cached matrix evaluation. However, when the point of interest changes repeatedly so that the interpolation matrix must be regenerated in the ‘standard’ approach, the barycentric method yields far greater performance, with a minimum speedup factor of. Furthermore, when derivatives of the solution evaluation are also required, the barycentric method in general slightly outperforms the cached interpolation matrix method across all elements and orders, with an up tospeedup. Finally we investigate a real-world example of scalar transport using a non-conformal discontinuous Galerkin simulation, in which we observe aroundspeedup in computational time for the barycentric method compared to the matrix-based approach. We also explore the complexity of both interpolation methods and show that the barycentric interpolation method requiresstorage compared to a best case space complexity offor the Lagrangian interpolation matrix method.