Evaluating the accuracy of hybrid finite element/particle-in-cell methods for modelling incompressible Stokes flow
Evaluating the accuracy of hybrid finite element/particle-in-cell methods for modelling incompressible Stokes flow
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评估混合有限元/单元内粒子方法模拟不可压缩斯托克斯流的准确性
DOI：
10.1093/gji/ggz405
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发表时间：
2019
影响因子：
2.8
通讯作者：
Puckett, Elbridge Gerry
中科院分区：
文献类型：
--
作者：
Gassmöller, Rene;Lokavarapu, Harsha;Bangerth, Wolfgang;Puckett, Elbridge Gerry
将不可压缩Stokes方程的有限元方法与质点方法相结合是计算地球动力学中的一项重要技术，在地幔对流、岩石圈动力学和地壳尺度模拟中得到了广泛应用。在这些应用中，粒子被用来沿着介质的性质(如温度、化学成分或其他材料性质)进行传输；因此，粒子方法被用来将平流方程简化为针对每个粒子的常微分方程组，从而产生了比原始方程更容易求解的问题，其中需要稳定技术来避免振荡。另一方面，用仅在粒子位置定义的量来代替基于场的描述会带来数值误差。这些错误以前已经被调查过了，但从理论和实践两方面到目前为止还缺乏完整的了解。此外，对于每个网格单元需要选择多少粒子才能达到一定的精度，我们还没有意识到系统的指导。在本文中，我们修改了现有的两个瞬时基准，并提出了两个新的含时不可压缩Stokes流的解析基准，以比较有限元、粒子平流和粒子内插方法的各种组合的收敛速度和精度。使用这些基准，我们发现，为了保持有限元公式的最佳精度，需要使用足够精确的粒子内插算法。此外，我们观察和解释了对于我们的高阶有限元方法，有必要随着网格分辨率的增加(即随着网格单元尺寸的减小)而增加每单元的粒子数，以避免收敛阶数的降低。我们的方法和结果允许设计具有特定收敛速度的新的单元中的粒子方法，并且还为公共积木和参数的选择提供了指导，例如每单元的粒子数。此外，我们新的与时间相关的基准测试提供了一个简单的测试，可用于比较不同的实现、算法以及评估用于粒子内插和平流的新的数值方法。我们提供了这个基准方面的一个参考实现(“地球对流问题的高级解算器”)，这是一个用于地球动力学建模的开放源代码。
Combining finite element methods for the incompressible Stokes equations with particle-in-cell methods is an important technique in computational geodynamics that has been widely applied in mantle convection, lithosphere dynamics and crustal-scale modelling. In these applications, particles are used to transport along properties of the medium such as the temperature, chemical compositions or other material properties; the particle methods are therefore used to reduce the advection equation to an ordinary differential equation for each particle, resulting in a problem that is simpler to solve than the original equation for which stabilization techniques are necessary to avoid oscillations.On the other hand, replacing field-based descriptions by quantities only defined at the locations of particles introduces numerical errors. These errors have previously been investigated, but a complete understanding from both the theoretical and practical sides was so far lacking. In addition, we are not aware of systematic guidance regarding the question of how many particles one needs to choose per mesh cell to achieve a certain accuracy.In this paper we modify two existing instantaneous benchmarks and present two new analytic benchmarks for time-dependent incompressible Stokes flow in order to compare the convergence rate and accuracy of various combinations of finite elements, particle advection and particle interpolation methods. Using these benchmarks, we find that in order to retain the optimal accuracy of the finite element formulation, one needs to use a sufficiently accurate particle interpolation algorithm. Additionally, we observe and explain that for our higher-order finite-element methods it is necessary to increase the number of particles per cell as the mesh resolution increases (i.e. as the grid cell size decreases) to avoid a reduction in convergence order.Our methods and results allow designing new particle-in-cell methods with specific convergence rates, and also provide guidance for the choice of common building blocks and parameters such as the number of particles per cell. In addition, our new time-dependent benchmark provides a simple test that can be used to compare different implementations, algorithms and for the assessment of new numerical methods for particle interpolation and advection. We provide a reference implementation of this benchmark inaspect(the ‘Advanced Solver for Problems in Earth’s ConvecTion’), an open source code for geodynamic modelling.
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