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首页> 外文期刊>Applied numerical mathematics >Reuse, Recycle, Reduce (3r) - Strategies For The Calculationof Transient Magnetic Fields
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Reuse, Recycle, Reduce (3r) - Strategies For The Calculationof Transient Magnetic Fields

机译：重用，回收，减少（3r）-瞬态磁场的计算策略

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摘要

The discretization of transient magneto-dynamic field problems with geometric discretization schemes such as the Finite Integration Technique or the Finite-Element Method based on Whitney form functions results in nonlinear differential-algebraic systems of equations of index 1. Their time integration with embedded s-stage singly diagonal implicit Runge-Kutta methods requires the solution of s nonlinear systems within one time step. Accelerated solution of these schemes is achieved with techniques following so-called 3R-strategies ("reuse, recycle, reduce"). This involves e.g. the solution of the linear(-ized) equations in each time step where the solution process of the iterative preconditioned conjugate gradient method reuses and recycles spectral information of linear systems from previous stages. Additionally, a combination of an error controlled spatial adaptivity and an error controlled implicit Runge-Kutta scheme is used to reduce the number of unknowns for the algebraic problems effectively and to avoid unnecessary fine grid resolutions both in space and time. First numerical results for 2D nonlinear magneto-dynamic problems validate the presented approach and its implementation. The space discretization in the numerical examples is done by Lagrangian nodal finite elements but the presented algorithms also work in combination with other discretization schemes for the Maxwell equations such as the Whitney vector finite elements.
机译：利用基于惠特尼形式函数的有限积分技术或有限元方法等几何离散方案对瞬态磁动力场问题进行离散，可得到指数为1的方程组的非线性微分代数系统。它们的时间积分与嵌入式s-单阶段对角隐式Runge-Kutta方法需要在一个时间步内求解s个非线性系统。这些方案的加速解决方案是通过遵循所谓3R策略（“重复使用，回收，减少”）的技术来实现的。这涉及例如迭代预条件共轭梯度法的求解过程在每个时间步中线性（-化）方程的解法，该方法重用和回收了先前阶段线性系统的光谱信息。另外，将误差控制的空间适应性与误差控制的隐式Runge-Kutta方案结合使用，可以有效地减少代数问题的未知数，并避免在空间和时间上不必要的精细网格分辨率。二维非线性磁动力问题的第一个数值结果验证了所提出的方法及其实现。数值示例中的空间离散化是通过拉格朗日节点有限元完成的，但所提出的算法也可以与麦克斯韦方程组的其他离散化方案（如惠特尼矢量有限元）结合使用。

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