Newton-Krylov-Multigrid Algorithms for Battery Simulation
Newton-Krylov-Multigrid Algorithms for Battery Simulation
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DOI：
10.1149/1.1505635
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发表时间：
2002-10
影响因子：
3.9
通讯作者：
Junqiao Wu;V. Srinivasan;Jinchao Xu;Chaoyang Wang
中科院分区：
文献类型：
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作者：
Junqiao Wu;V. Srinivasan;Jinchao Xu;Chaoyang Wang
偏微分方程的数值解形成了模拟各种电化学系统（特别是电池和燃料电池）行为的数学模型的支柱。在本文中，我们提出了一套数值算法，适用于有效地解决这个方程组。这些快速算法是通过充分理解问题的物理性质和识别控制方程之间的耦合强度来确定的。我们说明了这种耦合，特别是在两个潜在的方程，并证明了他们的同时使用牛顿法的解决方案的必要性。本文采用二维热-电耦合锂离子模型，利用Krylov迭代求解器（广义最小残差子程序（GMRES））代替直接求解器（高斯消去法），扩展了常见的Band（J）子程序，提高了求解大型非对称雅可比系统的效率。此外，我们使用一个非线性的高斯-赛德尔方法来提供牛顿迭代的初始猜测，并与块高斯-赛德尔和多重网格算法与基于三对角矩阵算法的平滑的GMRES求解器的预处理。这个过程中的每个阶段都被认为增加了最终计算机模拟的效率，最终结果是计算速度的大幅提高，即，对于45 × 32的网格大小，在不到10分钟的时间内模拟电池的完全放电。
Numerical solutions to partial differential equations form the backbone of mathematical models that simulate the behavior of various electrochemical systems, specifically, batteries and fuel cells. In this paper, we present a set of numerical algorithms applied to efficiently solve this system of equations. These fast algorithms are identified by fully understanding the physics of the problem and recognizing the strength of the coupling between the governing equations. We illustrate this coupling, specifically in the two potential equations, and demonstrate the need for their simultaneous solution using the Newton method. We take a 2D thermal and electrochemical coupled Li-ion model and extend the familiar Band(J) subroutine by utilizing a Krylov iterative solver, a generalized minimal residual subroutine (GMRES), instead of the direct solver (Gauss elimination), to improve the solution efficiency of the large, nonsymmetric Jacobian system. In addition, we use a nonlinear Gauss-Seidel method to provide the initial guess for the Newton iteration, and precondition the GMRES solver with a block Gauss-Seidel and multigrid algorithm with a smoother based on the tridiagonal matrix algorithm. Every stage in this process has been seen to add to the efficiency of the resulting computer simulation with the final result being a substantial improvement in computation speed, namely, simulating complete discharge of the cell in less than 10 min for grid size of 45 × 32.