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A mixed formulation of proper generalized decomposition for solving the Allen-Cahn and Cahn-Hilliard equations

机译：求解Allen-Cahn和Cahn-Hilliard方程的适当广义分解的混合制剂

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摘要

In this paper, we propose a new implementation strategy for solving the Allen-Cahn and Cahn-Hilliard equations with the proper generalized decomposition (PGD) method for parametric studies. As is common to all PGD methods, it is not necessary to do essentially repeating computations; instead, solutions for a range of parameters can be obtained in a single computation. The proposed implementation strategy includes a mixed formulation and a new data structure. The mixed formulation is applied with the staggered method which does not require the first variation of residual with the derivative of the bulk free energy and thereby avoid repeated matrix reassembling in the Newton-Raphson procedure. In addition, the proposed data structure stores pointwise values of the solution functions instead of the PGD representation for the residual of the Newton method, which is efficient for nonlinear time-dependent problems. Overall, the computational efficiency is significantly improved compared with most traditional PGD formulations and implementations.
机译：在本文中，我们提出了一种新的实施策略，用于求解艾伦-CAHN和CAHN-HILLIARD方程，具有适当的广义分解（PGD）参数研究方法。对于所有PGD方法常见，因此没有必要基本上重复计算;相反，可以在单个计算中获得一系列参数的解决方案。拟议的实施策略包括混合制定和新的数据结构。混合制剂用交错方法施加，该方法不需要具有散装自由能的衍生物的剩余物的第一变化，从而避免在牛顿-Raphson过程中重复的矩阵重新组装。另外，所提出的数据结构存储解决方案函数的点值，而不是用于牛顿方法的残差的PGD表示，这对于非线性时间依赖性问题是有效的。总体而言，与大多数传统的PGD配方和实施相比，计算效率显着提高。

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