Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media
Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media
复制标题
安德森加速固定应力分裂方案以固结不饱和多孔介质
DOI：
10.1016/j.camwa.2018.07.033
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
F. Radu
中科院分区：
文献类型：
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作者：
J. Both;Kundan Kumar;J. Nordbotten;F. Radu
本文研究非饱和材料非线性孔隙力学的鲁棒线性化问题。该模型将理查兹方程与线性弹性方程耦合，推广了经典的毕奥方程。在实践中，单片求解器并不总是可用的，定义了线性化方案的要求，以允许使用单独的模拟器。这是经典的牛顿法所不能满足的。我们提出了三种不同的线性化方案，结合固定应力分裂计划，再加上一个L-计划，修改Picard和牛顿线性化的流动方程。所有的计划允许有效的和强大的解耦的力学和流动方程。特别是，最简单的计划，固定应力-L-计划，只采用常数对角稳定，具有低成本的每次迭代，是非常强大的。在温和的物理假设下，它在理论上被证明是一种收缩。由于所有考虑的分裂格式可能会崩溃或收敛缓慢，安德森加速应用作为后处理。基于一个特殊的情况下，我们证明理论上的一般能力的安德森加速有效地加速收敛和稳定的基本计划，甚至允许非压缩不动点迭代收敛。据我们所知，这是第一个理论上的这种迹象。数值结果证实了理论研究结果。特别是，安德森加速度已被证明是非常有效的考虑皮卡德型方法。最后，结合安德森加速的固定应力牛顿格式显示了分裂格式中最好的性能。
In this paper, we study the robust linearization of nonlinear poromechanics of unsaturated materials. The model of interest couples the Richards equation with linear elasticity equations, generalizing the classical Biot equations. In practice a monolithic solver is not always available, defining the requirement for a linearization scheme to allow the use of separate simulators. It is not met by the classical Newton method. We propose three different linearization schemes incorporating the fixed-stress splitting scheme, coupled with an L-scheme, Modified Picard and Newton linearization of the flow equations. All schemes allow the efficient and robust decoupling of mechanics and flow equations. In particular, the simplest scheme, the Fixed-Stress-L-scheme, employs solely constant diagonal stabilization, has low cost per iteration, and is very robust. Under mild, physical assumptions, it is theoretically shown to be a contraction. Due to possible break-down or slow convergence of all considered splitting schemes, Anderson acceleration is applied as post-processing. Based on a special case, we justify theoretically the general ability of the Anderson acceleration to effectively accelerate convergence and stabilize the underlying scheme, allowing even non-contractive fixed-point iterations to converge. To our knowledge, this is the first theoretical indication of this kind. Theoretical findings are confirmed by numerical results. In particular, Anderson acceleration has been demonstrated to be very effective for the considered Picard-type methods. Finally, the Fixed-Stress-Newton scheme combined with Anderson acceleration shows the best performance among the splitting schemes.