Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems
Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems
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DOI：
10.1016/j.cam.2019.06.028
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发表时间：
2019-01
期刊：
影响因子：
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通讯作者：
E. Ahmed;J. Nordbotten;F. Radu
中科院分区：
文献类型：
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作者：
E. Ahmed;J. Nordbotten;F. Radu
在本文中，我们开发了自适应迭代耦合方案的毕奥系统模拟耦合孔隙力学问题。我们特别考虑了固定应力迭代方案的时空表述，在该方案中，我们首先解决整个时空区间上的流动问题，然后利用时空信息来求解力学。然后介绍了该算法的两种常用的离散方法，即空间上的两种耦合混合有限元方法和时间上的向后欧拉格式。由此，自适应固定应力算法建立在压力和位移的一致性重建以及平衡通量和应力重建的基础上。这些成分是用来获得后验误差估计的固定应力算法，区分不同的误差分量，即空间离散化，时间离散化，和固定应力迭代组件。准确地说，在自适应算法的迭代k≥ 1，我们证明了我们的估计给出了一个有保证的和完全可计算的能量型误差的上限测量之间的差异的准确和近似的压力和位移。这些误差分量被有效地用于设计自适应异步时间步进和自适应停止标准的固定应力算法。数值实验表明了我们的估计和自适应迭代耦合算法的性能的效率。
In this paper we develop adaptive iterative coupling schemes for the Biot system modeling coupled poromechanics problems. We particularly consider the space–time formulation of the fixed-stress iterative scheme, in which we first solve the problem of flow over the whole space–time interval, then exploiting the space–time information for solving the mechanics. Two common discretizations of this algorithm are then introduced based on two coupled mixed finite element methods in-space and the backward Euler scheme in-time. Therefrom, adaptive fixed-stress algorithms are build on conforming reconstructions of the pressure and displacement together with equilibrated flux and stresses reconstructions. These ingredients are used to derive a posteriori error estimates for the fixed-stress algorithms, distinguishing the different error components, namely the spatial discretization, the temporal discretization, and the fixed-stress iteration components. Precisely, at the iteration k≥ 1 of the adaptive algorithm, we prove that our estimate gives a guaranteed and fully computable upper bound on the energy-type error measuring the difference between the exact and approximate pressure and displacement. These error components are efficiently used to design adaptive asynchronous time-stepping and adaptive stopping criteria for the fixed-stress algorithms. Numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive iterative coupling algorithms.