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Uniformly well-posed hybridized discontinuous Galerkin/hybrid mixed discretizations for Biot’s consolidation model
Computer Methods in Applied Mechanics and Engineering ( IF 7.2 ) Pub Date : 2021-06-17 , DOI: 10.1016/j.cma.2021.113991
Johannes Kraus , Philip L. Lederer , Maria Lymbery , Joachim Schöberl

We consider the quasi-static Biot’s consolidation model in a three-field formulation with the three unknown physical quantities of interest being the displacement u of the solid matrix, the seepage velocity v of the fluid and the pore pressure p. As conservation of fluid mass is a leading physical principle in poromechanics, we preserve this property using an H(div)-conforming ansatz for u and v together with an appropriate pressure space. This results in Stokes and Darcy stability and exact, that is, pointwise mass conservation of the discrete model.

The proposed discretization technique combines a hybridized discontinuous Galerkin method for the elasticity subproblem with a mixed method for the flow subproblem, also handled by hybridization. The latter allows for a static condensation step to eliminate the seepage velocity from the system while preserving mass conservation. The system to be solved finally only contains degrees of freedom related to u and p resulting from the hybridization process and thus provides, especially for higher-order approximations, a very cost-efficient family of physics-oriented space discretizations for poroelasticity problems.

We present the construction of the discrete model, theoretical results related to its uniform well-posedness along with optimal error estimates and parameter-robust preconditioners as a key tool for developing uniformly convergent iterative solvers. Finally, the cost-efficiency of the proposed approach is illustrated in a series of numerical tests for three-dimensional test cases.



中文翻译:

Biot 固结模型的均匀适定混合不连续 Galerkin/混合混合离散化

我们在三场公式中考虑准静态 Biot 固结模型,其中三个未知的感兴趣物理量是位移 固体基质的渗流速度v 流体和孔隙压力的关系 . 由于流体质量守恒是多孔力学中的主要物理原理,我们使用H(div)-符合 ansatz 为 v以及适当的压力空间。这导致 Stokes 和 Darcy 稳定性和精确的,即离散模型的逐点质量守恒。

所提出的离散化技术将弹性子问题的混合不连续伽辽金方法与流动子问题的混合方法相结合,也由杂交处理。后者允许静态冷凝步骤以消除系统的渗流速度,同时保持质量守恒。最终要求解的系统只包含与 由混合过程产生,因此为多孔弹性问题提供了一个非常经济高效的面向物理的空间离散化系列,尤其是对于高阶近似。

我们介绍了离散模型的构建、与其均匀适定性相关的理论结果以及最佳误差估计和参数鲁棒性预处理器,作为开发均匀收敛迭代求解器的关键工具。最后,所提出的方法的成本效益在一系列三维测试用例的数值测试中得到了说明。

更新日期:2021-06-18
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