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A coupled multipoint stress-multipoint flux mixed finite element method for the Biot system of poroelasticity

机译:耦合多点应力 - 多点磁通混合有限元方法对孔弹性的生物系统

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We present a mixed finite element method for a five-field formulation of the Biot system of poroelasticity that reduces to a cell-centered pressure-displacement system on simplicial and quadrilateral grids. A mixed stress-displacement-rotation formulation for elasticity with weak stress symmetry is coupled with a mixed velocity-pressure Darcy formulation. The spatial discretization is based on combining the multipoint stress mixed finite element (MSMFE) method for elasticity and the multipoint flux mixed finite element (MFMFE) method for Darcy flow. It uses the lowest order Brezzi-Douglas-Marini mixed finite element spaces for the poroelastic stress and Darcy velocity, piecewise constant displacement and pressure, and continuous piecewise linear or bilinear rotation. A vertex quadrature rule is applied to the velocity, stress, and stress-rotation bilinear forms, which block-diagonalizes the corresponding matrices and allows for local velocity, stress, and rotation elimination. This leads to a cell-centered positive-definite system for pressure and displacement at each time step. We perform error analysis for the semidiscrete and fully discrete formulations, establishing first order convergence for all variables in their natural norms. The numerical tests confirm the theoretical convergence rates and illustrate the locking-free property of the method. (c) 2020 Elsevier B.V. All rights reserved.
机译:我们介绍了一种混合有限元法,用于孔弹性的Biot系统的五场配方,其减少了单独的和四边形网格上的细胞中心压力位移系统。具有弱应力对称性弹性的混合应力 - 位移旋转制剂与混合速度压力达锡配制偶联。空间离散化是基于组合的多点应力混合有限元(MSMFE)方法进行弹性和多点磁通混合有限元(MFMFE)方法,用于达西流动。它采用最低订单Brezzi-Douglas-Marini混合有限元空间,用于多孔弹性应力和达西速度,分段恒定的位移和压力,以及连续分段线性或双线性旋转。将顶点正交规则应用于速度,应力和应力旋转双线形式,其阻止对应矩阵并允许局部速度,应力和旋转消除。这导致细胞中心的正面定向系统,用于在每个时间步骤在压力和位移。我们对半同克雷特和完全离散的配方进行错误分析,为其自然规范中的所有变量建立一阶融合。数值测试证实了理论会聚速率,并说明了该方法的无锁定性能。 (c)2020 Elsevier B.v.保留所有权利。

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