论文标题
耦合的多点应力 - 多点通量混合有限元法，用于生物弹性的生物系统
A coupled multipoint stress -- multipoint flux mixed finite element method for the Biot system of poroelasticity
论文作者
论文摘要
我们提出了一种混合有限元方法，用于五场配方的毛弹性系统，该方法可在简单和四边形网格上降低到以细胞为中心的压力清位系统。用弱应力对称性弹性的混合应力 - 置换式旋转配方与混合速度压力达西制剂结合。空间离散化基于将弹性的多点应力混合有限元（MSMFE）方法组合为DARCY流量的多点通量混合有限元（MFMFE）方法。它使用最低级的brezzi-douglas-marini混合有限元空间用于毛弹性应力和达西速度，分段恒定位移和压力以及连续的分段线性或双线性旋转。顶点正交规则应用于速度，应力和应力旋转双线性形式，该形式将相应的矩阵构成障碍，并允许局部速度，应力和旋转消除。这导致以细胞为中心的阳性系统在每个时间步骤中进行压力和位移。我们对半分化和完全离散的配方进行误差分析，为其自然规范中的所有变量建立一阶收敛。数值测试证实了理论收敛速率，并说明了该方法的无锁定属性。
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.
