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  1. Free, publicly-accessible full text available November 1, 2023
  2. In this paper, we develop parameter-robust numerical algorithms for Biot model and apply the algorithms in brain edema simulations. By introducing an intermediate variable, we derive a multiphysics reformulation of the Biot model. Based on the reformulation, the Biot model is viewed as a generalized Stokes subproblem combining with a reaction–diffusion subproblem. Solving the two subproblems together or separately leads to a coupled or a decoupled algorithm. We conduct extensive numerical experiments to show that the two algorithms are robust with respect to the key physical parameters. The algorithms are applied to study the brain swelling caused by abnormal accumulation of cerebrospinal fluid in injured areas. The effects of the key physical parameters on brain swelling are carefully investigated. It is observed that the permeability has the biggest influence on intracranial pressure (ICP) and tissue deformation; the Young’s modulus and the Poisson ratio do not affect the maximum value of ICP too much but have big influence on the tissue deformation and the developing speed of brain swelling.
  3. In this paper, we develop a Finite Volume solver for a 3D incompressible Oldroyd-B model with infinity relaxation time. The Finite Volume solver is implemented by using a lead- ing open-source computational mechanics software OpenFOAM. We have imposed the di- vergence free condition as a constraint on the momentum equation to derive a pressure equation and a predictor-corrector procedure is applied when solving the velocity field. Both stability analysis and numerical experiments are given to show the robustness and accuracy of our algorithm. Two concrete examples on a cubical domain and a dumbbell are computed and illustrated.
  4. An H(div)-conforming finite element method for the Biot’s consolidation mo- del is developed, with displacements and fluid velocity approximated by elements from BDM_k space. The use of H(div)-conforming elements for flow variables ensures the local mass conservation. In the H(div)-conforming approximation of displacement, the tan- gential components are discretised in the interior penalty discontinuous Galerkin frame- work,and the normal components across the element interfaces are continuous. Having introduced a spatial discretisation, we develop a semi-discrete scheme and a fully dis- crete scheme,prove their unique solvability and establish optimal error estimates for each variable.
  5. In this paper, we study the normal mode solutions of 3D incompressible viscous fluid flow models. The obtained theoretical results are then applied to analyze several time-stepping schemes for the numerical solutions of the 3D incompressible fluid flow models.
  6. We investigate a multirate time step approach applied to decoupled meth- ods in fluid and structure interaction (FSI) computation, where two different time steps are employed for fluid and structure respectively. For illustration, the multirate technique is examined by applying the decoupled β scheme. Numerical experiments show that the proposed approach is stable and retains the same order of accuracy as the original single time step scheme, while with much less computational expense.
  7. In this paper, we aim at solving the Biot model under stabilized finite element discretizations. To solve the resulting generalized saddle point linear systems, some iterative methods are proposed and compared. In the first method, we apply the GMRES algorithm as the outer iteration. In the second method, the Uzawa method with variable relaxation parameters is employed as the outer iteration method. In the third approach, Uzawa method is treated as a fixed-point iteration, the outer solver is the so-called Anderson acceleration. In all these methods, the inner solvers are preconditioners for the generalized saddle point problem. In the preconditioners, the Schur complement approximation is derived by using Fourier analysis approach. These preconditioners are implemented exactly or inexactly. Extensive experiments are given to justify the performance of the proposed preconditioners and to compare all the algorithms.