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1. An iterative decoupled algorithm with unconditional stability for Biot model

   https://doi.org/10.1090/mcom/3809  

   Gu, Huipeng; Cai, Mingchao; Li, Jingzhi (May 2023, Mathematics of Computation)

   This paper is concerned with numerical algorithms for Biot model. By introducing an intermediate variable, the classical 2-field Biot model is written into a 3-field formulation. Based on such a 3-field formulation, we propose a coupled algorithm, some time-extrapolation based decoupled algorithms, and an iterative decoupled algorithm. Our focus is the analysis of the iterative decoupled algorithm. It is shown that the convergence of the iterative decoupled algorithm requires no extra assumptions on physical parameters or stabilization parameters. Numerical experiments are provided to demonstrate the accuracy and efficiency of the proposed method. 

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2. Parameter-robust multiphysics algorithms for Biot model with application in brain edema simulation

   https://doi.org/10.1016/j.matcom.2020.04.027  

   Ju, Guoliang; Cai, Mingchao; Li, Jingzhi; Tian, Jing (November 2020, Mathematics and computers in simulation)

   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. 

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3. A Robin–Robin strongly coupled partitioned method for fluid–poroelastic structure interaction

   https://doi.org/10.1515/jnma-2024-0057  

   Parrow, Connor; Bukač, Martina (February 2025, Journal of Numerical Mathematics)

   Abstract This work focuses on the development of a novel, strongly-coupled, second-order partitioned method for fluid–poroelastic structure interaction. The flow is assumed to be viscous and incompressible, and the poroelastic material is described using the Biot model. To solve this problem, a numerical method is proposed, based on Robin interface conditions combined with the refactorization of the Cauchy’s one-legged ‘ϑ-like’ method. This approach allows the use of the mixed formulation for the Biot model. The proposed algorithm consists of solving a sequence of Backward Euler–Forward Euler steps. In the Backward Euler step, the fluid and poroelastic structure problems are solved iteratively until convergence. Then, the Forward Euler problems are solved using equivalent linear extrapolations. We prove that the iterative procedure in the Backward Euler step is convergent, and that the converged method is stable whenϑ∈ [1/2, 1]. Numerical examples are used to explore convergence rates with varying parameters used in our scheme, and to compare our method to a monolithic method based on Nitsche’s coupling approach. 

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4. Schur Complement Based Preconditioners for Twofold and Block Tridiagonal Saddle Point Problems

   https://doi.org/10.4208/cmr.2023-0051  

   Cai, Mingchao; Ju, Guoliang; Li, Jingzhi (May 2024, Communications in Mathematical Research)

   In this paper, we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems. One type of the preconditioners are based on the nested (or recursive) Schur complement, the other is based on an additive type Schur complement after permuting the original saddle point systems. We analyze different preconditioners incorporating the exact Schur complements. We show that some of them will lead to positively stable preconditioned systems if proper signs are selected in front of the Schur complements. These positive-stable preconditioners outperform other preconditioners if the Schur complements are further approximated inexactly. Numerical experiments for a 3-field formulation of the Biot model are provided to verify our predictions. 

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5. Interval-Based Parameter Identification for Structural Static Problems

   3. Xiao, Naijia; Fedele, Francesco; Muhanna, Rafi (July 2016, Reliable computing)

   We present an interval-based approach for parameter identification in structural static problems. Our inverse formulation models uncertainties in measurement data as interval and exploits the Interval Finite Element Method (IFEM) combined with adjoint-based optimization. The inversion consists of a two-step algorithm: first, an estimate of the parameters is obtained by a deterministic iterative solver. Then, the algorithm switches to the interval extension of the previous solver, using the deterministic estimate of the parameters as an initial guess. The formulation is illustrated in solutions of various numerical examples showing how the guaranteed interval enclosures always contain Monte Carlo predictions. 

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