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1. 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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2. Handling Neumann and Robin boundary conditions in a fictitious domain volume penalization framework

   https://doi.org/10.1016/j.jcp.2021.110726  

   Thirumalaisamy, Ramakrishnan; Patankar, Neelesh A.; Bhalla, Amneet Pal (January 2022, Journal of Computational Physics)
3. An Embedded Robin Boundary Method for Incompressible Fluid-Structure Interaction Problems

   https://doi.org/10.2514/6.2017-3447  

   Cao, Shunxiang; Main, Alex; Wang, Kevin G. (June 2017, 23rd AIAA Computational Fluid Dynamics Conference)
4. A spatially varying robin interface condition for fluid‐structure coupled simulations

   https://doi.org/10.1002/nme.6386  

   Cao, Shunxiang; Wang, Guangyao; Wang, Kevin G. (August 2020, International Journal for Numerical Methods in Engineering)

   Summary We present a spatially varying Robin interface condition for solving fluid‐structure interaction problems involving incompressible fluid flows and nonuniform flexible structures. Recent studies have shown that for uniform structures with constant material and geometric properties, a constant one‐parameter Robin interface condition can improve the stability and accuracy of partitioned numerical solution procedures. In this work, we generalize the parameter to a spatially varying function that depends on the structure's local material and geometric properties, without varying the exact solution of the coupled fluid‐structure system. We present an algorithm to implement the Robin interface condition in an embedded boundary method for coupling a projection‐based incompressible viscous flow solver with a nonlinear finite element structural solver. We demonstrate the numerical effects of the spatially varying Robin interface condition using two example problems: a simplified model problem featuring a nonuniform Euler‐Bernoulli beam interacting with an inviscid flow and a generalized Turek‐Hron problem featuring a nonuniform, highly flexible beam interacting with a viscous laminar flow. Both cases show that a spatially varying Robin interface condition can clearly improve numerical accuracy (by up to two orders of magnitude in one instance) for the same computational cost. Using the second example problem, we also demonstrate and compare two models for determining the local value of the combination function in the Robin interface condition. 

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5. Nonconforming time discretization based on Robin transmission conditions for the Stokes–Darcy system

   https://doi.org/10.1016/j.amc.2021.126602  

   Hoang, Thi-Thao-Phuong; Kunwar, Hemanta; Lee, Hyesuk (January 2022, Applied mathematics and computation)

   We consider a space-time domain decomposition method based on Schwarz waveform relaxation (SWR) for the time-dependent Stokes-Darcy system. The coupled system is formulated as a time-dependent interface problem based on Robin-Robin transmission conditions, for which the decoupling SWR algorithm is proposed and proved for the convergence. In this approach, the Stokes and Darcy problems are solved independently and globally in time, thus allowing the use of different time steps for the local problems. Numerical tests are presented for both non-physical and physical problems with various mesh sizes and time step sizes to illustrate the accuracy and efficiency of the proposed method. 

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