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1. A partitioned numerical scheme for fluid–structure interaction with slip

   https://doi.org/10.1051/mmnp/2020051  

   Bukač, Martina; Čanić, Sunčica (January 2021, Mathematical Modelling of Natural Phenomena)

   Grandmont, C.; Hillairet, M.; Matin, S.; Muha, B.; Vergarra, Ch. (Ed.)

   We present a loosely coupled, partitioned scheme for solving fluid–structure interaction (FSI) problems with the Navier slip boundary condition. The fluid flow is modeled by the Navier–Stokes equations for an incompressible, viscous fluid, interacting with a thin elastic structure modeled by the membrane or Koiter shell type equations. The fluid and structure are coupled via two sets of coupling conditions: a dynamic coupling condition describing balance of forces, and a kinematic coupling condition describing fluid slipping tangentially to the moving fluid–structure interface, with no penetration in the normal direction. Problems of this type arise in, e.g. , FSI with hydrophobic structures or surfaces treated with a no-stick coating, and in biologic FSI involving rough surfaces of elastic tissues or tissue scaffolds. We propose a novel, efficient partitioned scheme where the fluid sub-problem is solved separately from the structure sub-problem, and there is no need for sub-iterations at every time step to achieve stability, convergence, and its first-order accuracy. We derive energy estimates, which prove that the proposed scheme is unconditionally stable for the corresponding linear problem. Moreover, we present convergence analysis and show that under a time-step condition, the method is first-order accurate in time and optimally convergent in space for a Finite Element Method-based spatial discretization. The theoretical rates of convergence in time are confirmed numerically on an example with an explicit solution using the method of manufactured solutions, and on a benchmark problem describing propagation of a pressure pulse in a two-dimensional channel. The effects of the slip rate and fluid viscosity on the FSI solution are numerically investigated in two additional examples: a 2D cylindrical FSI example for which an exact Navier slip Poiseuille-type solution is found and used for comparison, and a squeezed ketchup bottle example with gravity enhanced flow. We show that the Navier-slip boundary condition increases the outflow mass flow rate by 21% for a bottle angled at 45 degrees pointing downward, in the direction of gravity. 

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2. Partitioning method for the finite element approximation of a 3D fluid‐2D plate interaction system

   https://doi.org/10.1002/num.23132  

   Geredeli, Pelin G; Kunwar, Hemanta; Lee, Hyesuk (August 2024, Numerical Methods for Partial Differential Equations)

   Abstract We consider the finite element approximation of a coupled fluid‐structure interaction (FSI) system, which comprises a three‐dimensional (3D) Stokes flow and a two‐dimensional (2D) fourth‐order Euler–Bernoulli or Kirchhoff plate. The interaction of these parabolic and hyperbolic partial differential equations (PDE) occurs at the boundary interface which is assumed to be fixed. The vertical displacement of the plate dynamics evolves on the flat portion of the boundary where the coupling conditions are implemented via the matching velocities of the plate and fluid flow, as well as the Dirichlet boundary trace of the pressure. This pressure term also acts as a coupling agent, since it appears as a forcing term on the flat, elastic plate domain. Our main focus in this work is to generate some numerical results concerning the approximate solutions to the FSI model. For this, we propose a numerical algorithm that sequentially solves the fluid and plate subsystems through an effective decoupling approach. Numerical results of test problems are presented to illustrate the performance of the proposed method. 

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3. Fluid-poroviscoelastic structure interaction problem with nonlinear coupling

   Jeffrey Kuan, Suncica Canic (December 2024, Jorunal de Mathematiques Pures and Appliques)

   We prove the existence of a weak solution to a fluid-structure interaction (FSI) problem between the flow of an incompressible, viscous fluid modeled by the Navier-Stokes equations, and a poroviscoelastic medium modeled by the Biot equations. The two are nonlinearly coupled over an interface with mass and elastic energy, modeled by a reticular plate equation, which is transparent to fluid flow. The existence proof is constructive, consisting of two steps. First, the existence of a weak solution to a regularized problem is shown. Next, a weak-classical consistency result is obtained, showing that the weak solution to the regularized problem converges, as the regularization parameter approaches zero, to a classical solution to the original problem, when such a classical solution exists. While the assumptions in the first step only require the Biot medium to be poroelastic, the second step requires additional regularity, namely, that the Biot medium is poroviscoelastic. This is the first weak solution existence result for an FSI problem with nonlinear coupling involving a Biot model for poro(visco)elastic media. 

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4. A Robust Mesh Moving Method for Moving-Boundary Problems

   https://doi.org/10.2514/6.2024-1163  

   Tu, Shuangzhang; Jiang, Chao (January 2024, American Institute of Aeronautics and Astronautics)

   This paper presents a robust mesh moving solver developed to address moving boundary problems. Crucially, the resulting deformed mesh retains the same topology as the original mesh without being overly distorted. The mesh is treated as an elastic material, and the deformation of the computational domain resulting from moving boundaries is determined by solving the equilibrium linear elasticity equations. The linear elasticity equations are discretized by the classic Galerkin finite element method and solved by the block conjugate gradient iterative method. To maintain the quality of the mesh after motion, the Young's modulus of each element is weighted by the reciprocal of the distance between the element center and the moving boundaries. The effectiveness of this approach is demonstrated through a set of 2D and 3D test cases featuring prescribed translational and/or rotational motion of the embedded object. The method is now ready for integration into our existing in-house CFD solvers. 

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5. A Computational Analysis of Bubble-Structure Interaction in Near-Field Underwater Explosion

   https://doi.org/10.1115/IMECE2021-72854  

   Ma, Wentao; Ozenkoski, Timothy; Wang, Kevin (November 2021, Proceedings of ASME 2021 International Mechanical Engineering Congress and Exposition)

   Underwater explosion poses a significant threat to the structural integrity of ocean vehicles and platforms. Accurate prediction of the dynamic loads from an explosion and the resulting structural response is crucial to ensuring safety without overconservative design. When the distance between the explosive charge and the structure is relatively small (i.e., near-field explosion), the dynamics of the gaseous explosion product, i.e., the “bubble”, comes into play, rendering a multiphysics problem that features the interaction of the bubble, the surrounding liquid water, and the solid structure. The problem is highly nonlinear, as it involves shock waves, large deformation, yielding, contact, and possibly fracture. This paper investigates the two-way interaction between the cyclic expansion and collapse of an explosion bubble and the deformation of a thin-walled elastoplastic cylindrical shell in its vicinity. Intuitively, when a shock wave impinges on a thin cylindrical shell, the shell would collapse in the direction of shock propagation. However, some recent laboratory experiments have shown that under certain conditions the shell collapsed in a counter-intuitive mode in which the direction of collapse is perpendicular to that of shock propagation. In other words, the nearest point on the structural surface moved towards the explosion charge, despite being impacted by a compressive shock. This paper focuses on replicating this phenomenon through numerical simulation and elucidating the underlying mechanisms. A recently developed computational framework (“FIVER”) coupling a nonlinear finite element structural dynamics solver and a finite volume compressible fluid dynamics solver is used to complete this study. The solver utilizes an embedded boundary method to track the wetted surface of the structure (i.e. the fluid-structure interface), which is capable of handling large structural deformation and topological changes (e.g., fracture). The solver also adopts the level set method for tracking the bubble surface (i.e. the liquid-gas interface). The fluid-structure and liquid-gas interface conditions are enforced by constructing and solving one-dimensional multi-material Riemann problems, which naturally accommodates the propagation of shock waves across the interfaces. In this paper, mesh refinement study is made to examine the sensitivity of the results to various meshing parameters. The results show that the intermediate level of refinement is appropriate in terms of both the accuracy and the computation costs. Next, the deformation history of both the bubble and the structure are presented and analyzed to provide a detailed view of the counter-intuitive collapse mode mentioned above. We show that timewise, the structural collapse spans multiple cycles of bubble oscillation. Additional details about the time-histories of fluid pressure, structure displacement, and bubble size are presented to elucidate this dynamic bubble-structure interaction and the resulting structural failure. 

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