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Benchmarking the Immersed Finite Element Method for FluidStructure Interaction Problems
, 2014
"... We present an implementation of a fully variational formulation of an immersed methods for fluidstructure interaction problems based on the finite element method. While typical implementation of immersed methods are characterized by the use of approximate Dirac delta distributions, fully variation ..."
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We present an implementation of a fully variational formulation of an immersed methods for fluidstructure interaction problems based on the finite element method. While typical implementation of immersed methods are characterized by the use of approximate Dirac delta distributions, fully variational formulations of the method do not require the use of said distributions. In our implementation the immersed solid is general in the sense that it is not required to have the same mass density and the same viscous response as the surrounding fluid. We assume that the immersed solid can be either viscoelastic of differential type or hyperelastic. Here we focus on the validation of the method via various benchmarks for fluidstructure interaction numerical schemes. This is the first time that the interaction of purely elastic compressible solids and an incompressible fluid is approached via an immersed method allowing a direct comparison with established benchmarks.
An Efficient Parallel Immersed Boundary Algorithm using a PseudoCompressible Fluid Solver
, 2013
"... We propose an efficient algorithm for the immersed boundary method on distributedmemory architectures, with the computational complexity of a completely explicit method and excellent parallel scaling. The algorithm utilizes the pseudocompressibility method recently proposed by Guermond and Minev t ..."
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We propose an efficient algorithm for the immersed boundary method on distributedmemory architectures, with the computational complexity of a completely explicit method and excellent parallel scaling. The algorithm utilizes the pseudocompressibility method recently proposed by Guermond and Minev that uses a directional splitting strategy to discretize the incompressible NavierStokes equations, thereby reducing the linear systems to a series of onedimensional tridiagonal systems. We perform numerical simulations of several fluidstructure interaction problems in two and three dimensions and study the accuracy and convergence rates of the proposed algorithm. For these problems, we compare the proposed algorithm against other secondorder projectionbased fluid solvers. Lastly, the strong and weak scaling properties of the proposed algorithm are investigated.
Fully Eulerian finite element approximation of a fluidstructure interaction problem in cardiac cells
 Int. J. Numer. Methods Engrg
"... We propose in this paper an Eulerian finite element approximation of a coupled chemical fluidstructure interaction problem arising in the study of mesoscopic cardiac biomechanics. We simulate the active response of a myocardial cell (here considered as an anisotropic, hyperelastic, and incompressib ..."
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We propose in this paper an Eulerian finite element approximation of a coupled chemical fluidstructure interaction problem arising in the study of mesoscopic cardiac biomechanics. We simulate the active response of a myocardial cell (here considered as an anisotropic, hyperelastic, and incompressible material), the propagation of calcium concentrations inside it, and the presence of a surrounding Newtonian fluid. An active strain approach is employed to account for the mechanical activation, and the deformation of the cell membrane is captured using a level set strategy. We address in detail the main features of the proposed method, and we report several numerical experiments aimed at model validation. Copyright © 2013 John
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"... Abstract. In a recent paper Iyama and Yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal CohenMacaulay modules in terms of linear algebra data. In this paper we present two new approaches to these examples. In the fir ..."
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Abstract. In a recent paper Iyama and Yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal CohenMacaulay modules in terms of linear algebra data. In this paper we present two new approaches to these examples. In the first approach we give a relation with cluster categories. In the second
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
"... On hypersingular surface integrals in the symmetric Galerkin boundary element method: application to heat conduction in exponentially graded materials ..."
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On hypersingular surface integrals in the symmetric Galerkin boundary element method: application to heat conduction in exponentially graded materials
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"... Abstract. We develop numerical schemes for solving the isothermal compressible and incompressible equations of fluctuating hydrodynamics on a grid with staggered momenta. We develop a secondorder accurate spatial discretization of the diffusive, advective, and stochastic fluxes that satisfies a di ..."
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Abstract. We develop numerical schemes for solving the isothermal compressible and incompressible equations of fluctuating hydrodynamics on a grid with staggered momenta. We develop a secondorder accurate spatial discretization of the diffusive, advective, and stochastic fluxes that satisfies a discrete fluctuationdissipation balance and construct temporal discretizations that are at least secondorder accurate in time deterministically and in a weak sense. Specifically, the methods reproduce the correct equilibrium covariances of the fluctuating fields to the third (compressible) and second (incompressible) orders in the time step, as we verify numerically. We apply our techniques to model recent experimental measurements of giant fluctuations in diffusively mixing fluids in a microgravity environment [A. Vailati et al., Nat. Comm., 2 (2011), 290]. Numerical results for the static spectrum of nonequilibrium concentration fluctuations are in excellent agreement between the compressible and incompressible simulations and in good agreement with experimental results for all measured wavenumbers.
Simulating an Elastic Ring with Bend and Twist by anAdaptiveGeneralized Immersed BoundaryMethod
, 2011
"... Abstract. Many problems involving the interaction of an elastic structure and a viscous fluid can be solved by the immersed boundary (IB) method. In the IB approach to such problems, the elastic forces generated by the immersed structure are applied to the surrounding fluid, and the motion of the i ..."
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Abstract. Many problems involving the interaction of an elastic structure and a viscous fluid can be solved by the immersed boundary (IB) method. In the IB approach to such problems, the elastic forces generated by the immersed structure are applied to the surrounding fluid, and the motion of the immersed structure is determined by the local motion of the fluid. Recently, the IB method has been extended to treat more general elasticity models that include both positional and rotational degrees of freedom. For such models, force and torque must both be applied to the fluid. The positional degrees of freedom of the immersed structure move according to the local linear velocity of the fluid, whereas the rotational degrees of freedommove according to the local angular velocity. This paper introduces a spatially adaptive, formally secondorder accurate version of this generalized immersed boundary method. We use this adaptive scheme to simulate the dynamics of an elastic ring immersed in fluid. To describe the elasticity of the ring, we use an unconstrained version of Kirchhoff rod theory. We demonstrate empirically that our numerical scheme yields essentially secondorder convergence rates when applied to such problems. We also study dynamical instabilities of such fluidstructure systems, and we compare numerical results produced by our method to classical analytic results from elastic rod theory.
A Modular, Operator Splitting Scheme for FluidStructure Interaction Problems with Thick Structures
, 2013
"... We present an operatorsplitting scheme for fluidstructure interaction (FSI) problems in hemodynamics, where the thickness of the structural wall is comparable to the radius of the cylindrical fluid domain. The equations of linear elasticity are used to model the structure, while the NavierStokes ..."
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We present an operatorsplitting scheme for fluidstructure interaction (FSI) problems in hemodynamics, where the thickness of the structural wall is comparable to the radius of the cylindrical fluid domain. The equations of linear elasticity are used to model the structure, while the NavierStokes equations for an incompressible viscous fluid are used to model the fluid. The operator splitting scheme, based on Lie splitting, separates the elastodynamics structure problem, from a fluid problem in which structure inertia is included to achieve unconditional stability. We prove energy estimates associated with unconditional stability of this modular scheme for the full nonlinear FSI problem defined on a moving domain, without requiring any subiterations within time steps. Two numerical examples are presented, showing excellent agreement with the results of monolithic schemes. Firstorder convergence in time is shown numerically. Modularity, unconditional stability without temporal subiterations, and simple implementation are the features that make this operatorsplitting scheme particularly appealing for multiphysics problems involving fluidstructure interaction.
Variational Implementation of Immersed Finite Element Methods
"... Diracδ distributions are often crucial components of the solidfluid coupling operators in immersed solution methods for fluidstructure interaction (FSI) problems. This is certainly so for methods like the Immersed Boundary Method (IBM) or the Immersed Finite Element Method (IFEM), where Diracδ d ..."
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Diracδ distributions are often crucial components of the solidfluid coupling operators in immersed solution methods for fluidstructure interaction (FSI) problems. This is certainly so for methods like the Immersed Boundary Method (IBM) or the Immersed Finite Element Method (IFEM), where Diracδ distributions are approximated via smooth functions. By contrast, a truly variational formulation of immersed methods does not require the use of Diracδ distributions, either formally or practically. This has been shown in the Finite Element Immersed Boundary Method (FEIBM), where the variational structure of the problem is exploited to avoid Diracδ distributions at both the continuous and the discrete level. In this paper, we generalize the FEIBM to the case where an incompressible Newtonian fluid interacts with a general hyperelastic solid. Specifically, we allow (i) the mass density to be different in the solid and the fluid, (ii) the solid to be either viscoelastic of differential type or purely elastic, and (iii) the solid to be either compressible or incompressible. At the continuous level, our variational formulation combines the natural stability estimates of the fluid and elasticity problems. In immersed methods, such stability estimates do not transfer to the discrete level automatically due to the nonmatching nature of the finite dimensional spaces involved in the discretization. After presenting our general mathematical framework for the solution of FSI problems, we focus in detail on the construction of natural interpolation operators between the fluid and the solid discrete spaces, which guarantee semidiscrete stability estimates and strong consistency of our spatial discretization.