Robin–Robin preconditioned Krylov methods for fluid–structure interaction problems

Badia, Santiago; Nobile, Fabio; Vergara, Christian

doi:10.1016/j.cma.2009.04.004

Cited by 104 publications

(139 citation statements)

References 39 publications

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“…In the partitioned case the successive solution of the fluid and solid subproblems in an iterative framework is carried out (see, e.g., [9,11,39,46,55,105,134]). In this case, the schemes feature in general poor convergence properties due to the added mass effect, that predicts a breakdown of performances when the values of the densities of fluid and structure are close as it happens in hemodynamics [10,39,69,76,137]. Alternatively, one could consider space-time finite elements, see, e.g., [17,187], or the iso-geometric analysis, see [15].…”

Section: Numerical Discretizationmentioning

confidence: 99%

Geometric multiscale modeling of the cardiovascular system, between theory and practice

Quarteroni

Veneziani

Vergara

2016

Computer Methods in Applied Mechanics and Engineering

169

157

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This review paper addresses the so called geometric multiscale approach for the numerical simulation of blood flow problems, from its origin (that we can collocate in the second half of '90s) to our days. By this approach the blood fluid-dynamics in the whole circulatory system is described mathematically by means of heterogeneous featuring different degree of detail and different geometric dimension that interact together through appropriate interface coupling conditions.Our review starts with the introduction of the stand-alone problems, namely the 3D fluidstructure interaction problem, its reduced representation by means of 1D models, and the so-called lumped parameters (aka 0D) models, where only the dependence on time survives. We then address specific methods for stand-alone 3D models when the available boundary data are not enough to ensure the mathematical well posedness. These so-called "defective problems" naturally arise in practical applications of clinical relevance but also because of the interface coupling of heterogeneous problems that are generated by the geometric multiscale process. We also describe specific issues related to the boundary treatment of reduced models, particularly relevant to the geometric multiscale coupling. Next, we detail the most popular numerical algorithms for the solution of the coupled problems. Finally, we review some of the most representative works -from different research groups -which addressed the geometric multiscale approach in the past years.A proper treatment of the different scales relevant to the hemodynamics and their interplay is essential for the accuracy of numerical simulations and eventually for their clinical impact. This paper aims at providing a state-of-the-art picture of these topics, where the gap between theory and practice demands rigorous mathematical models to be reliably filled.

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Section: Numerical Discretizationmentioning

confidence: 99%

Geometric multiscale modeling of the cardiovascular system, between theory and practice

Quarteroni

Veneziani

Vergara

2016

Computer Methods in Applied Mechanics and Engineering

169

157

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“…Only few works have focused on this aspect. We mention [7,25,37] among the monolithic schemes, which build the whole non-linear system, and [33] among the partitioned schemes, which consist in the successive solution of the subproblems in an iterative framework (see also [15,5,11,4,10] in the case of the linear elasticity). In this work, we focus on partitioned strategies, whose main difficulties are:…”

Section: Introductionmentioning

confidence: 99%

Inexact accurate partitioned algorithms for fluid–structure interaction problems with finite elasticity in haemodynamics

Nobile

Pozzoli

Vergara

2014

Journal of Computational Physics

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In this paper we consider the numerical solution of the three-dimensional fluid-structure interaction problem in haemodynamics, in the case of real geometries, physiological data and finite elasticity vessel deformations. We study some new inexact schemes, obtained from semi-implicit approximations, which treat exactly the physical interface conditions while performing just one or few iterations for the management of the interface position and of the fluid and structure non-linearities. We show that such schemes allow to improve the efficiency while preserving the accuracy of the related exact (implicit) scheme. To do this we consider both a simple analytical test case and two real cases of clinical interest in haemodynamics. We also provide an error analysis for a simple differential model problem when a BDF method is considered for the time discretization and only few Newton iterations are performed at each temporal instant.

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“…This ensures a smooth ψ function and an accurate tracking of the free surface. Note that condition (8) enforces that the gradient of ψ in the the direction of the external boundary normal is null, except in those elements cuts by Γ free , where the essential boundary condition prevails. This approach yields a smooth initial field, which is the main objective we pursue.…”

Section: Initializationmentioning

confidence: 99%

“…In the original immersed boundary method [59] and in the fictitious domain method [43], the part of the computational mesh which is not covered by the physical domain is considered to be, and computed as if it was, made of the same material as the physical domain. Although this approach can be convenient and it has advantageous properties when dealing with fluid-structure interaction problems using a partitioned approach and facing the so called added-mass effect [41,9,8,21,32], it also introduces an error due to the fact that temporal derivatives in nodes close to the boundary are computed using velocity values from the non-physical, meaningless domain. This is also the case for other approaches like the ones presented in [61,52,51], where some kind of approximation needs to be done in order to compute the time derivative close to the interface.…”

Section: Introductionmentioning

confidence: 99%

An adaptive Fixed-Mesh ALE method for free surface flows

Baiges

Codina

Pont

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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In this work we present a Fixed-Mesh ALE method for the numerical simulation of free surface flows capable of using an adaptive finite element mesh covering a background domain. This mesh is successively refined and unrefined at each time step in order to focus the computational effort on the spatial regions where it is required. Some of the main ingredients of the formulation are the use of an Arbitrary-Lagrangian-Eulerian formulation for computing temporal derivatives, the use of stabilization terms for stabilizing convection, stabilizing the lack of compatibility between velocity and pressure interpolation spaces, and stabilizing the ill-conditioning introduced by the cuts on the background finite element mesh, and the coupling of the algorithm with an adaptive mesh refinement procedure suitable for running on distributed memory environments. Algorithmic steps for the projection between meshes are presented together with the algebraic fractional step approach used for improving the condition number of the linear systems to be solved. The method is tested in several numerical examples. The expected convergence rates both in space and time are observed. Smooth solution fields for both velocity and pressure are obtained (as a result of the contribution of the stabilization terms). Finally, a good agreement between the numerical results and the reference experimental data is obtained.

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Robin–Robin preconditioned Krylov methods for fluid–structure interaction problems

Cited by 104 publications

References 39 publications

Geometric multiscale modeling of the cardiovascular system, between theory and practice

Geometric multiscale modeling of the cardiovascular system, between theory and practice

Inexact accurate partitioned algorithms for fluid–structure interaction problems with finite elasticity in haemodynamics

An adaptive Fixed-Mesh ALE method for free surface flows

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