An Effective Fluid-Structure Interaction Formulation for Vascular Dynamics by Generalized Robin Conditions

Nobile, Fabio; Vergara, Christian

doi:10.1137/060678439

Cited by 182 publications

(246 citation statements)

References 37 publications

(85 reference statements)

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“…obtained by a linear combination of conditions (19) and (20); M is a given scalar function of time, and R is a parameter that can be either measured or set up to damp spurious reflections induced by the truncation of the domain of interest [98,133,136,202].…”

Section: Cross-sectional Lumen γ That Is the Real Number (Function Omentioning

confidence: 99%

“…, where ρ 1 (x) and ρ 2 (x) are the mean and Gaussian curvatures of Σ, respectively, [136], and f s the forcing term. The previous model is derived from the equations of the linear infinitesimal elasticity (Hooke law) under the assumptions of small thickness, plane stresses, and negligible elastic bending terms.…”

Section: Further Developments and Commentsmentioning

confidence: 99%

“…-for the fluid subproblem, the upstream velocity u up on the proximal boundaries and absorbing traction conditions T f · n = h on the distal boundaries, h being a suitable function [136];…”

Section: Modeling Blood Vascular Wall and Their Interactionmentioning

confidence: 99%

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

Quarteroni

Veneziani

Vergara

2016

Computer Methods in Applied Mechanics and Engineering

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159

153

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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: Cross-sectional Lumen γ That Is the Real Number (Function Omentioning

confidence: 99%

Section: Further Developments and Commentsmentioning

confidence: 99%

See 1 more Smart Citation

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

Quarteroni

Veneziani

Vergara

2016

Computer Methods in Applied Mechanics and Engineering

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159

153

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“…The RFSI model as presented in Colciago et al [1] is an unsteady Navier-Stokes model set on a fixed domain with generalized Robin boundary conditions (For similar models see e.g., [21][22][23][24] Although the RFSI model lives in a fixed domain, it is necessary to define an auxiliary variable which stands for the displacement of the arterial wall d s,h . Using a backward Euler finite difference method for the time derivatives, the fully discrete weak formulation of the RFSI problem is written as follows:…”

Section: Model Equationmentioning

confidence: 99%

Reduced Numerical Approximation of Reduced Fluid-Structure Interaction Problems With Applications in Hemodynamics

Colciago

Deparis

2018

Front. Appl. Math. Stat.

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This paper deals with fast simulations of the hemodynamics in large arteries by considering a reduced model of the associated fluid-structure interaction problem, which in turn allows an additional reduction in terms of the numerical discretisation. The resulting method is both accurate and computationally cheap. This goal is achieved by means of two levels of reduction: first, we describe the model equations with a reduced mathematical formulation which allows to write the fluid-structure interaction problem as a Navier-Stokes system with non-standard boundary conditions; second, we employ numerical reduction techniques to further and drastically lower the computational costs. The non standard boundary condition is of a generalized Robin type, with a boundary mass and boundary stiffness terms accounting for the arterial wall compliance. The numerical reduction is obtained coupling two well-known techniques: the proper orthogonal decomposition and the reduced basis method, in particular the greedy algorithm. We start by reducing the numerical dimension of the problem at hand with a proper orthogonal decomposition and we measure the system energy with specific norms; this allows to take into account the different orders of magnitude of the state variables, the velocity and the pressure. Then, we introduce a strategy based on a greedy procedure which aims at enriching the reduced discretization space with low offline computational costs. As application, we consider a realistic hemodynamics problem with a perturbation in the boundary conditions and we show the good performances of the reduction techniques presented in the paper. The results obtained with the numerical reduction algorithm are compared with the one obtained by a standard finite element method.The gains obtained in term of CPU time are of three orders of magnitude.

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“…The generalized string model had been utilized as the structure of blood flow in compliant vessels and arteries [9][10][11][12][13][14][15]. Causin et al [16] explained that the generalized string model is a structural model derived from the theory of linear elasticity for a cylindrical tube with small thickness.…”

Section: Introductionmentioning

confidence: 99%

Some Numerical Approaches to Solve Fluid Structure Interaction Problems in Blood Flow

Tang

Amin

2014

Abstract and Applied Analysis

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Some numerical approaches to solve fluid structure interaction problems in blood flow are reviewed. Fluid structure interaction is the interaction between a deformable structure with either an internal or external flow. A discussion on why the compliant artery associated with fluid structure interaction should be taken into consideration in favor of the rigid wall model being included. However, only the Newtonian model of blood is assumed, while various structure models which include, amongst others, generalized string models and linearly viscoelastic Koiter shell model that give a more realistic representation of the vessel walls compared to the rigid structure are presented. Since there exists a strong added mass effect due to the comparable densities of blood and the vessel wall, the numerical approaches to overcome the added mass effect are discussed according to the partitioned and monolithic classifications, where the deficiencies of each approach are highlighted. Improved numerical methods which are more stable and offer less computational cost such as the semi-implicit, kinematic splitting, and the geometrical multiscale approach are summarized, and, finally, an appropriate structure and numerical scheme to tackle fluid structure interaction problems are proposed.

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An Effective Fluid-Structure Interaction Formulation for Vascular Dynamics by Generalized Robin Conditions

Cited by 182 publications

References 37 publications

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

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

Reduced Numerical Approximation of Reduced Fluid-Structure Interaction Problems With Applications in Hemodynamics

Some Numerical Approaches to Solve Fluid Structure Interaction Problems in Blood Flow

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