The BFGS algorithm for a nonlinear least squares problem arising from blood flow in arteries

“…The wall of the vessel is described by the beam equation and the fluid is the blood which is modeled by the Stokes equations. The parameter values of the fluid and the structure are Mbaye and Murea [9,10]:…”

Section: Numerical Resultsmentioning

confidence: 99%

Controllability Approach for a Fluid Structure Interaction Problem

Mbaye¹

2012

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“…This is a benchmark problem for which the existence of solutions has been demonstrated in [18,19] and whose numerical solution has been previously considered e.g. in [29,35,54]. In Sect.…”

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confidence: 99%

“…We call this the parametric coupling approach. The parametric coupling approach was used in [35] to solve the same problem, with the exception that there the applied traction (depending on this case only on the pressure profile on the wall) was directly parametrized.…”

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confidence: 99%

A Reduced Basis Model with Parametric Coupling for Fluid-Structure Interaction Problems

Lassila¹,

Quarteroni²,

Rozza³

2012

SIAM J. Sci. Comput.

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Abstract. We present a new model reduction technique for steady fluid-structure interaction problems. When the fluid domain deformation is suitably parametrized, the coupling conditions between the fluid and structure can be formulated in the low-dimensional space of geometric parameters. Moreover, we apply the reduced basis method to reduce the cost of repeated fluid solutions necessary to achieve convergence of fluid-structure iterations. In this way a reduced order model with reliable a posteriori error bounds is obtained. The proposed method is validated with an example of steady Stokes flow in an axisymmetric channel, where the structure is described by a simple 1-d generalized string model. We demonstrate rapid convergence of the reduced solution of the parametrically coupled problem as the number of geometric parameters is increased. AMS subject classifications. 65N30, 74F10, 76D07Key words. fluid-structure interaction, model reduction, incompressible Stokes equations, reduced basis method, free-form deformation 1. Introduction. The numerical simulation of Fluid-Structure Interaction (FSI) problems is an important topic in wide areas of engineering and medical research. Concerning the latter, of great importance is the modelling of blood flow in the large arteries of the human cardiovascular system, where pulsatile flows combined with a high degree of deformability of the arterial walls together cause large displacement effects that cannot be neglected when attempting to accurately model the flow dynamics of the system. High fidelity computational fluid dynamics and structural mechanics solvers based on, for example, the Finite Element Method (FEM) need to be combined in a framework that is challenging both from a mathematical as well as implementation viewpoint. For an overview of cardiovascular modelling techniques we refer to [42,44] and the book [14]. The complexity and nonlinearity of FSI problems has until recently limited the scope of physically meaningful simulations to just small and isolated sections of arteries. When attempting to consider the entire cardiovascular system as a complex network of different time and spatial scales, Model Order Reduction (MOR) techniques can accurately and reliably reduce the nonlinear FSI models to computationally more cost-efficient ones.In the geometric multiscale approach to MOR [13] the flow network is decomposed to smaller parts that are joined together using physical coupling conditions, and each part of which is modelled at a level necessary to capture the relevant local dynamics of the system. The target for our proposed reduced model is those parts of the cardiovascular network, where a full fidelity 3-d Navier-Stokes solution is not necessary, but where fluid-structure interaction effects are still important. The reduced model should fulfill two conditions: (i) it should have certified a posteriori error bounds that can be tuned to the user's requirements, and (ii) it should have sufficiently low online computational memory requirements to fit on one pa...

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The BFGS algorithm for a nonlinear least squares problem arising from blood flow in arteries

Cited by 27 publications

References 14 publications

Conjugate Gradient Method to Solve Fluid Structure Interaction Problem

Conjugate Gradient Method to Solve Fluid Structure Interaction Problem

Controllability Approach for a Fluid Structure Interaction Problem

A Reduced Basis Model with Parametric Coupling for Fluid-Structure Interaction Problems

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