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

Abstract

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...

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

“…A domain decomposition approach [131,132], combined with a reduced basis method, has been successfully applied in [94,97,92] and further extensions discussed in [68,72,65]. A coupled multiphysics setting has been proposed for simple fluid-structure interaction problems in [85,84,80] and [106] for Stokes-Darcy.…”

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

“…Ad hoc reduced order modelling techniques have recently been proposed for optimal flow control problems [104,108,121], optimal shape design of devices related with fluid flows [6,58,23,88], and the treatment of fluid-structure interaction problems [76,78].…”

On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition

“…However, since in practice β f (μ) 1 and even negative for some finite number of parameter points μ, we can expect that the correction terms given in the previous section will be much more sensitive to the choice of δ c . This proposed method can be used also for vectorial elliptic noncoercive problems [22,25].…”

“…Our proposed approach is to limit the number of affine terms Q used in the SCM without sacrificing the accuracy of the lower bound β LB h (μ). This improvement will allow to treat problems with a more complex nonaffine parametrization, like the ones arising with geometrical parametrization of more complex shapes [22,25].…”

“…Therefore our interest is in providing numerical methods for obtaining efficient lower bounds for β h (μ). Some algorithms for deriving lower bounds for parametric stability factors have been proposed in literature, among them we mention the successive constraint method (SCM) [5,17,18,29,34] with recent applications in viscous flows characterized by an increasing complexity in terms of geometrical parametrization [22,25]. This algorithm (briefly detailed in Appendix A) is based on an offline-online computational approach, where the offline stage consists of the solution of a large number of eigenproblems of the underlying PDE at different parametric points.…”