Many problems in cardiovascular modelling can be expressed as fluid-structure interaction problems. The mathematical modelling of such coupled problems consists of four main parts: solution of the fluid equations given the current fluid geometry, solution of the structural displacements given the normal stresses exerted by the fluid, fulfillment of coupling constraints to achieve force balance across the interface, and transport of the fluid-structure interface. Even when both the fluid and structure equations are independently linear, the geometric variability of the fluid-structure interface can cause a significant nonlinearity in the coupled system. Solution methods for fluid-structure interaction problems are iterative in nature and involve the repeated solution of the fluid and structure equations in many different congurations.
A model reduction technique for steady fluid-structure interaction problem of an incompressible Stokes flow in a flexible 2-d channel is presented. A geometric reduction of the moving fluid-structure interface is performed using free-form deformations to reduce the free-boundary problem to a low-dimensional parameter space. Reduced basis methods (model reduction methods for parametric partial differential equations using well-chosen snapshot solutions to build a set of global basis functions) are then used to reduce the complexity of the resulting state equations. A least-squares parametric coupling formulation between the fluid and structure is given. The approximate balance between applied traction and structural displacement is formulated using the parameterized displacement of the interface and solved using nonlinear programming techniques. A low-dimensional parameterization with few parameters coming from a free-form deformation technique is sufficient to obtain approximate coupling.