A finite element method for the Navier-Stokes equations in moving domain with application to hemodynamics of the left ventricle

Abstract

The paper introduces a finite element method for the Navier-Stokes equations of incompressible viscous fluid in a time-dependent domain. The method is based on a quasi-Lagrangian formulation of the problem and handling the geometry in a time-explicit way. We prove that numerical solution satisfies a discrete analogue of the fundamental energy estimate. This stability estimate does not require a CFL time-step restriction. The method is further applied to simulation of a flow in a model of the left ventricle of a human heart, where the ventricle wall dynamics is reconstructed from a sequence of contrast enhanced Computed Tomography images.

A quasi-Lagrangian finite element method for the Navier–Stokes equations in a time-dependent domain

“…Thanks to the definition of P k h and boundary conditions for u k h , the vector field e k h vanishes on ∂Ω 0 . Thus, we can set ψ h = e k h in (47) and apply the same arguments that were used to show (21). We obtain the estimate…”

“…We deduce an energy stability estimate for the finite element method from the balance in (21) and a priori estimates in (17). For the sake of notation, we introduce · k := Ω 0 J k | · | 2 dx 1 2 , which defines a k-dependent norm uniformly equivalent to the L 2 -norm.…”

“…The analysis [20] imposes time step restriction, assumes zero velocity boundary condition and certain smoothness assumptions for the finite element displacement field. A closely related method to the one studied here was considered in [21]. However, that paper introduced an assumption that a divergence free extension of a boundary condition function to the computational domain is given.…”

A stable method for 4D CT-based CFD simulation in the right ventricle of a TGA patient

“…The mesh generation process is similar to the one proposed in our previous work [16] and we refer to that paper for any omitted details. The algorithm requires a reference domain, which is defined implicitly as an enclosed volume of the averaged distance function over all 90 frames.…”

A multigrid ghost-point level-set method for incompressible Navier-Stokes equations on moving domains with curved boundaries

“…Numerical methods for steady complex-shaped objects are mainly divided between fitted and unfitted boundary methods. In fitted boundary methods, the grid, either structured or unstructured (such as Finite Element Methods (FEM)), is adapted to the curved boundary [53,34,32,43,42,23,22]. Although they provide an extremely accurate representation of the domain, the generation of the mesh may become cumbersome and significantly expensive from a computational point of view, especially in moving domain problems where a new mesh generation is needed at each time step.…”