On the usage of NURBS as interface representation in free‐surface flows

Abstract

SUMMARY When simulating free‐surface flows using the finite element method, there are many cases where the governing equations require information which must be derived from the available discretized geometry. Examples are curvature or normal vectors. The accurate computation of this information directly from the finite element mesh often requires a high degree of refinement—which is not necessarily required to obtain an accurate flow solution. As a remedy and an option to be able to use coarser meshes, the representation of the free surface using non‐uniform rational B‐splines (NURBS) curves or surfaces is investigated in this work. The advantages of a NURBS parameterization in comparison with the standard approach are discussed. In addition, it is explored how the pressure jump resulting from surface tension effects can be handled using doubled interface nodes. Numerical examples include the computation of surface tension in a two‐phase flow as well as the computation of normal vectors as a basis for mesh deformation methods. For these examples, the improvement of the numerical solution compared with the standard approaches on identical meshes is shown. Copyright © 2011 John Wiley & Sons, Ltd.

Numerical Modeling and Experimental Validation of Free Surface Flow Problems

“…Boundary conditions on curved boundaries are addressed in [25] for DSD/SST in 3D, while in [237] a shape vector is used for a general ALE strategy. Elgeti et al [73] have used DSD/SST for simulating two-phase incompressible fluid flows by approximating the interface with NURBS with the aim of improving the precision on determining the normal vectors and the curvature calculation, i.e. for a better representation of the interface movement and the surface tension effects.…”

Space‐time NURBS‐enhanced finite elements for free‐surface flows in 2D

“…(22) leads to the complication that the kinematic rule is defined on the surface, whereas all quantities governing the shape of the NURBS curve are defined at the control points P j -whose geometrical position does usually not even coincide with the surface (cf. A first attempt into this direction has been explored in [9], where the DSD/SST approach was coupled with a NURBS free-surface representation without the NEFEM. Consequently, the surface displacement velocity has to be translated into a displacement velocity for the control points O v j .…”

“…This discontinuity cannot be captured with continuous shape functions, and a standard finite element analysis will lead to spurious velocities [7]. However, these methods lead to more involved finite element formulations, and recently, a simple and effective procedure for standard formulations has been proposed by Elgeti et al [9]. However, these methods lead to more involved finite element formulations, and recently, a simple and effective procedure for standard formulations has been proposed by Elgeti et al [9].…”

“…An intuitive approach to overcome this problem is to enrich the pressure approximation space with discontinuous functions, as applied in [7,33] and also known from the XFEM [34]. This approach is adopted in the current work, and an identical setting to [9] is chosen for the simulation. It exploits the a priori known position of the pressure jump, namely on the boundary separating the phases, and introduces an additional degree for the nodes describing it.…”

“…This geometrical information is additional to the actual degrees of freedom and usually requires a much finer discretization of the computational domain than the flow solution itself. This has been done by Ganesan et al using cubic splines [7] and Elgeti et al using non-uniform rational B-splines (NURBS) [8,9]. An example of such method is the NURBS-enhanced finite element method (NEFEM), recently proposed by Sevilla et al The current paper discusses the extension of the spatial NEFEM to space-time methods and investigates the application of space-time NURBS-enhanced elements to free-surface flows.…”