20150127
Two important influences on the accuracy of simulation results are the approximation properties of the numerical scheme with respect to, on the one hand, the unknown solution, and on the other hand, the geometry representation. The latter aspect poses a particular challenge in the context of deforming computational domains.
These deformations can either be externally prescribed, as for example in fluidstructure interaction or shape optimization, or occur in the frame of freeboundary problems, where the computational domain itself is part of the solution. In the context of fluid flow, freeboundary problems can, e.g., be found in freesurface flows. Considering the coupled problem of flow solution and domain deformation, our solution approach is based on the DeformingSpatial Domain/Stabilized SpaceTime (DSD/SST) finite element method. In DSD/SST, the variational form is written over the complete spacetime domain, thus easily incorporating deforming domains into the formulation. The deformation itself is treated with a boundary conforming interface tracking scheme. In order to further enhance the boundary conformation, the scheme employs NonUniform Rational BSplines (NURBS) as a support of the standard finite element representation of geometry and flow solution. As the basis of CAD systems, NURBS are closely connected to any engineering application, particularly since the concept of Isogeometric Analysis (IGA) [1] has introduced NURBS to the numerical analysis. However, the generation of complex threedimensional grids suited for IGA is still a challenge, hindering its use in the area of fluid mechanics. Nevertheless, methods for fluid simulation can profit immensely from the use of NURBS as a boundary description. Several stages of NURBS usage (here, all in the context of the finite element method) are possible:
1. Certain information needed for the computation (e.g., curvature or normals) is computed from a NURBS representing the boundary [2].
2. The computational domain is represented exactly using NURBS, but the solution is still interpolated using polynomials. This idea is, for example, realized in the NURBSenhanced finite element method [3].
3. The NURBS represent both the geometry and the unknown solution (IGA).
Stemming from both the exact boundary description and the superior approximation properties of NURBS, such approaches can have a variety of advantages. These involve both the computational accuracy reached with a specific computational cost as well as the efficiency and accuracy of coupling schemes. The advantages of the discussed approaches will be demonstrated through a variety of engineering applications. Furthermore, the use of NURBS in fluid flow connected shape optimization problems — especially from the area of production engineering — will be presented. Here, the focus is on the incorporation of shape constraints imposed through the manufacturing processes.
References
[1] T. J. R. Hughes, J. A. Cottrell,Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering 194 (2005), 4135–4195.
[2] S. Elgeti, H. Sauerland, L. Pauli, M. Behr. On the usage of NURBS as interface representation in freesurface flows. International Journal for Numerical Methods in Fluids 69 (2012), 73–87.
[3] R. Sevilla, S. FernandezMendez, A. Huerta. NURBSEnhanced Finite Element Method (NEFEM): A Seamless Bridge Between CAD and FEM. Archives of Computational Methods in Engineering 18 (2011), 441–484.
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