ISBN 978-3-8439-3281-3

978-3-8439-3281-3, Reihe Ingenieurwissenschaften

Matthias Sonntag
Shape Derivatives and Shock Capturing for the Navier-Stokes Equations in Discontinuous Galerkin Methods

195 Seiten, Dissertation Universität Stuttgart (2017), Softcover, A5

Zusammenfassung / Abstract

This work addresses two different topics, the shape derivatives for the compressible Navier-Stokes equations and the treatment of shocks in Discontinuous Galerkin (DG) methods. There is a strong demand for efficient methods for shape optimization in the aerospace industry. The use of gradient based optimization schemes requires derivatives of the cost function with respect to the shape of an object. With the shape derivatives presented in this work, these derivatives can be calculated independent of the parametrization of the object's shape, and, since the derivation takes place in the continuous space, they can be applied to almost any discretization. Nevertheless, one has to take the numerical scheme, which is later applied, into account. For methods based on the variational formulation a difference in the shape derivative, compared to the pointwise approach, arises, which cannot be neglected. Hence, one objective of this work is to derive the shape derivatives of the drag- and lift-coefficient for the Navier-Stokes equations in variational formulation and compare it with the pointwise approach. A discrepancy has to be expected, especially for flow phenomena with high gradients or discontinuities which do not fulfill the strong form of the governing equations.

These flow phenomena require a special treatment in numerical methods of high order. In the second part of this work, a shock capturing for the DG method is developed which prevents the oscillations originating from the approximation of discontinuities with high order polynomials. Therefore a hybrid approach is presented, where the original DG scheme is coupled with a second order Finite Volume (FV) method. In all elements containing shocks the operator of the DG method is replaced by the FV scheme which is well known for its strengths in handling shocks. However, using the same mesh for the FV method as for the DG scheme would lead to a big reduction in resolution. Hence, to compensate this loss the original elements of the mesh are divided into logical sub-cells. By associating exactly one FV sub-cell to each degree of freedom of a DG element, the same data structures can be used. This enables an efficient implementation of the outlined shock capturing designated for high performance computations. Therefore, not only the basic properties of this hybrid DG/FV sub-cell approach are investigated with several examples, but also studies regarding the parallel efficiency are performed.