2017

Sonntag, Matthias Shape derivatives and shock capturing for the NavierStokes equations in discontinuous Galerkin methods PhD Thesis 2017, ISBN: 9783843932813. Abstract  Links  BibTeX @phdthesis{sonntagPHD,
title = {Shape derivatives and shock capturing for the NavierStokes equations in discontinuous Galerkin methods},
author = {Matthias Sonntag},
url = {http://dx.doi.org/10.18419/opus9342},
doi = {10.18419/opus9342},
isbn = {9783843932813},
year = {2017},
date = {20170810},
abstract = {This work addresses two different topics, the shape derivatives for the compressible NavierStokes equations on the one hand and, on the other hand, the treatment of shocks or other flow discontinuities in Discontinuous Galerkin methods. There is a strong demand for very efficient methods for shape optimization in the aerospace industry, for example drag reduction or lift maximization of an aircraft. 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 liftcoefficient for the NavierStokes equations in variational formulation and compare it with the pointwise approach both analytically and numerically. 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 Discontinuous Galerkin 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 method. In all elements containing shocks or discontinuities the operator of the DG method is replaced by the Finite Volume scheme. This scheme is, due to the use of slope limiters, well known for its strengths in handling shocks. However, in regions where the flow is smooth the Finite Volume method requires a finer resolution for the same accuracy than the Discontinuous Galerkin scheme. 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 subcells. By associating exactly one Finite Volume subcell 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 subcell approach are investigated with several examples, but also studies regarding the parallel efficiency are performed.},
keywords = {},
pubstate = {published},
tppubtype = {phdthesis}
}
This work addresses two different topics, the shape derivatives for the compressible NavierStokes equations on the one hand and, on the other hand, the treatment of shocks or other flow discontinuities in Discontinuous Galerkin methods. There is a strong demand for very efficient methods for shape optimization in the aerospace industry, for example drag reduction or lift maximization of an aircraft. 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 liftcoefficient for the NavierStokes equations in variational formulation and compare it with the pointwise approach both analytically and numerically. 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 Discontinuous Galerkin 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 method. In all elements containing shocks or discontinuities the operator of the DG method is replaced by the Finite Volume scheme. This scheme is, due to the use of slope limiters, well known for its strengths in handling shocks. However, in regions where the flow is smooth the Finite Volume method requires a finer resolution for the same accuracy than the Discontinuous Galerkin scheme. 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 subcells. By associating exactly one Finite Volume subcell 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 subcell approach are investigated with several examples, but also studies regarding the parallel efficiency are performed. 
Sonntag, Matthias; Munz, ClausDieter Efficient Parallelization of a Shock Capturing for Discontinuous Galerkin Methods using Finite Volume Subcells Journal Article Journal of Scientific Computing, 70 (3), pp. 1262–1289, 2017, ISSN: 15737691. Abstract  Links  BibTeX @article{Sonntag2016,
title = {Efficient Parallelization of a Shock Capturing for Discontinuous Galerkin Methods using Finite Volume Subcells},
author = {Matthias Sonntag and ClausDieter Munz},
url = {http://dx.doi.org/10.1007/s1091501602875},
doi = {10.1007/s1091501602875},
issn = {15737691},
year = {2017},
date = {20170101},
journal = {Journal of Scientific Computing},
volume = {70},
number = {3},
pages = {12621289},
abstract = {We present a shock capturing procedure for high order Discontinuous Galerkin methods, by which shock regions are refined in subcells and treated by finite volume techniques. Hence, our approach combines the good properties of the Discontinuous Galerkin method in smooth parts of the flow with the perfect properties of a total variation diminishing finite volume method for resolving shocks without spurious oscillations. Due to the subcell approach the interior resolution on the Discontinuous Galerkin grid cell is nearly preserved and the number of degrees of freedom remains the same. This structure allows the interpretation of the data either as DG solution or as finite volume solution on the subgrid. In this paper we explain the efficient implementation of this coupled method on massively parallel computers and show some numerical results.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We present a shock capturing procedure for high order Discontinuous Galerkin methods, by which shock regions are refined in subcells and treated by finite volume techniques. Hence, our approach combines the good properties of the Discontinuous Galerkin method in smooth parts of the flow with the perfect properties of a total variation diminishing finite volume method for resolving shocks without spurious oscillations. Due to the subcell approach the interior resolution on the Discontinuous Galerkin grid cell is nearly preserved and the number of degrees of freedom remains the same. This structure allows the interpretation of the data either as DG solution or as finite volume solution on the subgrid. In this paper we explain the efficient implementation of this coupled method on massively parallel computers and show some numerical results. 
2016

Sonntag, Matthias; Schmidt, Stephan; Gauger, Nicolas R Shape derivatives for the compressible NavierStokes equations in variational form Journal Article Journal of Computational and Applied Mathematics, 296 , pp. 334–351, 2016, ISSN: 03770427. Links  BibTeX @article{Sonntag_shapederivative,
title = {Shape derivatives for the compressible NavierStokes equations in variational form },
author = {Matthias Sonntag and Stephan Schmidt and Nicolas R. Gauger},
url = {http://www.sciencedirect.com/science/article/pii/S0377042715004628},
doi = {10.1016/j.cam.2015.09.010},
issn = {03770427},
year = {2016},
date = {20160101},
journal = {Journal of Computational and Applied Mathematics},
volume = {296},
pages = {334351},
keywords = {},
pubstate = {published},
tppubtype = {article}
}

Boblest, Sebastian; Hempert, Fabian; Hoffmann, Malte; Offenhäuser, Philipp; Sonntag, Matthias; Sadlo, Filip; Glass, Colin W; Munz, ClausDieter; Ertl, Thomas; Iben, Uwe Toward a Discontinuous Galerkin Fluid Dynamics Framework for Industrial Applications Book Chapter Nagel, Wolfgang E; Kröner, Dietmar H; Resch, Michael M (Ed.): High Performance Computing in Science and Engineering textasciiacute15: Transactions of the High Performance Computing Center, Stuttgart (HLRS) 2015, pp. 531–545, Springer International Publishing, Cham, 2016, ISBN: 9783319246338. Links  BibTeX @inbook{Boblest2016,
title = {Toward a Discontinuous Galerkin Fluid Dynamics Framework for Industrial Applications},
author = {Sebastian Boblest and
Fabian Hempert
and Malte Hoffmann
and Philipp Offenhäuser
and Matthias Sonntag
and Filip Sadlo
and Colin W. Glass
and ClausDieter Munz
and Thomas Ertl
and Uwe Iben},
editor = { Wolfgang E. Nagel
and Dietmar H. Kröner
and Michael M. Resch},
url = {http://dx.doi.org/10.1007/9783319246338_34},
doi = {10.1007/9783319246338_34},
isbn = {9783319246338},
year = {2016},
date = {20160101},
booktitle = {High Performance Computing in Science and Engineering textasciiacute15: Transactions of the High Performance Computing Center, Stuttgart (HLRS) 2015},
pages = {531545},
publisher = {Springer International Publishing},
address = {Cham},
keywords = {},
pubstate = {published},
tppubtype = {inbook}
}

2015

Kaland, L; Sonntag, M; Gauger, N R Adaptive Aerodynamic Design Optimization for NavierStokes Using Shape Derivatives with Discontinuous Galerkin Methods Book Chapter Greiner, David ; Galván, Blas ; Périaux, Jacques ; Gauger, Nicolas ; Giannakoglou, Kyriakos ; Winter, Gabriel (Ed.): Advances in Evolutionary and Deterministic Methods for Design, Optimization and Control in Engineering and Sciences, pp. 143–158, Springer International Publishing, Cham, 2015, ISBN: 9783319115412. Links  BibTeX @inbook{Kaland2015,
title = {Adaptive Aerodynamic Design Optimization for NavierStokes Using Shape Derivatives with Discontinuous Galerkin Methods},
author = {Kaland, L. and Sonntag, M. and Gauger, N. R.},
editor = {Greiner, David and Galván, Blas and Périaux, Jacques and Gauger, Nicolas and Giannakoglou, Kyriakos and Winter, Gabriel},
doi = {10.1007/9783319115412_9},
isbn = {9783319115412},
year = {2015},
date = {20150101},
booktitle = {Advances in Evolutionary and Deterministic Methods for Design, Optimization and Control in Engineering and Sciences},
pages = {143158},
publisher = {Springer International Publishing},
address = {Cham},
keywords = {},
pubstate = {published},
tppubtype = {inbook}
}

2014

Sonntag, Matthias ; Munz, ClausDieter Shock Capturing for Discontinuous Galerkin Methods using Finite Volume Subcells Incollection Fuhrmann, Jürgen ; Ohlberger, Mario ; Rohde, Christian (Ed.): Finite Volumes for Complex Applications VIIElliptic, Parabolic and Hyperbolic Problems, 78 , pp. 945953, Springer International Publishing, 2014, ISBN: 9783319055909. Abstract  Links  BibTeX @incollection{,
title = {Shock Capturing for Discontinuous Galerkin Methods using Finite Volume Subcells},
author = {Sonntag, Matthias and Munz, ClausDieter},
editor = {Fuhrmann, Jürgen and Ohlberger, Mario and Rohde, Christian},
url = {http://dx.doi.org/10.1007/9783319055916_96},
doi = {10.1007/9783319055916_96},
isbn = {9783319055909},
year = {2014},
date = {20140516},
booktitle = {Finite Volumes for Complex Applications VIIElliptic, Parabolic and Hyperbolic Problems},
volume = {78},
pages = {945953},
publisher = {Springer International Publishing},
series = {Springer Proceedings in Mathematics & Statistics},
abstract = {We present a shock capturing procedure for high order discontinuous Galerkin methods, by which shock regions are refined and treated by the finite volume techniques. Hence, our approach combines the good properties of the discontinuous Galerkin method in smooth parts of the flow with the perfect properties of a total variation diminishing finite volume method for resolving shocks without spurious oscillations. Due to the subcell approach the interior resolution on the discontinuous Galerkin grid cell is preserved and the number of degrees of freedom remains the same. In this paper we focus on an implementation of this coupled method and show our first results.},
keywords = {},
pubstate = {published},
tppubtype = {incollection}
}
We present a shock capturing procedure for high order discontinuous Galerkin methods, by which shock regions are refined and treated by the finite volume techniques. Hence, our approach combines the good properties of the discontinuous Galerkin method in smooth parts of the flow with the perfect properties of a total variation diminishing finite volume method for resolving shocks without spurious oscillations. Due to the subcell approach the interior resolution on the discontinuous Galerkin grid cell is preserved and the number of degrees of freedom remains the same. In this paper we focus on an implementation of this coupled method and show our first results. 