Stability issues of entropy-stable and/or split-form high-order schemes -- Analysis of local linear stability

The focus of the present research is on the analysis of local linear stability of high-order (including split-form) summation-by-parts methods, with e.g. two-point entropy-conserving fluxes, approximating non-linear conservation laws. Our main finding is that local linear stability is not guaranteed even when the scheme is non-linearly stable and that this has grave implications for simulation results. We show that entropy-conserving two-point fluxes are inherently locally linearly unstable, as they can be dissipative or anti-dissipative. Unfortunately, these fluxes are at the core of many commonly used high-order entropy-stable extensions, including split-form summation-by-parts discontinuous Galerkin spectral element methods (or spectral collocation methods). For the non-linear Burgers equation, we demonstrate numerically that such schemes cause exponential growth of errors. Furthermore, we demonstrate a similar abnormal behaviour for the compressible Euler equations. Finally, we demonstrate numerically that other commonly used split-forms, such as the Kennedy and Gruber splitting, are also locally linearly unstable

An Entropy Stable Nodal Discontinuous Galerkin Method for the resistive MHD Equations. Part II: Subcell Finite Volume Shock Capturing

The second paper of this series presents two robust entropy stable shock-capturing methods for discontinuous Galerkin spectral element(DGSEM)discretizations of the compressible magneto-hydrodynamics (MHD) equations. Specifically, we use the resistive GLM-MHD equations, which include a divergence cleaning mechanism that is based on a generalized Lagrange multiplier (GLM). For the continuous entropy analysis to hold, and due to the divergence-free constraint on the magnetic field, the GLM-MHD system requires the use of non-conservative terms, which need special treatment. Hennemann et al. ["A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations". JCP, 2020] recently presented an entropy stable shock-capturing strategy for DGSEM discretizations of the Euler equations that blends the DGSEM scheme with a subcell first-order finite volume (FV) method. Our first contribution is the extension of the method of Hennemann et al. to systems with non-conservative terms, such as the GLM-MHD equations. In our approach, the advective and non-conservative terms of the equations are discretized with a hybrid FV/DGSEM scheme, whereas the visco-resistive terms are discretized only with the high-order DGSEM method. We prove that the extended method is entropy stable on three-dimensional unstructured curvilinear meshes. Our second contribution is the derivation and analysis of a second entropy stable shock-capturing method that provides enhanced resolution by using a subcell reconstruction procedure that is carefully built to ensure entropy stability. We provide a numerical verification of the properties of the hybrid FV/DGSEM schemes on curvilinear meshes and show their robustness and accuracy with common benchmark cases, such as the Orszag-Tang vortex and the GEM (Geospace Environmental Modeling) reconnection challenge. Finally, we simulate a space physics application: the interaction of Jupiter's magnetic field with the plasma torus generated by the moon Io

An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I: Theory and Numerical Verification

The first paper of this series presents a discretely entropy stable discontinuous Galerkin (DG) method for the resistive magnetohydrodynamics (MHD) equations on three-dimensional curvilinear unstructured hexahedral meshes. Compared to other fluid dynamics systems such as the shallow water equations or the compressible Navier-Stokes equations, the resistive MHD equations need special considerations because of the divergence-free constraint on the magnetic field. For instance, it is well known that for the symmetrization of the ideal MHD system as well as the continuous entropy analysis a non-conservative term proportional to the divergence of the magnetic field, typically referred to as the Powell term, must be included. As a consequence, the mimicry of the continuous entropy analysis in the discrete sense demands a suitable DG approximation of the non-conservative terms in addition to the ideal MHD terms. This paper focuses on the resistive MHD equations: Our first contribution is a proof that the resistive terms are symmetric and positive-definite when formulated in entropy space as gradients of the entropy variables, which enables us to show that the entropy inequality holds for the resistive MHD equations. This continuous analysis is the key for our DG discretization and guides the path for the construction of an approximation that discretely mimics the entropy inequality, typically termed entropy stability. Our second contribution is a detailed derivation and analysis of the discretization on three-dimensional curvilinear meshes. The discrete analysis relies on the summation-by-parts property, which is satisfied by the DG spectral element method (DGSEM) with Legendre-Gauss-Lobatto (LGL) nodes. Although the divergence- free constraint is included in the non-conservative terms, the resulting method has no particular treatment of the magnetic field divergence errors, which might pollute the solution quality. Our final contribution is the extension of the standard resistive MHD equations and our DG approximation with a divergence cleaning mechanism that is based on a generalized Lagrange multiplier (GLM). As a conclusion to the first part of this series, we provide detailed numerical validations of our DGSEM method that underline our theoretical derivations. In addition, we show a numerical example where the entropy stable DGSEM demonstrates increased robustness compared to the standard DGSEM

Experimental multi-level seismic performance assessment of 3D RC frame designed for damage avoidance

This paper experimentally investigates the application of damage avoidance design (DAD) philosophy to moment-resisting frames with particular emphasis on detailing of rocking interfaces. An 80% scale three-dimensional rocking beam-column joint sub-assembly designed and detailed based on damage avoidance principles is constructed and tested. Incremental dynamic analysis is used for selecting ground motion records to be applied to the sub-assembly for conducting a multi-level seismic performance assessment (MSPA). Analyses are conducted to obtain displacement demands due to the selected near- and medium-field ground motions that represent different levels of seismic hazard. Thus, predicted displacement time histories are applied to the sub-assembly for conducting quasi-earthquake displacement tests. The sub-assembly performed well reaching drifts up to 4.7% with only minor spalling occurring at rocking beam interfaces and minor flexural cracks in beams. Yielding of post-tensioning threaded bars occurred, but the sub-assembly did not collapse. The externally attached energy dissipators provided large hysteretic dissipation during large drift cycles. The sub-assembly satisfied all three seismic performance requirements, thereby verifying the superior performance of the DAD philosoph

Techniken zur Erstellung von gekrümmten Gittern für parallele Simulationen hoher Ordnung auf unstrukturierten Gittern

In this work, the generation of high order curved three-dimensional hybrid meshes and its application are presented. Meshes with linear edges are the standard of today's state-of-the-art meshing software. Industrial applications typically imply geometrically complex domains, mostly described by curved domain boundaries. To apply high order methods in this context, the geometry - in contrast to classical low order methods - has to be represented with a high order approximation, too. Therefore, a high order element mapping has to be used for the discretization of curved domain boundaries. The main idea here is to rely on existing linear mesh generators and provide additional information to produce high order curved elements. A very promising candidate for future numerical solvers in computational fluid dynamics is the family of high order discontinuous Galerkin (DG) schemes. They are locally conservative schemes, with a continuous polynomial representation within each element and allow a discontinuous solution across element faces. Elements couple only to direct face neighbors, and the discontinuity is resolved via numerical flux functions. As the main focus of this work are curved elements, the different formulations and possible implementations of the DG scheme with non-linear element mappings are discussed in detail. Especially, a highly efficient variant of the DG scheme for hexahedra, namely the discontinuous Galerkin spectral element method (DG-SEM), is presented. The main focus of this thesis is the generation of high order meshes. Several techniques to generate curved elements are described and their applicability to complex geometries is demonstrated. Starting from a linear mesh, the first step curves the element faces representing the curved geometry. Two approaches are presented, the first based on continuity conditions using surface normal vectors and the second based on interpolation of additionally generated surface points. The high order mapping of the volumetric element is computed as a blending of the curved element faces. In the case of boundary layer meshes, the blending may lead to inverted elements. As a remedy to this problem, an additional mesh deformation approach is proposed and validated. Independent thereof, another approach is presented, allowing one to directly generate curved volume mappings from the agglomeration of block-structured meshes. One of the reasons making high order DG schemes attractive for the simulation of fluid dynamics is their parallel efficiency. As future applications in fluid dynamics comprise the resolution of three-dimensional unsteady effects and are increasingly complex, the simulations require more and more computing resources, and weak and strong scalability of the numerical method becomes extremely important. In the last part of this thesis, the parallelization concept of the DG-SEM code Flexi is described in detail. A new domain decomposition strategy based on space filling curves is introduced, and is shown to be simple and flexible. A thorough parallel performance analysis confirms that the overall implementation scales perfectly. Ideal speed-up is maintained for high polynomial degrees, up to the limit of one element per core. As the DG scheme only communicates with direct neighbors, the same parallel efficiency is found on both cartesian meshes as well as fully unstructured meshes. The findings underline that the proposed Discontinuous Galerkin scheme exhibit a great potential for highly resolved simulations on current and future large scale parallel computer systems.Diese Arbeit befasst sich mit der Erstellung von dreidimensionalen hybriden Gittern hoher Ordnung und deren Verwendung. Die Standardelemente heutiger Vernetzungssoftware besitzen fast ausschließlich lineare Elementkanten. Bei industriellen Anwendung sind die zu vernetzenden Geometrien sehr komplex und weisen meist gekrümmte Gebietsgrenzen vor. Bei der Verwendung von Verfahren hoher Ordnung ist es im Unterschied zu klassischen Verfahren niedriger Ordnung notwendig, die Geometrie ebenfalls mit hoher Ordnung darzustellen. Die Diskretisierung der gekrümmten Randflächen erfolgt also durch eine höherwertige Abbildung der Elemente. Es wird die Grundidee verfolgt, dass vorhandene lineare Vernetzungssoftware weiterhin genutzt werden kann und Elemente hoher Ordnung mithilfe zusätzlich bereitgestellter Informationen generiert werden sollen. Innerhalb der Verfahren hoher Ordnung ist das discontinuous Galerkin (DG) Verfahren ein vielversprechender Kandidat für zukünftige Strömungslöser. Es handelt sich um ein lokal konservatives Verfahren. Die Lösung wird innerhalb des Elements als stetiges Polynom dargestellt und ist über die Elementgrenzen hinweg unstetig. Die Elemente koppeln nur mit direkten Nachbarelementen und aufgrund der Unstetigkeit an der Elementgrenze werden hierfür numerische Flüsse verwendet. Da ein Hauptaugenmerk dieser Arbeit auf der Behandlung von gekrümmten Elementen liegt, werden die unterschiedlichen Formulierungen und mögliche Implementierungen des DG Verfahrens auf Elementen mit nicht-linearen Abbildungen im Detail diskutiert. Insbesondere wird die Discontinuous Galerkin Spektrale Element Methode (DG-SEM) vorgestellt, eine besonders effiziente Implementierung für Hexaederelemente. Der Schwerpunkt dieser Arbeit liegt auf der Generierung von Gittern hoher Ordnung. Es werden unterschiedliche Techniken zur Erstellung von Elementen hoher Ordnung beschrieben und deren Anwendbarkeit auf komplexe Geometrien demonstriert. Ausgehend von einem linearen Netz erfolgt in einem ersten Schritt die Krümmung der Elementflächen, die an der gekrümmten Randbedingung anliegen. Hierbei wird zwischen zwei Ansätzen unterschieden, der erste basierend auf Normalenvektoren an den Oberflächenpunkten, der zweite auf Interpolation von zusätzlich generierten Oberflächenpunkten. Die volumetrische Abbildung des Elements wird dann durch eine Linearkombination der gekrümmten Elementflächen gebildet. Im Fall von Grenzschichtnetzen kann die Linearkombination zu ungültigen Elementabbildungen führen. Ein gültiges Gitter kann durch eine zusätzliche Gitterverformung generiert werden. Die gesamte Vorgehensweise wird anhand einem Grenzschichtnetz von einem Flügelprofil erläutert und validiert. Hiervon unabhängig wird ein weiterer Ansatz beschrieben, bei dem man die Volumenabbildung direkt durch Agglomeration block-strukturierter Gitter erhält. Einer der Gründe für die Attraktivität von DG Verfahren zur Simulation von Strömungen ist deren parallele Effizienz. Da zukünftige Anwendungen in der Strömungsmechanik die Auflösung von instationären dreidimensionalen Effekten umfassen und eine wachsende Komplexität aufweisen, ergibt sich ein steigender Bedarf an Rechenressourcen, und die schwache und starke Skalierbarkeit des numerischen Verfahrens spielt eine entscheidende Rolle. Daher wird im letzten Teil dieser Arbeit das Parallelisierungskonzept des G-SEM Lösers Flexi vorgestellt. Es wird eine neue Strategie zur Gebietszerlegung erläutert, die auf raum-füllenden Kurven basiert und daher besonders einfach und flexibel ist. Eine ausführliche Analyse der parallelen Performance bestätigt, dass die gesamte Implementierung perfekt skaliert und einen idealen Speed-up bis auf ein Element pro Kern für hohe Polynomgrade aufweist. Da das DG Verfahren nur mit den direkten Nachbarelementen kommunizieren muss, konnte gezeigt werden, dass die parallele Performance unabhängig davon ist, ob ein kartesisches oder voll unstrukturiertes Gitter verwendet wird. Die Ergebnisse zeigen, dass das hier vorgestellte Discontinuous Galerkin Verfahren ein großes Potential für hochaufgelöste Simulationen auf heutigen und zukünftigen Supercomputern aufweist

A space-time adaptive discontinuous Galerkin scheme

A discontinuous Galerkin scheme for unsteady fluid flows is described that allows a very high level of adaptive control in the space-time domain. The scheme is based on an explicit space-time predictor, which operates locally and takes the time evolution of the data within the grid cell into account. The predictor establishes a local space-time approximate solution in a whole space-time grid cell. This enables a time-consistent local time-stepping, by which the approximate solution is advanced in time in every grid cell with its own time step, only restricted by the local explicit stability condition. The coupling of the grid cells is solely accomplished by the corrector which is determined by the numerical fluxes. The considered discontinuous Galerldn scheme allows non-conforming meshes, together with p-adaptivity in 3 dimensions and h/p-adaptivity in 2 dimensions. Hence, we combine in this scheme all the flexibility that the discontinuous Galerkin approach provides. In this work, we investigate the combination of the local time-stepping with h- and p-adaptivity. Complex unsteady flow problems are presented to demonstrate the advantages of such an adaptive framework for simulations with strongly varying resolution requirements, e.g. shock waves, boundary layers or turbulence. (C) 2015 Elsevier Ltd. All rights reserved