ICOSAHOM 2020
International Conference on Spectral and High Order Methods
12th  16th July 2021  Vienna, Austria
Conference Agenda
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Please note that all times are shown in the time zone of the conference. The current conference time is: 30th Nov 2022, 08:35:02pm CET

Session Overview 
Session  
MS13a: Advancing Adaptive High Order Methods for Robustness
 
Session Abstract  
This minisymposium focusses on adaptive high order methods for the solution of PDEs. As computational power grows, so does the complexity of problems that can be simulated. For example, high order spectral element methods are now solving very high Reynolds number flows such as Re=O(10 5 ) flow past a wing section in Direct Numerical Simulation [1] and Re=O(10 6 ) flows in Large Eddy Simulation [2]. However, these are still being done with conforming fixed grids. There is a need to push the efficiency of adaptive high order methods in order to more effectively capture complex phenomena with fewer resources and to push the envelope of the problems that can be solved. High order adaptivity is highly unstructured which poses new challenges to high performance computing. The minisymposium will bring together researchers working on the development of adaptive high order methods with the aim to improve parallel efficiency and load balancing, while ensuring robustness for high fidelity calculations in a number of application fields. [1] Hosseini et al (2016) Int. J. Heat Fluid Flow, 61(A), 117128. [2] Vinuesa et al. (2018) Int. J. Heat Fluid Flow, 72, 8699.  
Presentations  
2:00pm  2:30pm
A padaptive Discontinuous Galerkin method for the underresolved simulation of incompressible turbulent flows ^{1}University of Bergamo, 24044 Dalmine, BG, Italy; ^{2}Polytechnic University of Marche, 60131 Ancona, Italy; ^{3}NASA Ames Research Center, Moffett Field, CA 94035, United States; ^{4}University of Brescia, 25123 Brescia, Italy Most of the industrial flow problems are characterized by a turbulent regime. The computational cost for a direct simulation (DNS) of all the spatial and temporal scales is, to date, well beyond the available computational power. To deal with such complex flows a certain level of modeling needs to be considered. The most popular approach relies on the solution of the Reynolds averaged NavierStokes (RANS) equations where all the scales of turbulence are modeled. RANS equations can solve attached turbulent flows around complex geometries but fail in some other cases, e.g., when dealing with large flow separations at high Reynolds numbers. Scaleresolving approaches like the Large Eddy Simulation (LES), where only the small scales of turbulence are modeled, are able to accurately predict massive separation but their computational cost is still too large for a routine use. In recent years, several research efforts have been focused on the reduction of the cost of highfidelity flow simulations to enable their use in industry, thus improving the design process in several engineering fields, e.g., turbomachinery. In the LES context an appealing choice is the socalled Implicit LES (ILES), where no explicit model is used and the dissipation of the numerical scheme plays the role of a subgrid scale model. Discontinuous Galerkin (DG) methods proved to be well suited for the ILES, [1], due to their highorder accuracy and good dispersion and dissipation properties. Together with a very high spatial accuracy, LES needs longterm time integration that results in large turnaround times. To reduce the computational time, we use highorder linearlyimplicit Rosenbrocktype time integration schemes that allow for much larger time steps than explicit integrators. However, when dealing with implicit schemes we run into: i) a large memory footprint; ii) a large CPU time for the operator assembly; iii) the difficulty to design effective and scalable preconditioners for the linear solver. To reduce the memory footprint and the cost of the assembly of operators we implement a spatially adaptive method that locally varies the polynomial degree of the solution (padaptation) [2]. The method is designed not to locally spoil the accuracy needed by ILES and uses a simple indicator to drive the prefinement. The estimator combines: i) a measure of the pressure jump at cell interfaces; ii) the decay rate of the modal coefficients of the polynomial expansion. To evenly distribute the computational load of the (nonuniformly) adapted spatial discretization, the solver takes advantage of a multiconstraint domain decomposition algorithm. An adaptive strategy for the degree of exactness of the quadrature rules is also implemented to further mitigate the cost of assembly when dealing with meshes with curved edges. For the linear systems solution we rely on an efficient matrixfree Flexible GMRES method coupled with a cheap and scalable pmultigrid preconditioner [3]. This technique proved to dramatically speed up the solution process when dealing with the peculiar DG discretization for the incompressible NavierStokes equations used in this work [4]. The performance and accuracy of the padaptive method and its building blocks are assessed by computing the flow around a rounded leading edge flat plate and the flow past a circular cylinder. The approach is able to deliver an almost meshindependent solution for many quantities of interest with results that are in reasonable agreement with the literature but with significant savings in terms of degrees of freedom. REFERENCES [1] Bassi, F., Botti, L., Colombo, A., Crivellini, A., Ghidoni, A., Massa, F. “On the development of an implicit highorder Discontinuous Galerkin method for DNS and implicit LES of turbulent flows”, Eur. J. Mech. B/Fluids 55, 367–379 (2016). [2] Bassi, F., Botti, L., Colombo, A., Crivellini, A., Franciolini, M., Ghidoni, A., Noventa, G. “A padaptive MatrixFree Discontinuous Galerkin Method for the Implicit LES of Incompressible Transitional Flows” Flow, Turbulence and Combustion, 105 (2), pp. 437470 (2020) [3] Franciolini, M., Botti, L., Colombo, M., Crivellini, A. “A pMultigrid matrixfree discontinuous Galerkin solution strategies for the underresolved simulation of incompressible turbulent flows”, Computers and Fluids, 206, art. no. 104558 (2020) [4] Bassi, F., Crivellini, A., Di Pietro, D.A., Rebay, S. “An implicit highorder discontinuous Galerkin method for steady and unsteady incompressible flows”, Computers and Fluids, 36 (10) pp. 15291546 (2007) 2:30pm  3:00pm
A Subcell Finite Volume ShockCapturing Method for hAdaptive Discontinuous Galerkin Discretizations of the Compressible Magnetohydrodynamics Equations ^{1}Department of Mathematics and Computer Science, University of Cologne; ^{2}German Aerospace Center (DLR); ^{3}Max Planck Institute for Plasma Physics; ^{4}Department of Mathematics, Computational Mathematics, Linköping University We present two robust entropy stable shockcapturing methods for discontinuous Galerkin spectral element (DGSEM) discretizations of the compressible magnetohydrodynamics (MHD) equations. In particular, we use the resistive GLMMHD equations, which include a divergence cleaning mechanism based on a generalized Lagrange multiplier (GLM). For the continuous entropy analysis to hold, and due to the divergencefree constraint on the magnetic field, the GLMMHD system requires the use of nonconservative terms, which need special treatment. Our first contribution is the extension of a recently developed subcell finite volume (FV) shockcapturing method [1] to systems with nonconservative terms, such as the GLMMHD equations. As the baseline scheme, we use the entropystable DGSEM for the resistive GLMMHD equations proposed by Bohm et al. [2], and continuously blend the highorder DGSEM discretization of the advective and nonconservative terms with a loworder FV scheme in an elementwise manner. The amount of FV method is adaptively chosen in each element with a troubled cell indicator to capture the shocks correctly. Our second contribution is the derivation and analysis of an additional entropy stable shockcapturing method that provides enhanced resolution by using a subcell reconstruction procedure that is carefully built to ensure entropy stability. Finally, we combine our hybrid FV/DGSEM scheme with an octtreebased hrefinement strategy that uses the parallel adaptive mesh refinement (AMR) library p4est [3]. The combination of highorder DGSEM discretizations, subcell FV shockcapturing methods, and AMR allows us to achieve enhanced accuracy and a sharper resolution of the shock profiles. 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 OrszagTang 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. [1] S Hennemann, AM RuedaRamírez, FJ Hindenlang, GJ Gassner. “A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations". JCP, 2020. [2] M Bohm, AR Winters, GJ Gassner, D Derigs, F Hindenlang, J Saur. “An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I: Theory and numerical verification”. JCP, 2018. [3] C Burstedde, LC Wilcox, Omar Ghattas. p4est: Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees. SIAM Journal on Scientific Computing, 2011. 3:00pm  3:30pm
A specificpurpose solver to approximate planar curves with superconvergent rates Barcelona Supercomputing CentreCentro Nacional de Computación BSCCNS, Spain We present a specificpurpose solver to approximate planar curves with superconvergent rates. To obtain superconvergence, we minimize a disparity measure in terms of a piecewise polynomial approximation and a reparametrization of the curve. We have numerical evidence that the disparity converges at 2p rate, p being the mesh polynomial degree. To meet these rates, we exploit the quadratic convergence of a globalized Newton's method with the help of two main ingredients. First, we employ a nonmonotone line search reducing the number of nonlinear iterations. The second ingredient is to introduce a logbarrier function preventing element inversion in the curve reparameterization. The results analyze the influence of these two ingredients. Furthermore, we demonstrate the superconvergent rates for several curves and polynomial degrees. We conclude that the solver is wellsuited to obtain superconvergent approximations to planar curves. We plan to use an analogous solver for curves and surfaces in the threedimensional space. 3:30pm  4:00pm
Spacetime adaptive meshing for tracking and capturing dynamic solution features ^{1}University of Tennessee, United States of America; ^{2}University of Illinois at UrbanaChampaign, United States of America Causal Spacetime Discontinuous Galerkin (cSDG) methods use robust adaptive meshing to construct unstructured grids on spacetime analysis domains with no global timestep constraints. Element durations in cSDG meshes are not subject to the global timestep constraints imposed by traditional timemarching schemes. While synchronous solvers perform adaptive remeshing only once per N time steps, asynchronous cSDG solvers may perform millions of adaptive operations per layer of spacetime elements to capture fastmoving wavefronts and track rapidlyevolving interfaces. We describe two applications of adaptive spacetime meshing for improving solution accuracy. In fronttracking methods, we align spacetime interelement boundaries with the trajectories of sharp solution features, such as shocks and contact discontinuities in fluid mechanics. In front capturing, we use heavy spacetime refinement along these trajectories to capture these features. When feasible, front tracking is much more efficient than front capturing. We present example applications of front capturing from fluid mechanics, solid mechanics [1], and electromagnetics [2] and discuss the advantages of hadaptive causal spacetime meshing schemes relative to nonadaptive causal meshing and to conventional time marching schemes. In the context of dynamic fracture mechanics, we demonstrate how spacetime adaptive operations resolve highgradient wave fronts and cracktip singular response, first for stationary crack tips and then for dynamic crack propagation where fractureprocesszone sizes tend to zero as cracktip speeds approach the Rayleigh wave speed. The latter problem is particularly challenging from a numerical perspective. It requires intense and rapid mesh refinement, in both space and time, within fracture process zones. Further, it is necessary to model continuous cracktip motion, as opposed to snapshots of the crack configurations at discrete time intervals, to obtain the correct cracktip solution fields. Synchronous adaptive meshing schemes based on conventional timemarching strategies struggle to meet these requirements. They are unable to refine and coarsen rapidly enough to keep up with fast cracktip motion; their synchronous structure imposes global timestep constraints based on the most refined part of the mesh that makes it difficult and expensive to resolve the finest physical space and time scales; and accurate projection of the solution from an old mesh to a newly adapted mesh that reflects a new cracktip position is problematic. We describe appropriate error indicators to drive adaptive meshing for dynamic fracture problems and demonstrate that the cSDG method’s asynchronous structure, spacetime format, and finegrained adaptive meshing procedures are able to meet these challenges. Finally, we present advanced fronttracking methods [3] capable of capturing complex dynamic fracture patterns, such as crackpath oscillation, microcracking, and crack branching. References: [1] R. Abedi, R.B. Haber, S. Thite, and J. Erickson, “An h–adaptive spacetime discontinuous Galerkin method for linearized elastodynamics”, Revue Européenne de Mécanique Numérique, special issue on adaptive analysis (ed.), 15(6):619 – 642, 2006. [2] R. Abedi, S. Mudaliar. “An Asynchronous Spacetime Discontinuous Galerkin Finite Element Method for Time Domain Electromagnetics”, Journal of Computational Physics, , 351:121144, 2017 [3] R. Abedi, R. Haber, and P. Clarke. “Effect of random defects on dynamic fracture in quasibrittle materials”, International Journal of Fracture, 208.12, 241268, 2017. 
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