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Session Overview
MS13a: Advancing Adaptive High Order Methods for Robustness
Wednesday, 14/July/2021:
2:00pm - 4:00pm

Session Chair: Catherine Mavriplis
Virtual location: Zoom 7

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), 117-128.

[2] Vinuesa et al. (2018) Int. J. Heat Fluid Flow, 72, 86-99.

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2:00pm - 2:30pm

A p-adaptive Discontinuous Galerkin method for the under-resolved simulation of incompressible turbulent flows

Alessandro Colombo1, Andrea Crivellini2, Matteo Franciolini3, Antonio Ghidoni4, Gianmaria Noventa4

1University of Bergamo, 24044 Dalmine, BG, Italy; 2Polytechnic University of Marche, 60131 Ancona, Italy; 3NASA Ames Research Center, Moffett Field, CA 94035, United States; 4University 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 Navier-Stokes (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. Scale-resolving 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 high-fidelity 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 so-called 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 high-order accuracy and good dispersion and dissipation properties. Together with a very high spatial accuracy, LES needs long-term time integration that results in large turn-around times. To reduce the computational time, we use high-order linearly-implicit Rosenbrock-type 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 (p-adaptation) [2]. The method is designed not to locally spoil the accuracy needed by ILES and uses a simple indicator to drive the p-refinement. 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 (non-uniformly) adapted spatial discretization, the solver takes advantage of a multi-constraint 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 matrix-free Flexible GMRES method coupled with a cheap and scalable p-multigrid preconditioner [3]. This technique proved to dramatically speed up the solution process when dealing with the peculiar DG discretization for the incompressible Navier-Stokes equations used in this work [4].

The performance and accuracy of the p-adaptive 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 mesh-independent 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.


[1] Bassi, F., Botti, L., Colombo, A., Crivellini, A., Ghidoni, A., Massa, F. “On the development of an implicit high-order 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 p-adaptive Matrix-Free Discontinuous Galerkin Method for the Implicit LES of Incompressible Transitional Flows” Flow, Turbulence and Combustion, 105 (2), pp. 437-470 (2020)

[3] Franciolini, M., Botti, L., Colombo, M., Crivellini, A. “A p-Multigrid matrix-free discontinuous Galerkin solution strategies for the under-resolved 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 high-order discontinuous Galerkin method for steady and unsteady incompressible flows”, Computers and Fluids, 36 (10) pp. 1529-1546 (2007)

2:30pm - 3:00pm

A Subcell Finite Volume Shock-Capturing Method for h-Adaptive Discontinuous Galerkin Discretizations of the Compressible Magnetohydrodynamics Equations

Andrés M Rueda-Ramírez1, Sebastian Hennemann2, Florian J Hindenlang3, Andrew R Winters4, Gregor J Gassner1

1Department of Mathematics and Computer Science, University of Cologne; 2German Aerospace Center (DLR); 3Max Planck Institute for Plasma Physics; 4Department of Mathematics, Computational Mathematics, Linköping University

We present two robust entropy stable shock-capturing methods for discontinuous Galerkin spectral element (DGSEM) discretizations of the compressible magneto-hydrodynamics (MHD) equations. In particular, we use the resistive GLM-MHD 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 divergence-free constraint on the magnetic field, the GLM-MHD system requires the use of non-conservative terms, which need special treatment.

Our first contribution is the extension of a recently developed subcell finite volume (FV) shock-capturing method [1] to systems with non-conservative terms, such as the GLM-MHD equations. As the baseline scheme, we use the entropy-stable DGSEM for the resistive GLM-MHD equations proposed by Bohm et al. [2], and continuously blend the high-order DGSEM discretization of the advective and non-conservative terms with a low-order FV scheme in an element-wise 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 shock-capturing 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 oct-tree-based h-refinement strategy that uses the parallel adaptive mesh refinement (AMR) library p4est [3]. The combination of high-order DGSEM discretizations, subcell FV shock-capturing 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 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.

[1] S Hennemann, AM Rueda-Ramírez, 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 specific-purpose solver to approximate planar curves with super-convergent rates

Julia Docampo-Sánchez, Eloi Ruiz-Gironés, Xevi Roca

Barcelona Supercomputing Centre-Centro Nacional de Computación BSC-CNS, Spain

We present a specific-purpose solver to approximate planar curves with super-convergent rates. To obtain super-convergence, we minimize a disparity measure in terms of a piece-wise polynomial approximation and a re-parametrization 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 non-monotone line search reducing the number of non-linear iterations. The second ingredient is to introduce a log-barrier function preventing element inversion in the curve re-parameterization. The results analyze the influence of these two ingredients. Furthermore, we demonstrate the super-convergent rates for several curves and polynomial degrees. We conclude that the solver is well-suited to obtain super-convergent approximations to planar curves. We plan to use an analogous solver for curves and surfaces in the three-dimensional space.

3:30pm - 4:00pm

Spacetime adaptive meshing for tracking and capturing dynamic solution features

Reza Abedi1, Robert Haber2

1University of Tennessee, United States of America; 2University of Illinois at Urbana-Champaign, 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 time-step constraints. Element durations in cSDG meshes are not subject to the global time-step constraints imposed by traditional time-marching 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 fast-moving wavefronts and track rapidly-evolving interfaces.

We describe two applications of adaptive spacetime meshing for improving solution accuracy. In front-tracking methods, we align spacetime inter-element 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 h-adaptive causal spacetime meshing schemes relative to non-adaptive causal meshing and to conventional time marching schemes.

In the context of dynamic fracture mechanics, we demonstrate how spacetime adaptive operations resolve high-gradient wave fronts and crack-tip singular response, first for stationary crack tips and then for dynamic crack propagation where fracture-process-zone sizes tend to zero as crack-tip 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 crack-tip motion, as opposed to snapshots of the crack configurations at discrete time intervals, to obtain the correct crack-tip solution fields. Synchronous adaptive meshing schemes based on conventional time-marching strategies struggle to meet these requirements. They are unable to refine and coarsen rapidly enough to keep up with fast crack-tip motion; their synchronous structure imposes global time-step 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 crack-tip 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 fine-grained adaptive meshing procedures are able to meet these challenges. Finally, we present advanced front-tracking methods [3] capable of capturing complex dynamic fracture patterns, such as crack-path oscillation, microcracking, and crack branching.


[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, , 351:121-144, 2017

[3] R. Abedi, R. Haber, and P. Clarke. “Effect of random defects on dynamic fracture in quasi-brittle materials”, International Journal of Fracture, 208.1-2, 241-268, 2017.

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