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Session Overview
MS13b: Advancing Adaptive High Order Methods for Robustness
Wednesday, 14/July/2021:
4:10pm - 6:10pm

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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4:10pm - 4:40pm

Spectral element simulations of turbulent flows with adaptive mesh refinement

Daniele Massaro, Nicholas Offermans, Adam Peplinski, Philipp Schlatter

KTH Royal Institute of Technology, Sweden

In this presentation, we review some of our latest results on adaptive mesh refinement implemented in the open-source spectral-element code Nek5000. This code is known for its excellent parallel scaling, and the flexibility of the solver. Motivated by large-scale simulations of turbulence in the fields of aeronautics, where typically the expected flow solution is only partically known, adaptive methods provide a natural way of increasing the resolution as the simulation progresses, and thus give high-fidelity results at a reduce grid count. There are however a number of challenges involved, including implementation, algorithmic adaptations and the choice of adequate error estimators and indicators that drive the adaptivity.

In the present talk, we discuss our extension of Nek5000 to non-conformal meshes coupled to adaptivity. Our approach is based on the library p4est to handle the mesh, and we use the parallel recursive spectral bisection (ParRSB) and the standard library parmetis for the partitioning. A few subtle adjustments to the internals of the spectral-element method had to be made, including an adapted definition of the preconditioner. The most important choice relates to the way the adaptivity is driven: We will show results for two classes of error quantifiers: an error indicator based on the spectral representation of the solution in each element and an error estimator based on the adjoint equation. The latter one is in principle goal-oriented, but suffers for growing adjoint solutions for longer time horizons.

Our results are discussed on three flow cases: The steady and unsteady flow around in cylinder, a turbulent pipe flow and the turbulent flow around a stepped cylinder. The cylinder case is used to illustrate some of the basic choices in our appraoch, including the various weights of the error functional. The turbulent pipe flow will then extend the results to fully turbulent flows, and discuss the requirements for the mesh and its implications on the flow solution. In particular, the appearance of wiggles in higher-order statistics is addressed. Finally, we discuss the adaptive solutions in the case of a stepped cylinder, where the solutions and thus the required solution is largely unknown.

4:40pm - 5:10pm

Parallel hp-Adaptive Discontinuous Galerkin Methods for the Incompressible Navier-Stokes Equations

Catherine Mavriplis1, Gbemeho Agbaglah2, Noel Chalmers3, Marcin Chrust4, Shiqi He1

1University of Ottawa, Canada; 2Wayne State University, USA; 3AMD Research, USA; 4European Centre for Medium Range Weather Forecasts

We present a parallel hp-adaptive high order discontinuous Galerkin spectral element method for the incompressible Navier-Stokes equations. Nonlinear convective terms are treated with local Lax-Friedrichs fluxes, while divergence and gradient operators use central fluxes. An interior penalty discontinuous Galerkin method to ensure stability and invertibility of the diffusion terms. An implicit-explicit Runge-Kutta time discretization method is paired with high-order splitting. p-Multigrid and pressure projection techniques are used to precondition the conjugate gradient linear solvers. We implement hp adaptive mesh refinement (AMR) using an inexpensive a posteriori error estimator to determine where and how to refine the grid. Numerical tests demonstrate the efficacy of the method, in reducing the number of degrees of freedom and allocating computing resources to regions of sharp variation in transient incompressible Navier-Stokes flows. The method scales well for increasing numbers of processors for fixed grids and adaptive grids with small load imbalances. However, a more robust load balancing algorithm is needed.

Parallel AMR poses significant challenges both in encoding and storing a distributed mesh and in equipartitioning the workload among processes. A hash table data structure is proposed to supplant the conventional tree-based data structure for parallel AMR. A space-filling Hilbert curve reduces the multi-dimensional workload partitioning problem to a 1D problem. The Hilbert curve is known for its natural compactness, which insures locality. We combine the fast Hilbert curve generation algorithm with 1D partitioning heuristics to develop a low-memory usage and good spatial properties partitioning algorithm. The algorithm is implemented and tested on an acoustic wave equation with a high-order quadrilateral discontinuous Galerkin spectral element method. Memory usage is kept low while parallel efficiency remains high as a result of efficient load balancing. Strong and weak scaling tests are presented, showing excellent parallel performance on cases of over 1 million elements (over 154 million degrees of freedom) on up to 16,384 processors.

5:10pm - 5:40pm

Adaptive mesh refinement for atmosphere and ocean in the Galerkin Numerical Modeling Environment (GNuME)

Michal A Kopera

Boise State University, United States of America

Galerkin Numerical Modeling Environment (GNuME) is an arbitrary order element-based continuous/discontinuous Galerkin method environment for solving various systems of nonlinear partial differential equations. It contains solvers for three-dimensional compressible Euler, incompressible Navier-Stokes, and shallow water equations on a plane and a sphere. Work is currently underway to develop a prototype for a next-generation hydrostatic ocean model.

One of the GNuME framework features is the ability to dynamically adapt the quadrilateral (2D) and hexahedral (3D) mesh. We achieve that by using the quadtree approach used in the p4est package, and careful construction of the non-conforming grid operators. In this talk, we will discuss the accuracy and conservation properties of the AMR methods in GNuMe.

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