International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
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Please note that all times are shown in the time zone of the conference. The current conference time is: 1st Dec 2022, 08:14:24am CET
Z5: adaptivity, meshing
4:00pm - 4:20pm
Comparison of implicit and explicit p-multigrid strategies for compressible Navier-Stokes discontinuous Galerkin solvers
1ETSIAE-UPM (School of Aeronautics - Universidad Politécnica de Madrid); 2Center for Computational Simulation - Universidad Politécnica de Madrid
In the last decades, multigrid methods have become an important component of the convergence acceleration techniques for viscous compressible flows. In the context of the high-order methods, p-multigrid is a natural choice since the coarse levels are obtained just by reducing the approximation order. We analyze the computational cost and memory footprint of several popular relaxation strategies for multigrid: explicit Runge-Kutta methods with local time stepping, GMRES and different element-block implicit smoothers (Block-Jacobi, Gauss-Seidel) on representative test cases including two-dimensional and three-dimensional flow simulations. Moreover, we compare the performance of the smoothers in the context of standard DG discretization, where the Gauss nodal points are selected and the entropy stable DG scheme with Gauss-Lobatto points employed. The former is of a main interest, as it has enhanced stability and is convenient for under-resolved simulations.
4:20pm - 4:40pm
A Direct Method of Generating High-order Tetrahedral Meshes Using an Advancing Front Approach
University of Kansas, United States of America
The use of high-order methods has attracted the interest of the scientific computing community, thanks to the ability to deliver highly accurate results at a low computational cost while solving partial differential equations (PDEs). However, while working with curved boundaries, meshes used with high-order PDE solvers need to be high-order so as to accurately capture the curvature of the geometric domain. A high-order mesh is composed of both straight-sided and curved elements depending on the curvature of the geometric domain. To date, there are a limited number of algorithms that can generate high-order meshes in a robust manner, i.e., without tangling the mesh.
In this talk, we propose a novel method for generating high-order curvilinear tetrahedral meshes using an advancing front approach . There are two categories of high-order mesh generation methods: a posteriori methods and direct methods. A posteriori methods are most commonly used; such methods deform an enriched linear mesh of the geometry into a high-order, curvilinear mesh. Since the mesh curving process can create tangled mesh elements, such methods often require a post-processing step, such as mesh untangling.
Our method uses a direct approach to generate meshes on geometries with curved boundaries. It can generate meshes based on various types of boundary representations, e.g., from computer-aided design (CAD) files and patient-specific boundary meshes obtained from medical images. Since our method uses a direct approach instead of an a posteriori approach, it can generate high-quality meshes of models obtained from medical images where no CAD representation is available. Our advancing front method generates high quality tetrahedral mesh elements on each iteration, thus omitting the need for post-processing steps.
We present several numerical examples of second-order tetrahedral meshes generated using our method based on input triangular surface meshes. Our results show that our high-order tetrahedral meshes are of high-quality according to the scaled Jacobian and equiangular skewness metrics. Examples are drawn from mechanical engineering and biomedical engineering.
We will conclude the talk by discussing the possibility of extending the current algorithm to generate anisotropic meshes in 3D.
 F. Mohammadi and S.M. Shontz, A direct method of generating high-order tetrahedral meshes using an advancing front approach, Proceedings of the 29th International Meshing Roundtable, Under review, March 2021.
4:40pm - 5:00pm
A second-order optimizer for parallel distributed mesh curving on virtual geometry
Barcelona Supercomputing Center - BSC, Spain
We present a mesh curving method for virtual geometry ready for parallel distributed computing. Our technique generates geometrically accurate large-scale meshes composed of high-quality elements on complex geometry. To this end, we advocate for a penalty-based second-order optimizer that uses global tight tolerances to converge the distortion residuals.
To curve larger meshes without further increasing the computational resources, we aim to reduce the memory footprint, waiting time, and energy consumption. To this end, we consider four main ingredients. First, we propose a degree continuation technique that reduces the number of linear iterations performed on higher polynomial degrees. Second, we adaptively update the penalty parameter to reduce the number of non-linear problems to solve. Third, we introduce a matrix-free GMRES solver pre-conditioned with a block-based successive over-relaxation method. This preconditioner reduces the memory footprint three times. Fourth, we consider an indicator to set a dynamic tolerance for the linear solver and thus, reduce the number of iterations.
Compared with our previous solver, we reduce the memory footprint, waiting time, and energy consumption. Using tight tolerances, we curve meshes composed of millions of quartic elements with high stretching for complex virtual geometries using thousands of processors. These capabilities are critical for high-fidelity simulations on complex domains using unstructured high-order methods.
5:00pm - 5:20pm
Sparse spectral methods for curvilinear domains in the Dedalus code
1Massachusetts Institute of Technology; 2University of Sydney; 3Northwestern University; 4Bates College; 5University of Colorado Boulder
Dedalus is an open-source Python framework for solving general partial differential equations at scale with modern spectral methods. Here we will describe recent additions to the code which support general tensor fields in curvilinear domains. This capability is based on recently developed polynomial bases that possess sparse operators for arbitrary tensor calculus in full disks and balls. We will detail our interface for the coordinate-free entry of systems of PDEs including prognostic equations and algebraic constraints. New fast direct solvers allow the inclusion of general boundary conditions, including nonlocal conditions, via generalized tau corrections that can also be specified symbolically. We will also describe ongoing extensions of this framework to support high-order spectral elements and direct couplings with other PDE and BIE solvers.
5:20pm - 5:40pm
A P-adaptive discontinuous Galerkin incompressible Navier-Stokes/Cahn-Hilliard solver for multiphase flows
1ETSIAE-UPM - School of Aeronautics, Universidad Politécnica de Madrid, Spain; 2Center for Computational Simulation, Universidad Politécnica de Madrid, Spain
In this work we present an extension of the works presented in  and  for the simulation of two phase immiscible fluid flows on p–non–conforming meshes. The solver consists of an incompressible Navier-Stokes/Cahn-Hilliard system of equations which uses the artificial compressibility method to enforce a divergence–free velocity field. The system is approximated with a high–order discontinuous Galerkin spectral element method. The inter-element coupling with different polynomial orders is treated through the standard mortar method  in such a way that the new scheme mimics the entropy analysis of the continuous system. Therefore, the entropy remains bounded even for p–non–conforming meshes. A novelty of this work is that the p–adaptation method is applied as a means of reducing the computational cost of a multiphase simulation and we suggest it as an alternative or compliment to h–adaptation, which has been thoroughly studied. Since multiphase flows are typically of unsteady nature, we introduce a methodology to perform dynamic p–adaptation based on the location of the interface between the two phases. The aim is to retain the accuracy of the uniform solver while reducing the associated cost. The validation and assessment of the proposed scheme is done though numerical experiments such as that of a rising bubble or a dam–break, which show that the computational cost can be reduced by 40% to 50% compared to a uniform solver for the same level of accuracy.
 Juan Manzanero, Gonzalo Rubio, David A Kopriva, Esteban Ferrer, and Eusebio Valero. Entropy–stable discontinuous Galerkin approximation with summation–by–parts property for the incompressible Navier–Stokes/Cahn–hilliard system. Journal of Computational Physics, 408:109363, 2020.
 Gerasimos Ntoukas, Juan Manzanero, Gonzalo Rubio, Eusebio Valero, and Esteban Ferrer. A free-energy stable p-adaptive nodal discontinuous Galerkin for the Cahn-Hilliard equation. Submitted toJournal of Computational Physics, 2020.
 David A Kopriva, Stephen L Woodruff, and M Yousuff Hussaini. Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method. International journal for numerical methods in engineering, 53(1):105–122, 2002
5:40pm - 6:00pm
A functional oriented truncation error adaptation method
Mesh adaptation allows to localize the numerical degrees of freedom in mesh regions that require increased accuracy. This is particularly important in Computational Fluid Dynamics (CFD) where millions of degrees of freedom are required. Adaptation can be performed by moving the nodes (r-adaptation) subdividing or merging elements (h-adaptation) or by enriching or reducing the polynomial order (p-adaptation).
Mesh adaptation algorithms require a sensor to identify the flow regions that need refinement or coarsening. Three classic approaches can be found in the literature: feature-based, local error-based and adjoint-based . Feature-based adaptation refines where high solution gradients are found. This low-cost approach does not have an explicit connection between the sensor and the error in the solution, and thus often results inefficiently adapted meshes. Adjoint-based adaptations define a functional target (e.g., energy, drag or lift) to generate an optimal mesh to reduce the error associated to the selected functional. This approach is efficient in reducing the selected functional error but cannot ensure the reduction of other functionals. Additionally, adjoint-based adaption has a high cost since an adjoint problem needs to be solved. Local error-based include truncation error adaptations, which aims at equi-distributing the local error in the mesh domain. For hyperbolic problems, truncation error-based sensors localize the elements where the discretization error is generated such that the degrees of freedom can be efficiently located at the source, and not wasted in elements polluted by the advected error. The computation of this sensor can be done using the tau-estimation approach, which does not require the solution of additional systems and is cheap (see [2-4] for the application of the tau-estimation approach to discontinuous Galerkin (DG) and spectral element method). The advantages of truncation errors have encouraged the development of adaptation algorithms. In particular, p-adaptation methods have been developed for DG techniques [5-7].
However, truncation error adaptation techniques have a substantial drawback: they are not related to functional errors. Therefore, the user needs to find arbitrary truncation thresholds to improve the mesh. We propose here to solve this inconvenience by linking the truncation error and the functional error, which enables a cheap truncation error adaptation with the advantages of adjoints, i.e. targeting a specific functional error.
 C. Roy, Strategies for driving mesh adaptation in CFD (invited), in: 47th AIAA aerospace sciences meeting including the new horizonsforum and aerospace exposition, 2009, p. 1302
 G. Rubio, F. Fraysse, J. de Vicente, E. Valero, The estimation of truncation error by tau-estimation for Chebyshev spectral collocation method, Journal of Scientific Computing 57 (2013) 146–173
 G. Rubio, F. Fraysse, D. A. Kopriva, E. Valero, Quasi-a priori truncation error estimation in the DGSEM, Journal of Scientific Computing (2014) 1–31
 A. M. Rueda-Ramírez, G. Rubio, E. Ferrer, E. Valero, Truncation error estimation in the p-anisotropic discontinuous Galerkin spectral element method, Journal of Scientific Computing 78 (2019) 433–466
 M. Kompenhans, G. Rubio, E. Ferrer, and E. Valero, Adaptation strategies for high order discontinuous Galerkin methods based on tau-estimation, Journal of Computational Physics 306 (2016) 216 – 236
 M. Kompenhans, G. Rubio, E. Ferrer, E. Valero, Comparisons of p-adaptation strategies based on truncation- and discretisation-errors for high order discontinuous Galerkin methods, Computers and Fluids 139 (2016) 36–46
 A. M. Rueda-Ramírez, J. Manzanero, E. Ferrer, G. Rubio, E. Valero, A p-multigrid strategy with anisotropic p-adaptation based on truncation errors for high-order discontinuous Galerkin methods, Journal of Computational Physics 378 (2019) 209–233
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