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Towards adaptive mesh refinement in Nek5000
KTH, School of Engineering Sciences (SCI), Mechanics. (Linné flow centre)ORCID iD: 0000-0002-1724-0188
2017 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

The development of adaptive mesh refinement capabilities in the field of computational fluid dynamics is an essential tool for enabling the simulation of larger and more complex physical problems. While such techniques have been known for a long time, most simulations do not make use of them because of the lack of a robust implementation. In this work, we present recent progresses that have been made to develop adaptive mesh refinement features in Nek5000, a code based on the spectral element method. These developments are driven by the algorithmic challenges posed by future exascale supercomputers. First, we perform the study of the strong scaling of Nek5000 on three petascale machines in order to assess the scalability of the code and identify the current bottlenecks. It is found that strong scaling limit ranges between 5, 000 and 220, 000 degrees of freedom per core depending on the machine and the case. The need for synchronized and low latency communication for efficient computational fluid dynamics simulation is also confirmed. Additionally, we present how Hypre, a library for linear algebra, is used to develop a new and efficient code for performing the setup step required prior to the use of an algebraic multigrid solver for preconditioning the pressure equation in Nek5000. Finally, the main objective of this work is to develop new methods for estimating the error on a numerical solution of the Navier–Stokes equations via the resolution of an adjoint problem. These new estimators are compared to existing ones, which are based on the decay of the spectral coefficients. Then, the estimators are combined with newly implemented capabilities in Nek5000 for automatic grid refinement and adaptive mesh adaptation is carried out. The applications considered so far are steady and two-dimensional, namely the lid-driven cavity at Re = 7, 500 and the flow past a cylinder at Re = 40. The use of adaptive mesh refinement techniques makes mesh generation easier and it is shown that a similar accuracy as with a static mesh can be reached with a significant reduction in the number of degrees of freedom.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2017. , 140 p.
Series
TRITA-MEK, ISSN 0348-467X ; 2017:18
Keyword [en]
Error estimators, mesh refinement, adaptivity, spectral element method, algebraic multigrid method
National Category
Fluid Mechanics and Acoustics
Research subject
Engineering Mechanics
Identifiers
URN: urn:nbn:se:kth:diva-217501ISBN: 978-91-7729-616-4 OAI: oai:DiVA.org:kth-217501DiVA: diva2:1156611
Presentation
2017-12-11, E3, Kungliga Tekniska Högskolan, Osquars Backe 14, Stockholm, 10:15 (English)
Opponent
Supervisors
Funder
Knut and Alice Wallenberg FoundationSwedish National Infrastructure for Computing (SNIC)EU, Horizon 2020, 671571
Note

QC 20171114

Available from: 2017-11-14 Created: 2017-11-13 Last updated: 2017-11-14Bibliographically approved
List of papers
1. On the strong scaling of the spectral element solver Nek5000 on petascale systems
Open this publication in new window or tab >>On the strong scaling of the spectral element solver Nek5000 on petascale systems
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2016 (English)In: Proceedings of the 2016 Exascale Applications and Software Conference (EASC2016): April 25-29 2016, Stockholm, Sweden, Association for Computing Machinery (ACM), 2016, a5Conference paper, Published paper (Refereed)
Abstract [en]

The present work is targeted at performing a strong scaling study of the high-order spectral element uid dynamics solver Nek5000. Prior studies such as [5] indicated a recommendable metric for strong scalability from a theoretical viewpoint, which we test here extensively on three parallel machines with different performance characteristics and interconnect networks, namely Mira (IBM Blue Gene/Q), Beskow (Cray XC40) and Titan (Cray XK7). The test cases considered for the simulations correspond to a turbulent ow in a straight pipe at four different friction Reynolds numbers Reτ = 180, 360, 550 and 1000. Considering the linear model for parallel communication we quantify the machine characteristics in order to better assess the scaling behaviors of the code. Subsequently sampling and profiling tools are used to measure the computation and communication times over a large range of compute cores. We also study the effect of the two coarse grid solvers XXT and AMG on the computational time. Super-linear scaling due to a reduction in cache misses is observed on each computer. The strong scaling limit is attained for roughly 5000 - 10; 000 degrees of freedom per core on Mira, 30; 000 - 50; 0000 on Beskow, with only a small impact of the problem size for both machines, and ranges between 10; 000 and 220; 000 depending on the problem size on Titan. This work aims at being a reference for Nek5000 users and also serves as a basis for potential issues to address as the community heads towards exascale supercomputers.

Place, publisher, year, edition, pages
Association for Computing Machinery (ACM), 2016
Series
ACM International Conference Proceeding Series
Keyword
Benchmarking, Computational fluid dynamics, Nek5000, Scaling, Degrees of freedom (mechanics), Reynolds number, Supercomputers, Computational time, Interconnect networks, Parallel communication, Parallel machine, Performance characteristics, Spectral element, Application programs
National Category
Mechanical Engineering
Identifiers
urn:nbn:se:kth:diva-207506 (URN)10.1145/2938615.2938617 (DOI)2-s2.0-85014776002 (Scopus ID)9781450341226 (ISBN)
Conference
2016 Exascale Applications and Software Conference, EASC 2016, Stockholm, Sweden, 25 April 2016 through 29 April 2016
Funder
Swedish e‐Science Research Center
Note

QC 20170814

Available from: 2017-06-07 Created: 2017-06-07 Last updated: 2017-11-13Bibliographically approved
2. Towards adaptive mesh refinement for the spectral element solver Nek5000
Open this publication in new window or tab >>Towards adaptive mesh refinement for the spectral element solver Nek5000
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2017 (English)Report (Refereed)
Abstract [en]

Hypre, a library for linear algebra, is used to replace a Matlab code for performing the setup step of an Algebraic Multigrid Method (AMG). The AMG method is used to compute part of the preconditioner in Nek5000, a code for Computational Fluid Dynamics based on the spectral element method. However, the solution of the AMG problem is not performed via Hypre but by Nek5000’s internal solver. The new AMG setup is shown to be faster by at least one order of magnitude, while it does not significantly impact the efficiency of the AMG solver, as is shown from its application to relevant test cases.

Publisher
6 p.
National Category
Fluid Mechanics and Acoustics
Research subject
Engineering Mechanics
Identifiers
urn:nbn:se:kth:diva-217497 (URN)
Available from: 2017-11-13 Created: 2017-11-13 Last updated: 2017-11-13
3. Adjoint error estimators and adaptive mesh refinement in Nek5000
Open this publication in new window or tab >>Adjoint error estimators and adaptive mesh refinement in Nek5000
2017 (English)Report (Other academic)
Abstract [en]

The development of adaptive mesh refinement capabilities in the field of computational fluid dynamics is an essential tool for enabling the simulation of larger and more complex physical problems. In this report, we describe recent developments that have been made to enable adaptive mesh refinement in Nek5000 and we validate the method on simple, two-dimensional, steady test cases.We start by describing the modifications brought to Nek5000 that enable the presence of hanging nodes in the mesh. Thanks to this new feature, we can use the h-refinement technique for mesh adaptation, where selected elements are split via quadtree (2D) or octree (3D) structures. Then, two methods are considered to estimate and control the error. The first method is local and based on the spectral properties of the solution on each element. The second method is goal-oriented and takes into account both the local properties of the solution and the global dependence of the error in the solution via the resolution of an adjoint problem. Finally, the use of automatic mesh refinement is demonstrated in Nek5000 on two test cases: the lid-driven cavity at Re = 7, 500 and the flow past a cylinder at Re = 40. Both error estimation methods are compared andare shown to efficiently reduce the number of degrees of freedom required to reach a given tolerance on the solution compared to conforming refinement. Moreover, the gains in terms of mesh generation, accuracy and computational cost are discussed by analysing the convergence of some functional of interest and the evolution of the mesh as refinement proceeds.

Keyword
Error estimators; mesh refinement; adaptivity; spectral element method
National Category
Fluid Mechanics and Acoustics
Research subject
Engineering Mechanics
Identifiers
urn:nbn:se:kth:diva-217498 (URN)
Funder
Knut and Alice Wallenberg Foundation
Available from: 2017-11-13 Created: 2017-11-13 Last updated: 2017-11-13

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