Dealiasing techniques for high-order spectral element methods on regular and irregular grids

High-order methods are becoming increasingly attractive in both academia and industry, especially in the context of computational fluid dynamics. However, before they can be more widely adopted, issues such as lack of robustness in terms of numerical stability need to be addressed, particularly when treating industrial-type problems where challenging geometries and a wide range of physical scales, typically due to high Reynolds numbers, need to be taken into account. One source of instability is aliasing effects which arise from the nonlinearity of the underlying problem. In this work we detail two dealiasing strategies based on the concept of consistent integration. The first uses a localised approach, which is useful when the nonlinearities only arise in parts of the problem. The second is based on the more traditional approach of using a higher quadrature. The main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods, thereby reducing aliasing-driven instabilities. We demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations, where, in the latter, both volumetric and interface approximations must be considered. We show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations

Interpolation error bounds for curvilinear finite elements and their implications on adaptive mesh refinement

This is the author accepted manuscript. The final version is available from Springer via the DOI in this record.Mesh generation and adaptive renement are largely driven by the objective of minimizing the bounds on the interpolation error of the solution of the partial di erential equation (PDE) being solved. Thus, the characterization and analysis of interpolation error bounds for curved, high-order nite elements is often desired to e ciently obtain the solution of PDEs when using the nite element method (FEM). Although the order of convergence of the projection error in L2 is known for both straight-sided and curved-elements [1], an L1 estimate as used when studying interpolation errors is not available. Using a Taylor series expansion approach, we derive an interpolation error bound for both straight-sided and curved, high-order elements. The availability of this bound facilitates better node placement for minimizing interpolation error compared to the traditional approach of minimizing the Lebesgue constant as a proxy for interpolation error. This is useful for adaptation of the mesh in regions where increased resolution is needed and where the geometric curvature of the elements is high, e.g, boundary layer meshes. Our numerical experiments indicate that the error bounds derived using our technique are asymptotically similar to the actual error, i.e., if our interpolation error bound for an element is larger than it is for other elements, the actual error is also larger than it is for other elements. This type of bound not only provides an indicator for which curved elements to re ne but also suggests whether one should use traditional h-re nement or should modify the mapping function used to de ne elemental curvature. We have validated our bounds through a series of numerical experiments on both straight-sided and curved elements, and we report a summary of these results.The first author acknowledges support from the EU Horizon 2020 project ExaFLOW( grant 671571) and the PRISM project under EPSRC grant EP/L000407/1. The work of the second author was supported in part by the NIH/NIGMS Center for Integrative Biomedical Computing grant 2P41 RR0112553-12 and DOE NET DE-EE0004449 grant. The work of the third author was supported in part by the DOE NET DE-EE0004449 grant and ARO W911NF1210375 (Program Manager: Dr. Mike Coyle) grant

Optimising the performance of the spectral/hp element method with collective linear algebra operations

This is the final version of the article. Available from Elsevier via the DOI in this record.As computing hardware evolves, increasing core counts mean that memory bandwidth is becoming the deciding factor in attaining peak performance of numerical methods. High-order finite element methods, such as those implemented in the spectral/hp framework Nektar++, are particularly well-suited to this environment. Unlike low-order methods that typically utilise sparse storage, matrices representing high-order operators have greater density and richer structure. In this paper, we show how these qualities can be exploited to increase runtime performance on nodes that comprise a typical high-performance computing system, by amalgamating the action of key operators on multiple elements into a single, memory-efficient block. We investigate different strategies for achieving optimal performance across a range of polynomial orders and element types. As these strategies all depend on external factors such as BLAS implementation and the geometry of interest, we present a technique for automatically selecting the most efficient strategy at runtime.We thank D. Ekelschot and M. Turner for their assistance in generating the mesh and parameters for the simulation of Section 6. We also thank F. Witherden for initial discussions motivating this study. This work was funded in part by support from the libHPC II EPSRC project under grant EP/K038788/1. DM additionally acknowledges support under the Laminar Flow Control Centre funded by Airbus/EADS and EPSRC under grant EP/I037946. SJS acknowledges Royal Academy of Engineering support under their research chair scheme. We thank the Imperial College High Performance Computing Service for computing time used to calculate the results seen in Section 6. We additionally acknowledge access to ARCHER with support from the UK Turbulence Consortium under EPSRC grant EP/L000261/1

An isoparametric approach to high-order curvilinear boundary-layer meshing

This is the final version of the article. Available from Elsevier via the DOI in this record.The generation of high-order curvilinear meshes for complex three-dimensional geometries is presently a challenging topic, particularly for meshes used in simulations at high Reynolds numbers where a thin boundary layer exists near walls and elements are highly stretched in the direction normal to flow. In this paper, we present a conceptually simple but very effective and modular method to address this issue. We propose an isoparametric approach, whereby a mesh containing a valid coarse discretization comprising of high-order triangular prisms near walls is refined to obtain a finer prismatic or tetrahedral boundary-layer mesh. The validity of the prismatic mesh provides a suitable mapping that allows one to obtain very fine mesh resolutions across the thickness of the boundary layer. We describe the method in detail for a high-order approximation using modal basis functions, discuss the requirements for the splitting method to produce valid prismatic and tetrahedral meshes and provide a sufficient criterion of validity in both cases. By considering two complex aeronautical configurations, we demonstrate how highly stretched meshes with sufficient resolution within the laminar sublayer can be generated to enable the simulation of flows with Reynolds numbers of 106 and above.This work was partly supported by EU Grant No. 265780 as part of the EU FP7 project “IDIHOM: Industrialization of High-Order Methods — A Top-Down Approach”. We would like to thank Dr. Tobias Leicht of DLR for asking a very pertinent question concerning the validity of the generated high-order mesh that we believe to have answered in this article. We also thank Jean-Eloi Lombard for his assistance in generating the mesh for Fig. 15

A Thermo-elastic Analogy for High-order Curvilinear Meshing with Control of Mesh Validity and Quality

This is the final version of the article. Available from Elsevier via the DOI in this record.In recent years, techniques for the generation of high-order curvilinear mesh have frequently adopted mesh deformation procedures to project the curvature of the surface onto the mesh, thereby introducing curvature into the interior of the domain and lessening the occurrence of self-intersecting elements. In this article, we propose an extension of this approach whereby thermal stress terms are incorporated into the state equation to provide control on the validity and quality of the mesh, thereby adding an extra degree of robustness which is lacking in current approaches

Automatic generation of 3D unstructured high-order curvilinear meshes

This is the final version of the article. Available from the publisher via the DOI in this record.The generation of suitable, good quality high-order meshes is a significant obstacle in the academic and industrial uptake of high-order CFD methods. These methods have a number of favourable characteristics such as low dispersion and dissipation and higher levels of numerical accuracy than their low-order counterparts, however the methods are highly susceptible to inaccuracies caused by low quality meshes. These meshes require significant curvature to accuratly describe the geometric surfaces, which presents a number of difficult challenges in their generation. As yet, research into the field has produced a number of interesting technologies that go some way towards achieving this goal, but are yet to provide a complete system that can systematically produce curved high-order meshes for arbitrary geometries for CFD analysis. This paper presents our efforts in that direction and introduces an open-source high-order mesh generator, NekMesh, which has been created to bring high-order meshing technologies into one coherent pipeline which aims to produce 3D high-order curvilinear meshes from CAD geometries in a robust and systematic way

Dealiasing techniques for high-order spectral element methods on regular and irregular grids

This is the final version of the article. Available from Elsevier via the DOI in this record.High-order methods are becoming increasingly attractive in both academia and industry, especially in the context of computational fluid dynamics. However, before they can be more widely adopted, issues such as lack of robustness in terms of numerical stability need to be addressed, particularly when treating industrial-type problems where challenging geometries and a wide range of physical scales, typically due to high Reynolds numbers, need to be taken into account. One source of instability is aliasing effects which arise from the nonlinearity of the underlying problem. In this work we detail two dealiasing strategies based on the concept of consistent integration. The first uses a localised approach, which is useful when the nonlinearities only arise in parts of the problem. The second is based on the more traditional approach of using a higher quadrature. The main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods, thereby reducing aliasing-driven instabilities. We demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations, where, in the latter, both volumetric and interface approximations must be considered. We show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations.This work was supported by the Laminar Flow Control Centre funded by Airbus/EADS and EPSRC under grant EP/I037946. We thank Dr. Colin Cotter for helpful discussions and Jean-Eloi Lombard for his assistance in the generation of results and figures for the NACA 0012 simulation. PV acknowledges the Engineering and Physical Sciences Research Council for their support via an Early Career Fellowship (EP/K027379/1). SJS additionally acknowledges Royal Academy of Engineering support under their research chair scheme. Data supporting this publication can be obtained on request from [email protected]

A Thermo-elastic Analogy for High-order Curvilinear Meshing with Control of Mesh Validity and Quality

AbstractIn recent years, techniques for the generation of high-order curvilinear mesh have frequently adopted mesh deformation procedures to project the curvature of the surface onto the mesh, thereby introducing curvature into the interior of the domain and lessening the occurrence of self-intersecting elements. In this article, we propose an extension of this approach whereby thermal stress terms are incorporated into the state equation to provide control on the validity and quality of the mesh, thereby adding an extra degree of robustness which is lacking in current approaches

Efficient vectorised Cuda kernels for high-order finite element flow solvers

In this work, we develop efficient kernels for elemental operators of matrix-free solvers of the Helmholtz equation, which are the core operations for more complete Navier-Stokes solvers. We consider straight-sided and deformed quadrilateral elements from unstructured high-order meshes. We investigate two types of efficient CUDA kernels for a range of polynomial orders; a first type which maps each elemental operation to a CUDA-thread, and a second that maps each element to a CUDA-block. Our results show that the first option is beneficial for small elements with low polynomial order, whereas the second option is beneficial for larger elements. For both options we show the importance of the right layout of data structures, and analyse the effect of utilising different memory spaces on the GPU