119 results found
Moxey D, Cantwell CD, Bao Y, et al., 2019, Nektar++: enhancing the capability and application of high-fidelity spectral/hp element methods
Nektar++ is an open-source framework that provides a flexible, performant and scalable platform for the development of solvers for partial differential equations using the high-order spectral/hp element method. In particular, Nektar++ aims to overcome the complex implementation challenges that are often associated with high-order methods, thereby allowing them to be more readily used in a wide range of application areas. In this paper, we present the algorithmic, implementation and application developments associated with our Nektar++ version 5.0 release. We describe some of the key software and performance developments, including our strategies on parallel I/O, on in-situ processing, the use of collective operations for exploiting current andemerging hardware, and interfaces to enable multi-solver coupling. Furthermore, we provide details on a newly developed Python interface that enable more rapid on-boarding of new users unfamiliar with spectral/$hp$ element methods, C++ and/or Nektar++. This release also incorporates a number of numerical method developments - in particular: the method of moving frames, which provides an additional approach for the simulation of equations on embedded curvilinear manifolds and domains; a means of handling spatially variable polynomial order; and a novel technique for quasi-3D simulations to permit spatially-varying perturbations to the geometry in the homogeneous direction. Finally, we demonstrate the new application-level features provided in this release, namely: a facility for generating high-order curvilinear meshes called NekMesh; a novel new AcousticSolver for aeroacoustic problems; our development of a 'thick' strip model for the modelling of fluid-structure interaction problems in the context of vortex-induced vibrations. We conclude by commenting some directions for future code development and expansion.
Green MD, Vacondio R, Peiro J, 2019, A smoothed particle hydrodynamics numerical scheme with a consistent diffusion term for the continuity equation, COMPUTERS & FLUIDS, Vol: 179, Pages: 632-644, ISSN: 0045-7930
Marcon J, Kopriva DA, Sherwin SJ, et al., 2019, A high resolution pde approach to quadrilateral mesh generation, Publisher: arXiv
We describe a high order technique to generate quadrilateral decompositions and meshes for complex two dimensional domains using spectral elements in a field guided procedure. Inspired by cross field methods, we never actually compute crosses. Instead, we compute a high order accurate guiding field using a continuous Galerkin (CG) or discontinuous Galerkin (DG) spectral element method to solve a Laplace equation for each of the field variables using the open source code Nektar++. The spectral method provides spectral convergence and sub-element resolution of the fields. The DG approximation allows meshing of corners that are not multiples of pi/2 in a discretization consistent manner, when needed. The high order field can then be exploited to accurately find irregular points, and accurately integrated using a high order streamline integration method to avoid features like limit cycles. The result is a mesh with naturally curved quadrilateral elements that do not need to be curved a posteriori to eliminate invalid elements. The mesh generation procedure is implemented in the open source mesh generation program NekMesh.
Marcon J, Peiro J, Moxey D, et al., 2019, A semi-structured approach to curvilinear mesh generation around streamlined bodies, AIAA Scitech 2019 Forum, Publisher: AIAA
We present an approach for robust high-order mesh generation specially tailored to streamlined bodies. The method is based on a semi-sructured approach which combines the high quality of structured meshes in the near-field with the flexibility of unstructured meshes in the far-field. We utilise medial axis technology to robustly partition the near-field into blocks which can be meshed coarsely with a linear swept mesher. A high-order mesh of the near-field is then generated and split using an isoparametric approach which allows us to obtain highly stretched elements aligned with the flow field. Special treatment of the partition is performed on the wing root juntion and the trailing edge --- into the wake --- to obtain an H-type mesh configuration with anisotropic hexahedra ideal for the strong shear of high Reynolds number simulations. We then proceed to discretise the far-field using traditional robust tetrahedral meshing tools. This workflow is made possible by two sets of tools: CADfix, focused on CAD system, the block partitioning of the near-field and the generation of a linear mesh; and NekMesh, focused on the curving of the high-order mesh and the generation of highly-stretched boundary layer elements. We demonstrate this approach on a NACA0012 wing attached to a wall and show that a gap between the wake partition and the wall can be inserted to remove the dependency of the partitioning procedure on the local geometry.
Turner M, Peiro J, Moxey D, 2018, Curvilinear mesh generation using a variational framework, Computer-Aided Design, Vol: 103, Pages: 73-91, ISSN: 0010-4485
We aim to tackle the challenge of generating unstructured high-order meshes of complex three-dimensional bodies, which remains a significant bottleneck in the wider adoption of high-order methods. In particular we show that by adopting a variational approach to the generation process, many of the current popular high-order generation methods can be encompassed under a single unifying framework. This allows us to compare the effectiveness of these methods and to assess the quality of the meshes they produce in a systematic fashion. We present a detailed overview of the theory and formulation of the variational framework, and we highlight how such formulation can be effectively exploited to yield a highly-efficient parallel implementation. The effectiveness of this approach is examined by considering a number of two- and three-dimensional examples, where we show how the proposed approach can be used for both mesh quality optimisation and untangling of invalid high-order meshes.
Green MD, Peiro J, 2018, Long duration SPH simulations of sloshing in tanks with a low fill ratio and high stretching, COMPUTERS & FLUIDS, Vol: 174, Pages: 179-199, ISSN: 0045-7930
Eichstaedt JR, Green M, Turner M, et al., 2018, Accelerating high-order mesh optimisation with an architecture-independent programming model, Computer Physics Communications, Vol: 229, Pages: 36-53, ISSN: 0010-4655
Heterogeneous manycore performance-portable programming models and libraries, such as Kokkos, have been developed to facilitate portability and maintainability of high-performance computing codes and enhance their resilience to architectural changes. Here we investigate the suitability of the Kokkos programming model for optimizing the performance of the high-order mesh generator NekMesh, which has been developed to efficiently generate meshes containing millions of elements for industrial problem involving complex geometries. We describe the variational approach for a posteriori high-order mesh optimisation employed within NekMesh and its parallel implementation. We discuss its implementation for modern manycore massively parallel shared-memory CPU and GPU platforms using Kokkos and demonstrate that we achieve increased performance on multicore CPUs and accelerators compared with a native Pthreads implementation. Further, we show that we achieve additional speedup and cost reduction by running on GPUs without any hardware-specific code optimisation.
Mengaldo G, Moura RC, Giralda B, et al., 2018, Spatial eigensolution analysis of discontinuous Galerkin schemes with practical insights for under-resolved computations and implicit LES, Computers and Fluids, Vol: 169, Pages: 349-364, ISSN: 0045-7930
The study focusses on the dispersion and diffusion characteristics of discontinuous spectral element methods - specifically discontinuous Galerkin (DG) - via the spatial eigensolution analysis framework built around a one-dimensional linear problem, namely the linear advection equation. Dispersion and diffusion characteristics are of critical importance when dealing with under-resolved computations, as they affect both the numerical stability of the simulation and the solution accuracy. The spatial eigensolution analysis carried out in this paper complements previous analyses based on the temporal approach, which are more commonly found in the literature. While the latter assumes periodic boundary conditions, the spatial approach assumes inflow/outflow type boundary conditions and is therefore better suited for the investigation of open flows typical of aerodynamic problems, including transitional and fully turbulent flows and aeroacoustics. The influence of spurious/reflected eigenmodes is assessed with regard to the presence of upwind dissipation, naturally present in DG methods. This provides insights into the accuracy and robustness of these schemes for under-resolved computations, including under-resolved direct numerical simulation (uDNS) and implicit large-eddy simulation (iLES). The results estimated from the spatial eigensolution analysis are verified using the one-dimensional linear advection equation and successively by performing two-dimensional compressible Euler simulations that mimic (spatially developing) grid turbulence.
Winters A, Moura RC, Mengaldo G, et al., A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations, Journal of Computational Physics, ISSN: 0021-9991
Cohen J, Marcon J, Turner M, et al., 2019, Simplifying high-order mesh generation for computational scientists, 10th International Workshop on Science Gateways, Publisher: CEUR Workshop Proceedings, ISSN: 1613-0073
Computational modelling is now tightly integrated into many fields of research in science and industry. Computational fluid dynamics software, for example, gives engineers the ability to model fluid flow around complex geometries defined in Computer-Aided Design (CAD) packages, without the expense of constructing large wind tunnel experiments. However, such modelling requires translation from an initial CAD geometry to a mesh of many small elements that modelling software uses to represent the approximate solution in the numerical method. Generating sufficiently high-quality meshes for simulation is a time-consuming, iterative and error-prone process that is often complicated by the need to interact with multiple command-line tools to generate and visualise the mesh data. In this paper we describe our approach to overcoming this complexity through the addition of a meshing console to Nekkloud, a science gateway for simplifying access to the functionality of the Nektar++ spectral/hp element framework. The meshing console makes use of the NekMesh tool in Nektar++ to help reduce the complexity of the mesh generation process. It offers a web-based interface for specifying parameters, undertaking meshing and visualising results. The meshing console enables Nekkloud to offer support for a full, end-to-end simulation pipeline from initial CAD geometry to simulation results.
Marcon J, Turner M, Peiro J, et al., 2018, High-order curvilinear hybrid mesh generation for CFD simulations, AIAA Aerospace Sciences Meeting
We describe a semi-structured method for the generation of high-order hybrid meshes suited for the simulation of high Reynolds number flows. This is achieved through the use of highly stretched elements in the viscous boundary layers near the wall surfaces. CADfix is used to first repair any possible defects in the CAD geometry and then generate a medial object based decomposition of the domain that wraps the wall boundaries with partitions suitable for the generation of either prismatic or hexahedral elements. The latter is a novel distinctive feature of the method that permits to obtain well-shaped hexahedral meshes at corners or junctions in the boundary layer. The medial object approach allows greater control on the “thickness” of the boundary-layer mesh than is generally achievable with advancing layer techniques. CADfix subsequently generates a hybrid straight-sided mesh of prismatic and hexahedral elements in the near-field region modelling the boundary layer, and tetrahedral elements in the far-field region covering the rest of the domain. The mesh in the near-field region provides a framework that facilitates the generation, via an isoparametric technique, of layers of highly stretched elements with a distribution of points in the direction normal to the wall tailored to efficiently and accurately capture the flow in the boundary layer. The final step is the generation of a high-order mesh using NekMesh, a high-order mesh generator within the Nektar++ framework. NekMesh uses the CADfix API as a geometry engine that handles all the geometrical queries to the CAD geometry required during the high-order mesh generation process. We will describe in some detail the methodology using a simple geometry, a NACA wing tip, for illustrative purposes. Finally, we will present two examples of application to reasonably complex geometries proposed by NASA as CFD validation cases: the Common Research Model and the Rotor 67.
Turner M, Moxey D, Peiro J, et al., 2017, A framework for the generation of high-order curvilinear hybrid meshes for CFD simulations, 26th International Meshing Roundtable (IMR), Publisher: ELSEVIER SCIENCE BV, Pages: 206-218, ISSN: 1877-7058
We present a pipeline of state-of-the-art techniques for the generation of high-order meshes that contain highly stretched elements in viscous boundary layers, and are suitable for flow simulations at high Reynolds numbers. The pipeline uses CADfix to generate a medial object based decomposition of the domain, which wraps the wall boundaries with prismatic partitions. The use of medial object allows the prism height to be larger than is generally possible with advancing layer techniques. CADfix subsequently generates a hybrid straight-sided (or linear) mesh. A high-order mesh is then generated a posteriori using NekMesh, a high-order mesh generator within the Nektar++ framework. During the high-order mesh generation process, the CAD definition of the domain is interrogated; we describe the process for integrating the CADfix API as an alternative backend geometry engine for NekMesh, and discuss some of the implementation issues encountered. Finally, we illustrate the methodology using three geometries of increasing complexity: a wing tip, a simplified landing gear and an aircraft in cruise configuration.
Rainbird J, Peiro J, Graham JM, 2017, Poststall Airfoil Performance and Vertical-Axis Wind Turbines, JOURNAL OF PROPULSION AND POWER, Vol: 33, Pages: 1053-1062, ISSN: 0748-4658
Marcon J, Turner M, Moxey D, et al., 2017, A variational approach to high-order r-adaptation, 26th International Meshing Roundtable
A variational framework, initially developed for high-order mesh optimisation, is being extended for r-adaptation. The method is based on the minimisation of a functional of the mesh deformation. To achieve adaptation, elements of the initial mesh are manipulated using metric tensors to obtain target elements. The nonlinear optimisation in turns adapts the final high-order mesh to best fit the description of the target elements by minimising the element distortion. Encouraging preliminary results prove that the method behaves well and can be used in the future for more extensive work which shall include the use of error indicators from CFD simulations.
Moura RC, Peiro J, Sherwin SJ, 2017, On the accuracy and robustness of implicit LES/under-resolved DNS approaches based on spectral element methods
We present a study on the suitability of under-resolved DNS (uDNS)-also called implicit LES (iLES)-approaches based on spectral element methods (SEM), with emphasis on high-order continuous and discontinuous Galerkin (i.e. CG and DG) schemes. Broadly speaking, these are model-free eddy-resolving approaches to turbulence which solve the governing equations in unfiltered form and rely on numerical stabilization techniques for small-scale regularization. Model problems in 1D, 2D and 3D are used in the assessment of solution quality and numerical stability. A rationale for the excellent potential of these methods for transitional and turbulent flows is offered on the basis of linear dispersion-diffusion analysis.
Turner M, Peiro J, Moxey D, 2016, A variational framework for high-order mesh generation, Procedia Engineering, Vol: 163, Pages: 340-352, ISSN: 1877-7058
The generation of sufficiently high quality unstructured high-order meshes remains a significant obstacle in the adoption of high-order methods. However, there is little consensus on which approach is the most robust, fastest and produces the ‘best’ meshes. We aim to provide a route to investigate this question, by examining popular high-order mesh generation methods in the context of an efficient variational framework for the generation of curvilinear meshes. By considering previous works in a variational form, we are able to compare their characteristics and study their robustness. Alongside a description of the theory and practical implementation details, including an efficient multi-threading parallelisation strategy, we demonstrate the effectiveness of the framework, showing how it can be used for both mesh quality optimisation and untangling of invalid meshes.
Moura RC, Mengaldo G, Peiro J, et al., 2016, On the eddy-resolving capability of high-order discontinuous Galerkin approaches to implicit LES / under-resolved DNS of Euler turbulence, Journal of Computational Physics, Vol: 330, Pages: 615-623, ISSN: 0021-9991
We present estimates of spectral resolution power for under-resolved turbulent Euler flows obtained with high-orderdiscontinuous Galerkin (DG) methods. The ‘1% rule’ based on linear dispersion–diusion analysis introduced byMoura et al. [J. Comput. Phys.298 (2015) 695–710] is here adapted for 3D energy spectra and validated throughthe inviscid Taylor–Green vortex problem. The 1% rule estimates the wavenumber beyond which numerical diusioninduces an artificial dissipation range on turbulent spectra. As the original rule relies on standard upwinding, dierentRiemann solvers are tested. Very good agreement is found for solvers which treat the dierent physical waves in aconsistent manner. Relatively good agreement is still found for simpler solvers. The latter however displayed spuriousfeatures attributed to the inconsistent treatment of dierent physical waves. It is argued that, in the limit of vanishingviscosity, such features might have a significant impact on robustness and solution quality. The estimates proposed areregarded as useful guidelines for no-model DG-based simulations of free turbulence at very high Reynolds numbers.
Ekelschot, Moxey D, Sherwin, et al., 2016, A p-adaptation method for compressible flow problems using a goal-based error indicator, Computers and Structures, Vol: 181, Pages: 55-69, ISSN: 1879-2243
An accurate calculation of aerodynamic force coefficients for a given geometry isof fundamental importance for aircraft design. High-order spectral/hp elementmethods, which use a discontinuous Galerkin discretisation of the compressibleNavier-Stokes equations, are now increasingly being used to improve the accuracyof flow simulations and thus the force coefficients. To reduce error in thecalculated force coefficients whilst keeping computational cost minimal, we proposea p-adaptation method where the degree of the approximating polynomialis locally increased in the regions of the flow where low resolution is identifiedusing a goal-based error estimator as follows.Given an objective functional such as the aerodynamic force coefficients, weuse control theory to derive an adjoint problem which provides the sensitivityof the functional with respect to changes in the flow variables, and assumethat these changes are represented by the local truncation error. In its finalform, the goal-based error indicator represents the effect of truncation error onthe objective functional, suitably weighted by the adjoint solution. Both flowgoverning and adjoint equations are solved by the same high-order method,where we allow the degree of the polynomial within an element to vary acrossthe mesh.We initially calculate a steady-state solution to the governing equations using a low polynomial order and use the goal-based error indicator to identify partsof the computational domain that require improved solution accuracy whichis achieved by increasing the approximation order. We demonstrate the costeffectivenessof our method across a range of polynomial orders by considering anumber of examples in two- and three-dimensions and in subsonic and transonicflow regimes. Reductions in both the number of degrees of freedom required toresolve the force coefficients to a given error, as well as the computational cost,are both observed in using the p-adaptive technique
Bianchini A, Balduzzi F, Rainbird JM, et al., 2016, An Experimental and Numerical Assessment of Airfoil Polars for Use in Darrieus Wind Turbines-Part I: Flow Curvature Effects, JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER-TRANSACTIONS OF THE ASME, Vol: 138, ISSN: 0742-4795
Bianchini A, Balduzzi F, Rainbird JM, et al., 2016, An Experimental and Numerical Assessment of Airfoil Polars for Use in Darrieus Wind Turbines-Part II: Post-stall Data Extrapolation Methods, JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER-TRANSACTIONS OF THE ASME, Vol: 138, ISSN: 0742-4795
Ang CDE, Rein G, Peiro J, et al., 2016, Simulating longitudinal ventilation flows in long tunnels: comparison of full CFD and multi-scale modelling approaches in FDS6, Tunnelling and Underground Space Technology, Vol: 52, Pages: 119-126, ISSN: 0886-7798
The accurate computational modelling of airflows in transport tunnels is needed for regulations compliance, pollution and fire safety studies but remains a challenge for long domains because the computational time increases dramatically. We simulate air flows using the open-source code FDS 6.1.1 developed by NIST, USA. This work contains two parts. First we validate FDS6’s capability for predicting the flow conditions in the tunnel by comparing the predictions against on-site measurements in the Dartford Tunnel, London, UK, which is 1200 m long and 8.5 m in diameter. The comparison includes the average velocity and the profile downstream of an active jet fan up to 120 m. Secondly, we study the performance of the multi-scale modelling approach by splitting the tunnel into CFD domain and a one-dimensional domain using the FDS HVAC (Heating, Ventilation and Air Conditioning) feature. The work shows the average velocity predicted by FDS6 using both the full CFD and multi-scale approaches is within the experimental uncertainty of the measurements. Although the results showed the prediction of the downstream velocity profile near the jet fan falls outside the on-site measurements, the predictions at 80 m and beyond are accurate. Our results also show multi-scale modelling in FDS6 is as accurate as full CFD but up to 2.2 times faster and that computational savings increase with the length of the tunnel. This work sets the foundation for the next step in complexity with fire dynamics introduced to the tunnel.
Moura RC, Sherwin SJ, Peiro J, 2015, Eigensolution analysis of spectral/hp continuous Galerkin approximations to advection-diffusion problems: insights into spectral vanishing viscosity, Journal of Computational Physics, Vol: 307, Pages: 401-422, ISSN: 1090-2716
This study addresses linear dispersion–diffusion analysis for the spectral/hp continuousGalerkin (CG) formulation in one dimension. First, numerical dispersion and diffusioncurves are obtained for the advection–diffusion problem and the role of multipleeigencurves peculiar to spectral/hp methods is discussed. From the eigencurves’ behaviour,we observe that CG might feature potentially undesirable non-smooth dispersion/diffusioncharacteristics for under-resolved simulations of problems strongly dominated by eitherconvection or diffusion. Subsequently, the linear advection equation augmented withspectral vanishing viscosity (SVV) is analysed. Dispersion and diffusion characteristics ofCG with SVV-based stabilization are verified to display similar non-smooth features inflow regions where convection is much stronger than dissipation or vice-versa, owing toa dependency of the standard SVV operator on a local Péclet number. First a modificationis proposed to the traditional SVV scaling that enforces a globally constant Péclet numberso as to avoid the previous issues. In addition, a new SVV kernel function is suggestedand shown to provide a more regular behaviour for the eigencurves along with aconsistent increase in resolution power for higher-order discretizations, as measured bythe extent of the wavenumber range where numerical errors are negligible. The dissipationcharacteristics of CG with the SVV modifications suggested are then verified to be broadlyequivalent to those obtained through upwinding in the discontinuous Galerkin (DG)scheme. Nevertheless, for the kernel function proposed, the full upwind DG scheme isfound to have a slightly higher resolution power for the same dissipation levels. Theseresults show that improved CG-SVV characteristics can be pursued via different kernelfunctions with the aid of optimization algorithms.
Rainbird JM, Bianchini A, Balduzzi F, et al., 2015, On the influence of virtual camber effect on airfoil polars for use in simulations of Darrieus wind turbines, ENERGY CONVERSION AND MANAGEMENT, Vol: 106, Pages: 373-384, ISSN: 0196-8904
Moxey D, Ekelschot D, Keskin U, et al., 2015, High-order curvilinear meshing using a thermo-elastic analogy, Computer Aided Design, Vol: 72, Pages: 130-139, ISSN: 0010-4485
With high-order methods becoming increasingly popular in both academia and industry, generating curvilinear meshes that align with the boundaries of complex geometries continues to present a significant challenge. Whereas traditional low-order methods use planar-faced elements, high-order methods introduce curvature into elements that may, if added naively, cause the element to self-intersect. Over the last few years, several curvilinear mesh generation techniques have been designed to tackle this issue, utilising mesh deformation to move the interior nodes of the mesh in order to accommodate curvature at the boundary. Many of these are based on elastic models, where the mesh is treated as a solid body and deformed according to a linear or non-linear stress tensor. However, such methods typically have no explicit control over the validity of the elements in the resulting mesh. In this article, we present an extension of this elastic formulation, whereby a thermal stress term is introduced to 'heat' or 'cool' elements as they deform. We outline a proof-of-concept implementation and show that the adoption of a thermo-elastic analogy leads to an additional degree of robustness, by considering examples in both two and three dimensions.
Rainbird JM, Peiro J, Graham JMR, 2015, Blockage-tolerant wind tunnel measurements for a NACA 0012 at high angles of attack, JOURNAL OF WIND ENGINEERING AND INDUSTRIAL AERODYNAMICS, Vol: 145, Pages: 209-218, ISSN: 0167-6105
Moura RC, Sherwin SJ, Peiro J, 2015, Linear dispersion-diffusion analysis and its application to under-resolved turbulence simulations using discontinuous Galerkin spectral/hp methods, Journal of Computational Physics, Vol: 298, Pages: 695-710, ISSN: 0021-9991
Peiro J, Rainbird JM, Ferrer E, et al., 2015, Vertical-axis wind turbine start-up modeled with a high-order numerical solver, CFD for Wind and Tidal Offshore Turbines, Publisher: Springer, ISBN: 9783319162027
Peiro J, Moxey D, Hazan M, et al., 2015, On the generation of curvilinear meshes through subdivision of isoparametric elements, New Challenges in Grid Generation and Adaptivity for Scientific Computing, Editors: Perotto, Formaggia, Publisher: Springer, ISBN: 9783319060538
This volume collects selected contributions from the “Fourth Tetrahedron Workshop on Grid Generation for Numerical Computations”, which was held in Verbania, Italy in July 2013.
© Springer International Publishing Switzerland 2015. Accurate visualization of high-order meshes and flow fields is a fundamental tool for the verification, validation, analysis and interpretation of high-order flow simulations. Standard visualization tools based on piecewise linear approximations can be used for the display of highorder fields but their accuracy is restricted by computer memory and processing time. More often than not, the accurate visualization of complex flows using this strategy requires computational resources beyond the reach of most users. This chapter describes ElVis, a truly high-order and interactive visualization system created for the accurate and interactive visualization of scalar fields produced by high-order spectral/hp finite element simulations. We show some examples that motivate the need for such a visualization system and illustrate some of its features for the display and analysis of simulation data.
Moxey D, Hazan M, Sherwin SJ, et al., 2015, Curvilinear mesh generation for boundary layer problems, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Vol: 128, Pages: 41-64, ISSN: 1612-2909
© Springer International Publishing Switzerland 2015. In this article, we give an overview of a new technique for unstructured curvilinear boundary layer grid generation, which uses the isoparametric mappings that define elements in an existing coarse prismatic grid to produce a refined mesh capable of resolving arbitrarily thin boundary layers. We demonstrate that the technique always produces valid grids given an initially valid coarse mesh, and additionally show how this can be extended to convert hybrid meshes to meshes containing only simplicial elements.
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