Abstract: In this article, cache blocking is implemented for the Navier Stokes equations with anti-aliasing support on mixed grids in PyFR for CPUs. In particular, cache blocking is used as an alternative to kernel fusion to eliminate unnecessary data movements between kernels at the main memory level. Specifically, kernels that exchange data are grouped together, and these groups are then executed on small sub-regions of the domain that fit in per-core private data cache. Additionally, cache blocking is also used to efficiently implement a tensor product factorisation of the interpolation operators associated with anti-aliasing. By using cache blocking, the intermediate results between application of the sparse factors are stored in per-core private data cache, and a significant amount of data movement from main memory is avoided. In order to assess the performance gains a theoretical model is developed, and the implementation is benchmarked using a compressible 3D Taylor-Green vortex test case on both hexahedral and prismatic grids, with third-, fourth-, and fifth-order solution polynomials. The expected performance gains based on the theoretical model range from 1.99 to 2.83, and the speedups obtained in practice range from 1.51 to 3.91 compared to PyFR v1.11.0.

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Abstract: Mars has a lower atmospheric density than Earth, and the speed of sound is lower due to its atmospheric composition and lower surface temperature. Consequently, Martian rotor blades operate in a low-Reynolds-number compressible regime that is atypical for terrestrial helicopters. Nonconventional airfoils with sharp edges and flat surfaces have shown improved performance under such conditions, and second-order-accurate Reynolds-averaged Navier-Stokes (RANS) and unsteady RANS (URANS) solvers have been combined with genetic algorithms to optimize them. However, flow over such airfoils is characterized by unsteady roll-up of coherent vortices that subsequently break down/transition. Accordingly, RANS/URANS solvers have limited predictive capability, especially at higher angles of attack where the aforementioned physics are more acute. To overcome this limitation, we undertake optimization using high-order direct numerical simulations (DNS). Specifically, a triangular airfoil is optimized using DNS. Multi-objective optimization is performed to maximize lift and minimize drag, yielding a Pareto front. Various quantities, including lift spectra and pressure distributions, are analyzed for airfoils on the Pareto front to elucidate flow physics that yield optimal performance. The optimized airfoils that form the Pareto front achieve up to a 48% increase in lift or a 28% reduction in drag compared to a reference triangular airfoil studied in the Mars Wind Tunnel at Tohoku University. The work constitutes the first use of DNS for aerodynamic shape optimization.

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Abstract: We present an extended range of stable flux reconstruction (FR) methods on triangles through the development and application of the summation-by-parts framework in two-dimensions. This extended range of stable schemes is then shown to contain the single parameter schemes of Castonguay et al. (J Sci Comput 51:224-256, 2011) on triangles, and our definition enables wider stability bounds to be developed for those single parameter families. Stable upwinded spectral difference (SD) schemes on triangular elements have previously been found using Fourier analysis. We used our extended range of FR schemes to investigate the linear stability of SD methods on triangles, and it was found that a only first order SD scheme could be recovered within this set of FR methods.

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Abstract: An extended range of energy stable flux reconstruction schemes, developed using a summation-by-parts approach, is presented on quadrilateral elements for various sets of polynomial bases. For the maximal order bases, a new set of correction functions which result in stable schemes is found. However, for a range of orders it is shown that only a single correction function can be cast as a tensor-product. Subsequently, correction functions are identified using a generalised analytic framework that results in stable schemes for total order and approximate Euclidean order polynomial bases on quadrilaterals - which have not previously been explored in the context of flux reconstruction. It is shown that the approximate Euclidean order basis can provide similar numerical accuracy as the maximal order basis but with fewer points per element, and thus lower cost.

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Abstract: Building facade systems can be subject to severe and highly localized wind suction pressures. Such pressures need to be accurately assessed in order to estimate the maximum loads on a typical cladding panel. Wind tunnel experiments studying turbulent flow over a model high-rise building have shown space–time localized peaks of extremely low pressure (Cp < -8) on the model building facade. Such low pressure values are unexpected and the potential implications for the cost and carbon intensity of cladding systems are significant. In this work, we use the open-source solver PyFR to carry out high-order Implicit Large Eddy Simulations (ILES) of this test case. The simulations capture, for the first time, the observed space–time localized peaks of extreme low pressure, replicating the experimental findings. The corresponding fluid structures are shown in detail. They are found to be relatively thin and long vortices spinning with an angular velocity approximately normal to the building wall.

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Abstract: The scale-resolving simulation of high speed compressible flow through direct numerical simulation (DNS) or large eddy simulation (LES) requires shock-capturing schemes to be more accurate for resolving broadband turbulence and robust for capturing strong shock waves. In this work, we develop a new paradigm of dissipation-adjustable, shock capturing scheme to resolve multi-scale flow structures in high speed compressible flow. The new scheme employs a polynomial of n-degree and non-polynomial THINC (Tangent of Hyperbola for INterface Capturing) functions of m-level steepness as reconstruction candidates. These reconstruction candidates are denoted as . From these candidates, the piecewise reconstruction function is selected through the boundary variation diminishing (BVD) algorithm. Unlike other shock-capturing techniques, the BVD algorithm effectively suppresses numerical oscillations without introducing excess numerical dissipation. Then, an adjustable dissipation (AD) algorithm is designed for scale-resolving simulations. This novel paradigm of shock-capturing scheme is named as . The proposed scheme has following desirable properties. First, it can capture large-scale discontinuous structures such as strong shock waves without obvious non-physical oscillations while resolving sharp contact, material interface and shear layer. Secondly, the numerical dissipation property of can be effectively adjusted between n+1 order upwind-biased scheme and non-dissipative n+2 order central scheme through a simple tunable parameter ?. Thirdly, with the scheme can recover to n+2 order non-dissipative central interpolation for smooth solution over all wavenumber, which is preferable for solving small-scale structures in DNS as well as resolvable-scale in explicit LES. Finally, the under-resolved small-scale can be solved with the dissipation adjustable algorithm through the so-called implicit LES (ILES) approach. Through simulating benchmark tests involving multi-scale flow structures and comparing with other central-upwind schemes, the superiority of the proposed scheme is evident. For instance, the simulation results of the supersonic planar jet show that schemes can achieve competitive results as schemes which utilize a higher degree of reconstruction polynomial. Thus, in comparison with the previous work, the proposed schemes have the benefit of a more compact stencil and lower cost. In summary, this work provides an alternative scheme for solving multi-scale problems in high speed compressible flows.

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Abstract: Martian conditions present various challenges when designing rotorcraft. Specifically, the thin atmosphere and low sound speed require Martian rotor blades to operate in a low-Reynolds-number (1000-10,000) compressible regime, for which conventional airfoils are not designed. Here, we use PyFR to undertake high-order direct numerical simulations (DNS) of flow over a triangular airfoil at a Mach number of 0.15 and Reynolds number of 3000. Initially, spanwise periodic DNS are undertaken. Extending the domain-span-to-chord ratio from 0.3 to 0.6 leads to better agreement with wind-tunnel data at higher angles of attack, when the flow is separated. This is because smaller domain spans artificially suppress three-dimensional breakdown of coherent structures above the suction surface of the airfoil. Subsequently, full-span DNS in a virtual wind tunnel are undertaken, including all wind-tunnel walls. These capture blockage and wall boundary-layer effects, leading to better agreement with wind-tunnel data for all angles of attack compared to spanwise periodic DNS. The results are important in terms of understanding discrepancies between previous spanwise periodic DNS and wind-tunnel data. They also demonstrate the utility of high-order DNS as a tool for accurately resolving flow over triangular airfoils under Martian conditions.

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Abstract: On modern hardware architectures, the performance of Flux Reconstructionr(FR) methods can be limited by memory bandwidth. In a typical implementation,rthese methods are implemented as a chain of distinct kernels. Often, ardataset which has just been written in the main memory by a kernel is read backrimmediately by the next kernel. One way to avoid such a redundant expenditurerof memory bandwidth is kernel fusion. However, on a practical level kernelrfusion requires that the source for all kernels be available, thus preventing callsrto certain third-party library functions. Moreover, it can add substantial complexityrto a codebase. An alternative to full kernel fusion is cache blocking. Butrfor this to be effective, CPU cache has to be meaningfully big. Historically, sizerof L1 and L2 caches prevented cache blocking for high-order CFD applications.rHowever in recent years, size of L2 cache has grown from around 0.25 MiB to 1.25rMiB, and made it possible to apply cache blocking for high-order CFD codes.rIn this approach, kernels remain distinct, and are executed one after another onrsmall chunks of data that can fit in the cache, as opposed to on full datasets.rThese chunks of data stay in the cache and whenever a kernel requests accessrto data that is already in the cache, memory bandwidth is conserved. In thisrstudy, a data structure that facilitates cache blocking is considered, and a rangerof kernel grouping configurations for an FR based Euler solver are examined. Artheoretical study is conducted for hexahedral elements with no anti-aliasing atrp = 3 and p = 4 in order to determine the predicted performance of a few kernel grouping configurations. Then, these candidates are implemented in the PyFR solver and the performance gains in practice are compared with the theoreticalrestimates that range between 2.05x and 2.50x. An inviscid Taylor-Green Vortexrtest case is used as a benchmark, and the most performant configuration leadsrto a speedup of approximately 2.81x in practice.

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Abstract: An extended version of the Synthetic Eddy Method for generation of synthetic turbulence has been developed via a source term formulation and implemented in the open-source cross platform solver PyFR. The method caters for the full space-dependent anisotropy of the target turbulent length scales and it is agnostic of the space and time discretization of the underlying solver, which can be incompressible or compressible. Moreover, the method does not requirereach solution point to communicate with nearest neighbours thus it is well suited for modern, massively parallel, high-order unstructured codes which support mixed and possibly curvedrelements. The method has been applied to two test cases: the classical incompressible flowrproblem of plane turbulent channel flow at Re = 180 and the compressible flow over the SD7003 wing at Re = 66000, Ma = 0.2 and alpha = 4. This test case has been run on three topologically different meshes composed of hexahedra, prisms and a combination of prismsrand tetrahedra, respectively. Almost identical results have been obtained on the three meshes.rResults also show that taking into account the anisotropy of the turbulent length scales canrreduce the development length. For the SD7003 wing, the injection of synthetic turbulence improves the agreement between numerical and experimental results.

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Abstract: Reynolds Averaged Navier-Stokes (RANS) simulations and wind tunnel testing have become the go-to tools for industrial design of Low-Pressure Turbine (LPT) blades. However, there is also an emerging interest in use of scale-resolving simulations, including Direct Numerical Simulations (DNS). These could generate insight and data to underpin development of improved RANS models for LPT design. Additionally, they could underpin a virtual LPT wind tunnel capability, that is cheaper, quicker, and more data-rich than experiments. The current study applies PyFR, a Python based Computational Fluid Dynamics (CFD) solver, to fifth-order accurate petascale DNS of compressible flow over a three-dimensional MTU-T161 LPT blade with diverging end walls at a Reynolds number of 200,000 on an unstructured mesh with over 11 billion degrees-of-freedom per equation. Various flow metrics, including isentropic Mach number distribution at mid-span, surface shear, and wake pressure losses are compared with available experimental data and found to be in agreement. Subsequently, a more detailed analysis of various flow features is presented. These include the separation/transition processes on both the suction and pressure sides of the blade, end-wall vortices, and wake evolution at various span-wise locations. The results, which constitute one of the largest and highest-fidelity CFD simulations ever conducted, demonstrate the potential of high-order accurate GPU-accelerated CFD as a tool for delivering industrial DNS of LPT blades.

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Abstract: The topology of isosurfaces changes at isovalues of critical points, making such points an important feature when building contour trees or Morse-Smale complexes. Hexahedral elements with linear inter- polants can contain additional off-vertex critical points in element bodies and on element faces. Moreover, a point on the face of a hexahedron which is critical in the element-local context is not necessarily critical in the global context. In "Exploring Scalar Fields Using Critical Isovalues", Weber et al. introduce a method to determine whether critical points on faces are also critical in the global context, based on the gradient of the asymptotic decider in each element that shares the face. However, as defined, the method of Weber et al. contains an error, and can lead to incorrect results. In this work we correct the error.

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Abstract: Nodal point sets, and associated collocation projections, play an important role in a range of high-order methods, including Flux Reconstruction (FR) schemes. Historically, efforts have focused on identifying nodal point sets that aim to minimise the L^&#8734 error of an associated interpolating polynomial. The present work combines a comprehensive review of known approximation theory results, with new results, and numerical experiments, to motivate that in fact point sets for FR should aim to minimise the L² error of an associated interpolating polynomial. New results include identification of a nodal point set that minimises the L² norm of an interpolating polynomial, and a proof of the equivalence between such an interpolating polynomial and an L² approximating polynomial with coefficients obtained using a Gauss-Legendre quadrature rule. Numerical experiments confirm that FR errors can be reduced by an order-of-magnitude by switching from popular point sets such as Chebyshev, Chebyshev-Lobatto and Legendre-Lobatto to Legendre point sets.

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Abstract: The potential for acute shortages of ventilators at the peak of the COVID-19 pandemic has raised the possibility of needing to support two patients from a single ventilator. To provide a system for understanding and prototyping designs we have developed a mathematical model of two patients supported by a mechanical ventilator. We propose a standard setup where we simulate the introduction of T-splitters to supply air to two patients and a modified setup where we introduce a variable resistance in each inhalation pathway and one-way valves in each exhalation pathway. Using the standard setup, we demonstrate that ventilating two patients with mismatched lung compliances from a single ventilator will lead to clinically-significant reductions in tidal volume in the patient with the lowest respiratory compliance. Using the modified setup, we demonstrate that it could be possible to achieve the same tidal volumes in two patients with mismatched lung compliances, and we show that the tidal volume of one patient can be manipulated independently of the other. The results indicate that, with appropriate modifications, two patients could be supported from a single ventilator with independent control of tidal volumes.n

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Abstract: This paper proposes a novel locally adaptive pseudo-time stepping convergence acceleration technique for dual time stepping which is a common integration method for solving unsteady low-Mach preconditioned/incompressible Navier-Stokes formulations. In contrast to standard local pseudo-time stepping techniques that are based on computing the local pseudo-time steps directly from estimates of the local Courant-Friedrichs-Lewy limit, the proposed technique controls the local pseudo-time steps using local truncation errors which are computed with embedded pair RK schemes. The approach has three advantages. First, it does not require an expression for the characteristic element size, which are difficult to obtain reliably for curved mixed-element meshes. Second, it allows a finer level of locality for high-order nodal discretisations, such as FR, since the local time-steps can vary between solution points and field variables. Third, it is well-suited to being combined with P-multigrid convergence acceleration. Results are presented for a laminar 2D cylinder test case at Re=100. A speed-up factor of 4.16 is achieved compared to global pseudo-time stepping with an RK4 scheme, while maintaining accuracy. When combined with P-multigrid convergence acceleration a speed-up factor of over 15 is achieved. Detailed analysis of the results reveals that pseudo-time steps adapt to element size/shape, solution state, and solution point location within each element. Finally, results are presented for a turbulent 3D SD7003 airfoil test case at Re=60,000. Speed-ups of similar magnitude are observed, and the flow physics is found to be in good agreement with previous studies.

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Abstract: Eigenmodes of averaged small-amplitude perturbations to a turbulent channel flow - which is one of the most fundamental canonical flows - are identified for the first time via an extensive set of high-fidelity GPU-accelerated direct numerical simulations. While the system governing averaged small-amplitude perturbations to turbulent channel flow remains unknown, the fact such eigenmodes can be identified constitutes direct evidence that it is linear. Moreover, while the eigenvalue associated with the slowest-decaying anti-symmetric eigenmode mode is found to be real, the eigenvalue associated with the slowest-decaying symmetric eigenmode mode is found to be complex. This indicates that the unknown linear system governing the evolution of averaged small-amplitude perturbations cannot be self-adjoint, even for the case of a uni-directional flow. In addition to elucidating aspects of the flow physics, the findings provide guidance for development of new unsteady Reynolds-averaged Navier-Stokes turbulence models, and constitute a new and accessible benchmark problem for assessing the performance of existing models, which are used widely throughout industry.

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Abstract: 3-D blood vector flow imaging is of great value in understanding and detecting cardiovascular diseases. Currently, 3-D ultrasound vector flow imaging requires 2-D matrix probes, which are expensive and suffer from suboptimal image quality. Our recent study proposed an interpolation algorithm to obtain a divergence-free reconstruction of the 3-D flow field from 2-D velocities obtained by high-frame-rate ultrasound particle imaging velocimetry (HFR echo-PIV, also known as HFR UIV), using a 1-D array transducer. The aim of this work was to significantly improve the accuracy and reduce the time-to-solution of our previous approach, thereby paving the way for clinical translation. More specifically, accuracy was improved by optimising the divergence-free basis to reduce Runge phenomena near domain boundaries, and time-to-solution was reduced by demonstrating that under certain conditions, the resulting system could be solved using widely available and highly optimised generalised minimum residual algorithms. To initially illustrate the utility of the approach, coarse 2-D subsamplings of an analytical unsteady Womersely flow solution and a steady helical flow solution obtained using computational fluid dynamics were used successfully to reconstruct full flow solutions, with 0.82% and 4.8% average relative errors in the velocity field, respectively. Subsequently, multiplane 2-D velocity fields were obtained through HFR UIV for a straight-tube phantom and a carotid bifurcation phantom, from which full 3-D flow fields were reconstructed. These were then compared with flow fields obtained via computational fluid dynamics in each of the two configurations, and average relative errors of 6.01% and 12.8% in the velocity field were obtained. These results reflect 15%-75% improvements in accuracy and 53- to 874-fold acceleration of reconstruction speeds for the four cases, compared with the previous divergence-free flow reconstruction method. In conclusion, the proposed method provides an effective and fast method to reconstruct 3-D 80 flow in arteries using a 1-D array transducer.

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Abstract: In this study we generate optimal Runge-Kutta (RK) schemes for converging the Artificial Compressibility Method (ACM) using dual time-stepping with high-order unstructured spatial discretizations. We present optimal RK schemes with between s=2 and s=7 stages for Spectral Difference (SD) and Discontinuous Galerkin (DG) discretizations obtained using the Flux Reconstruction (FR) approach with solution polynomial degrees of k=1 to k=8. These schemes are optimal in the context of linear advection with predicted speedup factors in excess of 1.80x relative to a classical RK44 scheme. Speedup factors of between 1.89x and 2.11x are then observed for incompressible Implicit Large Eddy Simulation (ILES) of turbulent flow over an SD7003 airfoil. Finally, we demonstrate the utility of the schemes for incompressible ILES of a turbulent jet, achieving good agreement with experimental data. The results demonstrate that the optimized RK schemes are suitable for simulating turbulent flows and can achieve significant speedup factors when converging the ACM using dual time-stepping with high-order unstructured spatial discretizations.

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Abstract: Quantification of 3D intravascular flow is valuable for studying arterial wall diseases but currently there is a lack of effective clinical tools for this purpose. Divergence Free Interpolation (DFI) using Radial Basis Function (RBF) is an emerging approach for full field flow reconstruction using experimental sparse flow field samples. Previous DFI reconstructs full field flow from scattered 3D velocity input obtained using Phase Contrast MRI with low temporal resolution. In this study, a new DFI algorithm is proposed to reconstruct full field flow from scattered 2D in-plane velocity vectors obtained using ultrafast contrast enhanced ultrasound (>1000 fps) and particle imaging velocimetry (Ultrasound PIV, or UIV). The full 3D flow field is represented by a sum of weighted divergence free RBFs in space. Due to the acquired velocity vectors being only in 2D and hence the problem being ill-conditioned, a regularized solution of the RBF weighting is achieved through Singular Value Decomposition (SVD) and the L-curve method. The effectiveness of the algorithm is demonstrated via numerical experiments for Poiseuille flow and helical flow with added noise, and it is shown that an accuracy of up to 95.6% can be achieved for Poiseuille flow (with 5% input noise). Experimental feasibility is also demonstrated by reconstructing full-field 3D flow from experimental 2D UIV measurements in a carotid bifurcation phantom. The method is typically faster for a range of problems compared to computational fluid dynamics, and has been demonstrated to be effective for the three flow cases.

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Abstract: Native arteriovenous fistulas (AVFs) for hemodialysis are susceptible to nonmaturation. Adverse features of local blood flow have been implicated in the formation of perianastomotic neointimal hyperplasia that may underpin nonmaturation. Whereas computational fluid dynamic simulations of idealized models highlight the importance of geometry on fluid and vessel wall interactions, little is known in vivo about AVF geometry and its role in adverse clinical outcomes. This study set out to examine the three-dimensional geometry of native AVFs and the geometric correlates of AVF failure.

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Abstract: Modern hardware architectures such as GPUs and manycore processors are characterised by an abundance of compute capability relative to memory bandwidth. This makes them well-suited to solving temporally explicit and spatially compact discretisations of hyperbolic conservation laws. However, classical pressure-projection-based incompressible Navier-Stokes formulations do not fall into this category. One attractive formulation for solving incompressible problems on modern hardware is the method of artificial compressibility. When combined with explicit dual time stepping and a high-order Flux Reconstruction discretisation, the majority of operations can be cast as compute bound matrix-matrix multiplications that are well-suited for GPU acceleration and manycore processing. In this work, we develop a high-order cross-platform incompressible Navier-Stokes solver, via artificial compressibility and dual time stepping, in the PyFR framework. The solver runs on a range of computer architectures, from laptops to the largest supercomputers, via a platform-unified templating approach that can generate/compile CUDA, OpenCL and C/OpenMP code at runtime. The extensibility of the cross-platform templating framework defined within PyFR is clearly demonstrated, as is the utility of P-multigrid for convergence acceleration. The platform independence of the solver is verified on Nvidia Tesla P100 GPUs and Intel Xeon Phi 7210 KNL manycore processors with a 3D Taylor-Green vortex test case. Additionally, the solver is applied to a 3D turbulent jet test case at Re=10,000, and strong scaling is reported up to 144 GPUs. The new software constitutes the first high-order accurate cross-platform implementation of an incompressible Navier-Stokes solver via artificial compressibility and P-multigrid accelerated dual time stepping to be published in the literature. The technology has applications in a range of sectors, including the maritime and automotive industries. Moreover, due to its cross-platform nature, the technology is well placed to remain relevant in an era of rapidly evolving hardware architectures.n

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Abstract: Arterio-Venous Fistulae (AVF) are regarded as the 'gold standard' method of vascular access for patients with End-Stage Renal Disease (ESRD) who require haemodialysis. However, a large proportion of AVF do not mature, and hence fail, as a result of various pathologies such as Intimal Hyperplasia (IH). Unphysiological flow patterns, including high-frequency flow unsteadiness, associated with the unnatural and often complex geometries of AVF are believed to be implicated in the development of IH. In the present study we employ a Mesh Adaptive Direct Search optimisation framework, computational fluid dynamics simulations, and a new cost-function, to design a novel non-planar AVF configuration that can suppress high-frequency unsteady flow. A prototype device for holding an AVF in the optimal configuration is then fabricated, and proof-of-concept is demonstrated in a porcine model. Results constitute the first use of numerical optimisation to design a device for suppressing potentially pathological high-frequency flow unsteadiness in AVF.

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Abstract: Arterio-Venous Fistulae (AVF) are regarded as the 'gold standard' method of vascular access for patients with End-Stage Renal Disease (ESRD) who require haemodialysis. However, up to 60% of AVF do not mature, and hence fail, as a result of Intimal Hyperplasia (IH). Unphysiological flow and oxygen transport patterns, associated with the unnatural and often complex geometries of AVF, are believed to be implicated in the development of IH. Previous studies have investigated the effect of arterial curvature on blood flow in AVF using idealised planar AVF configurations and non-pulsatile inflow conditions. The present study takes an important step forwards by extending this work to more realistic non-planar brachiocephalic AVF configurations with pulsatile inflow conditions. Results show that forming an AVF by connecting a vein onto the outer curvature of an arterial bend does not, necessarily, suppress unsteady flow in the artery. This finding is converse to results from a previous more idealised study. However, results also show that forming an AVF by connecting a vein onto the inner curvature of an arterial bend can suppress exposure to regions of low wall shear stress and hypoxia in the artery. This finding is in agreement with results from a previous more idealised study. Finally, results show that forming an AVF by connecting a vein onto the inner curvature of an arterial bend can significantly reduce exposure to high WSS in the vein. The results are important, as they demonstrate that in realistic scenarios arterial curvature can be leveraged to reduce exposure to excessively low/high levels of WSS and regions of hypoxia in AVF. This may in turn reduce rates of IH and hence AVF failure.

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Abstract: In this study the GPU-accelerated solver PyFR is used to simulate flow over a NACA0021 aerofoil in deep stall at a Reynolds number of 270,000 using the high-order Flux Reconstruction (FR) approach. Wall-resolved Implicit Large Eddy Simulations (ILES) are undertaken on unstructured hexahedral meshes at fourth- and fifth-order accuracy in space. It was found that either modal filtering, or anti-aliasing via an approximate L2 projection, is required in order to stabilise simulations. Time-span averaged pressure coefficient distributions on the aerofoil, and associated lift and drag coefficients, are seen to converge towards experimental data as the simulation setup is made more realistic by increasing the aerofoil span. Indeed, the lift and drag coefficients obtained by fifth-order ILES with anti-aliasing via an approximate L2 projection agree better with experimental data than a wide range of previous studies. Stabilisation via modal filtering, however, is found to reduce solution accuracy. Finally, performance of various PyFR simulations is compared, and it is found that fifth-order simulations with anti-aliasing via an L2 projection are the most efficient. Results indicate that high-order FR schemes with anti-aliasing via an L2 projection are a good candidate for underpinning accurate wall-resolved ILES of separated, turbulent flows over complex engineering geometries.

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Abstract: First- and second-order accurate numerical methods, implemented for CPUs, underpin the majority of industrial CFD solvers. Whilst this technology has proven very successful at solving steady-state problems via a Reynolds Averaged Navier-Stokes approach, its utility for undertaking scale-resolving simulations of unsteady flows is less clear. High-order methods for unstructured grids and GPU accelerators have been proposed as an enabling technology for unsteady scale-resolving simulations of flow over complex geometries. In this study we systematically compare accuracy and cost of the high-order Flux Reconstruction solver PyFR running on GPUs and the industry-standard solver STAR-CCM+ running on CPUs when applied to a range of unsteady flow problems. Specifically, we perform comparisons of accuracy and cost for isentropic vortex advection (EV), decay of the Taylor-Green vortex (TGV), turbulent flow over a circular cylinder, and turbulent flow over an SD7003 aerofoil. We consider two configurations of STAR-CCM+: a second-order configuration, and a third-order configuration, where the latter was recommended by CD-Adapco for more effective computation of unsteady flow problems. Results from both PyFR and Star-CCM+ demonstrate that third-order schemes can be more accurate than second-order schemes for a given cost e.g. going from second- to third-order, the PyFR simulations of the EV and TGV achieve 75x and 3x error reduction respectively for the same or reduced cost, and STAR-CCM+ simulations of the cylinder recovered wake statistics significantly more accurately for only twice the cost. Moreover, advancing to higher-order schemes on GPUs with PyFR was found to offer even further accuracy vs. cost benefits relative to industry-standard tools.

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Abstract: In this study we employ von Neumann analyses to investigate the dispersion, dissipation, group velocity, and error properties of several fully discrete flux reconstruction (FR) schemes. We consider three FR schemes paired with two explicit Runge-Kutta (ERK) schemes and two singly diagonally implicit RK (SDIRK) schemes. Key insights include the dependence of high-wavenumber numerical dissipation, relied upon for implicit large eddy simulation (ILES), on the choice of temporal scheme and time-step size. Also, the wavespeed characteristics of fully-discrete schemes and the relative dominance of temporal and spatial errors as a function of wavenumber and time-step size are investigated. Salient properties from the aforementioned theoretical analysis are then demonstrated in practice using a linear advection test cases. Finally, a Burgers turbulence test case is used to demonstrate the importance of the temporal discretisation when using FR schemes for ILES.

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Abstract: We begin by investigating the stability, order of accuracy, and dispersion and dissipation characteristics of the extended range of energy stable flux reconstruction (E-ESFR) schemes in the context of implicit large eddy simulation (ILES). We proceed to demonstrate that subsets of the E-ESFR schemes are more stable than collocation nodal discontinuous Galerkin methods recovered with the flux reconstruction approach (FRDG) for marginally-resolved ILES simulations of the Taylor-Green vortex. These schemes are shown to have reduced dissipation and dispersion errors relative to FRDG schemes of the same polynomial degree and, simultaneously, have increased Courant-Friedrichs-Lewy (CFL) limits. Finally, we simulate turbulent flow over an SD7003 aerofoil using two of the most stable E-ESFR schemes identified by the aforementioned Taylor-Green vortex experiments. Results demonstrate that subsets of E-ESFR schemes appear more stable than the commonly used FRDG method, have increased CFL limits, and are suitable for ILES of complex turbulent flows on unstructured grids.

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Abstract: This paper investigates the connections between many popular variants of the well-established discontinuous Galerkin method and the recently developed high-order flux reconstruction approach on irregular tensor-product grids. We explore these connections by analysing three nodal versions of tensor-product discontinuous Galerkin spectral element approximations and three types of flux reconstruction schemes for solving systems of conservation laws on irregular tensor-product meshes. We demonstrate that the existing connections established on regular grids are also valid on deformed and curved meshes for both linear and nonlinear problems, provided that the metric terms are accounted for appropriately. We also find that the aliasing issues arising from nonlinearities either due to a deformed/curved elements or due to the nonlinearity of the equations are equi- valent and can be addressed using the same strategies both in the discontinuous Galerkin method and in the flux reconstruction approach. In particular, we show that the discontinuous Galerkin and the flux reconstruction approach are equivalent also when using higher-order quadrature rules that are commonly employed in the context of over- or consistent-integration-based dealiasing methods. The connections found in this work help to complete the picture regarding the relations between these two numerical approaches and show the possibility of using over- or consistent-integration in an equivalent manner for both the approaches.

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Abstract: The Flux Reconstruction (FR) approach offers an efficient route ton high-order accuracy on unstructured grids. In this work we study then effect of solution point placement on the stability and accuracy of FRn schemes on tetrahedral grids. To accomplish this we generate a largen number of solution point candidates that satisfy various criteria atn polynomial orders P = 3,4,5. We then proceed to assess theirn properties by using them to solve the non-linear Euler equations onn both structured and unstructured meshes. The results demonstrate thatn the location of the solution points is important in terms of both then stability and accuracy. Across a range of cases it is possible ton outperform the solution points of Shunn and Ham for specificn problems. However, there appears to be a degree of problem-dependencen with regards to the optimal point set, and hence overall it isn concluded that the Shunn and Ham points offer a good compromise inn terms of practical utility.

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Abstract: Matrix multiplication is a fundamental linear algebra routine ubiquitous in all areas of science and engineering. Highly optimised BLAS libraries (cuBLAS and clBLAS on GPUs) are the most popular choices for an implementation of the General Matrix Multiply (GEMM) in software. In this paper we present GiMMiK - a generator of bespoke matrix multiplication kernels for the CUDA and OpenCL platforms. GiMMiK exploits a prior knowledge of the operator matrix to generate highly performant code. The performance of GiMMiK's kernels is particularly apparent in a block-bypanel type of matrix multiplication, where the block matrix is typically small (e.g. dimensions of 96 x 64). Such operations are characteristic to our motivating application in PyFR - an implementation of Flux Reconstruction schemes for high-order fluid flow simulations on mixed unstructured meshes. GiMMiK fully unrolls the matrix-vector product and embeds matrix entries directly in the code to benefit from the use of the constant cache and compiler optimisations. Further, it reduces the number of floating-point operations by removing multiplications by zeros. Together with the ability of our kernels to avoid the poorly optimisedrcleanup code, executed by library GEMM, we are able to outperform cuBLAS on two NVIDIA GPUs: GTX 780 Ti and Tesla K40c. We observe speedups of our kernels over cuBLAS GEMM of up to 9.98 and 63.30 times for a 294 x 1029 99% sparse PyFR matrix in double precision on the Tesla K40c and GTX 780 Ti correspondingly. In single precision, observed speedups reach 12.20 and 13.07 times for a 4 x 8 50% sparse PyFR matrix on the two aforementioned cards. Using GiMMiK as the matrix multiplication kernel provider allows us to achieve a speedup of up to 1.70 (2.19) for a simulation of an unsteady flow over a cylinder executed with PyFR in double (single) precision on the Tesla K40c. All results were generated with GiMMiK version 1.0.

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Abstract: The Flux Reconstruction (FR) approach offers an efficient route to achieving high-order accuracy on unstructured grids. Additionally, FR offers a flexible framework for defining a range of numerical schemes in terms of so-called FR correction functions. Recently, a one-parameter family of FR correction functions were identified that lead to stable schemes for 1D linear advection problems. In this study we develop a procedure for identifying an extended range of stable, symmetric, and conservative FR correction functions. The procedure is applied to identify ranges of such correction functions for various orders of accuracy. Numerical experiments are undertaken, and the results found to be in agreement with the theoretical findings.

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Abstract: PyFR is an open-source high-order accurate computational fluid dynamicsnsolver for unstructured grids. In this paper we detail how PyFR has beennextended to run on mixed element meshes, and a range of hardwarenplatforms, including heterogeneous multi-node systems. Performance ofnour implementation is benchmarked using pure hexahedral and mixednprismatic-tetrahedral meshes of the void space around a circularncylinder. Specifically, for each mesh performance is assessed at variousnorders of accuracy on three different hardware platforms; an NVIDIAnTesla K40c GPU, an Intel Xeon E5-2697 v2 CPU, and an AMD FirePro W9100nGPU. Performance is then assessed on a heterogeneous multi-node systemnconstructed from a mix of the aforementioned hardware. Resultsndemonstrate that PyFR achieves performance portability across variousnhardware platforms. In particular, the ability of PyFR to targetnindividual platforms with their 'native' language leads to significantlynenhanced performance c.f. targeting each platform with OpenCLnalone. PyFR is also found to be performant on the heterogeneousnmulti-node system, achieving a significant fraction of the availablenFLOP/s. Finally, each mesh is used to undertake nominally fifth-ordernaccurate long-time simulations of unsteady flow over a circular cylindernat a Reynolds number of 3,900 using a cluster of NVIDIA K20cnGPUs. Long-time dynamics of the wake are studied in detail, and resultsnare found to be in excellent agreement with previousnexperimental/numerical data. All results were obtained with PyFR v0.2.2,nwhich is freely available under a 3-Clause New Style BSD license (seenwww.pyfr.org).

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Abstract: In a high-pressure photoelectron spectrometer, the sample is positioned close to a differential pumping aperture, behind which the pressure is several orders of magnitude lower than the pressure in the analysis chamber. To find the optimal sample position, where the path length of the photoelectrons through the high pressure region is minimized as far as possible without compromising knowledge of the actual pressure at the sample surface, an understanding of the pressure variations near the sample and the aperture is required. A computational fluid dynamics study has been carried out to examine the pressure profiles, and the results are compared against experimental spectra whose intensities are analyzed using the Beer-Lambert law. The resultant pressure profiles are broadly similar to the one previously derived from a simplistic molecular flow model, but indicate that as the pressure in the analysis chamber is raised, the region over which the pressure drop occurs becomes progressively narrower.

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Abstract: There is increasing recognition of the influence of the flow field on functioning of vessels and their development of pathology. This study develops microbubble void imaging, a technique to non-invasively visualise and quantify mixing due to flow in large vessels using ultrasound and controlled destruction of microbubble contrast agents. The generation of microbubble void is safe and non-invasive and can be well controlled both spatially and temporally. Three different vessel geometries, straight, curved and helical, with known effects on mixing were chosen to evaluate the technique. An entropy measure was calculated to quantify the mixing. The experimental results were cross-compared and with Computational Fluid Dynamics (CFD). Results show that the proposed technique is able to quantify the degree of mixing within the different geometric configurations, with a helical geometry generating the highest mixing, and the straight geometry the lowest. The results agree well with CFD. Furthermore, the technique could also serve as a flow visualisation tool.

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Abstract: 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.

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Abstract: In this paper we describe a methodology for the identification of symmetric quadrature rules inside of quadrilaterals, triangles, tetrahedra, prisms, pyramids, and hexahedra. The methodology is free from manual intervention and is capable of identifying a set of rules with a given strength and a given number of points. We also present polyquad which is an implementation of our methodology. Using polyquad v1.0 we proceed to derive a complete set of symmetric rules on the aforementioned domains. All rules possess purely positive weights and have all points inside the domain. Many of the rules appear to be new, and an improvement over those tabulated in the literature.

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Abstract: Arterio-Venous Fistulae (AVF) are the preferred method of vascular access for patients with end stage renal disease who need hemodialysis. In this study simulations of blood flow and oxygen transport were undertaken in various idealized AVF configurations. The objective of the study was to understand how arterial curvature affects blood flow and oxygen transport patterns within AVF, with a focus on how curvature alters metrics known to correlate with vascular pathology such as Intimal Hyperplasia (IH). If one subscribes to the hypothesis that unsteady flow causes IH within AVF, then the results suggest that in order to avoid IH AVF should be formed via a vein graft onto the outer-curvature of a curved artery. However, if one subscribes to the hypothesis that low wall shear stress and/or low lumen-to-wall oxygen flux (leading to wall hypoxia) cause IH within AVF, then the results suggest that in order to avoid IH AVF should be formed via a vein graft onto a straight artery, or the inner-curvature of a curved artery. We note that the recommendations are incompatible - highlighting the importance of ascertaining the exact mechanisms underlying development of IH in AVF. Nonetheless, the results clearly illustrate the important role played by arterial curvature in determining AVF hemodynamics, which to our knowledge has been overlooked in all previous studies.

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Abstract: The flux reconstruction approach offers an efficient route to high-order accuracy on unstructured grids. The location of the solution points plays an important role in determining the stability and accuracy of FR schemes on triangular elements. In particular, it is desirable that a solution point set (i) defines a well conditioned nodal basis for representing the solution, (ii) is symmetric, (iii) has a triangular number of points and, (iv) minimises aliasing errors when constructing a polynomial representation of the flux. In this paper we propose a methodology for generating solution points for triangular elements. Using this methodology several thousand point sets are generated and analysed. Numerical performance is assessed through an Euler vortex test case. It is found that the Lebesgue constant and quadrature strength of the points are strong indicators of stability and performance. Further, at polynomial orders P=4,6,7 solution points with superior performance to those tabulated in literature are discovered.

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Abstract: High-order numerical methods for unstructured grids combine the superior accuracy of high-order spectral or finite difference methods with the geometric flexibility of low-order finite volume or finite element schemes. The Flux Reconstruction (FR) approach unifies various high-order schemes for unstructured grids within a single framework. Additionally, the FR approach exhibits a significant degree of element locality, and is thus able to run efficiently on modern streaming architectures, such as Graphical Processing Units (GPUs). The aforementioned properties of FR mean it offers a promising route to performing affordable, and hence industrially relevant, scale-resolving simulations of hitherto intractable unsteady flows within the vicinity of real-world engineering geometries. In this paper we present PyFR, an open-source Python based framework for solving advection-diffusion type problems on streaming architectures using the FR approach. The framework is designed to solve a range of governing systems on mixed unstructured grids containing various element types. It is also designed to target a range of hardware platforms via use of an in-built domain specific language based on the Mako templating engine. The current release of PyFR is able to solve the compressible Euler and Navier-Stokes equations on grids of quadrilateral and triangular elements in two dimensions, and hexahedral elements in three dimensions, targeting clusters of CPUs, and NVIDIA GPUs. Results are presented for various benchmark flow problems, single-node performance is discussed, and scalability of the code is demonstrated on up to 104 NVIDIA M2090 GPUs. The software is freely available under a 3-Clause New Style BSD license (see www.pyfr.org).

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Abstract: Popular high-order schemes with compact stencils for Computational Fluid Dynamics (CFD) include Discontinuous Galerkin (DG), Spectral Difference (SD), and Spectral Volume (SV) methods. The recently proposed Flux Reconstruction (FR) approach or Correction Procedure using Reconstruction (CPR) is based on a differential formulation and provides a unifying framework for these high-order schemes. Here we present a brief review of recent progress in FR/CPR research as well as some pacing items and future challenges.

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Abstract: With high-order methods becoming more widely adopted throughout the field of computational fluid dynamics, the development of new computationally efficient algorithms has increased tremendously in recent years. One of the most recent methods to be developed is the flux reconstruction approach, which allows various well-known high-order schemes to be cast within a single unifying framework. Whilst a connection between flux reconstruction and the more widely adopted discontinuous Galerkin method has been established elsewhere, it still remains to fully investigate the explicit connections between the many popular variants of the discontinuous Galerkin method and the flux reconstruction approach. In this work, we closely examine the connections between three nodal versions of tensor-product discontinuous Galerkin spectral element approximations and two types of flux reconstruction schemes for solving systems of conservation laws on quadrilateral meshes. The different types of discontinuous Galerkin approximations arise from the choice of the solution nodes of the Lagrange basis representing the solution and from the quadrature approximation used to integrate the mass matrix and the other terms of the discretization. By considering both linear and nonlinear advection equations on a regular grid, we examine the mathematical properties that connect these discretizations. These arguments are further confirmed by the results of an empirical numerical study.

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Abstract: Atherosclerosis is the underlying cause of most heart attacks and strokes. It is thereby the leading cause of death in the Western world, and it places a significant financial burden on healthcare systems. There is evidence that complex, multi-scale arterial mass transport processes play a key role in the development of atherosclerosis. Such processes can be controlled both by blood flow patterns and by properties of the arterial wall. This short review focuses on one vascular scale, flow-regulated arterial mass transport process, namely concentration polarization of Low Density Lipoprotein (LDL) at the luminal surface of the arterial endothelium, and on one cellular-scale, structural determinant of arterial wall mass transport, namely the Endothelial Glycocalyx Layer (EGL). Both have attracted significant attention in recent years. In addition to reviewing and appraising relevant literature, we propose various directions for future work.

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Abstract: High-order methods for unstructured grids provide a promising option for solving challenging problems in computational fluid dynamics. Flux reconstruction (FR) is a framework which unifies a number of these high-order methods, such as the spectral difference (SD) and collocation-based nodal discontinuous Galerkin (DG) methods, allowing for their more concise and flexible implementation. Additionally, the FR approach can be used to facilitate development of new numerical methods that offer arbitrary orders of accuracy on unstructured grids. In previous work, it has been shown that a particular range of FR schemes, referred to as Vincent-Castonguay-Jameson-Huynh (VCJH) schemes, are guaranteed to be stable for linear advection problems for all orders of accuracy. There have remained questions, however, regarding the stability of FR schemes for advection-diffusion problems. In this study a new class of VCJH schemes is developed for solving one-dimensional advection-diffusion problems. For the first time, it is shown that the schemes are linearly stable for linear advection-diffusion problems for all orders of accuracy on nonuniform grids. Linear and nonlinear numerical experiments are performed in 1D and 2D to investigate the accuracy and stability properties of the new schemes. The results indicate that certain VCJH schemes for advection?diffusion problems possess significantly higher explicit time-step limits than discontinuous Galerkin schemes, while still maintaining the expected order of accuracy.

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Abstract: The Flux Reconstruction (FR) approach unifies several well-known high-order schemes for unstructured grids, including a collocation-based nodal discontinuous Galerkin (DG) method and all types of Spectral Difference (SD) methods, at least for linear problems. The FR approach also allows for the formulation of new families of schemes. Of particular interest are the energy stable FR schemes, also referred to as the Vincent-Castonguay-Jameson-Huynh (VCJH) schemes, which are an infinite family of high-order schemes parameterized by a single scalar. VCJH schemes are of practical importance because they provide a stable formulation on triangular elements which are often required for numerical simulations over complex geometries. In particular, VCJH schemes are provably stable for linear advection problems on triangles, and include the collocation-based nodal DG scheme on triangles as a special case. Furthermore, certain VCJH schemes have Courant-Friedrichs-Lewy (CFL) limits which are approximately twice those of the collocation-based nodal DG scheme. Thus far, these schemes have been analyzed primarily in the context of pure advection problems on triangles. For the first time, this paper constructs VCJH schemes for advection-diffusion problems on triangles, and proves the stability of these schemes for linear advection-diffusion problems for all orders of accuracy. In addition, this paper uses numerical experiments on triangular grids to verify the stability and accuracy of VCJH schemes for linear advection-diffusion problems and the nonlinear Navier-Stokes equations.

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Abstract: The flux reconstruction (FR) approach allows various well-known high-order schemes, such as collocation based nodal discontinuous Galerkin (DG) methods and spectral difference (SD) methods, to be cast within a single unifying framework. Recently, the authors identified a new class of FR schemes for 1D conservation laws, which are simple to implement, efficient and guaranteed to be linearly stable for all orders of accuracy. The new schemes can easily be extended to quadrilateral elements via the construction of tensor product bases. However, for triangular elements, such a construction is not possible. Since numerical simulations over complicated geometries often require the computational domain to be tessellated with simplex elements, the development of stable FR schemes on simplex elements is highly desirable. In this article, a new class of energy stable FR schemes for triangular elements is developed. The schemes are parameterized by a single scalar quantity, which can be adjusted to provide an infinite range of linearly stable high-order methods on triangular elements. Von Neumann stability analysis is conducted on the new class of schemes, which allows identification of schemes with increased explicit time-step limits compared to the collocation based nodal DG method. Numerical experiments are performed to confirm that the new schemes yield the optimal order of accuracy for linear advection on triangular grids.

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Abstract: The flux reconstruction (FR) approach unifies various high-order schemes, including collocation based nodal discontinuous Galerkin (DG) methods, and all spectral difference methods (at least for a linear flux function), within a single framework. Recently a new range of linearly stable FR schemes have been identified, henceforth referred to as Vincent-Castonguay-Jameson-Huynh (VCJH) schemes. In this short note non-linear stability properties of FR schemes are elucidated via analysis of linearly stable VCJH schemes (so as to focus attention solely on issues of non-linear stability). It is shown that linearly stable VCJH schemes (at least in their standard form) may be unstable if the flux function is non-linear. This instability is due to aliasing errors, which manifest since FR schemes (in their standard form) utilize a collocation projection at the solution points to construct a polynomial approximation of the flux. Strategies for minimizing such aliasing driven instabilities are discussed within the context of the FR approach. In particular, it is shown that the location of the solution points will have a significant effect on non-linear stability. This result is important, since linear analysis of FR schemes implies stability is independent of solution point location. Finally, it is shown that if an exact L2 projection is employed to construct an approximation of the flux (as opposed to a collocation projection), then aliasing errors and hence aliasing driven instabilities will be eliminated. However, performing such a projection exactly, or at least very accurately, would be more costly than performing a collocation projection, and would certainly impact the inherent efficiency and simplicity of the FR approach. It can be noted that in all above regards, non-linear stability properties of FR schemes are similar to those of nodal DG schemes. The findings should motivate further research into the non-linear performance of FR schemes, which have hitherto been developed and analyzed solely in the context of a linear flux function.

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Abstract: The flux reconstruction approach to high-order methods is robust, efficient, simple to implement, and allows various high-order schemes, such as the nodal discontinuous Galerkin method and the spectral difference method, to be cast within a single unifying framework. Utilizing a flux reconstruction formulation, it has been proved (for one-dimensional linear advection) that the spectral difference method is stable for all orders of accuracy in a norm of Sobolev type, provided that the interior flux collocation points are located at zeros of the corresponding Legendre polynomials. In this article the aforementioned result is extended in order to develop a new class of one-dimensional energy stable flux reconstruction schemes. The energy stable schemes are parameterized by a single scalar quantity, which if chosen judiciously leads to the recovery of various well known high-order methods (including a particular nodal discontinuous Galerkin method and a particular spectral difference method). The analysis offers significant insight into why certain flux reconstruction schemes are stable, whereas others are not. Also, from a practical standpoint, the analysis provides a simple prescription for implementing an infinite range of energy stable high-order methods via the intuitive flux reconstruction approach.

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Abstract: The distribution of atherosclerotic lesions within the rabbit vasculature, particularly within the descending thoracic aorta, has been mapped in numerous studies. The patchy nature of such lesions has been attributed to local variation in the pattern of blood flow. However, there have been few attempts to model and characterize the flow. In this study, a high-order continuous Galerkin finite-element method was used to simulate blood flow within a realistic representation of the rabbit aortic arch and descending thoracic aorta. The geometry, which was obtained from computed tomography of a resin corrosion cast, included all vessels originating from the aortic arch (followed to at least their second generation) and five pairs of intercostal arteries originating from the proximal descending thoracic aorta. The simulations showed that small geometrical undulations associated with the ductus arteriosus scar cause significant deviations in wall shear stress (WSS). This finding highlights the importance of geometrical accuracy when analysing WSS or related metrics. It was also observed that two Dean-type vortices form in the aortic arch and propagate down the descending thoracic aorta (along with an associated skewed axial velocity profile). This leads to the occurrence of axial streaks in WSS, similar in nature to the axial streaks of lipid deposition found in the descending aorta of cholesterol-fed rabbits. Finally, it was observed that WSS patterns within the vicinity of intercostal branch ostia depend not only on local flow features caused by the branches themselves, but also on larger-scale flow features within the descending aorta, which vary between branches at different locations. This result implies that disease and WSS patterns in the vicinity of intercostal ostia are best compared on a branch-by-branch basis.

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Abstract: The flux reconstruction (FR) approach unifies various high-order schemes, including collocation based nodal discontinuous Galerkin methods, and all spectral difference methods (at least for a linear flux function), within a single framework. Recently, an infinite number of linearly stable FR schemes were identified, henceforth referred to as Vincent-Castonguay-Jameson-Huynh (VCJH) schemes. Identification of VCJH schemes offers significant insight into why certain FR schemes are stable (whereas others are not), and provides a simple prescription for implementing an infinite range of linearly stable high-order methods. However, various properties of VCJH schemes have yet to be analyzed in detail. In the present study one-dimensional (1D) von Neumann analysis is employed to elucidate how various important properties vary across the full range of VCJH schemes. In particular, dispersion and dissipation properties are studied, as are the magnitudes of explicit time-step limits (based on stability considerations). 1D linear numerical experiments are undertaken in order to verify results of the 1D von Neumann analysis. Additionally, two-dimensional non-linear numerical experiments are undertaken in order to assess whether results of the 1D von Neumann analysis (which is inherently linear) extend to real world problems of practical interest.

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Abstract: Theoretical studies and numerical experiments suggest that unstructured high-order methods can provide solutions to otherwise intractable fluid flow problems within complex geometries. However, it remains the case that existing high-order schemes are generally less robust and more complex to implement than their low-order counterparts. These issues, in conjunction with difficulties generating high-order meshes, have limited the adoption of high-order techniques in both academia (where the use of low-order schemes remains widespread) and industry (where the use of low-order schemes is ubiquitous). In this short review, issues that have hitherto prevented the use of high-order methods amongst a non-specialist community are identified, and current efforts to overcome these issues are discussed. Attention is focused on four areas, namely the generation of unstructured high-order meshes, the development of simple and efficient time integration schemes, the development of robust and accurate shock capturing algorithms, and finally the development of high-order methods that are intuitive and simple to implement. With regards to this final area, particular attention is focused on the recently proposed flux reconstruction approach, which allows various well known high-order schemes (such as nodal discontinuous Galerkin methods and spectral difference methods) to be cast within a single unifying framework. It should be noted that due to the experience of the authors the review is directed somewhat towards aerodynamic applications and compressible flow. However, many of the discussions have a wider applicability. Moreover, the tone of the review is intended to be generally accessible, such that an extended scientific community can gain insight into factors currently pacing the adoption of unstructured high-order methods.

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Abstract: It has been postulated that a flow-dependent (and hence spatially varying) low density lipoprotein (LDL) concentration polarisation layer forms on the luminal surface of the vascular endothelium. Such a layer has the potential to cause heterogeneity in the distribution of atherosclerotic lesions by spatially modulating the rate of LDL transport into the arterial wall. Theoretical analysis suggests that a transmural water flux which is spatially heterogeneous at the cellular scale can act to enhance LDL concentration polarisation in a shear dependent fashion. However, such an effect is only observed if a relevant Peclet number (i.e. the ratio of LDL convection to LDL diffusion) is of order unity or greater. Based on the diffusivity of LDL in blood plasma, such a Peclet number is found to be far less than unity, implying that the aforementioned enhancement and shear dependence will not occur. However, this conclusion ignores the existence of the endothelial glycocalyx layer (EGL), which may inhibit the diffusion of LDL near the luminal surface of the endothelium, and hence raise any Peclet number associated with the transport of LDL. The present study numerically investigates the effect of the EGL, as well as a heterogeneous transmural water flux, on arterial LDL concentration polarisation. Particular attention is paid to measures of LDL concentration polarisation thought relevant to the rate of transendothelial LDL transport. It is demonstrated that an EGL is unlikely to cause any additional shear dependence of such measures directly, irrespective of whether or not LDL can penetrate into the EGL. However, it is found that such measures depend significantly on the nature of the interaction between LDL and the EGL (parameterised by the height of the EGL, the depth to which LDL penetrates into the EGL, and the diffusivity of LDL in the EGL). Various processes may regulate the interaction of LDL with the EGL, possibly in a flow dependent and hence spatially non-uniform fashion. It is concluded that any such processes may be as important as vascular scale flow features in terms of spatially modulating transendothelial LDL transport via an LDL concentration polarisation mechanism.

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Abstract: Objectives: We used optical coherence tomography, which has a resolution of <20 &#956 m, to analyze thin layers of neointima in rapamycin-eluting coronary stents. Background: Lack of neointimal coverage has been implicated in the pathogenesis of drug-eluting coronary stent thrombosis. Angiography and intracoronary ultrasound lack the resolution to examine this. Methods: We conducted a randomized trial in patients receiving polymer-coated rapamycin-eluting stents (Cypher, Cordis, Johnson & Johnson, Miami, Florida) and nonpolymer rapamycin-eluting stents (Yukon, Translumina, Hechingen, Germany) to examine neointimal thickness, stent strut coverage, and protrusion at 90 days. Twenty-four patients (n = 12 for each group) underwent stent deployment and invasive follow-up at 90 days with optical coherence tomography. The primary end point was binary stent strut coverage. Coprimary end points were neointimal thickness and stent strut luminal protrusion. Results: No patient had angiographic restenosis. For polymer-coated and nonpolymer rapamycin-eluting stents, respectively, mean (SD), neointimal thickness was 77.2 (25.6) &#956 m versus 191.2 (86.7) &#956 m (p < 0.001). Binary stent strut coverage was 88.3% (11.8) versus 97.2% (6.1) (p = 0.030). Binary stent strut protrusion was 26.5% (17.5) versus 4.8% (8.6) (p = 0.001). Conclusions: Mean neointimal thickness for the polymer-coated rapamycin-eluting stent was significantly less than the nonpolymer rapamycin-eluting stent but as a result coverage was not homogenous, with >10% of struts being uncovered. High-resolution imaging allowed development of the concept of the protrusion index, and >25% of struts protruded into the vessel lumen with the polymer-coated rapamycin-eluting stent compared with <5% with the nonpolymer rapamycin-eluting stent. These findings may have important implications for the risk of stent thrombosis and, therefore, future stent design. (An optical coherence tomography study to determine stent coverage in polymer coated versus bare metal rapamycin eluting stents. ORCA 1, from the Optimal Revascularization of the Coronary Arteries group; ISRCTN42475919)

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Abstract: Uptake of low density lipoprotein (LDL) by the arterial wall is likely to play a key role in atherogenesis. A particular process that may cause vascular scale heterogeneity in the rate of transendothelial LDL transport is the formation of a flow-dependent LDL concentration polarization layer on the luminal surface of the arterial endothelium. In this study, the effect of a spatially heterogeneous transmural water flux (that traverses the endothelium only via interendothelial cell clefts) on such concentration polarization is investigated numerically. Unlike in previous investigations, realistic intercellular cleft dimensions are used here and several values of LDL diffusivity are considered. Particular attention is paid to the spatially averaged LDL concentration adjacent to different regions of the endothelial surface, as such measures may be relevant to the rate of transendothelial LDL transport. It is demonstrated in principle that a heterogeneous transmural water flux can act to enhance such measures, and cause them to develop a shear dependence (in addition to that caused by vascular scale flow features, affecting the overall degree of LDL concentration polarization). However, it is shown that this enhancement and additional shear dependence are likely to be negligible for a physiologically realistic transmural flux velocity of 0.0439 &#956 ms^-1 and an LDL diffusivity (in blood plasma) of 28.67 &#956 m²s^-1. Hence, the results imply that vascular scale studies of LDL concentration polarization are justified in ignoring the effect of a spatially heterogeneous transmural water flux.

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Abstract: An analytic series solution is presented for the shear driven flow of a viscous fluid over an infinite series of outflow slits covered by a Brinkman medium with an anisotropic Darcy permeability. The solution is used to model the cellular scale flow of water over and within the endothelial glycocalyx, when the transmural water flux through the vascular endothelium is only allowed to pass via interendothelial cell clefts. Results are presented illustrating the effect of both the glycocalyx properties and the applied shearing rate (imposed by vascular scale fluid dynamics) on several relevant measures of the velocity field, including the wall normal velocity and the shear rate evaluated at the luminal surface of the glycocalyx.

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Abstract: Experimental results are obtained for a roamx-0201 type airfoil and the clf5605 airfoil at highsubsonic, low Reynolds number conditions using the Tohoku University Mars Wind Tunnel, Japan. The tests are conducted at a Mach number of M = 0.60, and a Reynolds number of Re = 20,000 to reflect representative aerodynamics of a rotor blade for Mars exploration. The angle of attack is varied between alpha = -2.0 deg and alpha = 6.0 deg. The roamx-0201 type airfoil is an unconventional airfoil optimized for the chosen tunnel operating conditions using the Evolutionary aLgorithm for Iterative Studies of Aeromechanics (ELISA), developed under the Rotor Optimization for the Advancement of Mars eXploration (ROAMX) project. ELISA is utilized here to optimize aerodynamic airfoil performance using a Genetic Algorithm and two-dimensional high-fidelity CFD simulations, ultimately resulting in a Pareto-optimal airfoil set. The clf5605 airfoil is the outboard airfoil used on the Ingenuity Mars Helicopter and provides a baseline against which the roamx-0201, as well as possible future airfoil profiles for the compressible low Reynolds number regime, can be compared against. Lift and drag data are recorded using a balance, pressure distributions are obtained using Pressure Sensitive Paint (PSP) application, and Schlieren images are obtained to visualize the flowfield. The data is tabulated to aid future research.

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