Publications

Journal Papers

Under Review
Heterogeneity in the Non-Planarity and Arterial Curvature of Arteriovenous Fistulae In Vivo. R. Corbett, L. Grechy, F. Iori, J. S. Crane, P. E. Herbert, P. Di Cocco, W. Gedroyc, P. E. Vincent, C. G. Caro, N. D. Duncan. Submitted for publication in Clinical Journal of the American Society of Nephrology
2017
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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2016
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 to high-order accuracy on unstructured grids. In this work we study the effect of solution point placement on the stability and accuracy of FR schemes on tetrahedral grids. To accomplish this we generate a large number of solution point candidates that satisfy various criteria at polynomial orders P = 3,4,5. We then proceed to assess their properties by using them to solve the non-linear Euler equations on both structured and unstructured meshes. The results demonstrate that the location of the solution points is important in terms of both the stability and accuracy. Across a range of cases it is possible to outperform the solution points of Shunn and Ham for specific problems. However, there appears to be a degree of problem-dependence with regards to the optimal point set, and hence overall it is concluded that the Shunn and Ham points offer a good compromise in 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 optimised cleanup 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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2015
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 dynamics solver for unstructured grids. In this paper we detail how PyFR has been extended to run on mixed element meshes, and a range of hardware platforms, including heterogeneous multi-node systems. Performance of our implementation is benchmarked using pure hexahedral and mixed prismatic-tetrahedral meshes of the void space around a circular cylinder. Specifically, for each mesh performance is assessed at various orders of accuracy on three different hardware platforms; an NVIDIA Tesla K40c GPU, an Intel Xeon E5-2697 v2 CPU, and an AMD FirePro W9100 GPU. Performance is then assessed on a heterogeneous multi-node system constructed from a mix of the aforementioned hardware. Results demonstrate that PyFR achieves performance portability across various hardware platforms. In particular, the ability of PyFR to target individual platforms with their 'native' language leads to significantly enhanced performance c.f. targeting each platform with OpenCL alone. PyFR is also found to be performant on the heterogeneous multi-node system, achieving a significant fraction of the available FLOP/s. Finally, each mesh is used to undertake nominally fifth-order accurate long-time simulations of unsteady flow over a circular cylinder at a Reynolds number of 3,900 using a cluster of NVIDIA K20c GPUs. Long-time dynamics of the wake are studied in detail, and results are found to be in excellent agreement with previous experimental/numerical data. All results were obtained with PyFR v0.2.2, which is freely available under a 3-Clause New Style BSD license (see www.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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2014
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: 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: 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: 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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2013
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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2012
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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2011
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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2010
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.

Link:
2009
Abstract: Objectives: We used optical coherence tomography, which has a resolution of <20 μ 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) μ m versus 191.2 (86.7) μ 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)

Link:
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 μ ms-1 and an LDL diffusivity (in blood plasma) of 28.67 μ 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.

Link:
2008
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.

Link:

Conference Papers

2016
Towards Green Aviation with Python at Petascale. P. E. Vincent, F. D. Witherden, B. C. Vermeire, J. S. Park, A. Iyer. ACM Gordon Bell Finalist and Best Paper Finalist. Article 1. SC16 Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, 13-18 November 2016, Salt Lake City, Utah, USA
2015
Using the pyMIC Offload Module in PyFR. M. Klemm, F. D. Witherden, P. E. Vincent . 8th European Conference on Python in Science, 28-29 August 2015, Cambridge, UK
On the Utility of High-Order Methods for Unstructured Grids: A Comparison Between PyFR and Industry Standard Tools. B. C. Vermeire, F. D. Witherden, P. E. Vincent . Paper AIAA-2015-2743. 22nd AIAA Computational Fluid Dynamics Conference, 22-26 June 2015, Dallas, Texas, USA
PyFR: Next-Generation High-Order Computational Fluid Dynamics on Many-Core Hardware. P. E. Vincent, F. D. Witherden, A. M. Farrington, G. Ntemos, B. C. Vermeire, J. S. Park, A. S. Iyer . Paper AIAA-2015-3050. 22nd AIAA Computational Fluid Dynamics Conference, 22-26 June 2015, Dallas, Texas, USA
2014
A Guide to the Implementation of Boundary Conditions in Compact High-Order Methods for Compressible Aerodynamics. G. Mengaldo, D. De Grazia, J. Peiro, F. D. Witherden, A. M. Farrington, P. E. Vincent, S. J. Sherwin . Paper AIAA-2014-2923. 7th AIAA Theoretical Fluid Mechanics Conference, 16-20 June 2014, Atlanta, Georgia, USA
2013
High-Order Methods for Computational Fluid Dynamics: A Brief Review of Compact Differential Formulation on Unstructured Grids. H. T. Huynh, Z. J. Wang, P. E. Vincent. Paper AIAA-2013-2564. 21st AIAA Computational Fluid Dynamics Conference, 24-27 June 2013, San Diego, California, USA
2011
On the Development of a High-Order, Multi-GPU Enabled, Compressible Viscous Flow Solver for Mixed Unstructured Grids. P. Castonguay, D. Williams, P. E. Vincent, M. Lopez, A. Jameson. Paper AIAA-2011-3229. 20th AIAA Computational Fluid Dynamics Conference, 27-30 June 2011, Honolulu, Hawaii, USA
An Extension of Energy Stable Flux Reconstruction to Unsteady, Non-linear, Viscous Problems on Mixed Grids. D. Williams, P. Castonguay, P. E. Vincent, A. Jameson. Paper AIAA-2011-3405. 20th AIAA Computational Fluid Dynamics Conference, 27-30 June 2011, Honolulu, Hawaii, USA
Application of High-Order Energy Stable Flux Reconstruction Schemes to the Euler Equations. P. Castonguay, P. E. Vincent, A. Jameson. Paper AIAA-2011-686. 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 4-7 January 2011, Orlando, Florida, USA
High-Order Methods for Diffusion Equation with Energy Stable Flux Reconstruction Scheme. K. Ou, P. E. Vincent, A. Jameson. Paper AIAA-2011-46. 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 4-7 January 2011, Orlando, Florida, USA
Presenting Author

Conference Contributions

2017
High-Order Incompressible Computational Fluid Dynamics on GPUs and Co-Processors . N. Loppi, F. D. Witherden. A. Jameson, P. E. Vincent. Oral Presentation. United States National Congress on Computational Mechanics, July 7-20 2017. Montreal, Canada
Optimal Runge-Kutta Schemes for Pseudo Time Stepping with High-Order Methods. B. C. Vermeire, N. Loppi, P. E. Vincent. Oral Presentation. United States National Congress on Computational Mechanics, July 7-20 2017. Montreal, Canada
Three-Dimensional High-Order Simulations of Low-Pressure Turbine Blade Cascades. A. Iyer, F. D. Witherden, B. C. Vermeire, Y. Abe, R. D. Baier, A. Jameson, P. E. Vincent. Oral Presentation. United States National Congress on Computational Mechanics, July 7-20 2017. Montreal, Canada
Suppressing Unsteady Flow in Arterio-Venous Fistulae. L. Grechy, F. Iori, R. Corbett, W. Gedroyc, N. Duncan, C. G. Caro, P. E. Vincent. Poster Presentation. Summer Biomechanics Bioengineering and Biotransport Conference, 21-24 June 2017. Tucson, Arizona, USA
High-Order Accurate Scale-Resolving Simulations of Low-Pressure Turbine Linear Cascades using Python at Petascale. P. E. Vincent, A. S. Iyer, F. D. Witherden, B. C. Vermeire, Y. Abe, R. D. Baier, A. Jameson. Poster Presentation. OLCF Users Meeting, 23-25 May 2017. Oak Ridge, Tennessee, USA
High-Order Accurate Scale-Resolving Simulations of Low-Pressure Turbine Linear Cascades using Python at Petascale. P. E. Vincent, A. S. Iyer, F. D. Witherden, B. C. Vermeire, Y. Abe, R. D. Baier, A. Jameson. Oral Presentation. ParCFD, 15-17 May 2017. Glasgow, UK
Towards Green Aviation with Python at Petascale. P. E. Vincent. Oral Presentation. EPSRC Tier 2 Launch Event, 30 March 2017. Birmingham, UK
Eigenmodes of Small Perturbations to Turbulent Channel Flow. A. S. Iyer, S. Chernyshenko, P. E. Vincent. First SIG Meeting in UK Fluids Network, 29-30 March 2017. Southampton, UK
2016
Coupled Fluid-Wall Oxygen Transport In Arterio-Venous Fistulae: Impact of Mechanical Stresses. F. Iori, L. Grechy, R. Corbett, N. Duncan, C. G. Caro, P. E. Vincent. Oral Presentation. World Congress on Computational Mechanics, 24-29 July 2016. Seoul, South Korea
PyFR: Open-Source Software for Multi-Platform Computational Fluid Dynamics with High-Order Flux Reconstruction Schemes. P. E. Vincent. Oral Presentation. World Congress on Computational Mechanics, 24-29 July 2016. Seoul, South Korea
Anti-Aliasing of Flux Reconstruction Schemes in Rotating Curved Elements. J. S. Park, P. E. Vincent. Oral Presentation. World Congress on Computational Mechanics, 24-29 July 2016. Seoul, South Korea
On the Stability and Accuracy of Flux Reconstruction Schemes for Implicit Large Eddy simulation of Turbulent Flows. B. C. Vermeire, P. E. Vincent. Oral Presentation. World Congress on Computational Mechanics, 24-29 July 2016. Seoul, South Korea
From Arteries to Aircraft: Computational Fluid Dynamics Across Disciplines. P. E. Vincent. Keynote Oral Presentation. Osborne Reynolds Day, 14 July 2016. Manchester, UK
Towards the Industrial Adoption of GPU-Accelerated High-Order Computational Fluid Dynamics. P. E. Vincent. Oral presentation. European Congress on Computational Methods in Applied Sciences and Engineering, 6-10 June 2016. Crete Island, Greece
Coupled Fluid-Wall Oxygen Transport in Arteriovenous Fistulae: Impact of Mechanical Stresses. F. Iori, L. Grechy, R. Corbett, W. Gedroyc, N. Duncan, C. G. Caro, P. E. Vincent. Poster presentation. UK Kidney Week, 7-10 June 2016. Birmingham, UK
Investigating Flow Unsteadiness in Realistic and Optimised Arteriovenous Fistulae. L. Grechy, F. Iori, R. Corbett, W. Gedroyc, N. Duncan, C. G. Caro, P. E. Vincent. Poster presentation. UK Kidney Week, 7-10 June 2016. Birmingham, UK
The Effect of In-Plane Curvature on the Flow Field in Arteriovenous Fistulae. R. Corbett, L. Grechy, F. Iori, W. Gedroyc, P. E. Vincent, C. G. Caro, N. Duncan. Poster presentation. UK Kidney Week, 7-10 June 2016. Birmingham, UK
Investigating Flow Unsteadiness in Realistic and Optimised Arterio-Venous Fistulae. L. Grechy, F. Iori, R. Corbett, W. Gedroyc, N. Duncan, C. G. Caro, P. E. Vincent. Oral presentation. European Congress on Computational Methods in Applied Sciences and Engineering, 6-10 June 2016. Crete Island, Greece
Simulation of Compressible Flow over Rectangular Cavities using the Flux Reconstruction Approach. A. S. Iyer, P. E. Vincent. Oral presentation. European Congress on Computational Methods in Applied Sciences and Engineering, 6-10 June 2016. Crete Island, Greece
On the Stability and Accuracy of Flux Reconstruction Schemes for Implicit Large Eddy Simulation of Turbulent Flows. B. C. Vermeire, P. E. Vincent. Oral presentation. European Congress on Computational Methods in Applied Sciences and Engineering, 6-10 June 2016. Crete Island, Greece
Problem AS1: Flow Over a Cylinder at Re = 3900. B. C. Vermeire, J. S. Park, P. E. Vincent. Oral Presentation, 4th High-Order Workshop, 4-5 June 2016. Crete Island, Greece
PyFR. P. E. Vincent. Oral Presentation, 4th High-Order Workshop, 4-5 June 2016. Crete Island, Greece
Towards the Industrial Adoption of GPU Accelerated Computational Fluid Dynamics. P. E. Vincent. Oral Presentation, Nvidia GPU Technology Conference, 4-7 April 2016. San Jose, CA, USA
Petascale Computational Fluid Dynamics with Python on GPUs. F. D. Witherden, P. E. Vincent. Oral Presentation, Nvidia GPU Technology Conference, 4-7 April 2016. San Jose, CA, USA
Multi-Physics Modelling and Optimisation of Arterio-Venous Fistulae for Haemodialysis with STAR-CCM+. L. Grechy, F. Iori, R. Corbett, W. Gedroyc, N. Duncan, C. G. Caro, P. E. Vincent. Oral Presentation, STAR Global Conference, 7-9 March 2016. Prague, Czech Republic
2015
PyFR: The Flux Reconstruction Scheme and Next-Generation Computing. B. C. Vermeire, P. E. Vincent. Oral Presentation, Rolls Royce CFD Annual Review, 19-20 November 2015. Derby, UK
PyFR: High-Order Unstructured Simulations of Compressible Turbulent Flow. B. C. Vermeire, P. E. Vincent. Oral Presentation, Rolls Royce CFD Annual Review, 19-20 November 2015. Derby, UK
Beating cuBLAS: Automatically Generating Bespoke Matrix Multiplication Kernels Using GiMMiK. F. D. Witherden, B. D. Wozniak, F. P. Russell, P. E. Vincent, P. H. J. Kelly. Poster Presentation, Supercomputing 15, 15-20 December 2015. Austin, Texas, USA
Theoretical Aspects of High-Order Flux Reconstruction Schemes. P. E. Vincent, A. M. Farrington, F. D. Witherden, A. Jameson. Oral Presentation, Oberwolfach Workshop on Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, 14-18 September 2015. Oberwolfach, Germany
PyFR: Next Generation High-Order Computational Fluid Dynamics on Many-Core Hardware. P. E. Vincent. Oral Presentation, Oberwolfach Workshop on Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, 14-18 September 2015. Oberwolfach, Germany
PyFR: Next Generation High-Order Computational Fluid Dynamics on Many-Core Hardware. P. E. Vincent. Keynote Oral Presentation, IMA Conference on Numerical Methods for Simulation, 1-4 September 2015. Oxford, UK
PyFR: Heterogeneous Computing on Mixed Unstructured Grids with Python. F. D. Witherden, M. Klemm, P. E. Vincent. Oral Presentation, 8th European Conference on Python in Science, 28-29 August 2015. Cambridge, UK
PyFR: High-Order Accurate Computational Fluid Dynamics on Unstructured Grids. P. E. Vincent. Oral Presentation, 13th US National Congress on Computational Mechanics, 27-30 July 2015. San Diego, California, USA
Simulating Unsteady Flow over a NACA 0021 Airfoil in Deep Stall with PyFR. J. S. Park, F. D. Witherden, P. E. Vincent. Oral Presentation, 13th US National Congress on Computational Mechanics, 27-30 July 2015. San Diego, California, USA
PyFR: Next-Generation Computational Fluid Dynamics. P. E. Vincent, F. D. Witherden, A. M. Farrington, G. Ntemos, B. C. Vermeire, J. S. Park, A. S. Iyer, N. Loppi. Oral Presentation, EMiT Conference, 1 July 2015. Manchester, UK
Next-Generation Computational Fluid Dynamics: High-Order Methods and Many-Core Hardware. P. E. Vincent. Keynote Oral Presentation, CFD IMPACT Conference, 30 June 2015. Haifa, Israel
Theoretical Aspects of High-Order Flux Reconstruction Schemes. P. E. Vincent, A. M. Farrington, F. D. Witherden, A. Jameson. Invited Oral Presentation, 22nd AIAA Computational Fluid Dynamics Conference, 22-26 June 2015. Dallas, Texas, USA
The Effect of Vascular Curvature on Blood Flow and Oxygen Transport in Arterio-Venous Fistulae. F. Iori, L. Grechy, R. W. Corbett, W. Gedroyc, N. Duncan, C. G. Caro, P. E. Vincent. Poster Presentation, Summer Biomechanics Bioengineering and Biotransport Conference, 17-20 June 2015. Snowbird, Utah, USA
Microbubble Void Imaging - A Novel Technique For Flow Visualisation and Quantitative Assessment of Intravascular Mixing in Larger Vessels Using Ultrasound. C. Leow, F. Iori, R. W. Corbett, N. Duncan, C. G. Caro, P. E. Vincent, M. Tang. Poster Presentation, Summer Biomechanics Bioengineering and Biotransport Conference, 17-20 June 2015. Snowbird, Utah, USA
PyFR: Next Generation Computational Fluid Dynamics on GPU Platforms. P. E. Vincent, F. D. Witherden, A. M. Farrington, G. Ntemos, B. C. Vermeire, J. S. Park, A. S. Iyer. Oral Presentation, Nvidia GPU Technology Conference, 17-20 March 2015. San Jose, CA, USA
GiMMiK: Generating Bespoke Matrix Multiplication Kernels. F. D. Witherden, B. D. Wozniak, F. P. Russel, P. E. Vincent, P. H. J. Kelly. Oral Presentation, Nvidia GPU Technology Conference, 17-20 March 2015. San Jose, CA, USA
Heterogeneous Computing with a Homogeneous Codebase. F. D. Witherden, P. E. Vincent. Oral presentation, Society for Industrial and Applied Mathematics Computational Science and Engineering Conference, 14-18 March 2015. Salt Lake City, Utah, USA
On the Utility of High-Order Methods for Unstructured Grids: A Comparison Between PyFR and Industry Standard Tools. B. C. Vermeire, F. D. Witherden, P. E. Vincent. Oral presentation, Society for Industrial and Applied Mathematics Computational Science and Engineering Conference, 14-18 March 2015. Salt Lake City, Utah, USA
Theoretical Aspects of High-Order Flux Reconstruction Schemes. P. E. Vincent, F. D. Witherden, A. M. Farrington, A. Jameson. Oral presentation, Society for Industrial and Applied Mathematics Computational Science and Engineering Conference, 14-18 March 2015. Salt Lake City, Utah, USA
Problem C3.3: Direct Numerical Simulation of the Taylor Green Vortex. B. C. Vermeire, F. D. Witherden, P. E. Vincent. Oral Presentation, 3rd High-Order Workshop, 3-4 January 2015. Orlando, Florida, USA
High-Order Simulations of Turbulent Flow Over a Circular Cylinder. B. C. Vermeire, F. D. Witherden, P. E. Vincent. Oral Presentation, 3rd High-Order Workshop, 3-4 January 2015. Orlando, Florida, USA
2014
Heterogeneous Computing on Mixed Unstructured Grids with PyFR. F. D. Witherden, B. C. Vermeire, P. E. Vincent. Oral Presentation, UK Many-Core Developer Conference 2014, 15 December 2014. Cambridge, UK
PyFR: Heterogeneous Computing from a Homogeneous Codebase. P. E. Vincent. Invited oral presentation, AJ80th Symposium, 21-22 November 2014. Palo Alto, California, USA
PyFR: An Open Source Python Framework for High-Order CFD on Heterogeneous Platforms. F. D. Witherden, B. C. Vermeire, P. E. Vincent. Poster Presentation, Super Computing 2014, 16-21 November 2014. New Orleans, Louisiana, USA
Significant Heterogeneity Exists in the Conformation of Native Arteriovenous Fistulae. R. W. Corbett, L. Grechy, P. E. Herbert, J. Crane, F. Iori, W. M. Gedroyc, C. Caro, P. E. Vincent, N. D. Duncan. Poster Presentation, American Society of Nephrology Kidney Week, 11-16 November 2014. Philadelphia, Pennsylvania, USA
Intraoperative Measurement of Vessel Wall Oxygenation During Formation of Native Arteriovenous Fistulae. N. T. Clancy, R. W. Corbett, P. E. Herbert, J. Crane, P. E. Vincent, C. Caro, N. D. Duncan, D. Elson. Poster Presentation, American Society of Nephrology Kidney Week, 11-16 November 2014. Philadelphia, Pennsylvania, USA
The Effect of Vascular Curvature on Blood Flow and Oxygen Transport in Arterio-Venous Fistulae. L. Grechy, F. Iori, R. Corbett, W. Gedroyc, N. Duncan, C. G. Caro, P. E. Vincent. Poster Presentation, 2014 BMES Annual Meeting, 22-25 October 2014. San Antonio, Texas, USA
GPU-Accelerated High-Order Simulations of Flow over a Tandem Rod-Aerofoil Configuration using the Flux Reconstruction Approach. G. Ntemos, P. E. Vincent. Oral Presentation, 3rd International Workshop on Computational Engineering, 6-10 October 2014. Stuttgart, Germany
Heterogeneous Computing on Mixed Unstructured Grids with PyFR. B. C. Vermeire, F. D. Witherden, P. E. Vincent. Poster Presentation, UK Turbulence Consortium 2014 Workshop, 22-23 September 2014. London, UK
The Effect of Arterial Curvature on Blood Flow and Oxygen Transport in Arterio-Venous Fistulae. F. Iori, L. Grechy, R. Corbett, W. Gedroyc, N. Duncan, C. G. Caro, P. E. Vincent. Poster presentation, Medical Engineering Centres Annual Meeting and Bioengineering14, 10-11 September 2014. London, UK
Blood Flow and Oxygen Transport in Arterio-Venous Fistulae: Studying the Effect of Vascular Curvature. L. Grechy, F. Iori, R. Corbett, W. Gedroyc, N. Duncan, C. G. Caro, P. E. Vincent. Oral Presentation, IMA Conference on Mathematical Modelling of Fluid Systems, 10-12 September 2014. Bristol, UK
PyFR: Next Generation High-Order Computational Fluid Dynamics on Many-Core Hardware. P. E. Vincent. Oral Presentation, NIMS Hot Topics Workshop on Higher-Order Methods for Conservation Laws, 25-27 August. Daejeon, South Korea
PyFR: An Open Source Framework for High-Order Computational Fluid Dynamics on Streaming Architectures. P. E. Vincent, F. D. Witherden, A. M. Farrington. Oral Presentation, World Congress on Computational Mechanics, 21-25 July 2014. Barcelona, Spain
Equivalence Between Specific Discontinuous Galerkin Methods and High-Order Flux Reconstruction Schemes. D. De Grazia, G. Mengaldo, D. Moxey, P. E. Vincent, S. J. Sherwin. Oral Presentation, International Conference on Spectral and High Order Methods, 23-27 June 2014. Salt Lake City, Utah, USA
PyFR: An Open Source Framework for High-Order Computational Fluid Dynamics on Streaming Architectures. P. E. Vincent, F. D. Witherden, A. M. Farrington, G. Ntemos. Oral Presentation, International Conference on Spectral and High Order Methods, 23-27 June 2014. Salt Lake City, Utah, USA
PyFR: An Open Source Framework for High-Order Computational Fluid Dynamics on Streaming Architectures. P. E. Vincent, F. D. Witherden, A. M. Farrington, G. Ntemos. Oral Presentation, British Applied Mathematics Colloquium, 28-30 April 2014. Cardiff, UK
PyFR: Bringing Next Generation Computational Fluid Dynamics to GPU Platforms. P. E. Vincent. Oral Presentation, Nvidia GPU Technology Conference, 24-27 March 2014. San Jose, CA, USA
PyFR: Technical Challenges of Bringing Next Generation Computational Fluid Dynamics to GPU Platforms. F. D. Witherden. Oral Presentation, Nvidia GPU Technology Conference, 24-27 March 2014. San Jose, CA, USA
Using STAR-CCM+ to Design Arterio-Venous Fistulae. L. Grechy, F. Iori, R. Corbett, J. Crane, W. Gedroyc, N. Duncan, C. G. Caro, P. E. Vincent. Oral Presentation, Star Global Conference, 17-19 March 2014. Vienna, Austria
2013
PyFR: An Open Source Framework for Solving Advection-Diffusion Type Problems on Streaming Architectures. F. D. Witherden, A. M. Farrington, P. E. Vincent. Poster Presentation, 5th UK Manycore Developer Conference, 16 December 2013. Oxford, UK
PyFR: A High-Order Python-Based Flux Reconstruction Flow Solver for Many-Core Platforms. P. E. Vincent, F. D. Witherden, A. Farrington. Oral presentation. 12th US National Congress on Computational Mechanics, 22-25 July 2013. Raleigh, North Carolina, USA
Novel Approach to Inhibition of Neointimal Hyperplasia in Arteriovenous Fistulae. R. Corbett, N. Demicheli, L. Grechy, F. Iori, J. Crane, N. Duncan, P. E. Vincent, C. G. Caro. Poster presentation. International Union of Physiological Sciences Congress, 21-26 July 2013. Birmingham, UK
The Effect of Arterial Curvature on Flow and Oxygen Transport in Arterio-Venous Fistulae. F. Iori, L. Grechy, N. Duncan, C. G. Caro, P. E. Vincent. Poster presentation. Imperial College British Heart Foundation Centre Symposium, 13 June 2013. London, UK
The Effect of Offset Junctions on Flow and Oxygen Transport in Arterio-Venous Fistulae. L. Grechy, F. Iori, N. Duncan, C. G. Caro, P. E. Vincent. Poster presentation. Imperial College British Heart Foundation Centre Symposium, 13 June 2013. London, UK
Flow Visualisation in a Model Vascular Anastomosis. R. Corbett, N. Duncan, P. E. Vincent, C. G. Caro. Poster presentation. Imperial College British Heart Foundation Centre Symposium, 13 June 2013. London, UK
PyFR: An Open Source Python Framework for High-Order CFD on Many-Core Platforms. F. D. Witherden, A. Farrington, P. E. Vincent. Oral presentation. 4th International Congress on Computational Engineering and Sciences, 19-24 May 2013. Las Vegas, Nevada, USA
PyFR: An Open Source Python Framework for High-Order CFD on Many-Core Platforms. F. D. Witherden, A. Farrington, P. E. Vincent. Poster presentation. 4th International Congress on Computational Engineering and Sciences, 19-24 May 2013. Las Vegas, Nevada, USA
Novel Approach to Reduce the Burden of Neointimal Hyperplasia in Arteriovenous Fistulae. R. W. Corbett, N. Demicheli, L. Grechy, F. Iori, D. Ellis, W. Gedroyc, M. Hamady, J. Crane, N. Duncan, P. E. Vincent, C. G. Caro. Poster presentation. European Renal Association and European Dialysis and Transplant Association Congress, 18-21 May 2013. Istanbul, Turkey
Novel Approach to Reduce the Burden of Neointimal Hyperplasia in Arteriovenous Fistulae. R. W. Corbett, N. Demicheli, L. Grechy, F. Iori, D. Ellis, W. Gedroyc, M. Hamady, J. Crane, N. Duncan, P. E. Vincent, C. G. Caro. Poster presentation. Joint British Transplant Society and Renal Association Congress, 13-15 March 2013. Bournemouth, UK
Next Generation Computational Fluid Dynamics: High-Order Accurate Simulations on Many-Core Platforms. P. E. Vincent. Invited oral presentation. E-Infrastructure South Industry Day, 13 March 2013. Harwell Didcot, UK
High-Order Flux Reconstruction Schemes: Theory and Implementation. P. E. Vincent, F. Witherden, A. Farrington. Oral presentation. Society for Industrial and Applied Mathematics Computational Science and Engineering Conference, 25-28 February 2013. Boston, Massachusetts, USA
Separated Flow Over the T106c Turbine Aerofoil: A Comparison of High-Order Flux Reconstruction Methods and Current Industry Standard Technology. A. Farrington, F. D. Witherden, P. E. Vincent. Oral Presentation, International Workshop on Numerical Prediction of Detached Flows, 23-24 October 2013. Madrid, Spain
2012
A Generalized Matrix-Based Proceedure for Identifying Energy Stable Flux Reconstruction Schemes. P. E. Vincent, A. Jameson. Oral presentation. European Congress on Computational Methods in Applied Sciences and Engineering, 10-14 September 2012. Vienna, Austria
Blood Flow in the Rabbit Aortic Arch and Descending Thoracic Aorta. P. E. Vincent, A. M. Plata, A. A. E. Hunt, P. D. Weinberg, S. J. Sherwin.. Invited oral presentation. Institute of Biomedical Engineering Cardiovascular Technology Network 5th Symposium, 8 February 2012. London, UK
2011
The Flux Reconstruction Approach to High-Order Methods - Extension to Simplex Elements. P. Castonguay, P. E. Vincent, D. Williams, A. Jameson. Oral presentation. 11th United States National Congress on Computational Mechanics, 25-28 July 2011. Minneapolis, Minnesota, USA
The Flux Reconstruction Approach to High-Order Methods - One-Dimensional Theory. P. E. Vincent, P. Castonguay, A. Jameson. Oral presentation. 11th United States National Congress on Computational Mechanics, 25-28 July 2011. Minneapolis, Minnesota, USA
The Flux Reconstruction Approach to High-Order Methods - Theory and Application. P. E. Vincent, P. Castonguay, D. Williams, A. Jameson. Oral presentation. 7th International Congress on Industrial and Applied Mathematics, 18-22 July 2011. Vancouver, British Columbia, Canada
Advances in Bringing High-Order Methods to Practical Applications in Computational Fluid Dynamics. A. Jameson, P. E. Vincent. Oral presentation. 7th International Congress on Industrial and Applied Mathematics, 18-22 July 2011. Vancouver, British Columbia, Canada
Comparison of Flow Features in an Immature and a Mature Rabbit Aorta. V. Peiffer, A. M. Plata, P. E. Vincent, P. D. Weinberg, S. J. Sherwin. Oral presentation. Physiological Fluid Mechanics Conference, 14-15 July 2011. London, UK
Computational Study of Blood Flow Within a Multi-Branched Model of the Rabbit Thoracic Aorta. A. M. Plata, P. E. Vincent, A. A .E. Hunt, S. J. Sherwin and P. D. Weinberg. Oral presentation. 2nd International Conference on Mathematical and Computational Biomedical Engineering, 30 March - 1 April 2011. Washington, District of Columbia, USA
2010
The Flux Reconstruction Approach to High-Order Methods: Theory and Application. P. E. Vincent, P. Castonguay, D. Williams, A. Jameson. Poster presentation. Institute for Mathematics and its Applications Workshop: Numerical Solutions of Partial Differential Equations, 1-5 November 2010. Minneapolis, Minnesota, USA
A Fast, Scalable Unstructured Compressible Viscous Flow Solver on the GPU. P. Castonguay, D. Williams, P. E. Vincent, A. Jameson. Oral presentation. NVIDIA GPU Technology Conference, 20-23 September 2010. San Jose, California, USA
Multi-GPU High-Order Unstructured Solver for Compressible Navier-Stokes Equations. P. Castonguay, D. Williams, P. E. Vincent, A. Jameson. Poster presentation. NVIDIA GPU Technology Conference, 20-23 September 2010. San Jose, California, USA
Computational Modelling of Blood Flow in a Realistic Representation of the Rabbit Aorta. A. Plata Garcia, P. E. Vincent, A. Hunt, S. Sherwin, P. Weinberg. Oral presentation. 6th World Congress on Biomechanics, 1-6 August 2010. Singapore
A High-Order Flux Reconstruction Scheme Utilizing Triangular Raviart-Thomas Elements. P. E. Vincent, P. Castonguay, A. Jameson. Oral presentation. World Congress on Computational Mechanics, 19-23 July 2010. Sydney, Australia
Blood Flow Features In A Realistic Representation Of The Rabbit Aorta. A. Plata Garcia, P. E. Vincent, A. Hunt, S. Sherwin, P. Weinberg. Poster presentation. Cardiovascular Technology Symposium, 8 July 2010, London, UK
Blood Flow and Vascular Nitric Oxide Transport in the Rabbit Aorta. A. Plata Garcia, P. E. Vincent, A. Hunt, S. Sherwin, P. Weinberg. Oral presentation. Europa High Performance Computing Transnational Access Meeting, 15-17 June 2010. Helsinki, Finland
A Computational Study of Blood Flow and NO transport in a Realistic Representation of the Rabbit Aorta. S. J. Sherwin, A. Plata Garcia, P. E. Vincent, A. Hunt, R. Krams, P. Weinberg. Oral presentation. CECAM workshop on Trends in Computational Hemodynamics, 10-12 May 2010, Lausanne, Switzerland
2009
A Computational Study of Blood Flow In a Realistic Representation of the Rabbit Aorta. A. Plata Garcia, P. E. Vincent, A. Hunt, S. Sherwin, P. Weinberg. Oral presentation. Bioengineering 2009, 24-25 September 2009. Oxford, UK
Flow Features In a Realistic Representation of the Rabbit Aorta. A. Plata Garcia, P. E. Vincent, A. Hunt, S. Sherwin, P. Weinberg. Oral presentation, 12th International Symposium on Computer Simulation in Biomechanics, 2-4 July 2009. Cape Town, South Africa
A Realistic Representation of the Rabbit Aorta for use in Computational Haemodynamic Studies. P. E. Vincent, A. Hunt, L. Grinberg, S. Sherwin, P. Weinberg. Poster Presentation, American Society of Mechanical Engineering, Summer Bioengineering Conference, 17-21 June 2009. Lake Tahoe, California, USA. (Finalist in poster prize competition)
2008
Sub-Cellular Scale Features of Low Density Lipoprotein Concentration Polarisation Adjacent to the Arterial Endothelium. P. E. Vincent, S. Sherwin, P. Weinberg. Oral Presentation, Bioengineering 2008, 18-19 September 2008. London, UK
The Effect of Sub-Cellular Scale Endothelial Features on Low Density Lipoprotein Concentration Polarisation in Arteries. P. E. Vincent, S. Sherwin, P. Weinberg. Oral Presentation, UK Focus for Biomedical Engineering Futures Meeting Study and Treatment of Cardiovascular Disease: Devices and Fluidics, 15-17 September 2008. London, UK
Sub-Cellular Scale Features of Low Density Lipoprotein Concentration Polarisation in Arteries. P. E. Vincent, S. Sherwin, P. Weinberg. Oral Presentation, 16th International Conference on Mechanics in Medicine and Biology, 23-25 July 2008. Pittsburgh, Pennsylvania, USA
Effect of the Endothelial Glycocalyx Layer on Low Density Lipoprotein Concentration Polarisation Within Arteries. P. E. Vincent, S. Sherwin, P. Weinberg. Poster Presentation, 3rd International Symposium on Biomechanics in Vascualr Biology and Cardiovascualr Disease, 24-25 April 2008. Rotterdam, Netherlands
Arterial Low Density Lipoprotein Concentration Polarisation at the Sub-Cellular Scale. P. E. Vincent, S. Sherwin, P. Weinberg. Oral Presentation, 5th International Bio-Fluid Symposium and Workshop, 28-30 March 2008. Pasadena, Los Angeles, California, USA
Arterial Low Density Lipoprotein Concentration Polarisation at the Sub-Cellular Scale. P. E. Vincent, S. Sherwin, P. Weinberg. Poster Presentation, 5th International Bio-Fluid Symposium and Workshop, 28-30 March 2008. Pasadena, Los Angeles, California, USA
2007
Sub-Cellular Scale Variations in Low Density Lipoprotein Concentration Adjacent to the Endothelium. P. E. Vincent, S. Sherwin, P. Weinberg. Poster Presentation, BMES Annual Fall Meeting, 26-29 September 2007. Hollywood, Los Angeles, California, USA
The Effect of Sub-Cellular Scale Variations in Transmural Water Flux on LDL Buildup Adjacent to the Endothelium. P. E. Vincent, S. Sherwin, P. Weinberg. Poster Presentation, Fifth Physiological Flow Meeting: Size, Sex and Sight, 3-4 September 2007. London, UK
Sub Cellular Scale Variations in LDL Concentration Polarisation on the Luminal Side of the Endothelium. P. E. Vincent, S. Sherwin, P. Weinberg. Poster Presentation, Fourth Physiological Flow Meeting: Respiratory Biomechanics and Physiological Fluid-Structure Interaction Problems, 2-3 April 2007. Manchester, UK. (Runner up in poster prize competition)
2006
Computational Investigation of a Mechanism by Which Blood Flow Could Control Lipoprotein Uptake by the Arterial Wall.. P. E. Vincent, S. Sherwin, P. Weinberg. Oral Presentation, BAS and BSCR Joint Autumn Meeting, 21-22 September 2006. Cambridge, UK.
2D Computational Study of Cellular Scale Variations in LDL Concentration at an Endothelium with Physiologically Realistic Inter-Cellular Cleft Dimensions. P. E. Vincent, S. Sherwin, P. Weinberg. Poster Presentation, Third Physiological Flow Meeting: Imaging and Modelling for Interventional Planning, 18-19 April 2006. Oxford, UK
Presenting Author

Book Chapters

2016
Abstract: There is an increasing desire among industrial practitioners of computational fluid dynamics to undertake high-fidelity scale-resolving simulations of unsteady flows within the vicinity of complex geometries. Such simulations require numerical methods that can operate on unstructured meshes with low numerical dissipation. The flux reconstruction (FR) approach describes one such family of numerical methods, which includes a particular type of collocation-based nodal discontinuous Galerkin method, and spectral difference methods, as special cases. In this chapter we describe the current state-of-the-art surrounding research into FR methods. To begin, FR is described in one dimension for both advection and advection?diffusion problems. This is followed by a description of its extension to multidimensional tensor product and simplex elements. Stability and accuracy issues are then discussed, including an overview of energy-stability proofs, von Neumann analysis results, and stability characteristics when the flux function of the governing system is nonlinear. Finally, implementation aspects are outlined in the context of modern hardware platforms, and three example applications of FR are presented, demonstrating the potential utility of FR schemes for scale resolving simulation of unsteady flow problems.

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PhD Theses

2016
Abstract: Computational Studies of Unsteady Flow in Arterio-Venous Fistulae

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2015
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 eciently on modern streaming architectures, such as graphics processing units (GPUs). The aforementioned properties of FR mean it offers a promising route to performing affordable, and hence industrially relevant, scaleresolving simulations of hitherto intractable unsteady flows within the vicinity of realworld engineering geometries. In this thesis a formulation of the FR approach that is suitable for solving non-linear advection-diffusion type problems on mixed curvilinear grids is developed. Issues around aliasing are explored in detail and techniques for mitigation outlined. A methodology for identifying symmetric quadrature rules inside of a variety of domains is also presented and used to find several rules that appear to be an improvement over those in literature. This methodology is also used to obtain improved sets of solution points inside of triangular elements. PyFR, an open-source Python based framework for solving the compressible Navier?Stokes equations using the FR approach, is also developed. It is designed to target a range of hardware platforms via use of an in-built domain specific language based on the Mako templating engine. PyFR is able to operate on mixed grids in both two and three dimensions and can target NVIDIA GPUs, AMD GPUs, and Intel CPUs. Results are presented for various benchmark flow problems, single-node performance is discussed, heterogeneous multinode capabilities are analysed, and scalability is demonstrated on up to 2 000 NVIDIA K20X GPUs for a sustained performance of 1.3 PFLOP/s.

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2009
Abstract: Uptake of Low Density Lipoprotein (LDL) by the arterial wall is likely to play a key role in the process of atherogenesis, which occurs non-uniformly within the arterial vasculature. 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 polarisation layer adjacent to the luminal surface of the arterial endothelium. In this thesis the effects of cellular scale endothelial features on such LDL concentration polarisation are investigated using an idealised theoretical model. Specifically, the effect of a spatially heterogeneous transmural water flux is considered (flowing only through intercellular clefts), as well as the effect of the endothelial glycocalyx layer (EGL). The idealised model is implemented using both analytical techniques and the spectral/hp element method. A range of scenarios are considered, including those were no EGL is present, those where an EGL is present but LDL cannot penetrate into it, and finally those where an EGL is present and LDL can penetrate into it. For cases where no EGL is present, particular attention is paid to the spatially averaged LDL concentration adjacent to various 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 alone 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 polarisation). 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 microns per second and an LDL diffusivity in blood plasma of 28.67 microns squared per second. For cases where an EGL is present, measures of LDL concentration polarisation relevant to the rate of transendothelial LDL transport can also be defined. 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 within 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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Openings

PhD Studentship - Development of In-situ Visualisation and Analysis Technology for High-Fidelity Computational Fluid Dynamics
Summary: A PhD Studentship is currently available. The project, will involve addition of 'in-situ' visualisation, processing, and analysis technology to PyFR, an open-source high-order massively-parallel CFD platform, as well as its application to solve a range of challenging unsteady flow problems. Candidates should hold, or expect to obtain, an undergraduate degree in a numerate discipline. Previous programming experience is important (ideally Python, C++ and CUDA).

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