In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an ...automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require O(p2d) storage and O(p3d) computational work, where p is the degree of basis polynomials used, and d is the spatial dimension. Our SVD-based tensor-product preconditioner requires O(pd+1) storage, O(pd+1) work in two spatial dimensions, and O(pd+2) work in three spatial dimensions. Combined with a matrix-free Newton–Krylov solver, these preconditioners allow for the solution of DG systems in linear time in p per degree of freedom in 2D, and reduce the computational complexity from O(p9) to O(p5) in 3D. Numerical results are shown in 2D and 3D for the advection, Euler, and Navier–Stokes equations, using polynomials of degree up to p=30. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees p.
•A new tensor-product based preconditioner is developed for the implicit discontinuous Galerkin method.•This preconditioner greatly reduces the computational complexity required to solve the resulting linear systems.•Numerical examples demonstrate the effectiveness for a range of 2D and 3D test problems.
High-order CFD methods: current status and perspective Wang, Z.J.; Fidkowski, Krzysztof; Abgrall, Rémi ...
International journal for numerical methods in fluids,
20 July 2013, Letnik:
72, Številka:
8
Journal Article, Web Resource
We propose a method to generate high-order unstructured curved meshes using the classical Winslow equations. We start with an initial straight-sided mesh in a reference domain, and fix the position ...of the nodes on the boundary on the true curved geometry. In the interior of the domain, we solve the Winslow equations using a new continuous Galerkin finite element discretization. This formulation appears to produce high quality curved elements, which are highly resistant to inversion. In addition, the corresponding nonlinear equations can be solved efficiently using Picard iterations, even for highly stretched boundary layer meshes. Compared to several previously proposed techniques, such as optimization and approaches based on elasticity analogies, this can significantly reduce the computational cost while producing curved elements of similar quality. We show a number of examples in both two and three space dimensions, including complex geometries and stretched boundary layers, and demonstrate the high quality of the generated meshes and the performance of the nonlinear solver.
In this paper, we develop new techniques for solving the large, coupled linear systems that arise from fully implicit Runge–Kutta methods. This method makes use of the iterative preconditioned GMRES ...algorithm for solving the linear systems, which has seen success for fluid flow problems and discontinuous Galerkin discretizations. By transforming the resulting linear system of equations, one can obtain a method which is much less computationally expensive than the untransformed formulation, and which compares competitively with other time-integration schemes, such as diagonally implicit Runge–Kutta (DIRK) methods. We develop and test several ILU-based preconditioners effective for these large systems. We additionally employ a parallel-in-time strategy to compute the Runge–Kutta stages simultaneously. Numerical experiments are performed on the Navier–Stokes equations using Euler vortex and 2D and 3D NACA airfoil test cases in serial and in parallel settings. The fully implicit Radau IIA Runge–Kutta methods compare favorably with equal-order DIRK methods in terms of accuracy, number of GMRES iterations, number of matrix–vector multiplications, and wall-clock time, for a wide range of time steps.
Purpose
Three‐dimensional, time‐resolved blood flow measurement (4D‐flow) is a powerful research and clinical tool, but improved resolution and scan times are needed. Therefore, this study aims to ...(1) present a postprocessing framework for optimization‐driven simulation‐based flow imaging, called 4D‐flow High‐resolution Imaging with a priori Knowledge Incorporating the Navier‐Stokes equations and the discontinuous Galerkin method (4D‐flow HIKING), (2) investigate the framework in synthetic tests, (3) perform phantom validation using laser particle imaging velocimetry, and (4) demonstrate the use of the framework in vivo.
Methods
An optimizing computational fluid dynamics solver including adjoint‐based optimization was developed to fit computational fluid dynamics solutions to 4D‐flow data. Synthetic tests were performed in 2D, and phantom validation was performed with pulsatile flow. Reference velocity data were acquired using particle imaging velocimetry, and 4D‐flow data were acquired at 1.5 T. In vivo testing was performed on intracranial arteries in a healthy volunteer at 7 T, with 2D flow as the reference.
Results
Synthetic tests showed low error (0.4%‐0.7%). Phantom validation showed improved agreement with laser particle imaging velocimetry compared with input 4D‐flow in the horizontal (mean −0.05 vs −1.11 cm/s, P < .001; SD 1.86 vs 4.26 cm/s, P < .001) and vertical directions (mean 0.05 vs −0.04 cm/s, P = .29; SD 1.36 vs 3.95 cm/s, P < .001). In vivo data show a reduction in flow rate error from 14% to 3.5%.
Conclusions
Phantom and in vivo results from 4D‐flow HIKING show promise for future applications with higher resolution, shorter scan times, and accurate quantification of physiological parameters.
We aimed to identify determinants in acute stroke that are associated with falls during the stroke unit stay. In order to enable individualized preventive actions, this knowledge is fundamental. ...Based on local and national quality register data on an unselected sample of 5065 stroke patients admitted to a stroke unit at a Swedish university hospital, univariable and multivariable logistic regression analyses were performed. The dependent variable was any fall during stroke unit stay. The independent variables related to function, activity, personal factors, time to assessment, comorbidities and treatments. Determinants of falls were: being male (odds ratio (OR) 2.25, 95% confidence interval (95% CI) 1.79-2.84), haemorrhagic stroke (OR 1.39, 95% CI 1.05-1.86), moderate stroke symptoms according to the National Institutes of Health Stroke Scale (NIHSS score 2-5 vs. NIHSS score 0-1) (OR 1.43, 95% CI 1.08-1.90), smoking (OR 1.70, 95% CI 1.29-2.25), impaired postural control in walking (OR 4.61, 95% CI 3.29-6.46), impaired postural control in standing (OR 1.60, 95% CI 1.25-2.05), stroke-related arm- and hand problems, OR 1.45, 95% CI 1.11-1.91), impaired cognition (OR 1.43, 95% CI 1.04-1.95), and urinary tract infection (OR 1.91, 95% CI 1.43-2.56). The findings from this study are useful in clinical practice and might help to improve patient safety after stroke.
•(Non-linear) stability of high-order methods for conservation laws is an open issue.•This paper introduces a new discontinuous Galerkin method.•An approximation space is considered with high-order ...and sub-grid basis functions.•The high-order modes can additionally suppressed by penalty using a new sensor.
This article considers a new discretization scheme for conservation laws. The discretization setting is based on a discontinuous Galerkin scheme in combination with an approximation space that contains high-order polynomial modes as well as piece-wise constant modes on a sub-grid. The high-order modes can continuously be suppressed with a penalty function that is based on a sensor which is intertwined with the approximation space. Numerical tests finally illustrate the performance of this scheme.
We present a high-order accurate scheme for coupled fluid–structure interaction problems. The fluid is discretized using a discontinuous Galerkin method on unstructured tetrahedral meshes, and the ...structure uses a high-order volumetric continuous Galerkin finite element method. Standard radial basis functions are used for the mesh deformation. The time integration is performed using a partitioned approach based on implicit–explicit Runge–Kutta methods. The resulting scheme fully decouples the implicit solution procedures for the fluid and the solid parts, which we perform using two separate efficient parallel solvers. We demonstrate up to fifth order accuracy in time on a non-trivial test problem, on which we also show that additional subiterations are not required. We solve a benchmark problem of a cantilever beam in a shedding flow, and show good agreement with other results in the literature. Finally, we solve for the flow around a thin membrane at a high angle of attack in both 2D and 3D, and compare with the results obtained with a rigid plate.
We present a new line-based discontinuous Galerkin (DG) discretization scheme for first- and second-order systems of partial differential equations. The scheme is based on fully unstructured meshes ...of quadrilateral or hexahedral elements, and it is closely related to the standard nodal DG scheme as well as several of its variants such as the collocation-based DG spectral element method (DGSEM) or the spectral difference (SD) method. However, our motivation is to maximize the sparsity of the Jacobian matrices, since this directly translates into higher performance in particular for implicit solvers, while maintaining many of the good properties of the DG scheme. To achieve this, our scheme is based on applying one-dimensional DG solvers along each coordinate direction in a reference element. This reduces the number of connectivities drastically, since the scheme only connects each node to a line of nodes along each direction, as opposed to the standard DG method which connects all nodes inside the element and many nodes in the neighboring ones. The resulting scheme is similar to a collocation scheme, but it uses fully consistent integration along each 1-D coordinate direction which results in different properties for nonlinear problems and curved elements. Also, the scheme uses solution points along each element face, which further reduces the number of connections with the neighboring elements. Second-order terms are handled by an LDG-type approach, with an upwind/downwind flux function based on a switch function at each element face. We demonstrate the accuracy of the method and compare it to the standard nodal DG method for problems including Poisson’s equation, Euler’s equations of gas dynamics, and both the steady-state and the transient compressible Navier–Stokes equations. We also show how to integrate the Navier–Stokes equations using implicit schemes and Newton–Krylov solvers, without impairing the high sparsity of the matrices.
Abstract
The early identification of individuals at risk of fear of falling after stroke is crucial in order to individualise preventive actions and interventions. The aim of this study was to ...identify the incidence of, and baseline factors in acute stroke that are associated with fear of falling at 6 months after stroke. Fear of falling was assessed by one question, which was answered by 279 of 452 eligible individuals. Univariable and multivariable logistic regression analyses were performed to determine the factors that were associated with fear of falling. The dependent variable was fear of falling at 6 months after stroke. The independent variables were related to function, activity and participation, including personal and environmental factors. Fear of falling was reported by 117 (41.9%) individuals. Poor postural control in acute stroke, measured using the modified version of the Postural Assessment Scale for Stroke Patients (odds ratio OR: 2.60, 95% confidence interval CI: 1.26–5.36), and being physically inactive prior to the stroke, measured using the Saltin-Grimby Physical Activity Scale (OR: 2.04, 95% CI: 1.01–4.12), were found to be associated with fear of falling at 6 months after stroke. The findings in this study are useful in clinical practice to optimise rehabilitation after stroke.
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