Flux Reconstruction Research Articles

The Flux Reconstruction Method with Lax–Wendroff Type Temporal Discretization for Hyperbolic Conservation Laws

In this paper, we develop a Lax–Wendroff type time discretization method for high order Flux Reconstruction scheme to solve hyperbolic conservation laws. Through Cauchy–Kowalewski procedure, the resulting Lax–Wendroff Flux Reconstruction (LWFR) scheme is an alternative spatial–temporal coupling method to the popular Runge–Kutta Flux Reconstruction (RKFR) scheme. LWFR is a one-step explicit high order discontinuous finite element method and its discretization procedure is more compact and effective for certain problems than that of RKFR. Furthermore, aiming at accurate simulation of discontinuity, we propose a robust local artificial viscosity formulation of LWFR for the first time. A collection of successful numerical experiments show that LWFR can give essentially non-oscillatory and sharp solutions for discontinuity, and maintain designed order of accuracy for smooth regions, both in one-dimensional and two-dimensional Euler equations. In conclusion, LWFR scheme is cost effective and accuracy-preserving for certain problems and can be an attractive candidate to solve hyperbolic conservation laws.

ZEFR: A GPU-accelerated high-order solver for compressible viscous flows using the flux reconstruction method

In this work we present ZEFR, a GPU accelerated flow solver based around the high-order accurate flux reconstruction (FR) approach. Written in a combination of C++ and CUDA, ZEFR is designed to perform scale resolving simulations within the vicinity of complex geometrical configurations. A unique feature of ZEFR is its support for overset grids; a feature which greatly expands the addressable problem space compared with existing high-order codes. The C++ implementation of FR in a manner which is suitable for modern hardware platforms is described in detail. Additionally, an overview of the input deck used by ZEFR is included. Validation results are presented for a range of steady and unsteady flow problems including Couette flow, the Taylor–Green vortex, and flow around an SD7003 airfoil. Single node performance on a NVIDIA V100 GPU is analyzed where it is shown that all of the kernels in ZEFR attain a high proportion of peak memory bandwidth. Moreover, multi-node performance is also assessed with strong scalability being demonstrated from 60 to 3840 NVIDIA V100 GPUs. Program summaryProgram title: ZEFR v1.0Program files doi:http://dx.doi.org/10.17632/wzy83bscxd.1Licensing provisions: BSD 3-clauseProgramming language: C++ and CUDAExternal routines/libraries: Eigen, HDF5, METIS, MPI, and TIOGA.Nature of problem: Compressible Euler and Navier–Stokes equations.Solution method: High-order direct flux reconstruction approach suitable for curved, mixed, unstructured grids.Unusual features: Code incorporates support for overset grids.

Strong compact formalism for characteristic boundary conditions with discontinuous spectral methods

Characteristic boundary conditions for the Navier-Stokes equations (NSCBC) are implemented for the first time with discontinuous spectral methods, namely the spectral difference and flux reconstruction. The implementation makes use of the resolution by these methods of the strong form of the Navier-Stokes equations by applying these conditions through a flux balance regularization which takes the form of a generalized element-compact correction polynomial. It is shown to be at least as effective as similar implementations in finite volume solvers, and sustains arbitrarily-high orders of accuracy on hexahedral-based unstructured meshes. Further, Navier-Stokes time-domain impedance boundary conditions are derived and implemented as a NSCBC sub-class. They account for the diffusive process at the wall and are shown to properly resolve the broadband response of porous materials under normal and grazing flow conditions. The ability of these NSCBC in preventing the appearance of spurious reflections at the boundaries is demonstrated through a varied series of bench-marking simulations. They effectively shield the inner computational domain from any far-field unphysical contamination. Overall, this work enables the use of strong discontinuous spectral methods to study unsteady problems on complex geometries.

A posteriori error estimates for a compositional two-phase flow with nonlinear complementarity constraints

In this work, we develop an a-posteriori-steered algorithm for a compositional two-phase flow with exchange of components between the phases in porous media. As a model problem, we choose the two-phase liquid-gas flow with appearance and disappearance of the gas phase formulated as a system of nonlinear evolutive partial differential equations with nonlinear complementarity constraints. The discretization of our model is based on the backward Euler scheme in time and the finite volume scheme in space. The resulting nonlinear system is solved via an inexact semismooth Newton method. The key ingredient for the a posteriori analysis are the discretization, linearization, and algebraic flux reconstructions allowing to devise estimators for each error component. These enable to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver whenever the corresponding error components do not affect significantly the overall error. Numerical experiments are performed using the Newton-min algorithm as well as the Newton-Fischer-Burmeister algorithm in combination with the GMRES iterative linear solver to show the efficiency of the proposed adaptive method.

Lead contamination from gold mining in Yellowknife Bay (Northwest Territories), reconstructed using stable lead isotopes

The contributions of contaminant sources are difficult to resolve in the sediment record using concentration gradients and flux reconstruction alone. In this study, we demonstrate that source partitioning using lead isotopes provide complementary and unique information to concentration gradients to evaluate point-source releases, transport, and recovery of metal mining pollution in the environment. We analyzed eight sediment cores, collected within 24 km of two gold mines, for Pb stable isotopes, Pb concentration, and sediment chronology. Stable Pb isotope ratios (206Pb/207Pb, 208Pb/204Pb) of mining ore were different from those of background (pre-disturbance) sediment, allowing the use of a quantitative mixing model. As previously reported for some Arctic lakes, Pb isotope ratios indicated negligible aerosol inputs to sediment from regional or long-range pollution sources, possibly related to low annual precipitation. Maximum recorded Pb flux at each site reached up to 63 mg m−2 yr−1 in the period corresponding to early years of mining when pollution mitigation measures were at a minimum (1950s–1960s). The maximum contribution of mining-derived Pb to these fluxes declined with distance from the mines from 92 ± 8% to 8 ± 4% at the farthest site. Mining-derived Pb was still present at the sediment surface within 9 km of Giant Mine more than ten years after mine closure (5–26 km, 95% confidence interval) and model estimates suggest it could be present for another ∼50–100 years. These results highlight the persistence of Pb pollution in freshwater sediment and the usefulness of Pb stable isotopes to quantify spatial and temporal trends of contamination from mining pollution, particularly as concentrations approach background.

Sensitivity of functionals of the solution of a variational data assimilation problem with simultaneous reconstruction of heat fluxes and the initial state for the sea thermodynamics model

Sensitivity of functionals of the solution of a variational data assimilation problem with simultaneous reconstruction of heat fluxes and the initial state for the sea thermodynamics model

Adaptive Full State Observer for Nonsalient PMSM with Noised Measurements of the Current and Voltage

Abstract An algorithm of adaptive estimation of the magnetic flux for the non-salient permanent magnet synchronous motor (PMSM) for the case when measurable electrical signals are corrupted by a constant offset is presented. A new nonlinear parameterization of the electric drive model based on dynamical regressor extension and mixing (DREM) procedure is proposed. Due to this parameterization the problem of flux estimation is translated to the auxiliary task of identification of unknown constant parameters related to measurement errors. It is proved that when both current and voltage measurements are biased the proposed algorithm ensures convergence of the flux observation error to a bounded set. At the same time the position error converges to zero. The observer provides global exponential convergence if the corresponding regression function satisfies the persistent excitation condition. If the regression function is not square integrable the global asymptotic convergence is ensured. In comparison with known analogues this paper gives a constructive way of the flux reconstruction for a nonsalient PMSM with guaranteed performance (low oscillation, convergence rate regulation) and, from other hand, a straightforwardly easy implementation of the proposed observer to embedded systems.

A Semi-global Hybrid Sensorless Observer for Permanent Magnet Synchronous Machines with Unknown Mechanical Model

In this paper, we present a hybrid sensorless observer for Permanent Magnets Synchronous Machines, with no a priori knowledge of the mechanical dynamics and without the typical assumption of constant or slowly-varying speed. Instead, we impose the rotor speed to have a constant (unknown) sign and a non-zero magnitude at all times. For the design of the proposed scheme, meaningful Lie group formalism is adopted to describe the rotor position as an element of the unit circle. This choice, however, leads to a non-contractible state space, and therefore it introduces topological constraints that complicate the achievement of global/semi-global and robust results. In this respect it is shown that the proposed observer, which employs a clock to periodically reset the estimates, is semi-globally practically asymptotically stable, and thus it improves a continuous-time version designed under the same assumptions. As highlighted in the simulation results, the novel hybrid strategy leads to enhanced transient performance, notably without any modification of the gains employed in the continuous-time solution. These features motivate to augment the observer with a discrete-time identifier, leading to significantly faster rotor flux reconstruction.

Reduction of the uncertainty in the atmospheric neutrino flux prediction below 1 GeV using accurately measured atmospheric muon flux

We examine the uncertainty of the calculation of the atmospheric neutrino flux and present a way to reduce it using accurately measured atmospheric muon flux. Considering the difference of the hadronic interaction model and the real one as a variation of hadronic interaction, we find a quantitative estimation method for the error of the atmospheric neutrino flux calculation from the residual of the reconstruction of the atmospheric muon flux observed in a precision experiment, by the study of atmospheric neutrino and muon fluxes response to the variation of hadronic interaction. However, the efficiencty of this method is largely dependent on the observation site of the atmospheric muon flux, as the relation of the error of the atmospheric neutrino flux calculation and the residual of the reconstruction of the atmospheric muon flux is also largely dependent on the muon observation site, especially for the low energy neutrinos. We calculate several observation sites, near Kamioka at sea level, same but 2770m a.s.l.., Hanle India (4500m a.s.l.), and at Balloon altitude ($\sim$ 32km). Then we estimate how stringently can the atmospheric muon reduce the error in the calculation of the atmospheric neutrino flux. We also discuss on the source of error which is difficult to reduce by only the observation of atmospheric muon.

Three-dimensional transient inverse heat conduction problems in the enhanced pool boiling heat transfer

In this paper, the solution methods of three-dimensional transient inverse heat conduction problems (3D IHCP) arising from pool boiling applications that need to reconstruct Neumann boundary conditions are summarized. Due to their mathematical ill-posedness, it is still non-trivial to solve these 3D IHCP at a reasonable cost. Firstly, we introduce the optimization-based regularization methods, such as Tikhonov regularization and iterative regularization. In general, it requires to solve a large number of optimization problems constrained by 3D partial differential equations (PDE). In this case, an improved Tikhonov regularization method is developed to quickly obtain optimal Tikhonov regularization parameter and thereby can accelerate the entire inverse calculation process. Furthermore, an iterative regularization method that is based on the conjugate gradient method is proposed to handle the high-resolution measurement data and many pointwise measurement data. In pool boiling experiments, they can be obtained by an infrared camera and mirco-thermocouples, respectively. Secondly, a completely different approach, a direct two-step strategy is presented to solve the same 3D transient ICHP on thin-plates that commonly occur in engineering applications. The main idea of the approach is to transform the inverse problem into a forward problem that is easier to solve by replacing the unknown Neumann boundary condition with an approximated Dirichlet boundary condition. The method has great advantage in efficiency compared with the optimization-based method that often requires to solve dozens or hundreds of direct, adjoint and sensitivity PDE problems, and therefore has the potential to be developed as efficient online heat-flux soft sensors for many practical applications. Thirdly, to further speed up the numerical solution procedure we develop a time-space dependent adaptive mesh refinement strategy due to the fact that the heat-flux and temperature dynamics related to bubble structures rapidly change in space and time on the boiling surface. A few classical engineering benchmark problems are considered to validate the aforementioned solution approaches, and the proposed approaches are also successfully applied to reconstruct the transient heat-flux distributions on the boiling surface from the noisy experimental data obtained in a series of real pool boiling experiments along the entire boiling curve. On the one hand, the corresponding research works reviewed can be used to explore the heat-transfer mechanism and establish a more accurate heat transfer model enabling the fast, robust and accurate reconstruction of local heat fluxes, which can be considered as realistic boundary conditions on the boiling surface to further study the two-phase flow and heat transfer process of boiling fluid above the heater. On the other hand, it can be also applied to explore the local details of the enhanced pool boiling heat transfer for the design of new heaters such as micro/nano structured surfaces, provide predictions related to the boiling process and conduct optimal experimental design in a systematic way. In the future work, a large number of 3D IHCP for different test regularization parameters will be calculated through a high-throughput parallel computational strategy. Besides, for thick heaters the direct two-step strategy can be used to solve the 3D ICHP accurately with an initial heat-flux guess obtained by the proposed optimization-based inversion approach. This way, the computational efficiency can be further improved. Furthermore, other new inversion methods, for example, which are based on artificial neural network method are being developed to solve 3D transient IHCP for more complex structured heater surfaces such as the micro/nano structures.

On the mesh resolution of industrial LES based on the DNS of flow over the T106C turbine

A high-order Navier-Stokes solver based on the flux reconstruction (FR) or the correction procedure via reconstruction (CPR) formulation is employed to perform a direct numerical simulation (DNS) and large eddy simulations (LES) of a well-known benchmark problem – transitional flow over the low-pressure T106C turbine cascade. Hp-refinement studies are carried out to assess the resolution requirement. A 4th order (p3) simulation on the fine mesh is performed with a DNS resolution to establish a "converged" solution, including the mean pressure and skin-friction distributions, and the power spectral density in the wake. Then LES on the coarse and fine meshes with lower order schemes are conducted to assess the mesh and order dependence of the solution. In particular, we study the error in the transition location, the mean skin-friction distribution, and the mean lift and drag coefficients. These h- and p-refinement studies provide a much-needed guideline in h- and p- resolutions to achieve a certain level of accuracy for industrial LES applications.

On the Accuracy of Reconstructing Plasma Sheet Electron Fluxes From Temperature and Density Models

AbstractThe particle simulations of the inner magnetosphere require time‐dependent boundary conditions for the particle flux set in the transition region between dipolar and tail‐like configurations. Usually, the flux is reconstructed from particle density and temperature predicted by empirical models or magnetohydrodynamic simulations. However, this method requires assumptions about the energy spectra to be made. This uncertainty adds to the inaccuracy of the empirical models or magnetohydrodynamic predictions. We use electron flux measurements in the nightside at r=6–11RE in the 1–300 keV energy range to estimate the potential accuracy of the electron flux reconstruction from the macroscopic plasma parameter models. We use kappa and Maxwellian distribution functions as well as two population approximations to describe the electron spectra. It is found that this method works reasonably well in the thermal energy range (1–10 keV). However, the average difference between measured and predicted fluxes becomes as large as 1 order of magnitude at energies ≥40 keV. The optimal value of the kappa parameter is found to be between 3 and 4, but it depends strongly on magnetic local time and radial distance. We conclude that the development of the flux‐based models (model of differential flux at several reference energies) instead of density and temperature models can be considered as a promising direction.

Three-dimensional vortical structures and wall shear stress in a curved artery model

We numerically investigate spatial and temporal evolution of multiple three-dimensional vortex pairs in a curved artery model under a fully developed pulsatile inflow of a Newtonian blood-analog fluid. We discuss the connection along the axial direction between regions of organized vorticity observed at various cross sections of the model, extending previous two-dimensional analysis. We model a human artery with a simple, rigid 180° curved pipe with circular cross section and constant curvature, neglecting effects of taper, torsion, and elasticity. Numerical results are computed from a discontinuous high-order spectral element flow solver using the flux reconstruction scheme and compared to experimental results obtained using particle image velocimetry. The flow rate used in both the simulation and the experiment is physiological. Vortical structures resulting from secondary flow are observed in various cross sections of the curved pipe, in particular, during the deceleration phase of the physiological waveform. We provide side-by-side comparisons of the numerical and experimental velocity and vorticity fields during acceleration and deceleration, the latter during which multiple vortical structures of both Dean-type and Lyne-type coexist. Correlations and quantitative comparisons of the data at these cross sections are computed along with trajectories of Dean-type vortices. Comparing cross-sectional flow fields and vortices provides a means to validate wall shear stress values computed from these numerical simulations, since the evolution of interior flow structures is heavily dependent upon geometry curvature and inflow and boundary conditions.

Poincaré–Friedrichs inequalities of complexes of discrete distributional differential forms

We derive bounds for the constants in Poincare–Friedrichs inequalities with respect to mesh-dependent norms for complexes of discrete distributional differential forms. A key tool is a generalized flux reconstruction which is of independent interest. The results apply to piecewise polynomial de Rham sequences on bounded domains with mixed boundary conditions.

Applications of multi-dimensional schemes on unstructured grids for high-accuracy heat flux prediction

The simulation of hypersonic flows with fully unstructured (tetrahedral) grids has severe problems with respect to the prediction of stagnation region heating, due to the random face orientation without alignment to the bow shock. To improve the accuracy of aero-heating predictions, three multi-dimensional approaches on unstructured grids are coupled in our Reynolds-averaged Navier–Stokes (RANS) solver, including multi-dimensional upwind flux reconstruction (MUP), multi-dimensional limiter (MLP-u2) and multi-dimensional gradient reconstruction (MLR). The coupled multi-dimensional RANS solver is validated by several typical verification and validation (V&V) cases, including hypersonic flows over a cylinder, a blunt biconic, and a double-ellipsoid, with commonly used prism/tetrahedral hybrid grids. Finally, the coupled multi-dimensional solver is applied to simulating the heat flux distribution over a 3D engineering configuration, i.e. a Hermes-like space shuttle model. The obtained numerical results are compared with experimental data. The predicted results demonstrate that the coupled multi-dimensional approach has a good prediction capability on aerodynamic heating over a wide range of complex engineering configurations.

A locally conservative reduced flux reconstruction for elliptic problems

AbstractIn the context of model order reduction of parametric elliptic problems, we present a methodology to reconstruct a conforming flux from a given a reduced solution, that is locally conservative with respect to the underlying finite element grid. All components of the procedure depend separably on the parameter and allow for further use in offline/online decomposed computations, for instance in the context of a posterior error estimation or flow problems.

Hypersonic Heat Flux Reconstruction with Distributed Temperature Sensors

This paper presents a new method for the multidimensional reconstruction of the heat flux via distributed surface temperature measurements and its validation using hypersonic wind tunnel tests. The first innovation is the formulation of the reconstruction in terms of the Green’s function, approximated using a weak Galerkin technique together with the method of variation of parameters. When compared with previous research, the method does not require nullification of the first derivative of the polynomial basis at boundaries of the second kind, and it applies seamlessly to complex geometries. The second innovation is the use of homotopy to reduce the nonlinear conduction at high temperature to a series of linear subproblems. Our main contribution is the determination of the conditions for the quadrature formulas to yield a convergent homotopy sequence. The third innovation is the use of a multidimensional B-spline basis to reconstruct the heat flux on curved surfaces and in time. The inverse problem projected in the B-spline basis is well-posed and does not require regularization parameters. The fourth innovation is the introduction of a piecewise Green’s function on partitioned multimaterial domains by imposing continuity of heat flux and temperature. The partitioned approach improves the accuracy of the reconstruction in the linear and nonlinear regimes.

A technique of flux reconstruction at the interfaces of nonconforming grids for aeroacoustic simulations

SummaryA flux reconstruction technique is presented to perform aeroacoustic computations using implicit high‐order spatial schemes on multiblock structured grids with nonconforming interfaces. The use of such grids, with mesh spacing discontinuities across the block interfaces, eases local mesh refinements, simplifies the mesh generation process, and thus facilitates the computation of turbulent flows. In this work, the spatial discretization consists of sixth‐order finite‐volume implicit schemes with low‐dispersion and low‐dissipation properties. The flux reconstruction is based on the combination of noncentered schemes with local interpolations to define ghost cells and compute flux values at the grid interfaces. The flow variables in the ghost cells are calculated from the flow field in the grid cells using a meshless interpolation with radial basis functions. In this study, the flux reconstruction is applied to both plane and curved nonconforming interfaces. The performance of the method is first evaluated by performing two‐dimensional simulations of the propagation of an acoustic pulse and of the convection of a vortex on Cartesian and wavy grids. No significant spurious noise is produced at the grid interfaces. The applicability of the flux reconstruction to a three‐dimensional computation is then demonstrated by simulating a jet at a Mach number of 0.9 and a diameter‐based Reynolds number of 4×105 on a Cartesian grid. The nonconforming grid interface located downstream of the jet potential core does not appreciably affect the flow development and the jet sound field, while reducing the number of mesh points by a factor of approximately two.

Stable broken $H^1$ and $H(\mathrm {div})$ polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions

We study extensions of piecewise polynomial data prescribed on faces and possibly in elements of a patch of simplices sharing a vertex. In the $H^1$ setting, we look for functions whose jumps across the faces are prescribed, whereas in the ${\bf H}(\mathrm{div})$ setting, the normal component jumps and the piecewise divergence are prescribed. We show stability in the sense that the minimizers over piecewise polynomial spaces of the same degree as the data are subordinate in the broken energy norm to the minimizers over the whole broken $H^1$ and ${\bf H}(\mathrm{div})$ spaces. Our proofs are constructive and yield constants independent of the polynomial degree. One particular application of these results is in a posteriori error analysis, where the present results justify polynomial-degree-robust efficiency of potential and flux reconstructions.

A simplified finite volume lattice Boltzmann method for simulations of fluid flows from laminar to turbulent regime, Part I: Numerical framework and its application to laminar flow simulation

In this paper, an unstructured grid based finite volume lattice Boltzmann method (FVLBM) that can be used for the simulation of incompressible laminar flows is presented and studied in detail. This method is derived from a simple modification of the cell-vertex unstructured grid based FVLBM proposed by Stiebler et al. (2006). Compared with other complex flux reconstruction methods, the present scheme has a low computational cost and can also achieve second-order spatial accuracy. Furthermore, depending on the use of the different temporal discretization schemes, the temporal accuracy can be adjusted for both steady and unsteady flows. Besides, some comparisons of the computational cost and accuracy with another FVLBM scheme are also presented. Meanwhile, different boundary conditions are illustrated that are easy to implement on complex geometries. To validate the present method, four cases are carried out, including a Couette flow driven by one plate for an accuracy test, flow in a square cavity, flow around a single circular cylinder and a more complex flow around double circular cylinders. Numerical experiments show that the present scheme can simulate steady and unsteady flows at relatively high Reynolds number with relatively few grid cells, thus demonstrating the good capability of the present method.