Reconstruction Solver Research Articles

Gridder-HO: Rapid and efficient parallel software for high-order curvilinear mesh generation

The advancement in high-order computational methods is reshaping the landscape of mesh generation in Computational Fluid Dynamics (CFD), steering the focus towards curvilinear mesh techniques to meet the escalating accuracy demands. Gridder-HO, the software designed to generate high-order curvilinear mesh efficiently and rapidly, has been developed. Gridder-HO supports the elevation of meshes to P2 (quadratic-order) or P3 (cubic-order). It features a layered architecture and utilizes the concurrent hash table and the Alternating Digital Tree (ADT) data structure, supporting thread-level parallelism to convert straight-edge mesh into high-order curvilinear mesh seamlessly. Gridder-HO utilizes the projection method based on a thread pool to precisely preserve geometry, and employs a novel localized RBF method with ADT for volume node interpolation to untangle the mesh, which aims to achieve a satisfactory balance between efficiency and accuracy. Validated through CFD simulations using the GPU-accelerated Python Flux Reconstruction (PyFR) solver, the practicality of Gridder-HO is demonstrated across various Reynolds numbers in typical cases such as sphere, cylinder, and SD7003 airfoil. These results confirm the high-order curvilinear meshes generated by Gridder-HO meet the high-order requirements of emerging computational methods. Moreover, Gridder-HO exemplifies its effectiveness in generating large-scale, high-order curvilinear meshes for the DLR-F6 transport aircraft configuration standard test cases. It elevates a mesh with 5 million elements to P2 in 3 min 39 sec at 68% parallel efficiency on 16 threads, and another with 14 million elements to P3 in 52 min 39 sec at 60% efficiency, illustrating its efficiency and potential in satisfying the demands of complex geometries in engineering applications.

Development of an implicit high-order Flux Reconstruction solver for high-speed flows on simplex elements

In this study, a significant emphasis is placed on the extension of high-order Flux Reconstruction (FR) schemes to cater to high-speed flows, specifically those involving high Mach numbers. The paper delves into the development and implementation of these FR schemes on both straight and curved-edged 2D and 3D simplex elements within the open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid Dynamics) platform. The proposed FR solver provides more accurate detection of complex flow features over relatively coarser meshes when compared to their low-order peers (particularly in presence of shock waves), is fully implicit and able to simulate compressible flow problems governed by the Euler or Navier-Stokes equations on triangular and tetrahedral meshes. In order to simulate cases with shocks, a shock-capturing scheme previously developed for quadrilateral elements, was extended to tackle supersonic and hypersonic simulations. Extensive verification of the resulting FR solver (up to 7th order, a.k.a P6, for the solution polynomial and 3rd order a.k.a Q2 for the geometrical representation) has been performed on benchmark test cases with flow speeds ranging from subsonic to hypersonic, up to Mach 9.6. The obtained results have been favorably compared to those available in the literature.

An efficient implementation of compact third-order implicit reconstruction solver with a simple WBAP limiter for compressible flows on unstructured meshes

In this paper, we present the development of a third-order density-based solver within the OpenFOAM framework, tailored for handling compressible flows. The solver incorporates implicit variational reconstruction on three-dimensional unstructured meshes, as well as a novel technology coupling reconstruction and time integration for both steady and unsteady simulations. To address the challenge of achieving high-order accuracy for curved geometries, we introduce a new approach for curved wall boundary reconstruction, specifically designed for situations where high-order mesh information is not readily available in OpenFOAM. Furthermore, we propose a simple WBAP limiter capable of capturing shocks without necessitating the whole-domain successive limiting procedure. Numerical tests were conducted to assess the solver's performance. The results reveal that our established solver exhibits higher accuracy in smooth flow simulations, while maintaining an excellent balance between accuracy and robustness for problems involving strong shocks and other complex flow structures.

Development of an implicit high-order flux reconstruction solver for the Langtry-Menter Laminar-Turbulent Transition RANS model

This work presents the development of the first high-order Flux Reconstruction (FR) solver for flows governed by the compressible Reynolds-Averaged Navier-Stokes (RANS) equations including Laminar-Turbulent Transition (LTT). The solver is fully implicit and implemented in the open-source COOLFluiD platform. Both the k−ω˜ Shear Stress Transport (SST) and the Langtry-Menter Local Correlation-Based Transition Model (LCTM) are implemented. Some modifications are made to the original k−ω SST model in order to obtain more robust and stable simulations. In particular, the natural logarithm of ω is transported and the equations are transformed accordingly. The implementation of the solver and the physics model are discussed in detail, as well as the residual Jacobian assembly algorithm and source term Jacobian treatment. In order to obtain adequate computational efficiency at higher orders, a semi-analytical Jacobian for the FR method is derived and investigated, as well as a specific Jacobian treatment to maintain stability and obtain favorable convergence characteristics. The solver is verified both in fully turbulent flow and in several transitional flow cases. The results of the verification cases are compared to references from the literature, showing a good agreement in all cases. Furthermore, it is shown that for increasingly high orders and fine meshes, the solver solutions coincide with each other, verifying grid independence. It is also shown that the FR method obtains accurate results on relatively coarse meshes, with a considerably larger first element height (y1+), i.e. up to y1+≃25, than state-of-the-art second-order finite volume solvers without wall-extended functions. Finally, the performance of the solver is briefly discussed, focusing on the scalability of the solver in terms of order of accuracy.

R-adaptive algorithms for supersonic flows with high-order Flux Reconstruction methods

The present paper addresses the development and implementation of the first r-adaptive mesh refinement (r-AMR) algorithm for a high-order Flux Reconstruction solver. The r-refinement consists on nodal re-positioning while keeping the number of mesh nodes and their connectivity frozen. The developed algorithm is based on physics-driven spring-analogies, where the mesh can be seen as a network of fictitious springs. While AMR increases the local mesh density, the high-order Flux Reconstruction method potentially provides a more accurate detection of complex flow features over relatively coarser mesh, when compared to low-order methods. In this work, a concise overview of the Flux Reconstruction method and spring-based AMR techniques will be given, followed by some promising results of r-AMR applied to benchmark high-order steady-state supersonic flow simulations.

Convolutional neural network-based spectrum reconstruction solver for channeled spectropolarimeter.

Channeled spectropolarimetry is a snapshot technique for measuring the spectra of Stokes parameters of light by demodulating the measured spectrum. As an indispensable part of the channeled spectropolarimeter, the spectrometer module is far from being perfect to reflect the real modulation spectrum, which further reduces the polarimetric reconstruction accuracy of the channeled spectropolarimeter. Since the modulation spectrum is composed of many continuous narrow-band spectra with high frequency, it is a challenging work to reconstruct it effectively by existing methods. To alleviate this issue, a convolutional neural network (CNN)-based spectral reconstruction solver is proposed for channeled spectropolarimeter. The key idea of the proposed method is to first preprocess the measured spectra using existing traditional methods, so that the preprocessed spectra contain more spectral features of the real spectra, and then these spectral features are employed to train a CNN to learn a map from the preprocessed spectra to the real spectra, so as to further improve the reconstruction quality of the preprocessed spectra. A series of simulation experiments and real experiments were carried out to verify the effect of the proposed method. In simulation experiments, we investigated the spectral reconstruction accuracy and robustness of the proposed method on three synthetic datasets and evaluate the effect of the proposed method on the demodulation results obtained by the Fourier reconstruction method. In real experiments, system matrices are constructed by using measured spectra and reconstructed spectra respectively, and the spectra of Stokes parameters of incident light are estimated by the linear operator method. Several other advanced demodulation methods are also used to demodulate the measured spectrum in both simulation and real experiments. The results show that compared with other methods, the accuracy of the demodulation results can be much more improved by employing the CNN-based solver to reconstruct the measured spectrum.

AN IMPLICIT BLOCK JACOBI APPROACH FOR HIGH-ORDER FLUX RECONSTRUCTION METHOD

Recently, the flux reconstruction (FR) method has attracted more and more attentions for its simplicity and generality. However, it is still computationally expensive and time consuming when simulating the complex flow problems by FR method. There is a huge demand for developing appropriate efficient implicit solvers and parallel computing techniques for FR. This paper proposes an implicit high-order flux reconstruction solver on GPU platform based on the block Jacobi iteration method. As it is inefficient to solve the large global linear system resulting from spatial and implicit temporal discretization of FR directly. A block Jacobi approach is used to change the characteristics of the lift-hand matrix of the global linear system and this avoids the dependence of neighboring elements. Therefore, only the diagonal blocks of global matrix need to be stored and calculated. Then, the problem of solving the huge global linear system is transformed into solving a series of local linear equations simultaneously. Finally, these small local linear equations would be solved by the LU decomposition method in parallel on GPU platforms. Two typical cases, including subsonic flows over a bump and a NACA0012 airfoil, were simulated and compared with the multi-grid explicit Runge-Kutta scheme. The numerical results demonstrated that the present implicit method can reduce the iterations significantly. Meanwhile, the implicit solver has shown at least 10x speedup over the multi-grid Runge-Kutta scheme in all cases.

Solver-informed neural networks for spectrum reconstruction of colloidal quantum dot spectrometers.

Recently, the miniature spectrometer based on the optical filter array has received much attention due to its versatility. Among many open challenges, designing efficient and stable algorithms to recover the input spectrum from the raw measurements is the key to success. Of many existing spectrum reconstruction algorithms, regularization-based algorithms have emerged as practical approaches to the spectrum reconstruction problem, but the reconstruction is still challenging due to ill-posedness of the problem. To alleviate this issue, we propose a novel reconstruction method based on a solver-informed neural network (NN). This approach consists of two components: (1) an existing spectrum reconstruction solver to extract the spectral feature from the raw measurements (2) a multilayer perceptron to build a map from the input feature to the spectrum. We investigate the reconstruction performance of the proposed method on a synthetic dataset and a real dataset collected by the colloidal quantum dot (CQD) spectrometer. The results demonstrate the reconstruction accuracy and robustness of the solver-informed NN. In conclusion, the proposed reconstruction method shows excellent potential for spectral recovery of filter-based miniature spectrometers.

High-Order Overset Flux Reconstruction Method for Dynamic Moving Grids

Overset meshes have a unique advantage in handling moving boundary problems as remeshing is often unnecessary. Recently, overset Cartesian and strand meshes were used successfully to compute complex flow over rotorcraft. Although it is quite straightforward to deploy a high-order finite difference method on the Cartesian mesh, the near-body solver for the strand mesh is often limited to second-order accuracy. In the present study, a high-order flux reconstruction solver, hpMusic, is developed on both the near-body and background grids, and it is extended to handle moving boundary problems. Accuracy studies are carried out, and the designed order of accuracy is obtained for both inviscid and viscous flows. The method is then tested for a benchmark dynamic airfoil problem from the 4th International Workshop on High-Order CFD Methods. Mesh and order-convergent results are obtained and compared with those from other groups. Finally flow over a hovering rotor is simulated to compare with experimental data. In this case, the present high-order solver is capable of generating and propagating tip vortices with high resolution. Good agreement is achieved with experimental data in tip vortex core size, location, and the swirl velocity at third-order accuracy.

A high-order flux reconstruction method for 3D mixed overset meshes

The use of overset meshes can significantly simplify grid generation for complex configurations, and is particularly desired for moving boundary problems as remeshing is often unnecessary. In the present study, we develop a high-order flux reconstruction (FR) solver for mixed overset meshes including the near-body, and background meshes. The main objective is to achieve uniformly high order accuracy in the entire computational domain on both the near body and the background grids. Two different approaches to handle the overset interfaces are evaluated for accuracy, efficiency and robustness. Waves passing across the overset interfaces are tested with both smooth and discontinuous waves. In the present study, we focus on non-moving boundary problems, and demonstrate the overall methodology for steady and unsteady flow problems.

Implicit high-order flux reconstruction solver for high-speed compressible flows

The present paper addresses the development and implementation of the first high-order Flux Reconstruction (FR) solver for high-speed flows within the open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid Dynamics) platform. The resulting solver is fully implicit and able to simulate compressible flow problems governed by either the Euler or the Navier–Stokes equations in two and three dimensions. Furthermore, it can run in parallel on multiple CPU-cores and is designed to handle unstructured grids consisting of both straight and curved edged quadrilateral or hexahedral elements. While most of the implementation relies on state-of-the-art FR algorithms, an improved and more case-independent shock capturing scheme has been developed in order to tackle the first viscous hypersonic simulations using the FR method. Extensive verification of the FR solver has been performed through the use of reproducible benchmark test cases with flow speeds ranging from subsonic to hypersonic, up to Mach 17.6. The obtained results have been favorably compared to those available in literature. Furthermore, so-called “super-accuracy” is retrieved for certain cases when solving the Euler equations. The strengths of the FR solver in terms of computational accuracy per degree of freedom are also illustrated and its computation cost per degree of freedom is assessed for different orders. Finally, the influence of the characterizing parameters of the FR method as well as the influence of the novel shock capturing scheme on the accuracy of the developed solver is discussed.

Evaluation of Second- and High-Order Solvers in Wall-Resolved Large-Eddy Simulation

In the context of wall-resolved industrial large-eddy simulation, a comparison is made between a high-order flux reconstruction (FR)/correction procedure via reconstruction (CPR) solver (hpMusic) with refinement and a commercial second-order finite volume solver (Fluent) with mesh refinement ( refinement). A well-known benchmark problem in turbomachinery is employed: transonic flow over a von Karman Institute high-pressure turbine vane at a Reynolds number of . All of the meshes originated from the same coarse mesh, a mixed unstructured mesh, generated through global uniform refinement for the purpose of evaluating the solution dependence on mesh and polynomial order. Because the meshes used for hpMusic and Fluent belong to the same family, useful information about solution accuracy and efficiency can be obtained. Detailed comparisons are made in mean surface loading, heat transfer, power spectral density of pressure at selected monitor points, mean boundary-layer velocity and total temperature profiles, and wake loss. Numerical results are compared with experimental data, when available. The high-order FR/CPR method is shown to achieve a higher accuracy at a reduced cost than the second-order finite volume method.

A finite integration forward solver and a domain search reconstruction solver for electrical resistivity tomography (ERT)

This paper describes the development of a forward solver and an inverse solver for modelling the resistivity distribution in electrical resistivity tomography. The forward solver is based on the finite integration technique and was developed to simulate and model synthetic resistance data. The inverse solver, the domain search algorithm, reconstructs the resistivity distribution within the subsurface by searching for a model that gives an acceptable fit to the simulated data. To find an optimum model that fits the simulated data, the domain search algorithm searches for the minimum of a multi-variable objective function by using a step-wise refinement approach. The forward and inverse solvers were developed as necessary forward and inversion modelling tools to enable the optimisation of electrode positions in resistivity imaging to be explored. Numerical results from implementing these solvers show that they are successful for simulating and reconstructing the resistivity distribution in electrical resistivity tomography.

Influence of non-local thermodynamic equilibrium and Zeeman effects on magnetic equilibrium reconstruction using spectral motional Stark effect diagnostic.

The Motional Stark Effect (MSE) diagnostic is a well established technique to infer the local internal magnetic field in fusion plasmas. In this paper, the existing forward model which describes the MSE data is extended by the Zeeman effect, fine-structure, and relativistic corrections in the interpretation of the MSE spectra for different experimental conditions at the tokamak ASDEX Upgrade. The contribution of the non-Local Thermodynamic Equilibrium (non-LTE) populations among the magnetic sub-levels and the Zeeman effect on the derived plasma parameters is different. The obtained pitch angle is changed by 3°…4° and by 0.5°…1° including the non-LTE and the Zeeman effects into the standard statistical MSE model. The total correction is about 4°. Moreover, the variation of the magnetic field strength is significantly changed by 2.2% due to the Zeeman effect only. While the data on the derived pitch angle still could not be tested against the other diagnostics, the results from an equilibrium reconstruction solver confirm the obtained values for magnetic field strength.

On the utility of GPU accelerated high-order methods for unsteady flow simulations: A comparison with industry-standard tools

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 75× and 3× 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.

Fundamental Study of Wave Propagation on Liquid Surface Related IFE Mirror

A laser fusion reactor needs an optical mirror in its final optical system. This optical mirror is exposed to the neutron produced with fusion reaction. It is pointed out that the exposure to neutron will produce hydrogen gas in the mirror and cause swelling deformation of mirror. To avoid this swelling of mirror, a liquid-metal mirror is promising. High energy laser shots on the liquid mirror will cause the surface wave. These waves must be damped to under 1/10 of the laser wavelength in 250 ms or less. In this study, the hydrodynamic behavior of the liquid surface was investigated by experiment with water as surrogate liquid, the computational evaluation for the wave propagation with the MARS (Multi-interface Advection and Reconstruction Solver, 2001) was carried out, and the design window for the optical mirror based on the water experiment was discussed.

TH-CD-303-02: A GPU-Based Iterative Image Reconstruction Solver with 4D Regularization for Low-Dose Helical 4DCT

Purpose: 4DCT is an important technique for motion management in radiotherapy, but involves a significant amount of dose to patients. Low-dose helical 4DCT scans can be done by (a) lowering tube current and pulse duration, (b) increasing helical pitch, and (c) reducing data sampling. Conventional FDK methods may fail for these low-dose scan protocols due to noisy or incomplete data. We aim to develop an accurate and fast image reconstruction solver that enables dose reduction for helical 4DCT. Methods: For improved accuracy, the developed solver is based on a novel 4D regularization method for CT image reconstruction, which promotes both the smoothness in spatial variables for each phase and the similarity in temporal variables between phases, in contrast with the 3D regularization method with spatial regularization only. Both 3D and 4D methods here are based on total variation. The formulated optimization problem is solved by alternating direction method of multipliers (ADMM), which is chosen for its effectiveness for solving the convex problem, and more importantly its suitability for distributed computing.For improved speed, the developed solver utilizes an efficient parallel algorithm for X-ray transform and its adjoint with optimized computational cost per parallel thread O (1) that takes flying focal spots into account, and multi-GPU platforms for solving such a massive parallel computing problem in helical 4DCT. Results: The solver was validated using both simulations (4D NCAT phantoms) and experimental sinogram data (Siemens, Somatom 64-slices) with low-dose, high-pitch and sparse-view helical 4DCT scans. Both the reconstructed images and quantitative results demonstrate that the 4D method improved image quality compared with the 3D method, which is in turn better than FDK. Conclusion: We have developed a fast, accurate solver that synergizes state-of-art iterative reconstruction methods and parallel computing techniques that can potentially enable low-dose, high-pitch and sparse-view helical 4DCT scans. Minghao Guo and Hao Gao were partially supported by the NSFC (#11405105), the 973 Program (#2015CB856000) and the Shanghai Pujiang Talent Program (#14PJ1404500).

Numerical Study on Bubble Behaviour and Heat Transfer Characteristics of Subcooled Pool Boiling Based on Non-Empirical Boiling and Condensation Model

In this study, the transient three-dimensional numerical simulations based on the MARS (Multi-interface Advection and Reconstruction Solver) with the non-empirical boiling and condensation model have been conducted for isolated boiling bubble behaviour in a subcooled pool. The effects of the wettability on the heating surface for the subcooled bubble departure behaviour were investigated. The numerical results showed in very good agreement with the experimental results. Furthermore, resulting from the wall heat flux evaluation, it was found that the wall heat flux near the contact line at the bottom of the bubble just before the bubble departing from the heating surface increases with increases of the degree of subcooling.

Numerical Study on Mass Transfer of a Vapor Bubble Rising in Very High Viscous Fluid

This study focused on a bubble rising behavior in a molten glass because it is important to improve the efficiency of removal of bubbles from the molten glass. On the other hand, it is expected that some gas species which exists in a bubble are transferred into the molten glass through the bubble interface, i.e., the mass transfer, subsequently, it may cause a bubble contraction in the molten glass. In this paper, in order to understand the bubble rising behavior with its contraction caused by the mass transfer through the bubble interface in the very high viscous fluid such as the molten glass, a bubble contraction model has been developed. The direct numerical simulations based on the MARS (Multi-interface Advection and Reconstruction Solver) coupled with the mass transfer equation and the bubble contraction model regarding the mass transfer from the rising bubble in very high viscous fluid have been performed. Here, the working fluids were water vapor as the gas species and the molten glass as the very high viscous fluid. Also, the jump conditions at the bubble interface for the mass transfer were examined. Furthermore, the influence of the bubble contraction for the bubble rising compared to that in the water as a normal viscous fluid was investigated. From the result of the numerical simulations, it was found that the bubble rising behavior was strongly affected not only by the viscosity of the working fluid but also by the bubble contraction due to the mass transfer through the bubble interface.

Direct Numerical Simulation and Visualization of Subcooled Pool Boiling

A direct numerical simulation of the boiling phenomena is one of the promising approaches in order to clarify their heat transfer characteristics and discuss the mechanism. During these decades, many DNS procedures have been developed according to the recent high performance computers and computational technologies. In this paper, the state of the art of direct numerical simulation of the pool boiling phenomena during mostly two decades is briefly summarized at first, and then the nonempirical boiling and condensation model proposed by the authors is introduced into the MARS (MultiInterface Advection and Reconstruction Solver developed by the authors). On the other hand, in order to clarify the boiling bubble behaviors under the subcooled conditions, the subcooled pool boiling experiments are also performed by using a high speed and high spatial resolution camera with a highly magnified telescope. Resulting from the numerical simulations of the subcooled pool boiling phenomena, the numerical results obtained by the MARS are validated by being compared to the experimental ones and the existing analytical solutions. The numerical results regarding the time evolution of the boiling bubble departure process under the subcooled conditions show a very good agreement with the experimental results. In conclusion, it can be said that the proposed nonempirical boiling and condensation model combined with the MARS has been validated.