Three-dimensional Unstructured Meshes Research Articles

Implicit high-order gas-kinetic schemes for compressible flows on three-dimensional unstructured meshes I: Steady flows

In the previous studies, the high-order gas-kinetic schemes (HGKS) have achieved successes for unsteady flows on three-dimensional unstructured meshes. In this paper, to accelerate the rate of convergence for steady flows, the implicit non-compact and compact HGKSs are developed. For non-compact scheme, the simple weighted essentially non-oscillatory (WENO) reconstruction is used to achieve the spatial accuracy, where the stencils for reconstruction contain two levels of neighboring cells. Incorporate with the nonlinear generalized minimal residual (GMRES) method, the implicit non-compact HGKS is developed. In order to improve the resolution and parallelism of non-compact HGKS, the implicit compact HGKS is developed with Hermite WENO (HWENO) reconstruction, in which the reconstruction stencils only contain one level of neighboring cells. The cell averaged conservative variable is also updated with GMRES method. Simultaneously, a simple strategy is used to update the cell averaged gradient by the time evolution of spatial-temporal coupled gas distribution function. To accelerate the computation, the implicit non-compact and compact HGKSs are implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy, robustness and efficiency of both inviscid and viscous flows.

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.

A cache-efficient reordering method for unstructured meshes with applications to wall-resolved large-eddy simulations

Unstructured meshes provide a distinct advantage for handing complex geometries. However, the low cache utilization due to mesh-related data access patterns raises particular challenges in achieving a high computing efficiency for large-eddy simulations. We propose a geometrical-based mesh reordering method to improve cache utilization. The proposed method utilizes a Hilbert space-filling curve that passes through each cell once, guiding the reordering of the cells of the unstructured mesh. The reordering enables neighboring cells to be stored in contiguous areas of memory as much as possible. The performance of the proposed method is validated by two- and three-dimensional unstructured meshes. According to the memory and spatial distances, the proposed reordering method significantly improves data localities. Consequently, the cache hit rate is increased and the computing efficiency is improved. The proposed reordering method is then applied to large-eddy simulations of flows around an underwater vehicle model with ReL=1.2×106. To fully resolve the near-wall flows, an unstructured mesh consisting of 1.476 billion cells is used, which is partitioned into 12800 subdomains. The computed velocity profiles, pressure and friction coefficients are in good agreement with the experimental measurements. The computational costs are reduced by using the proposed reordering method.

A coupled level set and volume of fluid method for three dimensional unstructured polyhedral meshes for boiling flows

For the numerical modelling of boiling flows with accurate interface dynamics, a new coupled level set and volume of fluid (CLSVOF) method for three-dimensional unstructured polyhedral meshes has been presented in this paper. An unsplit multidirectional method is used for advecting volume fraction. In this method, the cell vertex velocities interpolated from the cell centres are used to create the flux polyhedron. A total variational diminishing (TVD) scheme is used for the advection of level set function. The level set field is geometrically reinitialized after every time step. The present CLSVOF method is tested against several two as well as three-dimensional advection test cases for both structured and unstructured polyhedral meshes to determine its accuracy and performance. In terms of advection test cases, the suggested method shows good agreement with results of previous studies that used interface reconstruction techniques of higher order. The proposed method is then used to simulate two as well as three-dimensional saturated film boiling over a flat planar surface for both unstructured polyhedral and structured meshes. The obtained results are in accordance with the results available in literature. The present CLSVOF method is also demonstrated for saturated film boiling on a horizontal cylinder in both two and three dimensions. In comparison to the earlier numerical studies, the present results are in good accordance with the available semi-empirical correlations. • A CLSVOF method is developed for boiling flows on 3D unstructured polyhedral meshes. • The proposed boiling flow solver can be used for any complex geometries. • The solver is validated for two-phase flows with and without phase change. • For boiling flows, the present results agree well with semi-empirical correlations.

High-Order CFD Solvers on Three-Dimensional Unstructured Meshes: Parallel Implementation of RKDG Method with WENO Limiter and Momentum Sources

In aerospace engineering, high-order computational fluid dynamics (CFD) solvers suitable for three-dimensional unstructured meshes are less developed than expected. The Runge–Kutta discontinuous Galerkin (RKDG) finite element method with compact weighted essentially non-oscillatory (WENO) limiters is one of the candidates, which might give high-order solutions on unstructured meshes. In this article, we provide an efficient parallel implementation of this method for simulating inviscid compressible flows. The implemented solvers are tested on lower-dimensional model problems and real three-dimensional engineering problems. Results of lower-dimensional problems validate the correctness and accuracy of these solvers. The capability of capturing complex flow structures even on coarse meshes is shown in the results of three-dimensional applications. For solving problems containing rotary wings, an unsteady momentum source model is incorporated into the solvers. Such a finite element/momentum source hybrid method eliminates the reliance on advanced mesh techniques, which might provide an efficient tool for studying rotorcraft aerodynamics.

Efficient tetrahedralization of multi-material images with quality, fidelity, and topological guarantees

An algorithm is proposed for generating three-dimensional unstructured tetrahedral meshes of multi-material images. The algorithm produces a mesh whose boundary is proved to be homeomorphic to the object surface. In addition, it provides a guaranteed dihedral angle bound of up to 19 . 4 7 ∘ for the output tetrahedra. Moreover, it allows for user-specified guaranteed bounds on the distance between the boundaries of the mesh and the boundaries of the materials. It produces a small number of mesh elements that comply with these guarantees, and is compatible in performance with other software. The algorithm offers all of the following properties: bounded dihedral angles, customized two-sided Hausdorff distance, topological faithfulness, a small number of elements, and run-time performance. The experimental evaluation on synthetic and real medical data illustrates the efficiency and effectiveness of the method. • The mesh offers a faithful topology to the materials. • Elements with arbitrarily small angles do not appear in the mesh. • The mesh offers a close representation (fidelity) of the underlying materials. • The mesh contains a small number of elements that comply with the guarantees above. • The mesh can be constructed within tight time constraints.

An Efficient Parallel Implementation of the Runge–Kutta Discontinuous Galerkin Method with Weighted Essentially Non-Oscillatory Limiters on Three-Dimensional Unstructured Meshes

In computational fluid dynamics, high-order solvers suitable for three-dimensional unstructured meshes are attractive but are less developed than other methods. In this article, we provide the formulation and a parallel implementation of the Runge–Kutta discontinuous Galerkin finite element method with weighted essentially non-oscillatory limiters, which are compact and effective for suppressing numerical oscillations near discontinuities. In our experiments, high-order solvers do outperform their low-order counterparts in accuracy and the efficient parallel implementation makes the time cost affordable for large problems. Such high-order parallel solvers are efficient tools for solving conservative laws including the Euler system that models inviscid compressible flows.

A sliding-mesh interface method for three dimensional high order spectral difference solver

In this study a sliding-mesh interface method for three-dimensional unstructured mesh is developed and implemented to the high order spectral difference (SD) solver. The mortar approach is utilized for data exchange on the sliding interface for non-uniform unstructured mesh. The factors affecting the accuracy of the mortar method are analyzed, including mortar number, polynomial degree of mortar, non-uniformity of mortars, and mortar omission in polygon clipping. The method is validated with both inviscid and viscous problems with rotating flows. The numerical results show that the present method is able to retain the high order accuracy of the SD method. The influence of mortar omission in polygon clipping for poor quality complex mesh is discussed and evaluated with several validation problems. The errors with and without mortar omission are compared. The results show that the developed method is reliable and accurate even with mortar omission occurred occasionally in the simulation if the maximum omitted area is less than about 15% of its connected element on the sliding interface.

High-order gas-kinetic scheme on three-dimensional unstructured meshes for compressible flows

In this paper, a high-order gas-kinetic scheme is developed on three-dimensional unstructured meshes for compressible Euler and Navier–Stokes equations. To achieve the high-order spatial accuracy, the three-dimensional weighted essentially non-oscillatory (WENO) reconstruction is extended to the unstructured tetrahedral and hexahedral meshes. A simple strategy is adopted for the selection of candidate stencils, and the topologically independent linear weights are used for the spatial reconstruction. The efficiency and robustness of the classical WENO reconstruction are improved. In addition to the two-stage fourth-order temporal discretization and lower–upper symmetric Gauss–Seidel method, the explicit and implicit high-order gas-kinetic schemes are developed for unsteady and steady problems. Accuracy tests on hexahedral and tetrahedral grids validate the third-order of accuracy, and various three-dimensional incompressible and compressible numerical experiments are also presented. The results validate the accuracy and robustness of the proposed scheme for both inviscid and viscous flows. In the future, the current scheme will be extended to the hybrid unstructured meshes and Reynolds-averaged Navier–Stokes simulation with high Reynolds numbers.

An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part II: Subcell finite volume shock capturing

The second paper of this series presents two robust entropy stable shock-capturing methods for discontinuous Galerkin spectral element (DGSEM) discretizations of the compressible magneto-hydrodynamics (MHD) equations. Specifically, we use the resistive GLM-MHD equations, which include a divergence cleaning mechanism that is based on a generalized Lagrange multiplier (GLM). For the continuous entropy analysis to hold, and due to the divergence-free constraint on the magnetic field, the GLM-MHD system requires the use of non-conservative terms, which need special treatment.Hennemann et al. (2020) [25] recently presented an entropy stable shock-capturing strategy for DGSEM discretizations of the Euler equations that blends the DGSEM scheme with a subcell first-order finite volume (FV) method. Our first contribution is the extension of the method of Hennemann et al. to systems with non-conservative terms, such as the GLM-MHD equations. In our approach, the advective and non-conservative terms of the equations are discretized with a hybrid FV/DGSEM scheme, whereas the visco-resistive terms are discretized only with the high-order DGSEM method. We prove that the extended method is semi-discretely entropy stable on three-dimensional unstructured curvilinear meshes. Our second contribution is the derivation and analysis of a second entropy stable shock-capturing method that provides enhanced resolution by using a subcell reconstruction procedure that is carefully built to ensure entropy stability.We provide a numerical verification of the properties of the hybrid FV/DGSEM schemes on curvilinear meshes and show their robustness and accuracy with common benchmark cases, such as the Orszag-Tang vortex and the GEM (Geospace Environmental Modeling) reconnection challenge. Finally, we simulate a space physics application: the interaction of Jupiter's magnetic field with the plasma torus generated by the moon Io.

Parametric Calculations of the Aerodynamics of a Descent Vehicle

We present the methodology and results of parametric aerodynamic studies of vehicles descending into the planet’s atmosphere. The proposed computational approach might serve as the basis for solving a number of problems such as predicting and optimizing the descent trajectory of the vehicle, the search for a rational aerodynamic layout of the vehicle, i.e., tasks requiring massive parametric calculations. The systematization of such calculations is the first step towards the creation of a specialized database that includes sets of input and output data (flight speed, angles of attack, drag and lift coefficients, aerodynamic pitching moment, etc.) and the corresponding three-dimensional fields of gas-dynamic quantities together with computational meshes of various granularity and parameters of the computational model. Additional information to each element of the database might be a set of variables, parameterizing the geometry of the vehicle, experimental data, etc. The probability of forming the information content of such a data-base using modern supercomputer systems is shown. The capabilities of the domestic supercomputer aerodynamic code NOISEtte are demonstrated in the field of multiparametric three-dimensional calculations of descent vehicles based on the numerical solution of the Navier --- Stokes equations on three-dimensional unstructured meshes

Three-dimensional magnetotelluric modeling in general anisotropic media using nodal-based unstructured finite element method

Magnetotelluric sounding is an efficient and economical method for detecting Earth's deep structure because of its large surveying depth. We use a finite element approach to solve magnetotelluric fields in three-dimensional general anisotropic media based on the vector–scalar potentials and unstructured meshes. The implementation of a three-dimensional unstructured mesh generator TetGen with high-quality local mesh refinement can easily handle complicated models with high performance. The electromagnetic field equations are expressed in terms of Coulomb-gauged vector–scalar potentials, and the nodal-based finite element method can be used to compute the electromagnetic fields with high efficiency. The symmetric successive over-relaxation preconditioned conjugate gradient solver is used to solve the resulting linear equation system. The magnetotelluric responses in one-, two-, and three-dimensional medias are computed through our proposed approach. These results are then compared with the analytical solutions, two- and three-dimensional adaptive finite element solutions. Numerical examples demonstrate that these results are in good agreement. Because an unstructured mesh is incorporated in this approach, it has great potential for simulating the magnetotelluric responses in geometrically complicated situations.

Multi-material hydrodynamics with algebraic sharp interface capturing

A finite volume method for Eulerian multi-material hydrodynamics with sharp interface capturing is presented. The pressure-temperature non-equilibrium multi-material system with finite-rate pressure relaxation in mixed-cells is considered here. This pressure closure facilitates material-property-dependent pressure relaxation, rather than instantaneous pressure equilibration, which in turn allows the use of unsplit high-order time-integrators. A modified tangent of hyperbola for interface capturing (THINC) method is used to reconstruct multi-material (>2) interfaces, on three-dimensional unstructured meshes. A simple modification which extends the THINC reconstruction to interfaces between more than two materials is proposed. It is demonstrated that the modified THINC can capture multi-material interfaces within 2–4 tetrahedral cells. Since no geometric reconstructions are required by the THINC method, the presented multi-material method is algorithmically simple, and computationally efficient. Consistent reconstructions of conserved quantities at material interfaces ensure that conservation and closure laws are satisfied at the discrete level. Through a suite of test problems solved on unstructured meshes, it is demonstrated that the presented method is a promising candidate for accurate and efficient multi-material hydrodynamics computations.

A three-dimensional unified gas-kinetic wave-particle solver for flow computation in all regimes

In this paper, the unified gas-kinetic wave-particle (UGKWP) method has been constructed on a three-dimensional unstructured mesh with parallel computing for multiscale flow simulation. Based on the direct modeling methodology, the unified gas-kinetic scheme (UGKS) models the flow dynamics directly on the numerical mesh size and time step scales, and it is able to capture the flow dynamics from the kinetic scale particle transport to the hydrodynamic wave propagation seamlessly according to the local cell Knudsen number. Instead of discretizing the particle velocity space in UGKS, the UGKWP method is composed of evolution of deterministic wave and stochastic particles. With dynamic wave-particle decomposition according to the cell Knudsen number, the UGKWP method is able to capture the continuum wave interaction and rarefied particle transport under a unified framework and achieves high efficiency in different flow regimes. The UGKWP flow solver is constructed in three-dimensional space and is validated by many test cases at different Mach and Knudsen numbers. The examples include a 3D shock tube problem, lid-driven cubic cavity flow, high-speed flow passing through a cubic object, and hypersonic flow around a space vehicle. The parallel performance has been tested on the Tianhe-2 supercomputer, and reasonable parallel performance has been observed up to 1000 cores. With the wave-particle formulation, the UGKWP method has great potential in solving three-dimensional multiscale transport problems with the co-existence of continuum and rarefied flow regimes, especially for the high-speed rarefied and continuum flow simulation around a space vehicle in near-space flight, where the local Knudsen number can vary significantly with five or six orders of magnitude differences.

Research on remote control algorithm for parallel implicit domain robot patrol inspection on 3D unstructured grid

Aiming at the disadvantages of structural meshes that are difficult to deal with complex shapes and unstructured meshes, it is impossible to calculate the viscous flow with boundary layers. A parallel implicit algorithm based on three-dimensional unstructured meshes is developed. The turbulence model adopts the k − π turbulence model. Due to the difficulty of the LU-SGS parallel equivalent model in time advancement, a three-dimensional unstructured grid DSMC parallel algorithm is proposed. Under the unstructured grid dynamic partitioning strategy, each sub-area is guaranteed. The number of molecules is roughly equal. Based on the implicit algorithm of DP-LUR (data-parallel lower–upper relaxation) format, the remote control combination model of robot inspection is calculated, and the mutual interference flow problem during the inspection process is accurately simulated. The operator realizes the remote control of the inspection robot through the live image transmitted by the network. The actual operation shows that the system can remotely access the service system and message transmission of the inspection robot in the state of the sub-structure grid, and successfully solve the robot tour. Check the complex shape viscous flow field calculation and efficiency problems that are difficult to handle in remote control systems.

A preconditioned lattice Boltzmann flux solver for steady flows on unstructured hexahedral grids

The lattice Boltzmann flux solver (LBFS), first introduced by Shu et al. (2014) on structured meshes, allows fluid flow problems to be solved on unstructured meshes discretised by the finite volume method. The solver calculates the macroscopic fluxes at the cell interfaces from a local reconstruction of the lattice Boltzmann solution. In this paper the LBFS is extended to three-dimensional unstructured hexahedral meshes and a preconditioned lattice Boltzmann flux solver (PLBFS) is presented. The PLBFS involves applying the preconditioning technique proposed by Guo (2004) to the LBFS and is achieved by modifying the equilibrium distribution function used to calculate the macroscopic fluxes at the cell interface. When the PLBFS is applied to steady flow problems, it is shown that convergence is significantly accelerated and the accuracy of predictions with unstructured grids is greatly improved when compared to the LBFS. This paper also introduces a strategy for choosing the optimal value of preconditioning factor with unstructured hexahedral meshes.

Evaluation and Application of Newly Designed Finite Volume Coastal Model FESOM-C, Effect of Variable Resolution in the Southeastern North Sea

A newly developed coastal model, FESOM-C, based on three-dimensional unstructured meshes and finite volume, is applied to simulate the dynamics of the southeastern North Sea. Variable horizontal resolution enables coarse meshes in the open sea with refined meshes in shallow areas including the Wadden Sea and estuaries to resolve important small-scale processes such as wetting and drying, sub-mesoscale eddies, and the dynamics of steep coastal fronts. Model results for a simulation of the period from January 2010 to December 2014 agree reasonably well with data from numerous regional autonomous observation stations with high temporal and spatial resolutions, as well as with data from FerryBoxes and glider expeditions. Analyzing numerical solution convergence on meshes of different horizontal resolutions allows us to identify areas where high mesh resolution (wetting and drying zones and shallow areas) and low mesh resolution (open boundary, open sea, and deep regions) are optimal for numerical simulations.

A Highly Parallel Multilevel Newton--Krylov--Schwarz Method with Subspace-Based Coarsening and Partition-Based Balancing for the Multigroup Neutron Transport Equation on Three-Dimensional Unstructured Meshes

The multigroup neutron transport equation is crucial for studying the motion of neutrons and their interaction with materials. Numerical simulation of the multigroup neutron transport equation is computationally challenging because the equation is defined on a high-dimensional phase space, the computational spatial domain is complex, and the materials are heterogeneous. A scalable parallel solver is required to address such a challenge. In this paper, we study a highly parallel Newton--Krylov--Schwarz (NKS) method consisting of a Newton-based eigenvalue solver, a Krylov subspace method, and a novel multilevel Schwarz preconditioner. The multilevel method is one of the most popular preconditioners for accelerating neutron transport calculations, but the construction of coarse spaces can be expensive and often unscalable when a large number of processors is used. We propose a novel matrix coarsening algorithm in which a multilevel hierarchy is constructed using a single-component matrix instead of the full matrix of the neutron transport equation. This new coarsening algorithm is referred to as “subspace-based coarsening.” Above 8,000 processors, we show a 13x enhancement in multilevel preconditioner setup time when using the subspace-based coarsening method. A partition-based balancing strategy is studied to enhance the parallel efficiency of the NKS algorithm by equalizing the work for each processor. A hierarchical mesh partitioning algorithm is employed to generate a large number of submeshes while minimizing off-node communication. We demonstrate that the proposed algorithm is scalable with more than 10,000 processors for a realistic application on three-dimensional unstructured meshes with a few billion degrees of freedom. Neutron transport calculations using the improved NKS algorithm are twice as fast as those based on the unmodified NKS solver when over 8,000 processors are employed.

A three-dimensional Fourier analysis of fine mesh rebalance acceleration of linear discontinuous sub-cell balance method for diffusion equation on tetrahedral meshes

This paper presents a three-dimensional Fourier analysis of the fine mesh rebalance (FMR) acceleration of the Linear Discontinuous Expansion Method with Subcell Balance (LDEM-SCB) discretization method for solving the neutron diffusion equation on three-dimensional unstructured tetrahedral meshes in order to theoretically understand the convergence characteristics. The three-dimensional Fourier analysis is performed on one and two cubes called the basic elements in which each element is divided into six tetrahedral meshes for translation and reflective boundary conditions on the external boundaries. The Fourier analysis shows that the FMR method significantly accelerates the Gauss-Seidel (GS)-like iteration to solve the LDEM-SCB discretized diffusion equation. The analytical results of the homogeneous test problem taking different aspect ratios show that the convergence of FMR with the translation boundary conditions is less sensitive on the aspect ratio than the one with the reflective boundary conditions. In addition, we identified the convergence characteristics of the GS-like iteration and its FMR acceleration of the heterogeneous problem.

Resource efficient finite element computing on multicore architectures

In this paper, we consider the construction of efficient finite element algorithms on three-dimensional unstructured meshes that take into account the complex parallel synchronization processes, the memory distribution problems and data storage. A layer-by-layer partitioning of the meshes into subdomains without branching internal boundaries is proposed to simplify the access to independent data and parallel computing at different stages of the finite element problem solving on unstructured meshes in multiply connected domains. The predictive capacity of the time efficiency and resource intensity for the proposed algorithmic solutions is analyzed. The analysis of the resource efficiency of the algorithms is given for the element-by-element scheme for forming and solving the system of linear algebraic equations of the finite element method. It is shown that the low arithmetic intensity of the algorithms considered results in the fact that their performance is limited by the bandwidth of the memory subsystem rather than by the processors'performance. The graphic memory has a larger bandwidth than the random-access memory. This allows a significant increase in the performance of the algorithm on GPU.