Cell-centered Finite Volume Research Articles

Comparison of Shallow Water Solvers: Applications for Dam-Break and Tsunami Cases with Reordering Strategy for Efficient Vectorization on Modern Hardware

We investigate in this paper the behaviors of the Riemann solvers (Roe and Harten-Lax-van Leer-Contact (HLLC) schemes) and the Riemann-solver-free method (central-upwind scheme) regarding their accuracy and efficiency for solving the 2D shallow water equations. Our model was devised to be spatially second-order accurate with the Monotonic Upwind Scheme for Conservation Laws (MUSCL) reconstruction for a cell-centered finite volume scheme—and be temporally fourth-order accurate using the Runge–Kutta fourth-order method. Four benchmark cases of dam-break and tsunami events dealing with highly-discontinuous flows and wet–dry problems were simulated. To this end, we applied a reordering strategy for the data structures in our code supporting efficient vectorization and memory access alignment for boosting the performance. Two main features are pointed out here. Firstly, the reordering strategy employed has enabled highly-efficient vectorization for the three solvers investigated on three modern hardware (AVX, AVX2, and AVX-512), where speed-ups of 4.5–6.5× were obtained on the AVX/AVX2 machines for eight data per vector while on the AVX-512 machine we achieved a speed-up of up to 16.7× for 16 data per vector, all with singe-core computation; with parallel simulations, speed-ups of up to 75.7–121.8× and 928.9× were obtained on AVX/AVX2 and AVX-512 machines, respectively. Secondly, we observed that the central-upwind scheme was able to outperform the HLLC and Roe schemes 1.4× and 1.25×, respectively, by exhibiting similar accuracies. This study would be useful for modelers who are interested in developing shallow water codes.

Validation of a 2D cell-centered Finite Volume method for elliptic equations

Following the approach in Gie and Temam(2010) and Gie and Temam(2015), we construct the Finite Volume (FV) approximations of a class of elliptic equations and perform numerical computations where a 2D domain is discretized by convex quadrilateral meshes. The FV method with Taylor Series Expansion Scheme (TSES), which is properly adjusted from a version widely used in engineering, is tested in a box, annulus, and in a domain which includes a topography at the bottom boundary. By comparing with other related convergent FV schemes in Sheng and Yuan(2008), Aavatsmark(2002), Hermeline(2000) and Faureet al. (2016), we numerically verify that our FV method is a convergent 2nd order scheme that manages well the complex geometry. The advantage of our scheme is on its simple structure which do not require any special reconstruction of dual type mesh for computing the nodal approximations or discrete gradients.

A Multi-GPU Parallel Algorithm in Hypersonic Flow Computations

Computational fluid dynamics (CFD) plays an important role in the optimal design of aircraft and the analysis of complex flow mechanisms in the aerospace domain. The graphics processing unit (GPU) has a strong floating-point operation capability and a high memory bandwidth in data parallelism, which brings great opportunities for CFD. A cell-centred finite volume method is applied to solve three-dimensional compressible Navier–Stokes equations on structured meshes with an upwind AUSM+UP numerical scheme for space discretization, and four-stage Runge–Kutta method is used for time discretization. Compute unified device architecture (CUDA) is used as a parallel computing platform and programming model for GPUs, which reduces the complexity of programming. The main purpose of this paper is to design an extremely efficient multi-GPU parallel algorithm based on MPI+CUDA to study the hypersonic flow characteristics. Solutions of hypersonic flow over an aerospace plane model are provided at different Mach numbers. The agreement between numerical computations and experimental measurements is favourable. Acceleration performance of the parallel platform is studied with single GPU, two GPUs, and four GPUs. For single GPU implementation, the speedup reaches 63 for the coarser mesh and 78 for the finest mesh. GPUs are better suited for compute-intensive tasks than traditional CPUs. For multi-GPU parallelization, the speedup of four GPUs reaches 77 for the coarser mesh and 147 for the finest mesh; this is far greater than the acceleration achieved by single GPU and two GPUs. It is prospective to apply the multi-GPU parallel algorithm to hypersonic flow computations.

A new multipoint flux approximation method with a quasi-local stencil (MPFA-QL) for the simulation of diffusion problems in anisotropic and heterogeneous media

In this paper, we present a new linear cell-centered finite volume multipoint flux approximation (MPFA-QL) scheme for discretizing diffusion problems on general polygonal meshes. This scheme uses a quasi-local stencil, based upon the conormal decomposition, to approximate the control face flux when solving the steady state diffusion problem, being able to reproduce piecewise linear solutions exactly and it is very robust when dealing with heterogeneous and highly anisotropic media and severely distorted meshes. In our linear scheme, we first construct the one-sided fluxes on each control surface independently and then a unique flux expression is obtained by a convex combination of the one-sided fluxes. The unknown values at the vertices that define a control surface are interpolated by means of a linearity-preserving interpolation procedure, considering control volumes surrounding these vertices. To show the potential of the MPFA-QL scheme, we solve some benchmark using triangular and quadrilateral meshes and we compare our scheme with other numerical formulations found in literature.

A Finite Volume Scheme Preserving Maximum Principle for the System of Radiation Diffusion Equations with Three-Temperature

We propose a cell-centered nonlinear finite volume scheme for the nonequilibrium three-temperature equations, where both the Dirichlet and Neumann boundary conditions are considered, and prove that the discrete solutions of the scheme satisfy the discrete maximum principle. In the construction of the flux we use the nonlinear combination of two single fluxes and the interpolation technique for the auxiliary unknowns. Some new interpolation methods are introduced, especially when the Neumann boundary condition is considered. Based on the bounded estimation of the discrete solution, we prove that there exists at least a solution for our scheme by using the Brouwer's fixed point theorem. Numerical results show that our scheme has second-order accuracy and good conservation and can preserve the discrete maximum principle.

Development of a robust solver to model the flow inside the engines for high-speed propulsion

The demand for discovering new commercial routes as well as the possibility to shortening civilian long-haul flights boosted the interest of civil hypersonic vehicle designs. Among all the multiple projects started by the various nations, the European community funded project STRATOFLY aims at refining the baseline LAPCAT II-MR2.4 design for further improvements. The new aircraft would enable a flight shorter that 3 hours from Brussels to Sydney, carrying 300-passengers above the already crowed atmosphere. The wide Mach range operability, up to Mach 8, demands the use of multiple engines, leading to a highly integrated propulsion system. The current study is focused on the development of new CFD platform to estimate the performance of the combined propulsion system during the supersonic to hypersonic transition. In order to control the complex flow physics, highfidelity CFD simulations remain the fundamental tools for the preliminary investigations. On the current framework, an advanced robust compressible solver has been develop d in order to handle the different flow regimes. The new tool solves Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations by employing cell-centered Finite Volume Method constructed on openFoam toolbox. Two innovative high-order discretization schemes, with different abilities, based on approximated Riemann solvers were developed for capturing the flow physics within high-speed propulsion systems. Advanced time discretization has been taken into account to increase the temporal accuracy. At the end, the whole implementation has been validated in multiple test cases, ranging from incompressible to hypersonic regimes, confirming its excellent stability, robustness and accuracy.

Assessment of unsteady Brinkman’s model for flow in karst aquifers

The Brinkman’s equation simplifies the numerical modeling of karst aquifers by allowing the use of a single transport equation to model the flow of fluids in both the free-flow and porous regions, in effect reducing the error arising from improper modeling of the interface between the two regions. Most equations available to model flow within karst aquifers deal with steady flow conditions. This may not be accurate in aquifers where unsteady conditions exist. We considered the effects of unsteady flow conditions in karst aquifers by assessing the addition of an unsteady flow term to the Brinkman’s equation. We solved the coupled mass conservation-transport equations that models unsteady fluid transport in karst aquifers and studied the effects of unsteady flow conditions on tracer transport in two different sample aquifers and compared to the results obtained from the steady flow Brinkman’s equation. The solution method adopted is sequential and it involves solving the unsteady Brinkman’s model first, followed by the advection-diffusion-adsorption equation using the cell-centered finite volume approach. The first example presented here is a simple aquifer model consisting of a single conduit surrounded by porous regions. The second example is a complicated structure consisting of complex geometrical caves embedded in a highly heterogeneous porous media. The results show that, inside the caves, the unsteady Brinkman’s model yielded lower tracer concentrations at early times when compared to the steady flow model. At longer times, both models produced almost similar results. In particular, the results obtained from the simplified example case (Example 1) indicate that the velocity profiles for unsteady flow within open conduits do not instantly yield a parabolic shape expected from the Brinkman’s equation, but gradually develops into one starting from a linear profile. Results obtained also show that the addition of unsteady flow term to the Brinkman’s model does not affect the flow of tracer within porous media in any significantly observable manner.

Simulation of Transonic Buffet Using a Time-Spectral Method

This paper extends the use of frequency-domain flow solvers to transonic buffet in turbulent flow regime. To do so, a time-spectral method with frequency adaptation is developed for the simulation of periodic flows of self-induced frequency. The time-spectral method is implemented into a cell-centered finite volume unsteady Reynolds-averaged Navier–Stokes (URANS) flow solver. This method uses the Fourier series to transform the URANS equations into a steady problem in the frequency domain, for a given frequency. This steady problem is recast in the time domain to allow the use of classic computational fluid dynamics techniques. Because transonic buffet is a self-induced phenomenon of unknown frequency before the computation, two algorithms are implemented in the time-spectral solver to determine its frequency. The first one relies on a gradient-based algorithm to minimize the residual of the system of equations. The second uses the phase shift per iteration of the frequency-domain solution. The accuracy of the time-spectral scheme for turbulent flows is first verified for the simulation of a pitching airfoil with a forced frequency. Then, the efficiency of the frequency-adaptation algorithms is assessed for the case of the laminar vortex shedding of a cylinder. Finally, the proposed method is applied to the solution of the transonic buffet over the OAT15A airfoil, and the results are compared to numerical and experimental results from the literature. It can be concluded that an accurate solution of the transonic buffet can be obtained with as few as five modes.

Continuum Dynamic Traffic Assignment Model for Autonomous Vehicles in a Polycentric Urban City with Environmental Consideration

This paper proposed a continuum dynamic model for autonomous vehicles in a polycentric urban city by considering the environment impact of traffic emission. The model assumes that homogeneous autonomous vehicles are continuously distributed over the urban areas which tend to choose a path to minimize their total travel cost from origin to destination. To describe the path choice behavior of travelers, we presented the continuum dynamic traffic assignment model which consists of a two-dimensional hyperbolic system of nonlinear conservation laws with source terms and an Eikonal-type equation. The elastic demand is considered using a function which associating each copy of flow with its total instantaneous travel cost. For the environmental impacts, here we consider the influence of CO emission and include the cost of emission into the actual transportation cost. A solution algorithm for the model is designed as a cell-centered finite volume method for conservation law equations and a fast sweeping method for Eikonal-type equations on unstructured grids. Numerical examples are given to demonstrate the model and the proposed solution algorithm. Further, the results of the travel cost considering CO emissions and not considering CO emissions are compared.

A Cell-Elimination method for solving steady and unsteady Navier–Stokes equations

In this study, a highly efficient approach for solving steady and unsteady incompressible flows (with\\without cavitation) is introduced to reduce the numerical computational cost for the first time. This approach is called Cell-Elimination Method (CEM). CEM is organized based on the concept of sparse matrix solvers and it removes useless cells in a computational domain (i.e. cells which are reached their suitable value). In the present study, CEM is applied to the combination of well-known Jameson's cell-centered finite volume numerical method and the progressive power-law preconditioning approach. To achieve the steady state solution, the equations are integrated in time using an explicit four-stage Runge–Kutta scheme with local time step. For unsteady problems a dual-time implicit algorithm is applied to obtain time-accurate solutions. Since for the use of CEM, it is essential to understand the Local Error variation through iteration, an error study is done over a typical test case. The simulations focus on steady and unsteady flows over hydrofoils (with\\without cavitation). Despite the simplicity of CEM's concept, results show acceptable accuracy and a remarkable computational cost reduction.

Grid Average and Nodal Value in High Order Scheme

ABSTRACTThe concepts of nodal value and grid average in cell centered finite volume method (FVM) are clarified in this work, strict distinction between the two concepts in constructing numerical schemes is made, and common fault in misidentifying the two concepts is pointed out. The expansion based on grid average, similar to Taylor’s expansion, is deduced to construct correct scheme in terms of grid average and to obtain modified partial differential equation (MPDE) which determines the order of accuracy of numerical scheme theoretically. Correct high order scheme, taking QUICK (Quadratic Upstream Interpolation for Convective Kinematics) scheme as an example, is constructed in different approaches. Furthermore, the property of interpolation coefficients is analyzed. We also pointed out that for high order schemes, round-off error dominates the absolute error in fine grid and truncation error dominates the absolute error in coarse grid.

Iterative inflow-implicit outflow-explicit finite volume scheme for level-set equations on polyhedron meshes

In this paper, we propose a cell-centered finite volume method for advective and normal flows on polyhedron meshes which is second-order accurate in space and time for smooth solutions. In order to overcome a time restriction caused by CFL condition, an implicit time discretization of inflow fluxes and an explicit time discretization of outflow fluxes are used in an iterative procedure. For an efficient computation, an 1-ring face neighborhood structure is introduced. Since it is limited to access unknown variables in an 1-ring face neighborhood structure, an iterative procedure is proposed to resolve the limitation of assembled linear system. Two types of gradient approximations, an inflow-based gradient and an average-based gradient, are studied and compared from the point of numerical accuracy. Numerical schemes are tested for an advective and a normal flow of level-set functions illustrating a behavior of the proposed method for an implicit tracking of a smooth and a piecewise smooth interface.

An upwind cell centred Total Lagrangian finite volume algorithm for nearly incompressible explicit fast solid dynamic applications

The paper presents a new computational framework for the numerical simulation of fast large strain solid dynamics, with particular emphasis on the treatment of near incompressibility. A complete set of first order hyperbolic conservation equations expressed in terms of the linear momentum and the minors of the deformation (namely the deformation gradient, its co-factor and its Jacobian), in conjunction with a polyconvex nearly incompressible constitutive law, is presented. Taking advantage of this elegant formalism, alternative implementations in terms of entropy-conjugate variables are also possible, through suitable symmetrisation of the original system of conservation variables. From the spatial discretisation standpoint, modern Computational Fluid Dynamics code “OpenFOAM” [http://www.openfoam.com/] is here adapted to the field of solid mechanics, with the aim to bridge the gap between computational fluid and solid dynamics. A cell centred finite volume algorithm is employed and suitably adapted. Naturally, discontinuity of the conservation variables across control volume interfaces leads to a Riemann problem, whose resolution requires special attention when attempting to model materials with predominant nearly incompressible behaviour (κ∕μ≥500). For this reason, an acoustic Riemann solver combined with a preconditioning procedure is introduced. In addition, a global a posteriori angular momentum projection procedure proposed in Haider et al. (2017) is also presented and adapted to a Total Lagrangian version of the nodal scheme of Kluth and Després (2010) used in this paper for comparison purposes. Finally, a series of challenging numerical examples is examined in order to assess the robustness and applicability of the proposed methodology with an eye on large scale simulation in future works.

A Novel Multislope MUSCL Scheme for Solving 2D Shallow Water Equations on Unstructured Grids

Within the framework of the two-dimensional cell-centered Godunov-type finite volume (CCFV) method, this paper presents a novel multislope scheme on the basis of the monotone upstream scheme for conservation law (MUSCL) for numerically solving nonlinear shallow water equations on two-dimensional triangular grids. The Riemann states of the considered edge are calculated by an edge-based reconstructing procedure, where a limited scalar slope is employed to prevent potential numerical oscillations. The novel aspect of the new scheme is that it takes advantage of the geometrical characteristics of triangular grids in the reconstructing and limiting procedures, which effectively reduces the cost of computation and provides higher resolution and accuracy compared with classical MUSCL schemes. Seven tests are adopted to verify the scheme, and the results indicate that this scheme is efficient, accurate, robust, and high-resolution, and can be an ideal alternative for solving shallow water problems over uneven and frictional topography.

A new smoothing approach for accelerating the convergence of power-law preconditioning method in steady and unsteady flows simulation

A new explicit smoothing approach is presented to improve the convergence acceleration of laminar viscous steady-state and unsteady flows simulation over airfoils (with/without considering surface air injection). This approach is developed based on Jameson's residual smoothing technique, and it is called here as a Solution Smoothing Method (SSM). This scheme is accompanied by the progressive locally power-law preconditioning method. The steady/unsteady preconditioned governing equations are solved using Jameson's cell-centered finite volume method in which the stabilization is achieved by means of the second- and fourth-order artificial dissipation terms. An implicit dual-time algorithm is used for simulation of unsteady cases and an explicit four-stage Runge–Kutta time integration with local time step is applied to achieve the steady-state condition in pseudo-time in an inner loop. Stability of SSM is analyzed on a one-dimensional linear model equation, and the effects of different Courant numbers, artificial dissipation coefficients, and smoothing parameters are investigated. Numerical results in comparison with other classic preconditioning methods show a reduction of the iteration number required to compute the steady-state and unsteady solution. Results show about 55 ∼ 77 and 40 ∼ 72% reduction in iteration number for steady and unsteady flows simulation, respectively.

A finite volume procedure for fluid flow, heat transfer and solid-body stress analysis

ABSTRACTA unified cell-centered unstructured mesh finite volume procedure is presented for fluid flow, heat transfer and solid-body stress analysis. An in-house procedure (A. W. Date, Solution of Transport Equations on Unstructured Meshes with Cell-Centered Colocated Variables. Part I: Discretization, International Journal of Heat and Mass Transfer, vol. 48 (6), 1117-1127, 2005) is extended to include the solid-body stress analysis. The transport terms for a cell-face are evaluated in a structured grid-like manner. The Cartesian gradients at the center of each cell-face are evaluated using the coordinate transformation relations. The accuracy of the procedure is demonstrated by solving several benchmark problems involving different boundary conditions, source terms, and types of loading.

A reconstruction-based cell-centered high-order finite volume method for incompressible viscous flow simulation on unstructured meshes

A new high-order (> 2nd order) cell-centered finite volume method is presented for incompressible flow simulation on unstructured meshes. Artificial compressibility is employed to couple the continuity and momentum equations in a manner that allows them to be solved simultaneously. A new numerical stencil, a so-called wrapping stencil, is utilized for linear and quadratic solution reconstruction in order to achieve more accurate and robust solution reconstruction not only for the interior cells, but also for the cells on the boundary, where fewer neighboring cells typically exist. The effectiveness of the current algorithm is demonstrated by various test cases, including an analytical solution reconstruction test, Kovasznay flow simulations with various Reynolds numbers, a driven cavity flow, and flow past a square cylinder. Based on the comparison with the standard low order scheme, the proposed second and third order schemes, based on linear and quadratic solution reconstruction, show superior accuracy, which sheds light on the method's applicability in solving more challenging incompressible flow problems.

A novel stencil selection method for the gradient reconstruction on unstructured grid based on OpenFOAM

For the unstructured finite volume algorithms, the stencil utilized for gradient reconstruction is important for the accuracy, efficiency and robustness. In this paper, we present a novel stencil selection method for the gradient reconstruction of the cell-centered finite volume methodology. In the novel stencil selection method, two local directions are generated for each control volume, and the stencil is generated based on the local directions. In this way, the characteristics of the structured grid are able to be reflected on the unstructured grid. To obtain the suitable local directions, an advancing front technique (AFT) is utilized to guide their generation. Through AFT, the direction of the boundary normal vector, along which the flow fields change dramatically, is obtained and propagated from the boundaries to the internal regions. Finally, the novel stencil extends in two directions, one of which is approximately aligned with the direction of the boundary normal vector. Hence, the gradient reconstruction methods with the new stencil are able to use the information along the boundary normal. In order to verify the validity of the new stencil, we present a new gradient reconstruction method, the structured least squares (struLSQR) method, by applying the new stencil to the least squares (LSQR) method. The performance of struLSQR, and two existing gradient reconstruction methods in OpenFOAM is tested by adopting the 2D channel flow and the 2D circular cylinder flow on both the quadrilateral and the triangular grids. A laminar flow past a flat plate is also adopted to verify the applicability of struLSQR to the viscous flow. The numerical results illustrate that the new stencil is able to solve the blow-up problem of LSQR. It can also obtain a higher order of accuracy compared to extLSQR. Moreover, the convergence speed of struLSQR can be up to 1.64 ∼ 2.60x of that of extLSQR.

A finite-volume discretization for deformation of fractured media

Simulating the deformation of fractured media requires the coupling of different models for the deformation of fractures and the formation surrounding them. We consider a cell-centered finite-volume approach, termed the multi-point stress approximation (MPSA) method, which is developed in order to discretize coupled flow and mechanical deformation in the subsurface. Within the MPSA framework, we consider fractures as co-dimension one inclusions in the domain, with the fracture surfaces represented as line pairs in 2D (face pairs in 3D) that displace relative to each other. Fracture deformation is coupled to that of the surrounding domain through internal boundary conditions. This approach is natural within the finite-volume framework, where tractions are defined on surfaces of the grid. The MPSA method is capable of modeling deformation, considering open and closed fractures with complex and nonlinear relationships governing the displacements and tractions at the fracture surfaces. We validate our proposed approach using both problems, for which analytical solutions are available, and more complex benchmark problems, including comparison with a finite-element discretization.

Extending slide-slip mesh update method to finite volume method

For simulations of flows around rotating bodies, usually sliding mesh method or overset grid method is used. But both of them have to perform inter-mesh interpolation which introduces much numerical error and usually violates conservation. Shear-slip mesh update method (SSMUM) is another method for such flows. In each time step of SSMUM, a mesh slipping step follows a mesh deforming step to undo the deformation and results in a new mesh of good quality. Each vertex on the slipping interface can only move from one node to the next node in the circumference, which makes the interface always conformal and no need for inter-mesh interpolation. However, SSMUM didn't get enough attention in the community of finite volume method. In this paper, SSMUM is extended to cell center finite volume method. To guarantee conservation and obtain high order accuracy on a slipping interface, a remapping procedure is needed to transfer flow field from an old mesh to a new mesh. This was achieved by solving a linear convective PDE with one or two explicit steps, thus only resulting in little extra computing cost. Oscillating NACA0012 airfoil was simulated with the improved SSMUM. The results showed excellent agreement with the data by rigid rotating mesh. And the flow field was always smooth. It suggests that this improved SSMUM has advantages in getting conservative, smooth and high accuracy solutions for rotating problems.