Cell-centered Finite Volume Research Articles

Overlapping domain decomposition methods for finite volume discretizations

Two-level additive overlapping domain decomposition methods are applied to solve the linear system arising from the cell-centered finite volume discretization methods (FVMs) for the elliptic problems. The conjugate gradient (CG) methods are used to accelerate the convergence. To analyze the preconditioned CG algorithm, a discrete L2 norm, an H1 norm, and an H1 semi-norm are introduced to connect the matrices resulting from the FVMs and related bilinear forms. It has been proved that, with a small overlap, the condition number of the preconditioned systems does not depend on the number of the subdomains. The result is similar to that for the conforming finite element. Numerical experiments confirm the theory.

Electromagnetic force calculation within finite volume framework

ABSTRACT In this research, the comparison of force calculation methods in magnetostatic simulations within a cell-centered finite volume framework is presented. Various procedures for postprocessing of the magnetic vector potential field are laid out to obtain the surface and volume integrals of Maxwell’s stress tensor and the Lorentz force density. Two different magnetostatic problems with multiple operating points were modeled with the AVL FIRETM M. The force results have been verified by comparing them to the Ansys Maxwell and were validated on a benchmark case from a Testing Electromagnetic Analysis Methods (TEAM) workshop. The interpolation of the magnetic flux density on a multi-material interface is presented, and it is used in conjunction with the surface integral of Maxwell’s stress tensor for an efficient global force computation. For all observed cases, the calculated results correspond well to the results found in the literature. It was found that the accuracy of the volume integral of Maxwell’s stress tensor is dependent on the mesh quality and gradient calculation scheme. Employing the semi-structured mesh with Gauss’s gradient scheme results in force discrepancies of less than 3% between the surface and volume integral of Maxwell’s stress tensor. The convergence of the iterative solver is observed to be slower in the case of the materials with permeability jump, where discontinuities between the tangential component of the magnetic flux density and the normal component of the magnetic field intensity occur.

GPU parallel implementation of a finite volume lattice Boltzmann method for incompressible flows

This work presents a graphics processing units (GPU) parallel algorithm of a cell-centered finite volume lattice Boltzmann method (FVLBM) on unstructured meshes. In the present GPU parallel algorithm, the parallelization is performed in the physical space. To reduce the frequency of GPU memory accesses, this algorithm develops coalesced access to GPU memory. In addition, to avoid the race for resources leading to data anomalies, such as dirty read or phantom read etc., and the double counting for flux calculation, the efficient face-based data structure often used for flux calculation in cells in the central processing unit (CPU) version of FVLBM is modified into a face-based data structure used for the fluxes on all faces, followed by a cell-based loop for the final residuals in all cells. Therefore, the proposed GPU parallel algorithm does not need to use the resource lock and retains the high efficiency of the face-based data structure in the fluxes computation to enhance its’ parallel efficiency. Additionally, to demonstrate the computational efficiency of the proposed GPU parallel algorithm, various benchmark studies are performed in this work by the proposed parallel scheme on a double precision NVIDIA GeForce RTX 3090Ti GPU card, including (a) the lid-driven flow in a two-dimensional (2D) square cavity, (b) a 2D flow past a cylinder, and (c) the lid-driven flow in a three-dimensional (3D) cubic cavity. The numerical results show that the proposed GPU parallel algorithm can be as accurate as the original CPU serial scheme with 1 to 2 orders of speedup.

GPU accelerated Staggered Update Procedure (SUP)

The advancement in programmable capability of graphics hardware has paved new opportunities in the domain of high performance computing (HPC). The computational fluid dynamics (CFD) community, being a significant user of HPC, has started exploiting the inherent data parallelism in the numerical solvers to be able to make efficient use of these many-core, high throughput accelerator based processors. In the present work, we examine the process of accelerating our CPU based Staggered Update Procedure (SUP) solver, i.e., a higher order accurate cell-centred finite volume solver by off-loading the computationally most expensive region of the code pertaining to the explicit residual computation. We have adopted OpenACC, a directive based programming model to expose parallelism in the code. The framework evolved for GPU porting in the context of SUP is also of value to those intending to port their CFD solvers based on classical finite volume methodology. The performance analysis is conducted using scalar convection–diffusion equations in both two- and three-dimensions. The findings demonstrate a speedup factor of 9 (in case of 2D) and 28 (in case of 3D) when considering the explicit residual alone, achieved with a single NVIDIA Tesla V100 GPU card. In addition, we could establish superior algorithmic scalability by the way of recovering near perfect serial performance, on the heterogeneous CPU+GPU architecture. Further, overall code acceleration can be achieved by porting other parts of the solver on GPU.

The cell-centered positivity-preserving finite volume scheme for 3D convection–diffusion equation on distorted meshes

PurposeThis paper aims to construct positivity-preserving finite volume schemes for the three-dimensional convection–diffusion equation that are applicable to arbitrary polyhedral grids.Design/methodology/approachThe cell vertices are used to define the auxiliary unknowns, and the primary unknowns are defined at cell centers. The diffusion flux is discretized by the classical nonlinear two-point flux approximation. To ensure the fully discrete scheme has positivity-preserving property, an improved discretization method for the convection flux was presented. Besides, a new positivity-preserving vertex interpolation method is derived from the linear reconstruction in the discretization of convection flux. Moreover, the Picard iteration method may have slow convergence in solving the nonlinear system. Thus, the Anderson acceleration of Picard iteration method is used to solve the nonlinear system. A condition number monitor of matrix is employed in the Anderson acceleration method to achieve better robustness.FindingsThe new scheme is applicable to arbitrary polyhedral grids and has a second-order accuracy. The results of numerical experiments also confirm the positivity-preserving of the discretization scheme.Originality/value1. This article presents a new positivity-preserving finite volume scheme for the 3D convection–diffusion equation. 2. The new discretization scheme of convection flux is constructed. 3. A new second-order interpolation algorithm is given to eliminate the auxiliary unknowns in flux expressions. 4. An improved Anderson acceleration method is applied to accelerate the convergence of Picard iterations. 5. This scheme can solve the convection–diffusion equation on the distorted meshes with second-order accuracy.

Numerical Seakeeping Analysis for a Floating Helicopter After Ditching in Water

Abstract This paper investigates the seakeeping behavior of helicopters after an emergency landing in water, focusing on a Northern North Sea wave climate and considering a realistic helicopter geometry. Computational fluid dynamics techniques, including the cell-centered finite volume method and boundary element methods, were utilized to analyze motion responses and load distribution. The study ensures numerical result reliability through recommended simulation practices. Results indicate that the inviscid model produces similar outcomes to the viscous model in decay tests with roll, pitch, and heave motions. Natural periods for roll, pitch, and heave motions were obtained. Linearity between incident wave amplitude and pitch/heave response was noted for regular waves, while roll linearity was limited for small angles. In irregular wave conditions, helicopters tended to align perpendicular to waves over time, resulting in increased peak roll angles with higher significant wave heights. Exceedance rates of maximum roll peaks, useful for the assessment of capsizing probability, were quantified for different significant wave heights.

Patient-specific modelling of contact characteristics in the ankle joint following triple arthrodesis in valgus, neutral and varus hindfoot positions

The aim of the current study was to understand the importance of the joint alignment following triple arthrodesis by analysing the contact characteristics in a normal and arthritic ankle joint using a patient-specific numerical model developed using open source software. The alignment of the hindfoot with respect to tibia is calculated from CT scans and the ankle joint model was numerically analysed for neutral, valgus and varus positions in both normal and arthritic conditions. The contact area, the magnitude and distribution of the contact pressure on the articular surface of the talar dome was evaluated using a cell-centred Finite Volume Method implemented in open-source software OpenFOAM. It was found that all positions of the hindfoot predict higher lateral pressures during heel strike. The varus position predicts the maximum increase in lateral pressures. Comparing the valgus and neutral positions, although the neutral position predicts 9.1 % higher increase in lateral pressures during heel strike than valgus, it predicts 33.6 % decrease in pressures during heel-rise and the distribution is more medial during toe-off. In the case of arthritic ankle, it could be observed that the neutral and varus hindfoot fusion positions result in a concentrated increase of lateral pressures in heel strike and flat-foot. In the case of toe-off, the neutral alignment results in an increase of 62.3 % in the contact pressures compared to the arthritic pressure of the unfused foot and is 20.8 % higher than the valgus alignment. The study helps to conclude that the fusion is more beneficial at the neutral position of the hindfoot for the patient specific ankle. However, the 5° valgus position of hindfoot alignment could be more beneficial in the arthritic ankle. Patient-specific approach to the placement of the hindfoot with the help of numerical analysis could help address the issue of ankle degradation following arthrodesis.

A finite volume adaptation of beam-to-beam contact interactions implemented for geometrically exact Simo–Reissner beams

AbstractThis paper presents an adaption of the finite-element based beam-to-beam contact interactions into a finite volume numerical framework. A previous work of the same authors, where a cell-centred based finite volume implementation of geometrically exact nonlinear Simo–Reissner beams was developed, is used as an underlying mathematical model. An implicit contact procedure is developed for both point-to-point and line-to-line beam frictionless contact interactions, and is implemented using the cell-centred finite volume method. To enforce the contact constraint, both penalty method and augmented-Lagrangian based techniques are used. A total of six numerical benchmark cases for point and line beam-to-beam contact interactions based on the finite element method are used to verify the numerical results, accuracy and robustness of the developed contact procedure.

Corrected Linear-Galerkin Schemes to Preserve Second-Order Accuracy for Cell-Centered Unstructured Finite Volume Methods

Corrected Linear-Galerkin Schemes to Preserve Second-Order Accuracy for Cell-Centered Unstructured Finite Volume Methods

Coupled flow and energy models with phase change in permafrost from pore- to Darcy scale: Modeling and approximation

We consider coupled models of flow and energy with phase change in thawing soils in permafrost at the pore- and Darcy scales. We develop the underlying models from first principles and define robust and conservative discretization algorithms based on cell-centered finite volume discretization on Cartesian grids and implicit time stepping for individual models. To couple the models we consider sequential approaches as well as time-lagging of some of the nonlinear properties. We also derive practical new surrogate models to obtain Darcy scale data from the pore-scale simulations. Finally, we study the sensitivity of models to the data in realistic scenarios. In particular we find that for the dynamics at the time scale of days, the nonlinear flow parameters have most significant impact on pressure, while temperature is less sensitive to data.

Time-Resolved Local Loss Analysis of Single- and Two-Blade Pump Flow

Abstract A method for the evaluation of time-resolved entropy production in isothermal and incompressible flow is presented. It is applied as a postprocessing of the three-dimensional (3D) flow field obtained by time-resolved computational fluid dynamics (CFD) with scale adaptive turbulence modeling. Wall functions for direct and turbulent entropy production are presented for a cell-centered finite volume method, implemented in the open-source software OpenFOAM and validated on channel, asymmetric diffuser, and periodic hill flow. Single- and two-blade centrifugal pump flow is considered for a wide range of load conditions. Results are compared to experimental data. Time-averaged analysis shows essentially the same loss density distribution among pump components for both pumps, with the impeller and volute region contributing the most, especially in off-design conditions. For both pumps, the losses exhibit significant fluctuations due to impeller–volute interactions. The fluctuation magnitude of loss density is in the same range as flowrate fluctuations and much smaller than pressure fluctuation magnitude. For the two-blade pump (2BP), loss fluctuation magnitude is smaller than for the single-blade pump (1BP). Distinct loss mechanisms are identified for different load conditions. Upon blade passage, a promoted or attenuated volute tongue separation is imposed at part or overload, respectively. In between blade passages, a direct connection from pump inlet to the discharge leads to enhanced flowrate and loss density fluctuations. Future work aims at extending this analysis to stronger off-design conditions in multiblade pumps, where stochastic cycle fluctuations occur.

Aircraft Simulations Using the New CFD Software from ONERA, DLR, and Airbus

This paper presents a thorough comparison between RANS simulations performed with the computational fluid dynamics by ONERA, DLR, and Airbus (CODA) new-generation flow solver and reference legacy codes TAU (DLR) and elsA (ONERA) for high-speed cruising NASA Common Research Model (CRM) configurations that were considered in the context of the 5th, 6th, and 7th Drag Prediction Workshops. The solver’s accuracy is assessed with several meshing strategies, including block-structured, hybrid-structured/unstructured, and fully unstructured tetrahedral meshes. This solver features both a cell-centered finite-volume (FV) scheme suited to arbitrary meshes as well as a modern high-order discontinuous Galerkin (DG) scheme. We show that for all cases considered, the FV component recovers an equivalent accuracy compared to elsA (cell-centered FV) on block-structured meshes and TAU (node-centered FV) on hybrid and unstructured meshes. The high-order DG scheme (third-order accurate) is found to significantly enhance the drag prediction on coarse meshes compared to legacy FV methods, both for structured and unstructured meshes.

A mimetic interpolation-free cell-centered finite volume scheme for the 2D and 3D heterogeneous anisotropic diffusion equations

We propose a mimetic interpolation-free cell-centered finite volume scheme for diffusion equations on arbitrary two-dimensional (2D) and three dimensional (3D) unstructured, possibly non star-shaped or nonplanar meshes. This scheme borrows ideas from the well known mimetic finite difference methods. This interpolation-free means that we do not need to introduce auxiliary unknowns in the process of constructing the scheme, and the final scheme has only cell-centered unknowns. After carefully handling continuity of solutions and normal fluxes, we construct piecewise linear approximations on each cell. This makes our scheme applicable to arbitrary discontinuities and arbitrary mesh topologies. Interpolation-free makes the program easy to implement and the 2D and 3D programs have a unified framework. It is worth mentioning that the traditional star-shaped mesh assumption has been abolished, and the new scheme has more relaxed restrictions on meshes. Some representative and even challenging numerical examples verify that the scheme is robust and almost has the optimal convergence order on some benchmark polygonal and polyhedral meshes.

Laplacian regularized eikonal equation with Soner boundary condition on polyhedral meshes

In this paper, we propose a numerical algorithm based on a cell-centered finite volume method to compute a distance from given objects on a three-dimensional computational domain discretized by polyhedral cells. Inspired by the vanishing viscosity method, a Laplacian regularized eikonal equation is solved and the Soner boundary condition is applied to the boundary of the domain to avoid a non-viscosity solution. As the regularization parameter depending on a characteristic length of the discretized domain is reduced, a corresponding numerical solution is calculated. A convergence to the viscosity solution is verified numerically as the characteristic length becomes smaller and the regularization parameter accordingly becomes smaller. From the numerical experiments, the second experimental order of convergence in the L1 norm error is confirmed for smooth solutions. Compared to solve a time-dependent form of the eikonal equation, the Laplacian regularized eikonal equation has the advantage of reducing computational cost dramatically when a more significant number of cells is used or a region of interest is far away from the given objects. The implementation of parallel computing using domain decomposition with 1-ring face neighborhood structure can be done straightforwardly by a standard cell-centered finite volume code.

Development and Numerical Testing of a Model of Equiaxed Alloy Solidification Using a Phase Field Formulation

A computational framework is developed to understand the transient behavior of isothermal and non-isothermal transformation between liquid and solid phases in a binary alloy using a phase-field method. The non-isothermal condition was achieved by applying a thermal gradient along the computational domain. The bulk solid and liquid phases were treated as regular solutions, along with introducing an order parameter (phase field) as a function of space and time to describe the interfacial region between the two phases. An antitrapping flux term was integrated into the present phase-field model to mitigate the amount of solute trapping, which is characterized by the non-equilibrium partitioning of the solute. The governing equations for the phase field and the solute composition were solved by the cell-centered finite volume method using the open-source computational tool OpenFOAM. Simulations were carried out for the evolution of equiaxed dendrites inside an undercooled melt of a binary alloy, considering the effect of various computational parameters such as interface thickness, strength of crystal anisotropy, stochastic noise amplitude, and initial orientation. The simulated results show that the solidification morphology is sensitive to the magnitude of anisotropy as well as the amplitude of noise. A strong influence of interface thickness on the growth morphology and solute redistribution during solidification was observed. Incorporating antitrapping flux resulted in the solute partitioning close to the equilibrium value. Simulations show that the grain shape is unaffected by changes to crystallographic orientation with respect to the Cartesian computational grid. Thermal gradients exerted discernible effects on the solute distribution and the dendritic growth pattern. Starting with multiple nucleation events the model predicted realistic polycrystalline solidification and as-solidified microstructure.

A Finite Volume Method for a 2D Dam-Break Simulation on a Wet Bed Using a Modified HLLC Scheme

This study proposes a numerical model for depth-averaged Reynolds equations (shallow-water equations) to investigate a dam-break problem, based upon a two-dimensional (2D) second-order upwind cell-centre finite volume method. The transportation terms were modelled using a modified approximate HLLC Riemann solver with the first-order accuracy. The proposed 2D model was assessed and validated through experimental data and analytical solutions for several dam-break cases on a wet and dry bed. The results showed that the error values of the model are lower than those of existing numerical methods at different points. Our findings also revealed that the dimensionless error parameters decrease as the wave propagates downstream. In general, the new model can model the dam-break problem and captures the shock wave superbly.

An efficient 3D cell-based discrete fracture-matrix flow model for digitally captured fracture networks.

Complex hydraulic fracture networks are critical for enhancing permeability in unconventional reservoirs and mining industries. However, accurately simulating the fluid flow in realistic fracture networks (compared to the statistical fracture networks) is still challenging due to the fracture complexity and computational burden. This work proposes a simple yet efficient numerical framework for the flow simulation in fractured porous media obtained by 3D high-resolution images, aiming at both computational accuracy and efficiency. The fractured rock with complex fracture geometries is numerically constructed with a cell-based discrete fracture-matrix model (DFM) having implicit fracture apertures. The flow in the complex fractured porous media (including matrix flow, fracture flow, as well as exchange flow) is simulated with a pipe-based cell-centered finite volume method. The performance of this model is validated against analytical/numerical solutions. Then a lab-scale true triaxial hydraulically fractured shale sample is reconstructed, and the fluid flow in this realistic fracture network is simulated. Results suggest that the proposed method achieves a good balance between computational efficiency and accuracy. The complex fracture networks control the fluid flow process, and the opened natural fractures behave as primary fluid pathways. Heterogeneous and anisotropic features of fluid flow are well captured with the present model.

Flow simulation in 3D fractured porous medium using a generalized pipe-based cell-centered finite volume model with local grid refinement

The presence of fractures in porous media strongly influences the fluid flow behavior. This work presents an efficient and simple numerical model to address the hydrodynamic behavior of complex fractured porous media, aiming at balancing computational efficiency and accuracy. Fractures are mapped to a set of cells having implicit aperture, where local grid refinement (LGR) is adopted around the fracture to improve accuracy with lower computational burden. The fluid flow in the matrix, fracture, and exchange between fracture-matrix, are described by a generalized equivalent pipe-network model. The numerical scheme and implementation are verified against established numerical solutions. Results show that this method is efficient in capturing main phenomena of the fluid transport in fractured porous media, without sacrificing computational accuracy. Heterogeneous and anisotropic features of fluid flow are well captured. Particularly, the convergence of the LGR is in a higher order than the global grid refinement (GGR). This model also provides great flexibility in handling a multitude of fracture input data, e.g., planar fractures obtained from statistical data and non-planar fractures from digital images represented with point clouds. Balancing computational accuracy and efficiency, this method provides an effective approach for simulating flow transport in fracture porous medium.

The cell-centered Finite-Volume self-consistent approach for heterostructures: 1D electron gas at the Si–SiO2 interface

Achieving self-consistent convergence with the conventional effective-mass approach at ultra-low temperatures (below 4.2 K) is a challenging task, which mostly lies in the discontinuities in material properties (e.g. effective-mass, electron affinity, dielectric constant). In this article, we develop a novel self-consistent approach based on cell-centered finite-volume discretization of the Sturm–Liouville form of the effective-mass Schrödinger equation and generalized Poisson’s equation (FV-SP). We apply this approach to simulate the one-dimensional electron gas formed at the Si–SiO2 interface via a top gate. We find excellent self-consistent convergence from high to extremely low (as low as 50 mK) temperatures. We further examine the solidity of FV-SP method by changing external variables such as the electrochemical potential and the accumulative top gate voltage. Our approach allows for counting electron–electron interactions. Our results demonstrate that FV-SP approach is a powerful tool to solve effective-mass Hamiltonians.

An unstructured preconditioned central difference finite volume multiphase Euler solver for computing inviscid cavitating flows over arbitrary two- and three-dimensional geometries

In the present work, a numerical method is adopted and applied for simulating the inviscid cavitating flows around two- and three-dimensional geometries on unstructured meshes. The algorithm uses the preconditioned multiphase Euler equations discretized by a cell-centered central difference finite volume scheme with suitable dissipation terms. The interface capturing method with three transport equation-based cavitation models, namely the Merkle et al., Singhal et al. and Kunz et al. models are employed for the mass transfer between the liquid and vapor phases to be calculated. The simulations of the steady inviscid cavitating flows are performed around different two- and three-dimensional geometries, namely the NACA0012, NACA66(MOD) and two-element NACA4412-4415 hydrofoils, the hemispherical head shape body and the twisted NACA0009 hydrofoil, and the results are obtained over these geometries with the three cavitation models used for different flow conditions. The effects of different numerical parameters on the accuracy of the solution are also examined by a sensitivity study. The present results are compared with those of performed by other researchers which exhibit good agreement. It is indicated that the solution method adopted based on the preconditioned finite volume multiphase Euler flow solver on unstructured meshes is capable of accurately predicting the surface pressure distribution and the cavity shape over the arbitrary two- and three-dimensional geometries.