Cell-centered Finite Volume Method Research Articles

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.

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.

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.

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.

Pragmatic modelling of axial flow-induced vibration (FIV) for nuclear fuel rods

Flow-induced vibration (FIV) at the spacer grid in the fuel assembly of a Light Water Reactor (LWR) is the leading cause of fuel failure. This project aims to produce a simulation benchmark on the experimental campaign at MACE on axial FIV of a cantilever beam in an annular tube, that mimics the configuration and environment of a typical LWR. The nuclear fuel rod, which consists of fuel pellets filled in Zirconium alloy cladding is modelled in the experiment as a steel rod filled with lead shots that closely approximates the filling density of the fuel pellet. To reduce the complexity and increase the efficiency of the simulation, further simplification is applied by on the geometry and assuming the solid domain as a single material instead of multiple materials. The first mode of frequency of the rod vibrating in quiescent water, which had been validated via the Euler-Bernoulli beam theory against experimental measurements, was used to design the single-material solid domain. Two models were proposed, firstly a solid rod with lower density and stiffness (SLE), and secondly an empty cladding with high density and low stiffness (EHD). Both solid and fluid domains were discretised using the cell-centred finite volume (FV) method and coupled with strong two-way fluid-structure interaction (FSI). Results on the frequency of vibration in quiescent water and in axial flow showed good agreement with experimental measurement, and the computational efficiency is analyzed for different rod models and changes in parameters of the solid domains.

An efficient quadratic interpolation scheme for a third-order cell-centered finite-volume method on tetrahedral grids

In this paper, we propose an efficient quadratic interpolation formula utilizing solution gradients computed and stored at nodes and demonstrate its application to a third-order cell-centered finite-volume discretization on tetrahedral grids. The proposed quadratic formula is constructed based on an efficient formula of computing a projected derivative. It is efficient in that it completely eliminates the need to compute and store second derivatives of solution variables or any other quantities, which are typically required in upgrading a second-order cell-centered unstructured-grid finite-volume discretization to third-order accuracy. Moreover, a high-order flux quadrature formula, as required for third-order accuracy, can also be simplified by utilizing the efficient projected-derivative formula, resulting in a numerical flux at a face centroid plus a curvature correction not involving second derivatives of the flux. Similarly, a source term can be integrated over a cell to high-order in the form of the source term evaluated at the cell centroid plus a curvature correction, again, not requiring second derivatives of the source term. The discretization is defined as an approximation to an integral form of a conservation law but the numerical solution is defined as a point value at a cell center, leading to another feature that there is no need to compute and store geometric moments for a quadratic polynomial to preserve a cell average. Third-order accuracy and improved second-order accuracy are demonstrated and investigated for simple but illustrative test cases in three dimensions.

Direct numerical simulation of acoustic wave propagation in ocean waveguides using a parallel finite volume solver

Numerical simulations based on the Finite Volume Method (FVM) solve the wave equation directly to provide highly accurate predictions of the sound field in ocean waveguides without model simplification and loss of generality. However, the huge computational complexity severely limits the application range of direct numerical solvers for underwater acoustic waves. In this paper, a highly scalable parallel algorithm for solving acoustic models using the cell-centred finite volume method was proposed, and a corresponding direct numerical solver was developed and validated. The paper has three main findings as follows. Firstly, the acoustic solver was implemented based on a generic framework for solving partial differential equations (PDEs), so that more complicated models such as fluid–sound coupling and sound–structure coupling could be easily extended. Secondly, the large spatial scales of underwater sound propagation can be efficiently calculated by parallel computing. Thirdly, our approach demonstrated high computational accuracy and the ability to support complex scenarios. We verify the accuracy and efficiency of the solver through experiments, and prove the solving ability of the solver in large-scale, complex structures and complex configurations.

A numerical solver for investigating the space charge effect on theelectric field in liquid argon time projection chambers

This paper reports the development of a numerical solver aimed to simulate the interaction between the space charge (i.e. ions) distribution and the electric field in liquid argon time projection chamber (LArTPC) detectors. The ion transport equation is solved by a time-accurate, cell-centered finite volume method and the electric potential equation by a continuous finite element method. The electric potential equation updates the electric field which provides the drift velocity to the ion transport equation. The ion transport equation updates the space charge density distribution which appears as the source term in the electric potential equation. The interaction between the space charge distribution and the electric field is numerically simulated within each physical time step. The convective velocity in the ion transport equation can include the background flow velocity in addition to the electric drift velocity. The numerical solver has been parallelized using the Message Passing Interface (MPI) library. Numerical tests show and verify the capability and accuracy of the current numerical solver. It is planned that the developed numerical solver, together with a Computational Fluid Dynamics (CFD) package which provides the flow velocity field, can be used to investigate the space charge effect on the electric field in large-scale particle detectors.

Numerical Simulation of Ice Melting Using the Finite Volume Method

The Aim of this paper is to investigate numerically the simulation of ice melting in one and two dimension using the cell-centered finite volume method. The mathematical model is based on the heat conduction equation associated with a fixed grid, latent heat source approach. The fully implicit time scheme is selected to represent the time discretization. The ice conductivity is chosento be the value of the approximated conductivity at the interface between adjacent ice and water control volumes. The predicted temperature distribution, percentage melt fraction, interface location and its velocity is compared with those obtained from the exact analytical solution. A good agreement is obtained when comparing the numerical results of one dimensional temperaturedistribution with the analytical results.

A total energy-based adaptive mesh refinement technique for the simulation of compressible flow

In this paper, an adaptive mesh refinement technique is presented for simulation of compressible flows, which can effectively refine the mesh in the regions with shock waves and vortices. The present approach uses the total energy per unit volume as an indicator to capture the shock waves and vortical structures. In the approach, an h-refinement strategy is adopted. To save the computational effort, the flow variables on the new mesh are obtained from the previous step by interpolation, which ensures that the problem is always solved on the refined mesh. Both inviscid and viscous compressible flows are considered in this work. Their governing equations are, respectively, Euler equations and Navier–Stokes equations associated with the implementation of the Spalart–Allmaras turbulence model. The cell-centered finite volume method and Jameson scheme are chosen to carry out spatial discretization, and the five-stage Runge–Kutta scheme is applied to discretize the temporal derivative. The present approach is applied to simulate three test problems for its validation. Numerical results show that it can effectively capture the shock waves and vortices with improvement in solution accuracy.

A modified zonal method to solve coupled conduction-radiation physics within highly porous large scale digitized cellular porous materials

Due to their distinct textures, porous cellular materials have been of interest to many engineering applications. Energy conversion through such materials, especially at high temperatures, is governed by naturally coupled conduction-radiation physics. Therefore, a thorough understanding of the conduction-radiation behavior within these materials is important to properly design and optimize these materials. Several numerical methods developed over the last few decades allow to solve and study the influence of different textural parameters on coupled conductive-radiative heat transfers. These numerical methods require 3D digitized images of the sample of interest obtained either using 3D imagery technique or generated using numerical algorithms. To better represent the complex geometry, the 3D image of the sample requires high spatial resolution, and for a sample which is representative of involved physics might contain hundreds of millions of voxels which is complex to solve and computationally expensive. To cope with this issue, we developed a parallelized discrete scale numerical approach using cell centered Finite Volume Method (FVM) and deterministic ray tracing to solve coupled heat transfer within highly porous large scale complex cellular materials. At the heart of this method rests a decomposition approach based on modified zonal method which significantly reduces the coupling efforts, memory requirements, and computation time. The results of two test cases presented in this study are cross-verified with those presented in literature and computed using Star-CCM+ software. Finally, the method is applied to virtual fibrous sample. These results present the ability of the solver to consider different boundary conditions, at large temperature gradient across boundaries, and to deal with large samples consisting hundreds of million of voxels while providing accurate results. We also present and discuss the sensitivity of other associated parameters on the final results.

A constrained boundary gradient reconstruction method for unstructured finite volume discretization of the Euler equations

We propose a novel boundary gradient and solution reconstruction approach for the second-order cell-centered unstructured finite volume method for solving the inviscid Euler equations. The proposed constrained reconstruction method involving ghost cells is shown to be more accurate than several conventional boundary reconstruction methods and linear-exact in the near-boundary areas. In the present procedure, the solutions at the ghost cell centroids are given by solving constraint equations, which are established based on boundary conditions. At the same time, the solution gradients in boundary cells and the solutions in boundary surfaces are obtained altogether, while guaranteeing the compatibility of the boundary conditions with the reconstructed results. A variety of numerical test cases show that the new method effectively reduces the errors of near-boundary approximations and improves the overall numerical accuracy. Furthermore, because the constrained reconstruction is only used for the boundary cells, the extra computational cost barely changes the overall computational efficiency.

Accelerating FVM-Based Parallel Fluid Simulations with Better Grid Renumbering Methods

Grid renumbering techniques have been shown to be effective in improving the efficiency of computational fluid dynamics (CFD) numerical simulations based on the finite volume method (FVM). However, with the increasing complexity of real-world engineering scenarios, there is still a huge challenge to choose better sequencing techniques to improve parallel simulation performance. This paper designed an improved metric (MDMP) to evaluate the structure of sparse matrices. The metric takes the aggregation of non-zero elements inside the sparse matrix as an evaluation criterion. Meanwhile, combined with the features of the cell-centered finite volume method supporting unstructured grids, we proposed the cell quotient (CQ) renumbering algorithm to further reduce the maximum bandwidth and contours of large sparse matrices with finite volume discretization. Finally, with real-world engineering cases, we quantitatively analyzed the evaluation effect of MDMP and the optimization effect of different renumbering algorithms. The results showed that the classical greedy algorithm reduces the maximum bandwidth of the sparse matrix by at most 60.34% and the profile by 95.38%. Correspondingly, the CQ algorithm reduced them by at most 92.94% and 98.70%. However, in terms of MDMP, the CQ algorithm was 83.43% less optimized than the Greedy algorithm. In terms of overall computational speed, the Greedy algorithm was optimized by a maximum of 38.19%, and the CQ algorithm was optimized by a maximum of 27.31%. The above is in accordance with the evaluation results of the MDMP metric. Thus, our new metric can more accurately evaluate the renumbering method for numerical fluid simulations, which is of great value in selecting a better mesh renumbering method in engineering applications of CFD.

A high-fidelity solver based on hybrid numerical methods on unstructured grids for incompressible multiphase flows

A high-fidelity solver is developed on the unstructured grids to simulate incompressible multiphase flows with free surface based on volume of fluid (VOF) method. In this framework, the velocity is defined by both volume integrated average (VIA) and point value (PV) as computational variables which are updated simultaneously by volume-average/point-value multi-moment (VPM) method at each time step. Compared with our previous work, a node-based finite element (FE) method is used instead of cell-centered finite volume (FV) method for the discretization of pressure Poisson equation, which provides a new pattern that can naturally treat with the velocity and pressure coupling. It does not require Rhie–Chow interpolation or the use of a Petrov–Galerkin finite element formulation with tunable parameters to circumvent the well-known instability conditions. Moreover, we propose a novel balanced-force formulation with FE discretization to accommodate the surface tension force, which effectively eliminates the large numerical errors associated with the non-orthogonality of unstructured grids. In VOF method, the volume fraction is defined as VIA to identify the moving interface which is updated by solving advection equation through the FV formulation. The resulting model that combines hybrid discretizations, rigorously preserves the mass and momentum conservation, substantially improves the numerical accuracy and robustness of pressure field, and significantly suppresses the spurious velocity to nearly machine error even for multiphase flows with large density ratios. Various numerical examples have been presented to demonstrate the excellent capabilities of present solver which can provide high-fidelity solutions for practical applications in presence of largely deformed interface and highly complex geometry.

A Pressure-Based Fully-Coupled Flow Algorithm for the Control Volume Finite Element Method

A pressure-based fully-coupled flow algorithm is developed for the control volume finite element method (CVFEM), which covers speeds up to the transonic regime. The CVFEM is used because it presents a number of advantages as compared to the popular cell-centered finite volume method (CCFVM), while retaining the properties of the finite volume method (FVM) in terms of flux conservation and numerical integration simplicity. The implementation presents a novel weak formulation of Dirichlet boundary conditions to resolve the disadvantages emerging from the strong formulation, by mimicking the methods followed in the CCFVM. Derivation and implementation details are presented, and a number of test cases are used to evaluate the accuracy and performance of this approach.

Finite volume method for coupled subsurface flow problems, II: Poroelasticity

The article introduces a cell-centered finite-volume method for the Biot problem in heterogeneous anisotropic media, characterized with full-tensor properties. We derive the expression for the coupled flux and the interpolation method across the discontinuity of the properties. The obtained flux expression consists of a two-point part, a transversal part and an additional contribution due to gravity. The interpolation method is the generalization of the harmonic averaging point concept to coupled problems. The method is stable despite collocation of both pressure and displacement at cell centers due to eigensplitting of the matrix coefficients in the flux expression and upstream approximation. A general type of boundary condition is integrated without introduction of auxiliary degrees of freedom. Our flux discretization method is a realization of our more general concept of stable flux discretization for saddle-point systems with vector of several unknowns. We demonstrate the applicability of the method on a set of challenging numerical benchmarks.

A very high-order flux reconstruction approach coupled to the MPFA-QL finite volume method for the numerical simulation of oil-water flows in 2D petroleum reservoirs

In recent years, high-order methods for the discretization of the advection dominated saturation equation, and that results from the modelling of the 2D Oil-Water displacement through porous formations, have received renewed attention from the CFD community, as a suitable alternative to the traditional first-order upwind (FOU) finite volume (FV) scheme, which is the standard in industry. A key aspect of high-order methods applied to numerical simulation of petroleum reservoirs is the lower computational cost for a comparable degree of accuracy against their lower-order counterparts. In this context, in the present paper, for the first time in literature, we propose the use of the very high order Flux Reconstruction (FR) formulation for the discretization of the advective terms in the saturation equation using unstructured quadrilateral meshes. To solve the elliptic pressure equation, we use the recently developed Multipoint Flux Approximation method with a Quasi-Local Stencil (MPFA-QL). This method is very robust and able to handle highly heterogeneous and anisotropic permeability tensors. The system of equations is solved in a segregated manner by using the classical Implicit Pressure Explicit Saturation (IMPES) procedure. To properly couple the cell-centered MPFA-QL method with the FR formulation, it is necessary to obtain an adequate velocity reconstruction throughout the control volumes of the mesh. As the cell-centered finite volume method naturally delivers fluxes across control surfaces of the primal grid, then, we use a reconstruction operator based on the first-order Brezzi–Douglas–Marini (BDM) space, and the Piola transformation, to get the complete knowledge of the velocity field throughout the domain. To eliminate spurious oscillations around the discontinuities, which are typical in high-order methods, we have used a hierarchical multidimensional limiting process (hMLP). To compute fluxes over the control surfaces we have used the Roe-E approximate Riemann solver. Finally, through some representative examples, we show the accuracy, efficiency and robustness of our new high order numerical methodology to model oil-water displacements in highly heterogeneous and anisotropic petroleum reservoirs.

Numerical investigation of full helicopter with and without the ground effect

In the present work, the aerodynamic performance of the full helicopter PSP in hover flight is investigated using a simplified concept of multiple reference frame (MRF) technique in the context of high-order Monotone Upstream Centred Scheme for Conservation Laws (MUSCL) cell-centred finite volume method. The predictions were obtained for two ground distances and several collective pitch angle at tip Mach number of 0.585. The calculations were made for both out-of-ground-effect (OGE) and in-ground-effect (IGE) cases and compared with experimental data in terms of pressure distribution and integrated thrust and torque and vortex system.

A Cell-Centered Semi-Lagrangian Finite Volume Method for Solving Two-Dimensional Coupled Burgers’ Equations

A cell-centered finite volume semi-Lagrangian method is presented for the numerical solution of two-dimensional coupled Burgers’ problems on unstructured triangular meshes. The method combines a modified method of characteristics for the time integration and a cell-centered finite volume for the space discretization. The new method belongs to fractional-step algorithms for which the convection and the viscous parts in the coupled Burgers’ problems are treated separately. The crucial step of interpolation in the convection step is performed using two local procedures accounting for the element where the departure point is located. The resulting semidiscretized system is then solved using a third-order explicit Runge-Kutta scheme. In contrast to the Eulerian-based methods, we apply the new method for each time step along the characteristic curves instead of the time direction. The performance of the current method is verified using different examples for coupled Burgers’ problems with known analytical solutions. We also apply the method for simulation of an example of coupled Burgers’ flows in a complex geometry. In these test problems, the new cell-centered finite volume semi-Lagrangian method demonstrates its ability to accurately resolve the two-dimensional coupled Burgers’ problems.