Velocity-pressure System Research Articles

An effective preconditioning strategy for volume penalized incompressible/low Mach multiphase flow solvers

The volume penalization (VP) or the Brinkman penalization (BP) method is a diffuse interface method for simulating multiphase fluid-structure interaction (FSI) problems in ocean engineering and/or phase change problems in thermal sciences and engineering. The method relies on a penalty factor (which is inversely related to body's permeability κ) that must be large to enforce rigid body velocity in the solid domain. When the penalty factor is large, the discrete system of equations becomes stiff and difficult to solve numerically. In this paper, we propose a projection method-based preconditioning strategy for solving volume penalized (VP) incompressible and low-Mach Navier-Stokes equations. The projection preconditioner enables the monolithic solution of the coupled velocity-pressure system in both single phase (uniform density and viscosity) and multiphase (variable density and viscosity) flow settings. In this approach, the penalty force is treated implicitly, which is allowed to take arbitrary large values without affecting the solver's convergence rate or causing numerical stiffness. It is made possible by including the penalty term in the pressure Poisson equation (PPE), which was not included in previous works that solved VP incompressible Navier-Stokes equations using the projection method. We show how and where the Brinkman penalty term enters the PPE by re-deriving the projection algorithm for the VP method. Solver scalability under grid refinement is demonstrated, i.e., convergence is achieved with the same number of iterations regardless of the problem size. A manufactured solution in a single phase setting is used to determine the spatial accuracy of the penalized solution. Various values of body's permeability κ are considered. Second-order pointwise accuracy is achieved for both velocity and pressure solutions for reasonably small values of κ. Error saturation occurs when κ is extremely small, but the convergence rate of the solver does not degrade. The solver converges faster as κ decreases, contrary to prior experience. Two multiphase fluid-structure interaction (FSI) problems from the ocean engineering literature are also simulated to evaluate the solver's robustness and performance (in terms of its number of iterations). The proposed solver also allows us to investigate the effect of κ on the motion of the contact line over the surface of the immersed body. It also allows us to investigate the dynamics of the free surface of a solidifying metal.

Global Existence and Uniqueness of The Inviscid Velocity-Vorticity Model of The g-Navier-Stokes Equations

In this paper, we prove the global existence and uniqueness of the weak solutions to the inviscid velocity-vorticity model of the g-Navier-Stokes equations. The system is performed by entegrating the velocity-pressure system which is involved by using the rotational formulation of the nonlinearity and the vorticity equation for the g-Navier-Stokes equations without viscosity term. In this study we particularly interest the inviscid velocity-vorticity system of the g-Navier-Stokes equations over the two dimensional periodic box Ω=(0,1)^2⊂R^2.

An Efficient B-Spline Lagrangian/Eulerian Method for Compressible Flow, Shock Waves, and Fracturing Solids

This study presents a new method for modeling the interaction between compressible flow, shock waves, and deformable structures, emphasizing destructive dynamics. Extending advances in time-splitting compressible flow and the Material Point Methods (MPM), we develop a hybrid Eulerian and Lagrangian/Eulerian scheme for monolithic flow-structure interactions. We adopt the second-order WENO scheme to advance the continuity equation. To stably resolve deforming boundaries with sub-cell particles, we propose a blending treatment of reflective and passable boundary conditions inspired by the theory of porous media. The strongly coupled velocity-pressure system is discretized with a new mixed-order finite element formulation employing B-spline shape functions. Shock wave propagation, temperature/density-induced buoyancy effects, and topology changes in solids are unitedly captured.

Parallelisation of selective algebraic multigrid for block–pressure–velocity system in OpenFOAM

In the world of computational fluid dynamics (CFD), solving the governing equations of incompressible, turbulent, single-phase fluid flow still represents the basis of many industrial and academic applications. The implicitly coupled (monolithic) solution approach is still being developed and investigated for industrial-size applications. A parallel selection algebraic multigrid algorithm (AMG) based on the domain decomposition method is presented, applied for the solution of the linearised implicitly coupled pressure–velocity system discretised by the finite volume method, implemented in OpenFOAM. Since the setup phase of the selection AMG, i.e.sorting the equations into coarse and fine subsets is inherently sequential, it was decided to perform the setup phase locally on each processing unit. The prolongation matrix for transferring the correction from coarse to fine level and restriction matrix for transferring the residual from fine to coarse level are assembled locally as well. Parallel communication is necessary only for the calculation of the coarse level matrix, i.e.the matrix elements which describe the cross-coupling of equations located on different processing units. A localised version of the ILU factorisation based on Crout’s algorithm is used as a smoother in the multigrid cycle. A detailed analysis of the coarse level matrix complexity is conducted in the context of the finite volume method in domain decomposition mode. The performance and scaling of our parallel implementation is investigated for two test cases and the possible drawbacks of the method are given.

A preconditioned, multilevel Mimetic Finite Difference method for solving the pressure-velocity system

A Mimetic Finite Difference (MFD) implementation is proposed to solve the pressure-velocity system iteratively. The recursive procedure only solves for the velocity variable, while the pressure is calculated by direct substitution, simplifying the problem. The symmetric, positive definite system obtained is solved with an innovative implementation of a pre-conditioned, multilevel, conjugate gradient algorithm, where coarse algebraic objects are easily computed from the corresponding fine level ones, based on fundamental geometric properties of the MFD technology. A succinct eigen-analysis shows second order convergence on the velocity variable for Cartesian grids, which is corroborated numerically. Also, results consider different topologies and different medium properties, showing the applicability of the method and optimal convergence rates for the velocity on Cartesian and h2-uniform curved grids.

Parallel block–selective algebraic multigrid in foam‐extend

AbstractWe present an extension of a well–known classic algebraic multigrid algorithm, where coarsening is based on selection of equations, for an implicitly coupled pressure–velocity system discretised by the finite volume (FV) method. The resulting coefficient matrix of the linear system is point–ordered, and there are 4 equations for each FV cell: three components of velocity and pressure. A significant challenge is posed in terms of parallelisation of the sequential coarsening process and calculation of coarse level matrix, using the Galerkin projection. In this paper we present the parallelisation strategy and performance of the block–selection AMG.

Preconditioning a Newton-Krylov solver for all-speed melt pool flow physics

In this paper, we introduce a multigrid block-based preconditioner for solving linear systems arising from a Discontinuous Galerkin discretization of the all-speed Navier-Stokes equations with phase change. The equations are discretized in conservative form with a reconstructed Discontinuous Galerkin (rDG) method and integrated with fully-implicit time discretization schemes. To robustly converge the numerically stiff systems, we use the Newton-Krylov framework with a primitive-variable formulation (pressure, velocity, and temperature), which is better conditioned than the conservative-variable form at low-Mach number. In the limit of large acoustic CFL number and viscous Fourier number, there is a strong coupling between the velocity-pressure system and the linear systems become non-diagonally dominant. To effectively solve these ill-conditioned systems, an approximate block factorization preconditioner is developed, which uses the Schur complement to reduce a 3×3 block system into a sequence of two 2×2 block systems: velocity-pressure, vP, and velocity-temperature, vT. We compare the performance of the vP-vT Schur complement preconditioner to classic preconditioning strategies: monolithic algebraic multigrid (AMG), element-block SOR, and primitive variable block Gauss-Seidel. The performance of the preconditioned solver is investigated in the limit of large CFL and Fourier numbers for low-Mach lid-driven cavity flow, Rayleigh-Bénard melt convection, compressible internally heated convection, and 3D laser-induced melt pool flow. Numerical results demonstrate that the vP-vT Schur complement preconditioned solver scales well both algorithmically and in parallel, and is robust for highly ill-conditioned systems, for all tested rDG discretization schemes (up to 4th-order).

A fully mass-conservative iterative IMPEC method for multicomponent compressible flow in porous media

In this paper we consider efficient and fully mass-conservative numerical methods for the multicomponent compressible single-phase Darcy flow in porous media. Compared with the classical IMplicit Pressure Explicit Concentration (IMPEC) scheme by which one of the components may be not mass-conservative, the new scheme enjoys an appealing feature that the conservation of mass is retained for each of the components. The pressure–velocity system is obtained by the summation of the discrete conservation equation for each component multiplying an unknown parameter which is nonlinearly dependent of the molar concentrations. This approach is quite different from the conventional method which is used in the classical IMPEC scheme. We utilize a fully mass-conservative iterative IMPEC method to solve the nonlinear system for molar concentration, pressure and velocity fields. The upwind mixed finite element methods are used to solve the pressure–velocity system. Although the Peng–Robinson equation of state (EOS) is utilized to describe the pressure as a function of the molar concentrations, our method is suitable for any type of EOS. Under some reasonable conditions, the iterative scheme can be proved to be convergent, and the molar concentration of each component is positivity-preserving. Several interesting examples of multicomponent compressible flow in porous media are presented to demonstrate the robustness of the new algorithm.

Influences of geometrical parameters on the heat transfer characteristics through symmetry trapezoidal-corrugated channel using SiO2-water nanofluid

Enhancing the geometrical parameters design of thermal devices leads to promote the thermal performance and boost design plan of these devices and make it more compact. In the current study, heat transfer and flow characteristics of the symmetry trapezoidal-corrugated channel with silicon dioxide (SiO2) - water as nanofluid was performed numerically over Reynolds number ranges of 10,000–30,000. The influence of geometrical parameters including height-to-width ratio (h/W) and pitch-to-length ratio (p/L) on the thermal and hydraulic characteristics are evaluated. A numerical simulation covers nanofluid with SiO2 volume fractions 8% and carried out by employing the finite volume method (FVM) and SIMPLE algorithm for discretization of the governing equations and coupling of the pressure-velocity system while the k−ε turbulence model was employed to compute the turbulent flow. The outcomes revealed that the (h/W) ratio has a more influence on the promotion of heat transfer compared with the (p/L) ratio. At Reynolds number 30000, there is 16.63% increment in Nuav due to a decrease of the (p/L) ratio from 0.175 to 0.075, while the increment about 99.45% due to an increase of the (h/W) ratio from 0.0 to 0.05. The numerical results indicate that the h/W of 0.05 with a p/L of 0.075 are the optimum parameters and have shown significant improvement in thermal performance factor. Furthermore, new correlations for Nusselt number and friction factor are developed and reported.

Block-selective algebraic multigrid for implicitly coupled pressure-velocity system

The experience with the implicitly coupled pressure–velocity solver has proven that multigrid methods are the most robust and efficient in solving the system. Until now the usual choice of the linear solver for the block–variable has been the Aggregative Algebraic Multigrid which has a relatively simple implementation: Coarsening is achieved by joining the cells into groups, coarse level operators are calculated by summing the coefficients from the fine level matrix, and restriction and prolongation is done by injection. However, our experience has shown that this type of AMG was inefficient for the implicitly coupled solver: In a single non–linear iteration, the residual would stall, not reaching the prescribed convergence tolerance.To remedy the problem, we have implemented a Block–Selective Algebraic Multigrid in foam-extend which is a community driven fork of OpenFOAM, an open–source CFD software library for arbitrary polyhedral Finite Volume Method. Coarsening is done by selecting a set of equations which will be solved on the coarse level, based on a heuristic algorithm. Prolongation operator is calculated from the approximation of the algebraically smooth error. Restriction and coarse level matrix are formed based on a variational principle for a symmetric positive–definite matrix. To apply the algorithm on the matrix arising from the discretisation of the implicitly coupled pressure–velocity system, a primary matrix must be chosen by calculating a norm of the block–coefficient.

Optimal vorticity accuracy in an efficient velocity–vorticity method for the 2D Navier–Stokes equations

We study a velocity–vorticity scheme for the 2D incompressible Navier–Stokes equations, which is based on a formulation that couples the rotation form of the momentum equation with the vorticity equation, and a temporal discretization that stably decouples the system at each time step and allows for simultaneous solving of the vorticity equation and velocity–pressure system (thus if special care is taken in its implementation, the method can have no extra cost compared to common velocity–pressure schemes). This scheme was recently shown to be unconditionally long-time $$H^1$$ stable for both velocity and vorticity, which is a property not shared by any common velocity–pressure method. Herein, we analyze the scheme’s convergence, and prove that it yields unconditional optimal accuracy for both velocity and vorticity, thus making it advantageous over common velocity–pressure schemes if the vorticity variable is of interest. Numerical experiments are given that illustrate the theory and demonstrate the scheme’s usefulness on some benchmark problems.

Analysis of two decoupled time-stepping finite-element methods for incompressible fluids with microstructure

ABSTRACTIn this article, two full discrete schemes of the micropolar Navier–Stokes equations (MNSE) are proposed and analysed, namely a fully decoupled scheme and a decoupled penalty-projection scheme. The fully decoupled scheme decouples the MNSE system into two sub-problems, one is the velocity–pressure system, the other is the angular velocity system, but still keeps unconditional stability with respect to the time step size. The stability and the optimal rate of convergence of the scheme are proved. Moreover, the decoupled penalty-projection method is derived based on the Chorin/Temam projection methods for the velocity–pressure system equipped with grad-div stabilization with parameter γ, we also prove that this scheme is unconditionally stable. In particular, we show that the solution of the decoupled penalty-projection method converges to the associated full decoupled method solution as . These time-stepping discretization schemes can be implemented efficiently in practice. Numerical experiments are given to validate the convergence rate predictions and demonstrate the efficiency of the new methods.

Perfectly matched layers for convex truncated domains with discontinuous Galerkin time domain simulations

This paper deals with the design of perfectly matched layers (PMLs) for transient acoustic wave propagation in generally-shaped convex truncated domains. After reviewing key elements to derive PML equations for such domains, we present two time-dependent formulations for the pressure–velocity system. These formulations are obtained by using a complex coordinate stretching of the time-harmonic version of the equations in a specific curvilinear coordinate system. The final PML equations are written in a general tensor form, which can easily be projected in Cartesian coordinates to facilitate implementation with classical discretization methods. Discontinuous Galerkin finite element schemes are proposed for both formulations. They are tested and compared using a three-dimensional benchmark with an ellipsoidal truncated domain. Our approach can be generalized to domains with corners.

A nodal discontinuous Galerkin method for reverse-time migration on GPU clusters

Improving both accuracy and computational performance of numerical tools is a major challenge for seismic imaging and generally requires specialized implementations to make full use of modern parallel architectures. We present a computational strategy for reverse-time migration (RTM) with accelerator-aided clusters. A new imaging condition computed from the pressure and velocity fields is introduced. The model solver is based on a high-order discontinuous Galerkin time-domain (DGTD) method for the pressure-velocity system with unstructured meshes and multi-rate local time-stepping. We adopted the MPI+X approach for distributed programming where X is a threaded programming model. In this work we chose OCCA, a unified framework that makes use of major multi-threading languages (e.g. CUDA and OpenCL) and offers the flexibility to run on several hardware architectures. DGTD schemes are suitable for efficient computations with accelerators thanks to localized element-to-element coupling and the dense algebraic operations required for each element. Moreover, compared to high-order finite-difference schemes, the thin halo inherent to DGTD method reduces the amount of data to be exchanged between MPI processes and storage requirements for RTM procedures. The amount of data to be recorded during simulation is reduced by storing only boundary values in memory rather than on disk and recreating the forward wavefields. Computational results are presented that indicate that these methods are strong scalable up to at least 32 GPUs for a three-dimensional RTM case.

Efficient Variable-Coefficient Finite-Volume Stokes Solvers

AbstractWe investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variable-coefficient Stokes equations on a uniform staggered grid. Building on the success of using the classical projection method as a preconditioner for the coupled velocity pressure system [B. E. Griffith, J. Comp. Phys., 228 (2009), pp. 7565-7595], as well; established techniques for steady and unsteady Stokes flow in the finite-element literature, we construct preconditioners that employ independent generalized Helmholtz and Poisson solvers for the velocity and pressure subproblems. We demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these subproblems. Contrary to traditional wisdom, we find that the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems, making the overall cost of solving the Stokes system comparable to the cost of classical projection or fractional step methods for incompressible flow, even for steady flow and in the presence of large density and viscosity contrasts. Two of the five preconditioners considered here are found to be robust to GMRES restarts and to increasing problem size, making them suitable for large-scale problems. Our work opens many possibilities for constructing novel unsplit temporal integrators for finite-volume spatial discretizations of the equations of low Mach and incompressible flow dynamics.

Entropically damped form of artificial compressibility for explicit simulation of incompressible flow

An alternative artificial compressibility (AC) scheme is proposed to allow the explicit simulation of the incompressible Navier-Stokes (INS) equations. Traditional AC schemes rely on an artificial equation of state that gives the pressure as a function of the density, which is known to enforce isentropic behavior. This behavior is nonideal, especially in viscously dominated flows. An alternative, the entropically damped artificial compressibility (EDAC) method, is proposed that employs a thermodynamic constraint to damp the pressure oscillations inherent to AC methods. The EDAC method converges to the INS in the low-Mach limit, and is consistent in both the low- and high-Reynolds-number limits, unlike standard AC schemes. The proposed EDAC method is discretized using a simple finite-difference scheme and is compared with traditional AC schemes as well as the lattice-Boltzmann method for steady lid-driven cavity flow and a transient traveling-wave problem. The EDAC method is shown to be beneficial in damping pressure and velocity-divergence oscillations when performing transient simulations. The EDAC method follows a similar derivation to the kinetically reduced local Navier-Stokes (KRLNS) method [Borok et al., Phys. Rev. E 76, 066704 (2007)]; however, the EDAC method does not rely on the grand potential as the thermodynamic variable, but instead uses the more common pressure-velocity system. Additionally, a term neglected in the KRLNS is identified that is important for accurately approximating the INS equations.

Assessment of a vorticity based solver for the Navier–Stokes equations

We investigate numerically a recently proposed vorticity based formulation of the incompressible Navier–Stokes equations. The formulation couples a velocity–pressure system with a vorticity–helicity system, and is intended to provide a numerical scheme with enhanced accuracy and superior conservation properties. For a few benchmark problems, we study the performance of a finite element method for this formulation and compare it with the commonly used velocity–pressure based finite element method. It is shown that both steady and unsteady discrete problems in the new formulation admit simple decoupling strategies followed by the application of iterative solves to auxiliary subproblems. Further, we compare several iterative strategies to solve the discrete problems and study the interplay between the choice of stabilization parameters in the finite element method and the efficiency of linear algebra solvers.

A comparison between a symmetric and a non-symmetric Galerkin finite element—boundary integral equation coupling for the two-dimensional exterior Stokes problem

Symmetric and non-symmetric Galerkin formulations are presented for the coupling of a finite element modelled interior region to a boundary integral supported exterior region for the two-dimensional steady state exterior Stokes problem. Both single and double-layer hydrodynamic potentials are used allowing a well conditioned symmetric matrix structure for the entire interior–exterior, velocity–pressure system when the exterior velocity boundary integral equation (VBIE) is augmented by a traction boundary integral equation (TBIE) with the pressure determined everywhere purely through the imposition of the divergence-free velocity condition. Corresponding non-symmetric formulations are obtained by additionally discretizing an associated pressure boundary integral equation (PBIE), where the associated kernel functions have singularities an order higher than in the VBIE, with a simple regularization of the new hyper-singular pressure kernel. Comparable solution convergence with mesh refinement for the symmetric and non-symmetric schemes is shown for stabilized and mixed velocity–pressure conforming finite element pairs using Lagrangian shape functions.

Toward large scale F.E. computation of hot forging process using iterative solvers, parallel computation and multigrid algorithms

AbstractThe industrial simulation code Forge3® is devoted to three‐dimensional metal forming applications. This finite element software is based on an implicit approach. It is able to carry out the large deformations of viscoplastic incompressible materials with unilateral contact conditions. The finite element discretization is based on a stable mixed velocity–pressure formulation and tetrahedral unstructured meshes. Central to the Newton iterations dealing with the non‐linearities, a preconditioned conjugate residual method (PCR) is used. The parallel version of the code uses an SPMD programming model and several results on complex applications have been published. In order to reduce the CPU time computation, a new solver has been developed which is based on multigrid theory. A detailed presentation of the different elements of the method is given: the geometrical approach based on embedded meshes, the direct resolution of the velocity–pressure system, the use of PCR method as an original smoother and for solving the coarse problem, the full multigrid method and the required preconditioning by an incomplete Cholesky factorization for problems with complex contact conditions. By considering different forging cases, the theoretical properties of the multigrid method are numerically verified, optimizations of the solver are presented and finally, the results obtained on several industrial problems are given, showing the efficiency of the new solver that provides speed‐up larger than 5. Copyright © 2001 John Wiley & Sons, Ltd.

Parallel Computation of Flow in Heterogeneous Media Modelled by Mixed Finite Elements

In this paper we describe a fast parallel method for solving highly ill-conditioned saddle-point systems arising from mixed finite element simulations of stochastic partial differential equations (PDEs) modelling flow in heterogeneous media. Each realisation of these stochastic PDEs requires the solution of the linear first-order velocity–pressure system comprising Darcy's law coupled with an incompressibility constraint. The chief difficulty is that the permeability may be highly variable, especially when the statistical model has a large variance and a small correlation length. For reasonable accuracy, the discretisation has to be extremely fine. We solve these problems by first reducing the saddle-point formulation to a symmetric positive definite (SPD) problem using a suitable basis for the space of divergence-free velocities. The reduced problem is solved using parallel conjugate gradients preconditioned with an algebraically determined additive Schwarz domain decomposition preconditioner. The result is a solver which exhibits a good degree of robustness with respect to the mesh size as well as to the variance and to physically relevant values of the correlation length of the underlying permeability field. Numerical experiments exhibit almost optimal levels of parallel efficiency. The domain decomposition solver (DOUG, http://www.maths.bath.ac.uk/~parsoft) used here not only is applicable to this problem but can be used to solve general unstructured finite element systems on a wide range of parallel architectures.