Solver Scheme Research Articles

A robust 2D shallow water model for solving flow over complex topography using homogenous flux method

SUMMARYA robust Godunov‐type numerical scheme solver is proposed for solving 2D SWEs and is applied to simulate flow over complex topography with wetting and drying. In reality, the topography is usually complex and irregular; therefore, to avoid the numerical errors generated by such features, a Homogenous Flux Method is used to handle the bed slope term in the SWEs. The method treats the bed slope term as a flux to be incorporated into the flux gradient and so maintains the balance between the two in a Godunov‐type shock‐capturing scheme. The main advantages of the method are: first, it is simple and easy to implement; second, numerical experiments demonstrate that it can handle discontinuous or vertical bed topography without any special treatment and third, it is applicable to both steady and unsteady flows. It is demonstrated how the approach set out here can be applied to the nonlinear hyperbolic system of the SWEs. The two‐dimensional hyperbolic system is then solved by use of a second‐order total‐variation‐diminishing version of the weighted average flux method in conjunction with a Harten‐Lax‐van Leer‐Contract approximate Riemann solver incorporating the new flux gradient term. Several benchmark tests are presented to validate the model and the approach is verified against experimental measurements from the European Union Concerted Action on Dam Break Modelling project. These show very good agreement. Finally, the method is applied to a volcano‐induced outburst flood over an initially dry channel with complex irregular topography to demonstrate the technique's capability in simulating a real flood. Copyright © 2013 John Wiley & Sons, Ltd.

Efficient linear solvers for incompressible flow simulations using Scott‐Vogelius finite elements

AbstractRecent research has shown that in some practically relevant situations like multiphysics flows (Galvin et al., Comput Methods Appl Mech Eng, to appear) divergence‐free mixed finite elements may have a significantly smaller discretization error than standard nondivergence‐free mixed finite elements. To judge the overall performance of divergence‐free mixed finite elements, we investigate linear solvers for the saddle point linear systems arising in ((Pk)d,P k‐1disc) Scott‐Vogelius finite element implementations of the incompressible Navier–Stokes equations. We investigate both direct and iterative solver methods. Due to discontinuous pressure elements in the case of Scott‐Vogelius (SV) elements, considerably more solver strategies seem to deliver promising results than in the case of standard mixed finite elements such as Taylor‐Hood elements. For direct methods, we extend recent preliminary work using sparse banded solvers on the penalty method formulation to finer meshes and discuss extensions. For iterative methods, we test augmented Lagrangian and \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{H}\end{align*} \end{document} ‐LU preconditioners with GMRES, on both full and statically condensed systems. Several numerical experiments are provided that show these classes of solvers are well suited for use with SV elements and could deliver an interesting overall performance in several applications.© 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Analysis of dual secondary injection for thrust vectoring

PurposeThe purpose of this paper is to analyze the flowfield structure and performance of dual secondary injection system for thrust vectoring in a convergent‐divergent nozzle and to compare it with a single secondary injection system.Design/methodology/approachDual secondary injection for thrust vectoring in a convergent‐divergent nozzle is studied by solving three‐dimensional Reynolds‐averaged Navier‐Stokes equations by the means of Fluent. Realizable k‐ε turbulent model with enhanced wall‐treatment approach is used for viscous model. Density‐based solver and explicit scheme are employed in the computational model. In order to study the effect of injection location on the flowfield, distance between ports is considered as the key variable.FindingsResults show that under some circumstances, dual secondary injection system is more effective than a single injection system with the same mass flow rate. The study shows that when the distance between two ports is 8.5 times of the injection port's diameter (or more) and in the same time the first injection port is at least 1 throat diameter far from the nozzle throat, this system will show a better performance. In addition, this system reduces the probability of bow shock impingement to the opposite wall and consequently, the side force production has less limitation.Practical implicationsDual secondary injection for thrust vector control (SITVC) needs less secondary flow and therefore it makes less reduction in the primary thrust. It means that for a specific primary thrust, less mass fuel is needed which makes it more economic regarding the traditional SITVC systems.Originality/valueThe paper's value lies in using a three‐dimensional model to study the effect of two ports distance on SITVC performance and comparison among the performance of dual and single injections when there is an impingement.

A Factorization of the Spectral Galerkin System for Parameterized Matrix Equations: Derivation and Applications

Recent work has explored solver strategies for the linear system of equations arising from a spectral Galerkin approximation of the solution of PDEs with parameterized (or stochastic) inputs. We consider the related problem of a matrix equation whose matrix and right hand side depend on a set of parameters (e.g. a PDE with stochastic inputs semidiscretized in space) and examine the linear system arising from a similar Galerkin approximation of the solution. We derive a useful factorization of this system of equations, which yields bounds on the eigenvalues, clues to preconditioning, and a flexible implementation method for a wide array of problems. We complement this analysis with (i) a numerical study of preconditioners on a standard elliptic PDE test problem and (ii) a fluids application using existing CFD codes; the MATLAB codes used in the numerical studies are available online.

Towards a scalable fully-implicit fully-coupled resistive MHD formulation with stabilized FE methods

This paper explores the development of a scalable, nonlinear, fully-implicit stabilized unstructured finite element (FE) capability for 2D incompressible (reduced) resistive MHD. The discussion considers the implementation of a stabilized FE formulation in context of a fully-implicit time integration and direct-to-steady-state solution capability. The nonlinear solver strategy employs Newton–Krylov methods, which are preconditioned using fully-coupled algebraic multilevel preconditioners. These preconditioners are shown to enable a robust, scalable and efficient solution approach for the large-scale sparse linear systems generated by the Newton linearization. Verification results demonstrate the expected order-of-accuracy for the stabilized FE discretization. The approach is tested on a variety of prototype problems, including both low-Lundquist number (e.g., an MHD Faraday conduction pump and a hydromagnetic Rayleigh–Bernard linear stability calculation) and moderately-high Lundquist number (magnetic island coalescence problem) examples. Initial results that explore the scaling of the solution methods are presented on up to 4096 processors for problems with up to 64M unknowns on a CrayXT3/4. Additionally, a large-scale proof-of-capability calculation for 1 billion unknowns for the MHD Faraday pump problem on 24,000 cores is presented.

An Efficient Algebraic Multigrid Solver Strategy for Adaptive Implicit Methods in Oil-Reservoir Simulation

Summary We propose a new, efficient, adaptive algebraic multigrid (AMG) solver strategy for the discrete systems of partial-differential equations (PDEs) arising from structured or unstructured grid models in reservoir simulation. The proposed strategy has been particularly tailored to linear systems of equations arising in adaptive implicit methods (AIMs). The coarsening process of the AMG method designed automatically employs information on the physical structure of the models; as a smoother, an adaptive incomplete LU factorization with thresholding (ILUT) method is employed, taking care of an efficient solution of the hyperbolic parts while providing adequately smooth errors for the elliptic parts. To achieve a good compromise of high efficiency and robustness for a variety of problem classes—ranging from simple, small black-oil to challenging, large compositional models—an automatic, adaptive ILUT parameter and AMG solver switching strategy, α-SAMG. has been developed. Its efficiency is demonstrated for eight industrial benchmark cases by comparison against standard one-level and AMG solvers, including constraint pressure residual (CPR), as well as the pure one-level variant of the proposed new strategy. In addition, very promising results of first parallel runs are shown.

Model-Based Integration and Optimization—Gas-Cycling Benchmark

Summary A benchmark for computational integration of petroleum operations has been constructed. The benchmark consists of two gas/ condensate reservoirs producing to a common process facility. A fraction of the processed gas is distributed between the two reservoirs for gas injection. Total project economics is calculated from the produced streams and process-related costs. This benchmark may be used to compare different computational integration frameworks and optimization strategies. Even though this benchmark aims to integrate all parts of a petroleum operation, from upstream to downstream, certain simplifications are made. For example, pipe flow from reservoir to process facility is not included in the integrated model. The methods of model integration and optimization discussed in this paper are applicable to complex petroleum operations where it is difficult to quantify cause and effect without comprehensive model-based integration. A framework for integration of models describing petroleum operations has been developed. An example test problem is described and studied in detail. Substantial gains in full-field development may be achieved by optimizing over the entire production system. All models and data in the benchmark problem are made available so that different software platforms can study the effects of alternative integration methods and optimization solver strategy. The project itself can, and probably should, be extended by others to add more complexity (realism) to the reservoir, process, and economics modeling.

Scalable robust solvers for unstructured FE geodynamic modeling applications: Solving the Stokes equation for models with large localized viscosity contrasts

The development of scalable robust solvers for unstructured finite element applications related to viscous flow problems in earth sciences is an active research area. Solving high‐resolution convection problems with order of magnitude 108 degrees of freedom requires solvers that scale well, with respect to both the number of degrees of freedom as well as having optimal parallel scaling characteristics on computer clusters. We investigate the use of a smoothed aggregation (SA) algebraic multigrid (AMG)‐type solution strategy to construct efficient preconditioners for the Stokes equation. We integrate AMG in our solver scheme as a preconditioner to the conjugate gradient method (CG) used during the construction of a block triangular preconditioner (BTR) to the Stokes equation, accelerating the convergence rate of the generalized conjugate residual method (GCR). We abbreviate this procedure as BTA‐GCR. For our experiments, we use unstructured grids with quadratic finite elements, making the model flexible with respect to geometry and topology and O(h3) accurate. We find that AMG‐type methods scale linearly (O(n)), with respect to the number of degrees of freedom, n. Although not all parts of AMG have preferred parallel scaling characteristics, we show that it is possible to tune AMG, resulting in parallel scaling characteristics that we consider optimal, for our experiments with up to 100 million degrees of freedom. Furthermore, AMG‐type methods are shown to be robust methods, allowing us to solve very ill‐conditioned systems resulting from strongly varying material properties over short distances in the model interior.

A coupled finite volume method for the computational modelling of mould filling in very complex geometries

Simulation of shape casting processes involves the simultaneous capture of the rapidly developing metal–air free surface and the liquid–solid moving boundary in the metal phase. Although a wide range of techniques have been developed to address this problem, there are occasions where existing techniques are limited, especially in very complex geometries where the mesh quality may inevitably be relatively poor at some key locations. Cell-centred finite volume methods, typical of most commercial CFD tools, are computationally efficient, but can lead to convergence problems on meshes that feature cells with highly non-orthogonal shapes. The control volume–finite element method uses a vertex-based approach and handles distorted meshes with relative ease, but is computationally expensive. A combined vertex-based–cell-centre technique, described in this paper, solves the flow field at the cell or element vertex, and all other variables are solved at the cell centre (as in conventional CFD tools). This flow solver strategy is much more tolerant of meshes with poor quality elements, allowing solutions on distorted meshes where purely cell-centred solutions procedures fail. The combined method enables the solution of simultaneous free surface flows coupled with solidification in very complex meshes, thus enabling the capture of such behavior in arbitrarily complex geometrical structures. It is relatively straightforward to embed within generic CC based CFD tools allowing it to be employed in a wide variety of processing applications.

High performance computing for the application of molecular theories to biological systems

In this paper we summarize our work to bring modern numerical methods to bear in computations for density functional theories (DFTs) of inhomogeneous fluid systems. We present the general mathematical structure of the problem, and briefly discuss different strategies for solving the problems. Finally, we present a few recent results from calculations on complex peptide assemblies in lipid bilayers to demonstrate the application of the methods in one complex 3-dimensional system. We find that while solver strategies developed and optimized for partial differential equations (PDEs) can be applied to these systems of equations, they do not provide optimal solutions with respect to speed, memory use, or parallel partitioning.

Inexact inverse iteration for symmetric matrices

In this paper we analyse inexact inverse iteration for the real symmetric eigenvalue problem Av = λ v. Our analysis is designed to apply to the case when A is large and sparse and where iterative methods are used to solve the shifted linear systems ( A − σ I) y = x which arise. We present a convergence theory that is independent of the nature of the inexact solver used, and, though the use of the Rayleigh quotient is emphasised, our analysis also extends to quite general choices for shift and inexact solver strategies. Additionally, the convergence framework allows us to treat both standard preconditioning and to present a new analysis of the variation introduced by Simoncini and Eldén (BIT, vol. 42, pp.159–182, 2002). Also, we provide an analysis of the performance of inner iteration solves when preconditioned MINRES is used as the inexact solver. This analysis provides descriptive bounds which are shown to predict well the actual behaviour observed in practice. Also, it explains the improvement in performance of the modification introduced by Simoncini and Eldén over the standard preconditioned form. Importantly, our analysis shows that letting the shift tend to the eigenvalue, as is the case if the Rayleigh quotient is used, does not harm significantly the performance of the iterative method for the shifted systems. Throughout the paper numerical results are given to illustrate the theory.

An implicit compact scheme solver for two-dimensional multicomponent flows

A 2D implicit compact scheme solver has been implemented for the vorticity–velocity formulation in the case of nonreacting, multicomponent, axisymmetric, low Mach number flows. To stabilize the discrete boundary value problem, two sets of boundary closures are introduced to couple the velocity and vorticity fields. A Newton solver is used for solving steady-state and time-dependent equations. In this solver, the Jacobian matrix is formulated and stored in component form. To solve the system of linearized equations within each iteration of Newton’s method, preconditioned Bi-CGSTAB is used in combination with a matrix–vector product computed in component form. The almost dense Jacobian matrix is approximated by a partial Jacobian. For the preconditioner equation, the partial Jacobian is approximately factored using several methods. In a detailed study of several preconditioning techniques, a promising method based on ILUT preconditioning in combination with reordering and double scaling using the MC64 algorithm by Duff and Koster is selected. To validate the implicit compact scheme solver, several nonreacting model problems have been considered. At least third order accuracy in space is recovered on nonuniform grids. A comparison of the results of the implicit compact scheme solver with the results of a traditional implicit low order solver shows an order of magnitude reduction of computer memory and time using the compact scheme solver in the case of time-dependent mixing problems.

A fully implicit, nonlinear adaptive grid strategy

We propose an efficient, fully implicit, nonlinear solver for the Beltrami grid generation equation. The Beltrami equation is obtained using Harmonic map theory, and therefore the existence and uniqueness of a solution is guaranteed. The nonlinear solver strategy is based on Newton–Krylov methods, preconditioned here with a multigrid-based method for scalability. Numerical experiments performed for both grid adaptation and grid alignment are presented and demonstrate optimal scaling under grid refinement. We therefore conclude that such a fully nonlinear approach is indeed feasible and efficient.

An implicit compact scheme solver with application to chemically reacting flows

A novel, stable, implicit compact scheme solver that is higher order in space, suitable for modeling steady-state and time-dependent phenomena on nonuniform grids for one-dimensional configurations, is presented. Several properties of compact scheme discretizations are introduced to develop efficient algorithms for Jacobian matrix generation and Jacobian-vector multiplication using a new component form for Jacobian operations. Composite nonuniform grids are introduced that enable the implicit compact scheme solver to achieve sixth order accuracy. A robust Newton’s method is employed with explicit generation of Jacobian matrices. Superior resolution characteristics of the implicit compact scheme solver are demonstrated with several steady-state and time-dependent problems for the Burgers equation. The example of the solution of stiff flame problem is given. An analysis of spectral properties of Jacobian matrices is presented, which shows that the condition number and the eigenvalue distributions behave similarly to those found in Jacobians associated with low-order discretizations. Two sparsification strategies are developed for the systematic approximation of a dense Jacobian aimed at the practical implementation of linear system preconditioning through partial Jacobians.

Efficient preconditioning of the linearized Navier–Stokes equations for incompressible flow

We outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of Navier–Stokes equations. We describe a very general strategy for preconditioning that has two basic building blocks; a multigrid V-cycle for the scalar convection–diffusion operator, and a multigrid V-cycle for a pressure Poisson operator. We present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid, robust with respect to the time-step, and only deteriorates very slowly as the Reynolds number is increased.

A PARALLEL THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES SOLVER WITH A PARALLEL MULTIGRID KERNEL

The development and applications of a parallel, time-dependent, three-dimensional incompressible Navier-Stokes flow solver and a parallel multigrid elliptic kernel are described. The flow solver is based on a second-order projection method applied to a staggered finite-difference grid. The multigrid algorithms implemented in the parallel elliptic kernel, which is used by the flow solver, are V-cycle and full V-cycle schemes. A grid-partition strategy is used in the parallel implementations of both the flow solver and the multigrid kernel on all fine and coarse grids. Numerical experiments and parallel performance measurements show the parallel solver package is numerically stable, physically robust and computationally efficient. Both the multigrid kernel and the flow solver scale well to a large number of processors on Intel Paragon and Cray T3D/T3E for two-and three-dimensional problems with moderate granularity. The solver package has been carefully designed and implemented so that it can be easily adapted to solve a variety of interesting scientific and engineering flow problems. The code is portable to parallel computers that support MPI, PVM and NX for interprocessor communications.

Approximate Riemann Solvers in FVM for 2D Hydraulic Shock Wave Modeling

This paper presents three approximate Riemann solver schemes, namely: the flux vector splitting (FVS), the flux difference splitting (FDS), and the Osher scheme. Originally used to solve the Euler equations in aerodynamic problems, these Riemann solvers based on the characteristic theory are used in the finite volume method (FVM) for solving the two-dimensional shallow water equations. The three solvers are compared in this paper according to theoretical development, difference schemes, practical applications to shock wave problems, and sensitivity analysis on the computational stability of the methods. The effects of changes in bed elevations on the solutions are also investigated. Comparison of numerical and analytical solutions indicates that very good agreement can be obtained by all three approximate Riemann solvers. Differences in accuracy, computer time, and numerical stability among the three schemes are not significant. For practical purposes, all of them can satisfactorily simulate the hydraulic phenomena in subcritical and supercritical flows as well as in smooth and discontinuous flows, especially shock wave modeling. These solvers are useful for studying levee failure or dam break due to extreme flood events, or the sudden opening or closing of sluice gates in a channel.

A Parallel Multigrid Preconditioned Conjugate Gradient Algorithm for Groundwater Flow Simulations

The numerical simulation of groundwater flow through heterogeneous porous media is discussed. The focus is on the performance of a parallel multigrid preconditioner for accelerating convergence of conjugate gradients, which is used to compute the pressure head. The numerical investigation considers the effects of boundary conditions, coarse grid solver strategy, increasing the grid resolution, enlarging the domain, and varying the geostatistical parameters used to define the subsurface realization. Scalability is also examined. The results were obtained using the PARFLOW groundwater flow simulator on the CRAY T3D massively parallel computer.

Implementation of a fully balanced periodic tridiagonal solver on a parallel distributed memory architecture

AbstractWhile parallel computers offer significant computational performance, it is generally necessary to evaluate several programming strategies. Two programming strategies for a fairly common problem—a periodic tridiagonal solver—are developed and evaluated. Simple model calculations as well as timing results are presented to evaluate these strategies.The particular tridiagonal solver evaluated is used in many computational fluid dynamic simulation codes. The feature that makes this algorithm unique is that these simulation codes usually require simultaneous solution for multiple right‐hand‐sides (RHS) of the system of equations. Each RHS solutions is independent and thus can be computed in parallel. Thus, a Gaussian‐elimination‐type algorithm can be used in a parallel computation and more complicated approaches such as cyclic reduction are not required.The two strategies are a transpose strategy and a distributed solver strategy. For the transpose strategy, the data are moved so that a subset of all the RHS problems is solved on each of the several processors. This usually requires significant data movement between processor memories across a network. The second strategy attempts to have the algorithm follow the data across processor boundaries in a chained manner. This usually requires significantly less data movement. An approach to accomplish this second strategy in a near‐perfect load‐balanced manner is developed. In addition, an algorithm will be shown to directly transform a sequential Gaussian‐elimination‐type algorithm into the parallel, chained, load‐balanced algorithm.

A study on vibration analysis of symmetrically laminated composite shells. Eigenvalue analysis by use of flat triangular shell elements.

A method for FEM eigenvalue problems of symmetrically laminated composite shells is developed. A 15 d. o. f. flat triangular shell element is formulated considering the anisotropic behavior of composite materials with the assumption that the transverse normal strain is linear in symmetrical lamination. The present method enables us to analyze large scale problems by only a desktop microcomputer, using a subspace iteration algorithm and a skyline solver scheme. In order to estimate the accuracy of the program, eigenvalue problems of Donnel-type circular cylinders are calculated and compared with exact solutions.