Nodal operator splitting adaptive finite element algorithms for the Navier–Stokes equations

Solution algorithms for solving the Navier–Stokes equations without storing equation matrices are developed. The algorithms operate on a nodal basis, where the finite element information is stored as the co-ordinates of the nodes and the nodes in each element. Temporary storage is needed, such as the search vectors, correction vectors and right hand side vectors in the conjugate gradient algorithms which are limited to one-dimensional vectors. The nodal solution algorithms consist of splitting the Navier–Stokes equations into equation systems which are solved sequencially. In the pressure split algorithm, the velocities are found from the diffusion–convection equation and the pressure is computed from these velocities. The computed velocities are then corrected with the pressure gradient. In the velocity–pressure split algorithm, a velocity approximation is first found from the diffusion equation. This velocity is corrected by solving the convection equation. The pressure is then found from these velocities. Finally, the velocities are corrected by the pressure gradient. The nodal algorithms are compared by solving the original Navier–Stokes equations. The pressure split and velocity–pressure split equation systems are solved using ILU preconditioned conjugate gradient methods where the equation matrices are stored, and by using diagonal preconditioned conjugate gradient methods without storing the equation matrices. © 1998 John Wiley & Sons, Ltd.     

A parallel finite element sliding mesh technique for the simulation of viscous flows in agitated tanks

A parallel sliding mesh algorithm for the finite element simulation of viscous fluid flows in agitated tanks is presented. Lagrange multipliers are used at the sliding interfaces to enforce the continuity between the fixed and moving subdomains. The novelty of the method consists of the coupled solution of the resulting velocity–pressure‐Lagrange multipliers system of equations by an ILU(0)‐QMR solver. A penalty parameter is introduced for both the interface and the incompressibility constraints to avoid pivoting problems in the ILU(0) algorithm. To handle the convective term, both the Newton–Raphson scheme and the semi‐implicit linearization are tested. A penalty parameter is introduced for both the interface and the incompressibility constraints to avoid the failure of the ILU(0) algorithm due to the lack of pivoting. Furthermore, this approach is versatile enough so that it allows partitioning of sliding and fixed subdomains if parallelization is required. Although the sliding mesh technique is fairly common in CFD, the main advantage of the proposed approach is its low computational cost due to the inexpensive and parallelizable calculations that involve preconditioned sparse iterative solvers. The method is validated for Couette and coaxial stirred tanks. Copyright © 2011 John Wiley & Sons, Ltd.     

Reordering and incomplete preconditioning in serial and parallel adaptive mesh refinement and coarsening flow solutions

The effects of reordering the unknowns on the convergence of incomplete factorization preconditioned Krylov subspace methods are investigated. Of particular interest is the resulting preconditioned iterative solver behavior when adaptive mesh refinement and coarsening (AMR/C) are utilized for serial or distributed parallel simulations. As representative schemes, we consider the familiar reverse Cuthill–McKee and quotient minimum degree algorithms applied with incomplete factorization preconditioners to CG and GMRES solvers. In the parallel distributed case, reordering is applied to local subdomains for block ILU preconditioning, and subdomains are repartitioned dynamically as mesh adaptation proceeds. Numerical studies for representative applications are conducted using the object‐oriented AMR/C software system libMesh linked to the PETSc solver library. Serial tests demonstrate that global unknown reordering and incomplete factorization preconditioning can reduce the number of iterations and improve serial CPU time in AMR/C computations. Parallel experiments indicate that local reordering for subdomain block preconditioning associated with dynamic repartitioning because of AMR/C leads to an overall reduction in processing time. Copyright © 2011 John Wiley & Sons, Ltd.     

A preconditioned alternating inner-outer iterative solution method for the mixed finite element formulation of the Navier-Stokes equations

In the present work a new iterative method for solving the Navier-Stokes equations is designed. In a previous paper a coupled node fill-in preconditioner for iterative solution of the Navier-Stokes equations proved to increase the convergence rate considerably compared with traditional preconditioners. The further development of the present iterative method is based on the same storage scheme for the equation matrix as for the coupled node fill-in preconditioner. This storage scheme separates the velocity, the pressure and the coupling of pressure and velocity coefficients in the equation matrix. The separation storage scheme allows for an ILU factorization of both the velocity and pressure unknowns. With the inner-outer solution scheme the velocity unknowns are eliminated before the resulting equation system for the pressures is solved iteratively. After the pressure unknown has been found, the pressures are substituted into the original equation system and the velocities are also found iteratively. The behaviour of the inner-outer iterative solution algorithm is investigated in order to find optimal convergence criteria for the inner iterations and compared with the solution algorithm for the original equation system. The results show that the coupled node fill-in preconditioner of the original equation system is more efficient than the coupled node fill-in preconditioner of the reduced equation system. However, the solution technique of the reduced equation system revals properties which may be advantageous in future solution algorithms.     

基于AFT-Delaunay的二维解耦并行网格生成算法

面向平面任意几何区域网格生成,提出了一种将波前法AFT(Advancing Front Technique)与Delaunay法相结合的解耦并行网格生成算法。算法主要思想是沿着求解几何区域惯性轴,采用扩展的AFT-Delaunay算法生成高质量三角形网格墙,递归地将几何区域动态划分成多个彼此解耦的子区域;采用OpenMP多线程并行技术,将子区域分配给多个CPU并行生成子区域网格;子区域内部的网格生成复用AFT-Delaunay算法,保证了生成网格的质量、效率和一致性要求。本算法优先生成几何边界与交界面网格,有利于提高有限元计算精度;各个子区域的网格生成彼此完全解耦,因此并行网格生成过程无需通信。该方法克服了并行交界面网格质量恶化难题,且具有良好的并行加速比,能够全自动、高效率地并行生成高质量的三角网格。     

A NON-LINEAR ADAPTIVE FULL TRI-TREE MULTIGRID METHOD FOR THE MIXED FINITE ELEMENT FORMULATION OF THE NAVIER–STOKES EQUATIONS

The full adaptive multigrid method is based on the tri-tree grid generator. The solution of the Navier–Stokes equations is first found for a low Reynolds number. The velocity boundary conditions are then increased and the grid is adapted to the scaled solution. The scaled solution is then used as a start vector for the multigrid iterations. During the multigrid iterations the grid is first recoarsed a specified number of grid levels. The solution of the Navier–Stokes equations with the multigrid residual as right-hand side is smoothed in a fixed number of Newton iterations. The linear equation system in the Newton algorithm is solved iteratively by CGSTAB preconditioned by ILU factorization with coupled node fill-in. The full adaptive multigrid algorithm is demonstr ated for cavity flow. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 1037ndash;1047, 1997.     

A comparative study of pressure correction and block-implicit finite volume algorithms on parallel computers

A comparison of a new parallel block-implicit method and the parallel pressure correction procedure for the solution of the incompressible Navier–Stokes equations is presented. The block-implicit algorithm is based on a pressure equation. The system of non-linear equation s is solved by Newton's method. For the solution of the linear algebraic systems the Bi-CGSTAB algorithm with incomplete lower–upper (ILU) decomposition of the matrix is applied. Domain decomposition serves as a strategy for the parallelization of the algorithms. Different algorithms for the parallel solution of the linear system of algebraic equations in conjunction with the pressure correction procedure are proposed. Three different flows are predicted with the parallel algorithms. Results and efficiency data of the block-implicit method are compared with the parallel version of the pressure correction algorithm. The block-implicit method is characterized by stable convergence behaviour, high numerical efficiency, insensitivity to relaxation parameters and high spatial accuracy. © 1997 John Wiley & Sons, Ltd.     

A control volume finite‐element method for numerical simulating incompressible fluid flows without pressure correction

This paper presents a numerical model to study the laminar flows induced in confined spaces by natural convection. A control volume finite‐element method (CVFEM) with equal‐order meshing is employed to discretize the governing equations in the pressure–velocity formulation. In the proposed model, unknown variables are calculated in the same grid system using different specific interpolation functions without pressure correction. To manage memory storage requirements, a data storage format is developed for generated sparse banded matrices. The performance of various Krylov techniques, including Bi‐CGSTAB (Bi‐Conjugate Gradient STABilized) with an incomplete LU (ILU) factorization preconditioner is verified by applying it to three well‐known test problems. The results are compared to those of independent numerical or theoretical solutions in literature. The iterative computer procedure is improved by using a coupled strategy, which consists of solving simultaneously the momentum and the continuity equation transformed in a pressure equation. Results show that the strategy provides useful benefits with respect to both reduction of storage requirements and central processing unit runtime. Copyright © 2006 John Wiley & Sons, Ltd.     

A composite multigrid method for calculating unsteady incompressible flows in geometrically complex domains

A time-accurate, finite volume method for solving the three-dimensional, incompressible Navier-Stokes equations on a composite grid with arbitrary subgrid overlapping is presented. The governing equations are written in a non-orthogonal curvilinear co-ordinate system and are discretized on a non-staggered grid. A semi-implicit, fractional step method with approximate factorization is employed for time advancement. Multigrid combined with intergrid iteration is used to solve the pressure Poisson equation. Inter-grid communication is facilitated by an iterative boundary velocity scheme which ensures that the governing equations are well-posed on each subdomain. Mass conservation on each subdomain is preserved by using a mass imbalance correction scheme which is secondorder-accurate. Three test cases are used to demonstrate the method's consistency, accuracy and efficiency.     

A segregated implicit solution algorithm for 2D and 3D laminar incompressible flows

A segregated algorithm for the solution of laminar incompressible, two- and three-dimensional flow problems is presented. This algorithm employs the successive solution of the momentum and continuity equations by means of a decoupled implicit solution method. The inversion of the coefficient matrix which is common for all momentum equations is carried out through an approximate factorization in upper and lower triangular matrices. The divergence-free velocity constraint is satisfied by formulating and solving a pressure correction equation. For the latter a combined application of a preconditioning technique and a Krylov subspace method is employed and proved more effecient than the approximate factorization method. The method exhibits a monotonic convergence, it is not costly in CPU time per iteration and provides accurate solutions which are independent of the underrelaxation parameter used in the momentum equations. Results are presented in two- and three-dimensional flow problems.     

Parallelization of the ILU(0) preconditioner for CFD problems on shared‐memory computers

The use of ILU(0) factorization as a preconditioner is quite frequent when solving linear systems of CFD computations. This is because of its efficiency and moderate memory requirements. For a small number of processors, this preconditioner, parallelized through coloring methods, shows little savings when compared with a sequential one using adequate reordering of the unknowns. Level scheduling techniques are applied to obtain the same preconditioning efficiency as in a sequential case, while taking advantage of parallelism through block algorithms. Numerical results obtained from the parallel solution of the compressible Navier–Stokes equations show that this technique gives interesting savings in computational times on a small number of processors of shared‐memory computers. In addition, it does this while keeping all the benefits of an ILU(0) factorization with an adequate reordering of the unknowns, and without the loss of efficiency of factorization associated with a more scalable coloring strategy. Copyright © 1999 John Wiley & Sons, Ltd.     

Fully coupled finite volume solutions of the incompressible Navier–Stokes and energy equations using an inexact Newton method

An inexact Newton method is used to solve the steady, incompressible Navier–Stokes and energy equation. Finite volume differencing is employed on a staggered grid using the power law scheme of Patankar. Natural convection in an enclosed cavity is studied as the model problem. Two conjugate-gradient -like algorithms based upon the Lanczos biorthogonalization procedure are used to solve the linear systems arising on each Newton iteration. The first conjugate-gradient-like algorithm is the transpose-free quasi-minimal residual algorithm (TFQMR) and the second is the conjugate gradients squared algorithm (CGS). Incomplete lower-upper (ILU) factorization of the Jacobian matrix is used as a right preconditioner. The performance of the Newton- TFQMR algorithm is studied with regard to different choices for the TFQMR convergence criteria and the amount of fill-in allowed in the ILU factorization. Performance data are compared with results using the Newton-CGS algorithm and previous results using LINPACK banded Gaussian elimination (direct-Newton). The inexact Newton algorithms were found to be CPU competetive with the direct-Newton algorithm for the model problem considered. Among the inexact Newton algorithms, Newton-CGS outperformed Newton- TFQMR with regard to CPU time but was less robust because of the sometimes erratic CGS convergence behaviour.     

Fast boundary‐domain integral algorithm for the computation of incompressible fluid flow problems

The paper deals with the numerical solution of fluid dynamics using the boundary‐domain integral method (BDIM). A velocity–vorticity formulation of the Navier–Stokes equations is adopted, where the kinematic equation is written in its parabolic form. Computational aspects of the numerical simulation of two‐dimensional flows is described in detail. In order to lower the computational cost, the subdomain technique is applied. A preconditioned Krylov subspace method (PKSM) is used for the solution of systems of linear equations. Level‐based fill‐in incomplete lower upper decomposition (ILU) preconditioners are developed and their performance is examined. Scaling of stopping criteria is applied to minimize the number of iterations for the PKSM. The effectiveness of the proposed method is tested on several benchmark test problems. Copyright © 1999 John Wiley & Sons, Ltd.     

求解接触问题的一种新的实验误差法

提出了一种带松弛因子的UZAW算法求解实验误差法中给定状态下的位移和接触力满足的等式方程，并证明了该算法是R超线性收敛的。整个区域被划分为多个子区域，不同子区域位移场的求解是独立的。还提出了一种带参数的以不完全因子分解为基础的预条件子共轭梯度法求解不同子区域位移场，该算法在块体规模较大时更加有效。     

An efficient parallel and fully implicit algorithm for the simulation of transient free-surface flows of multimode viscoelastic liquids

The parallelization of a fully implicit and stable finite element algorithm with relative low memory requirements for the accurate simulation of time-dependent, free-surface flows of multimode viscoelastic liquids is presented. It is an extension of our multi-stage sequential solution procedure which is based on the mixed finite element method for the velocity and pressure fields, an elliptic grid generator for the deformation of the mesh, and the discontinuous Galerkin method for the viscoelastic stresses [Dimakopoulos and Tsamopoulos [12], [14]]. Each one of the above subproblems is solved with the Newton–Rapshon technique according to its particular characteristics, while their coupling is achieved through Picard cycles. The physical domain is graphically partitioned into overlapping subdomains. In the process, two different kinds of parallel solvers are used for the solution of the distributed set of flow and mesh equations: a multifrontal, massively parallel direct one (MUMPS) and a hierarchical iterative parallel one (HIPS), while viscoelastic stress components are independently calculated within each finite element. The parallel algorithm retains all the advantages of its sequential predecessor, related with the robustness and the numerical stability for a wide range of levels of viscoelasticity. Moreover, irrespective of the deformation of the physical domain, the mesh partitioning remains invariant throughout the simulation. The solution of the constitutive equations, which constitutes the largest portion of the system of the governing, non-linear equations, is performed in a way that does not need any data exchange among the cluster's nodes. Finally, indicative results from the simulation of an extensionally thinning polymeric solution, demonstrating the efficiency of the algorithm are presented.     

A sliding interface method for unsteady unstructured flow simulations

The objective of this work is to develop a sliding interface method for simulations involving relative grid motion that is fast and efficient and involves no grid deformation, remeshing, or hole cutting. The method is implemented into a parallel, node‐centred finite volume, unstructured viscous flow solver. The rotational motion is accomplished by rigidly rotating the subdomain representing the moving component. At the subdomain interface boundary, the faces along the interface are extruded into the adjacent subdomain to create new volume elements forming a one‐cell overlap. These new volume elements are used to compute a flux across the subdomain interface. An interface flux is computed independently for each subdomain. The values of the solution variables and other quantities for the nodes created by the extrusion process are determined by linear interpolation. The extrusion is done so that the interpolation will maintain information as localized as possible. The grid on the interface surface is arbitrary. The boundary between the two subdomains is completely independent from one another; meaning that they do not have to connect in a one‐to‐one manner and no symmetry or pattern restrictions are placed on the grid. A variety of numerical simulations were performed on model problems and large‐scale applications to examine conservation of the interface flux. Overall solution errors were found to be comparable to that for fully connected and fully conservative simulations. Excellent agreement is obtained with theoretical results and results from other solution methodologies. Copyright © 2006 John Wiley & Sons, Ltd.     

A coupled multigrid-domain-splitting technique for simulating incompressible flows in geometrically complex domains

This paper describes a domain decomposition numerical procedure for solving the Navier-Stokes equations in regions with complex geometries. The numerical method includes a modified version of QUICK (quadratic upstream interpolation convective kinematics) for the formulation of convective terms and a central difference scheme for the diffusion terms. A second-order-accurate predictor-corrector scheme is employed for the explicit time stepping. Although the momentum equations are solved independently on each subdomain, the pressure field is computed simultaneously on the entire flow field. A multigrid technique coupled with a Schwarz-like iteration method is devised to solve the pressure equation over the composite domains. The success of this strategy depends crucially on appropriate methods for specifying intergrid pressure boundary conditions on subdomains. A proper method for exchanging information among subdomains during the Schwarz sweep is equally important to the success of the multigrid solution for the overall pressure field. These methods are described and subsequently applied to two forced convection flow problems involving complex geometries to demonstrate the power and versatility of the technique. The resulting pressure and velocity fields exhibit excellent global consistency. The ability to simulate complex flow fields with this method provides a powerful tool for analysis and prediction of mixing and transport phenomenon.     

A parallel solver based on the dual Schur decomposition of general finite element matrices

A parallel solver based on domain decomposition is presented for the solution of large algebraic systems arising in the finite element discretization of mechanical problems. It is hybrid in the sense that it combines a direct factorization of the local subdomain problems with an iterative treatment of the interface system by a parallel GMRES algorithm. An important feature of the proposed solver is the use of a set of Lagrange multipliers to enforce continuity of the finite element unknowns at the interface. A projection step and a preconditioner are proposed to control the conditioning of the interface matrix. The decomposition of the finite element mesh is formulated as a graph partitioning problem. A two-step approach is used where an initial decomposition is optimized by non-deterministic heuristics to increase the quality of the decomposition. Parallel simulations of a Navier–Stokes flow problem carried out on a Convex Exemplar SPP system with 16 processors show that the use of optimized decompositions and the preconditioning step are keys to obtaining high parallel efficiencies. Typical parallel efficiencies range above 80%. © 1998 John Wiley & Sons, Ltd.     

Four-point combined DE/FE algorithm for brittle fracture analysis of laminated glass

A four-point combined DE/FE algorithm is proposed to constrain the rotation of a discrete element about its linked point and analyze the cracks propagation of laminated glass. In this approach, four linked points on a discrete element are combined with four nodes of the corresponding surface of a finite element. The penalty method is implemented to calculate the interface force between the two subdomains, the finite element (FE) and the discrete element (DE) subdomains. The sequential procedure of brittle fracture is described by an extrinsic cohesive fracture model only in the DE subdomain. An averaged stress tensor for granular media, which is automatically symmetrical and invariant by translations, is used to an accurate calculation of the averaged stress of the DE. Two simple cases in the elastic range are given to certify the effectiveness of the combined algorithm and the averaged stress tensor by comparing with the finite element method and the mesh-size dependency of the combined algorithm and the cohesive model is also investigated. Finally, the impact fracture behavior of a laminated glass beam is simulated, and the cracks propagation is compared with experimental results showing that the theory in this work can be used to predict some fracture characteristics of laminated glass.     

Dynamic recursive decoupling method for closed-loop flexible mechanical systems

In this paper, a new method for the dynamic analysis of a closed-loop flexible kinematic mechanical system is presented. The kinematic and force models are developed using absolute reference, joint relative, and elastic coordinates as well as joint reaction forces. This recursive formulation leads to a system of loosely coupled equations of motion. In a closed-loop kinematic chain, cuts are made at selected auxiliary joints in order to form spanning tree structures. Compatibility conditions and reaction force relationships at the auxiliary joints are adjoined to the equations of open-loop mechanical systems in order to form closed-loop dynamic equations. Using the sparse matrix structure of these equations and the fact that the joint reaction forces associated with elastic degrees of freedom do not represent independent variables, a method for decoupling the joint and elastic accelerations is developed. Unlike existing recursive formulations, this method does not require inverse or factorization of large non-linear matrices. It leads to small systems of equations whose dimensions are independent of the number of elastic degrees of freedom. The application of dynamic decoupling method in dynamic analysis of closed-loop deformable multibody systems is also discussed in this paper. The use of the numerical algorithm developed in this investigation is illustrated by a closed-loop flexible four-bar mechanism.