Parallel numerical simulation for the coupled problem of continuous flow electrophoresis

The performance of parallel subdomain method with overlapping is analysed in the case of the 3D coupled boundary‐value problem of continuous flow electrophoresis which is governed by Navier–Stokes equations coupled with convection–diffusion and potential equations. Convergence of parallel synchronous and asynchronous iterative algorithms is studied. Comparison between implemented explicit and implicit schemes for the transport equation is made using these algorithms and shows that both methods provide similar results and comparable performances. Copyright © 2007 John Wiley & Sons, Ltd.     

Segregated finite element algorithms for the numerical solution of large-scale incompressible flow problems

This paper presents results of an ongoing research program directed towards developing fast and efficient finite element solution algorithms for the simulation of large-scale flow problems. Two main steps were taken towards achieving this goal. The first step was to employ segregated solution schemes as opposed to the fully coupled solution approach traditionally used in many finite element solution algorithms. The second step was to replace the direct Gaussian elimination linear equation solvers used in the first step with iterative solvers of the conjugate gradient and conjugate residual type. The three segregated solution algorithms developed in step one are first presented and their integrity and relative performance demonstrated by way of a few examples. Next, the four types of iterative solvers (i.e. two options for solving the symmetric pressure type equations and two options for solving the non-symmetric advection–diffusion type equations resulting from the segregated algorithms) together with the two preconditioning strategies employed in our study are presented. Finally, using examples of practical relevance the paper documents the large gains which result in computational efficiency, over fully coupled solution algorithms, as each of the above two main steps are introduced. It is shown that these gains become increasingly more dramatic as the complexity and size of the problem is increased.     

A fully implicit scheme for simulating ionized gas flows using the gas dynamics electrodynamics coupled system

The flow of ionized gases under the influence of electromagnetic fields is governed by the coupled system of the compressible flow equations and the Maxwell equations. In this system, coupling of the flow with the electromagnetic field is obtained through nonlinear and stiff source terms, which may cause difficulties with the numerical solution of the coupled system. The discontinuous Galerkin finite element method is used for the numerical solution of this system. For the magnetic field vector, discontinuous Galerkin discretization is performed using a divergence‐free vector base for the magnetic field to preserve zero divergence in the element and retain the implicit constraint of a divergence‐free magnetic field vector down to very low level both globally and locally. To circumvent difficulties resulting from the presence of the stiff source terms, implicit time marching is used for the fully coupled system to avoid wrong wave shapes and propagation speeds that are obtained when the coupling source terms are lagged in time or by using splitting iterative schemes. Numerical solutions for benchmark problems computed on collocated meshes for the flow and electromagnetic field variables with this fully coupled monolithic approach showed good agreement with other numerical solutions and exact results. Copyright © 2014 John Wiley & Sons, Ltd.     

Hybrid Parallel Linear System Solvers

This paper presents a new approach to the solution of nonsymmetric linear systems that uses hybrid techniques based on both direct and iterative methods. An implicitly preconditioned modified system is obtained by applying projections onto block rows of the original system. Our technique provides the flexibility of using either direct or iterative methods for the solution of the preconditioned system. The resulting algorithms are robust, and can be implemented with high efficiency on a variety of parallel architectures. The algorithms are used to solve linear systems arising from the discretization of convection-diffusion equations as well as those systems that arise from the simulation of particulate flows. Experiments are presented to illustrate the robustness and parallel efficiency of these methods.     

Large deflections of rectangular membranes under uniform pressure

The Föppl large deflection equations for laterally loaded membranes are solved for uniform load and for edges which are fixed normal to the edge but are free to move parallel to the edge. Both the deflection function and the Airy stress function are expanded in Fourier series. The resulting coupled non-linear cubic equations for the deflection function coefficients are truncated and solved by means of an iterative procedure. Results for the center normal deflection and the stress resultants at selected points are calculated with the use of 100 or more equations and are found to differ significantly from the previously accepted approximate results.     

A general numerical method for the solution of gravity wave problems. Part 1: 2D steep gravity waves in shallow water

In this paper, 2D steep gravity waves in shallow water are used to introduce and examine a new kind of numerical method for the solution of non-linear problems called the finite process method (FPM). On the basis of the velocity potential function and the FPM, a numerical method for 2D non-linear gravity waves in shallow water is described which can be applied to solve 3D problems, e.g. the wave resistance of a ship moving in deep or shallow water. The convergence is examined and a comparison with the results of other authors is made. The FPM can successfully avoid the use of iterative methods and therefore can overcome the disadvantages and limitations of such methods. In contrast to iterative methods, the FPM is insensitive to the selection of the initial solution and the number of unknowns. The basic idea of the FPM can be used to solve other non-linear problems. Its disadvantage is that much more CPU time is needed to obtain a sufficiently accurate result.     

A discussion and comparison of numerical techniques used to solve the Navier–Stokes and Euler equations

This paper is intended to provide some background to a number of widely used methods for solving the Navier–Stokes and Euler equations. The difference between coupled and uncoupled iterative schemes is discussed together with methods for solving the equations. Methods covered include time marching (both explicit and implicit), pressure correction and a Newton–Raphson technique. The relationship between the methods is illustrated.     

Analytical formulation of the Jacobian matrix for non-linear calculation of the forced response of turbine blade assemblies with wedge friction dampers

A fundamental issue in turbomachinery design is the dynamical stress assessment of turbine blades. In order to reduce stress peaks in the turbine blades at engine orders corresponding to blade natural frequencies, friction dampers are employed. Blade response calculation requires the solution of a set of non-linear equations originated by the introduction of friction damping.

Such a set of non-linear equations is solved using the iterative numerical Newton–Raphson method. However, calculation of the Jacobian matrix of the system using classical numerical finite difference schemes makes frequency domain solver prohibitively expensive for structures with many contact points. Large computation time results from the evaluation of partial derivatives of the non-linear equations with respect to the displacements.

In this work a methodology to compute efficiently the Jacobian matrix of a dynamic system having wedge dampers is presented. It is exact and completely analytical.

The proposed methods have been successfully applied to a real intermediate pressure turbine (IPT) blade under cyclic symmetry boundary conditions with underplatform wedge dampers. Its implementation showed to be very effective, and allowed to achieve relevant time savings without loss of precision.     

Representation of Normal Cone Inclusion Problems in Dynamics Via Non-linear Equations

Analyzing non-smooth mechanical systems requires often the solution of inclusion problems of normal cone type. These problems arise for example in the event-driven or time-stepping simulation approaches. Such inclusion problems can be written as non-linear equations, which can be solved iteratively. In this paper we discuss three different methods to derive the non-linear equations representing the inclusions arising in the event-driven simulation approach. First, we formulate inclusions describing the individual non-smooth constraints and solve them successively. Secondly, we interpret the non-linear equations as the conditions for the saddle point of the augmented Lagrangian function. As a third possibility we discuss the exact regularization of set-valued force laws. All three methods lead to the same numerical scheme, but give different insight into the problem. Especially the factor r occurring in the non-linear equations is discussed. Two iterative methods for solving the non-linear equations are presented together with some remarks on convergence.     

Convergence acceleration of continuous adjoint solvers using a reduced‐order model

A novel acceleration technique using a reduced‐order model is presented to speed up convergence of continuous adjoint solvers. The acceleration is achieved by projecting to an improved solution within an iterative process solely using early solution results. This is achieved by forming basis vectors from early iteration adjoint solutions to perform model order reduction of the adjoint equations. The reduced‐order model of the adjoint equations is then substituted into the full‐order discretized governing equations to determine weighting coefficients for each basis vector. With these coefficients, a linear combination of the basis vectors is used to project to an improved solution. The method is applied to 3 inviscid quasi‐1D nozzle flow cases including fully subsonic flow, subsonic inlet to supersonic outlet flow, and transonic flow with a shock. Significant cost reductions are achieved for a single application as well as repeated applications of the convergence acceleration technique.     

A novel statistical linearization solution for randomly excited coupled bending-torsional beams resting on non-linear supports

A novel statistical linearization technique is developed for computing stationary response statistics of randomly excited coupled bending-torsional beams resting on non-linear elastic supports. The key point of the proposed technique consists in representing the non-linear coupled response in terms of constrained linear modes. The resulting set of non-linear equations governing the modal amplitudes is then replaced by an equivalent linear one via a classical statistical error minimization procedure, which provides algebraic non-linear equations for the second-order statistics of the beam response, readily solved by a simple iterative scheme. Data from Monte Carlo simulations, generated by a pertinent boundary integral method in conjunction with a Newmark numerical integration scheme, are used as benchmark solutions to check accuracy and reliability of the proposed statistical linearization technique.

     

Non-linear dynamic phenomena in the behaviour of a railway wheelset model

A dicone moving on a pair of cylindrical rails can be considered as a simplified model of a railway wheelset. Taking into account the non-linear friction laws of rolling contact, the equations of motion for this non-linear mechanical system result in a set of differential-algebraic equations. Previous simulations performed with the differential-algebraic solver DASSL, [2], and experiments, [7], indicated non-linear phenomena such as limit-cycles, bifurcations as well as chaotic behaviour. In this paper the non-linear phenomena are investigated in more detail with the aid of special in-house software and the path-following algorithm PATH [10]. We apply Poincaré sections and Poincaré maps to describe the structure of periodic, quasiperiodic and chaotic motions. The analyses show that part of the chaotic behaviour of the non-linear system can be fully understood as a non-linear iterative process. The resulting stretching and folding processes are illustrated by series of Poincaré sections.     

A 3D incompressible Navier–Stokes velocity–vorticity weak form finite element algorithm

The velocity–vorticity formulation is selected to develop a time‐accurate CFD finite element algorithm for the incompressible Navier–Stokes equations in three dimensions.The finite element implementation uses equal order trilinear finite elements on a non‐staggered hexahedral mesh. A second order vorticity kinematic boundary condition is derived for the no slip wall boundary condition which also enforces the incompressibility constraint. A biconjugate gradient stabilized (BiCGSTAB) sparse iterative solver is utilized to solve the fully coupled system of equations as a Newton algorithm. The solver yields an efficient parallel solution algorithm on distributed‐memory machines, such as the IBM SP2. Three dimensional laminar flow solutions for a square channel, a lid‐driven cavity, and a thermal cavity are established and compared with available benchmark solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

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 semi-analytical approach for the non-linear large deflection analysis of laminated rectangular plates under general out-of-plane loading

A semi-analytical approach for the geometrically non-linear analysis of rectangular laminated plates with general inplane and out-of-plane boundary conditions under a general distribution of out-of-plane loads is developed. The analysis is based on the elastic thin plate theory with geometrically non-linear von Kármán strains. The solution of the non-linear partial differential equations is reduced to an iterative sequential solution of non-linear ordinary differential equations using the multi-term extended Kantorovich method. The efficiency, accuracy, and convergence of the proposed method are examined through a comparison with other semi-analytical methods and with finite element analyses. The capabilities of the approach and its applicability to the non-linear large deflection analysis of plate structures are demonstrated through various numerical examples. Emphasis is placed on combinations of lamination, boundary, and loading conditions that cannot be analyzed using alternative semi-analytical methods.     

Numerical simulation of fully non-linear steady free surface flow

The fully non-linear free surface potential flow past a 2D non-lifting body is computed. The numerical method is based on the simple layer integral formulation; the non-linear solution is obtained by means of an iterative procedure. Under some hypotheses, viscosity effects at the free surface are considered. All the numerical results obtained have been tested against analytical solutions and experimental results.     

A new fully coupled solution of the Navier-Stokes equations

A fully coupled method for the solution of incompressible Navier-Stokes equations is investigated here. It uses a fully implicit time discretization of momentum equations, the standard linearization of convective terms, a cell-centred colocated grid approach and a block-nanodiagonal structure of the matrix of nodal unknowns. The Method is specific in the interpolation used for the flux reconstruction problem, in the basis iterative method for the fully coupled system and in the acceleration means that control the global efficiency of the procedure. The performance of the method is discussed using lid-driven cavity problems, both for two and three-dimensional geometries, for steady and unsteady flows.     

Dynamic response of various von-Kármán non-linear plate models and their 3-D counterparts

Dynamic von-Kármán plate models consist of three coupled non-linear, time-dependent partial differential equations. These equations have been recently solved numerically [Kirby, R., Yosibash, Z., 2004. Solution of von-Kármán dynamic non-linear plate equations using a pseudo-spectral method. Comp. Meth. Appl. Mech. Eng. 193 (6–8) 575–599 and Yosibash, Z., Kirby, R., Gottlieb, D., 2004. Pseudo-spectral methods for the solution of the von-Kármán dynamic non-linear plate system. J. Comp. Phys. 200, 432–461] by the Legendre-collocation method in space and the implicit Newmark-β scheme in time, where highly accurate approximations were realized.Due to their complexity, these equations are often reduced by discarding some of the terms associated with time derivatives which are multiplied by the plate thickness squared (being a small parameter). Because of the non-linearities in the system of equations we herein quantitatively investigate the influence of these a-priori assumption on the solution for different plate thicknesses. As shown, the dynamic solutions of the so called “simplified von-Kármán” system do not differ much from the complete von-Kármán system for thin plates, but may have differences of few percent for plates with thicknesses to length ratio of about 1/20. Nevertheless, when investigating the modeling errors, i.e. the difference between the various von-Kármán models and the fully three-dimensional non-linear elastic plate solution, one realizes that for relatively thin plates (thickness is 1/20 of other typical dimensions), this difference is much larger. This implies that the simplified von-Kármán plate model used frequently in the literature is as good as an approximation as the complete (and more complicated) model. As a side note, it is shown that the dynamic response of any of the von-Kármán plate models, is completely different compared to the linearized plate model of Kirchhoff–Love for deflections of an order of magnitude as the plate thickness.     

An efficient parallel algebraic multigrid method for 3D injection moulding simulation based on finite volume method

Elapsed time is always one of the most important performance measures for polymer injection moulding simulation. Solving pressure correction equations is the most time-consuming part in the mould filling simulation using finite volume method with SIMPLE-like algorithms. Algebraic multigrid (AMG) is one of the most promising methods for this type of elliptic equations. It, thus, has better performance by contrast with some common one-level iterative methods, especially for large problems. And it is also suitable for parallel computing. However, AMG is not easy to be applied due to its complex theory and poor generality for the large range of computational fluid dynamics applications. This paper gives a robust and efficient parallel AMG solver, A1-pAMG, for 3D mould filling simulation of injection moulding. Numerical experiments demonstrate that, A1-pAMG has better parallel performance than the classical AMG, and also has algorithmic scalability in the context of 3D unstructured problems.