A Newton-like finite element scheme for compressible gas flows

Semi-implicit and Newton-like finite element methods are developed for the stationary compressible Euler equations. The Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable. To suppress numerical oscillations, the spatial discretization is performed by a high-resolution finite element scheme based on algebraic flux correction. A multidimensional limiter of TVD type is employed. An important goal is the efficient computation of stationary solutions in a wide range of Mach numbers, which is a challenging task due to oscillatory correction factors associated with TVD-type flux limiters. A semi-implicit scheme is derived by a time-lagged linearization of the nonlinear residual, and a Newton-like method is obtained in the limit of infinite CFL numbers. Special emphasis is laid on the numerical treatment of weakly imposed characteristic boundary conditions. Numerical evidence for unconditional stability is presented. It is shown that the proposed approach offers higher accuracy and better convergence behavior than algorithms in which the boundary conditions are implemented in a strong sense.     

Numerical simulation of viscous flows in hydraulic turbomachinery by the finite element method

Finite element simulations of 2-D blade-to-blade channel flows and of swirling flows in a conical diffuser are described. Algorithms use state-of-the-art features of the FEM and are shown to provide helpful capabilities for the design of hydraulic turbomachinery. Results of the computation include a flow pattern with laminar instabilities characteristic of the onset of turbulence. Numerical experiments are conducted up to a Reynolds number of 2000.     

A variational multiscale higher-order finite element formulation for turbomachinery flow computations

The Variational MultiScale approach for finite elements addresses the inclusion of the effect of fine scales of the solution in the coarse problem. In this framework advective–diffusive–reactive equations modelling turbomachinery flows feature both advection and reaction induced instabilities to be tackled. To this end, this work deals with a new method called V-SGS (Variable-SubGrid Scale), designed for quadratic elements, with a variable ‘intrinsic time’ parameter. Two-dimensional tests have been considered to compare V-SGS against SUPG and other stabilization devices, including the flow around a NACA 4412 airfoil to assess its reliability in the handling of advanced turbulence closures on cruder meshes.     

A mesh-refinement scheme for finite element computations

The paper considers the use of unorthodox grids where rapid transition from refined zones to coarser zones is effected, thus introducing exposed nodal freedoms at the zone interfaces. A technique for automated mesh enrichment of finite element discretizations is devised. Suitable convergence criteria are designed to delineate those subregions of a domain where refinement is necessary. Enriched and unaltered regions are separated by a fringe of semirefined elements. A valid finite element theory is provided for the fringe elements. A pilot numerical study illustrates the practical implementation of the automated refinement strategy. The principal features of the program are described.     

A parallel finite element scheme for thermo-hydro-mechanical (THM) coupled problems in porous media

Many applied problems in geoscience require knowledge about complex interactions between multiple physical and chemical processes in the sub-surface. As a direct experimental investigation is often not possible, numerical simulation is a common approach. The numerical analysis of coupled thermo-hydro-mechanical (THM) problems is computationally very expensive, and therefore the applicability of existing codes is still limited to simplified problems. In this paper we present a novel implementation of a parallel finite element method (FEM) for the numerical analysis of coupled THM problems in porous media. The computational task of the FEM is partitioned into sub-tasks by a priori domain decomposition. The sub-tasks are assigned to the CPU nodes concurrently. Parallelization is achieved by simultaneously establishing the sub-domain mesh topology, synchronously assembling linear equation systems in sub-domains and obtaining the overall solution with a sub-domain linear solver (parallel BiCGStab method with Jacobi pre-conditioner). The present parallelization method is implemented in an object-oriented way using MPI for inter-processor communication. The parallel code was successfully tested with a 2-D example from the international DECOVALEX benchmarking project. The achieved speed-up for a 3-D extension of the test example on different computers demonstrates the advantage of the present parallel scheme.     

A parallel finite element procedure for contact-impact problems

An efficient parallel finite element procedure for contact-impact problems is presented within the framework of explicit finite element analysis with thepenalty method. The procedure concerned includes a parallel Belytschko-Lin-Tsay shell element generation algorithm and a parallel contact-impact algorithm based on the master-slave slideline algorithm. An element-wise domain decomposition strategy and a communication minimization strategy are featured to achieve almost perfect load balancing among processors and to show scalability of the parallel performance. Throughout this work, a prototype code, named GT-PARADYN, is developed on the IBM SP2 to implement the procedure presented, under message-passing paradigm. Some examples are provided to demonstrate the timing results of the algorithms, discussing the accuracy and efficiency of the code.     

A parallel finite element solution method

New parallel computer architectures have revolutionized the design of computer algorithms, and promise to have significant influence on algorithms for structural engineering computations. In this paper, a parallel finite element solution method is presented. The solution method proposed does not require the formation of global system equations, but computes directly the element distortions, as opposed to solving a system of nodal equations. An element or substructure is mapped on to a processor of an MIMD multiprocessing system. Each processor stores only the information relevant to the element or substructure for which the processor represents. The finite element computations can be performed in parallel, in that a processor generates the local stiffness, computes the element distortions and determines the stress-strain characteristics for the element or substructure associated with the processor.     

A 3D finite element model for free-surface flows

The present work deals with the validation of 3D finite element model for free-surface flows. The model uses the non-hydrostatic pressure and the eddy viscosities from the conventional linear turbulence model are modified to account for the secondary effects generated by strong channel curvature in the natural rivers with meandering open channels. The unsteady Reynolds-averaged Navier–Stokes equations are solved on the unstructured grid using the Raviart–Thomas finite element for the horizontal velocity components, and the common P1 linear finite element in the vertical direction. To provide the accurate resolution at the bed and the free-surface, the governing equations are solved in the multi-layers system (the vertical plane of the domain is subdivided into fixed thickness layers). The up-to-date k–ε turbulence solver is implemented for computing eddy coefficients, the Eulerian–Lagrangian–Galerkin (ELG) temporal scheme is performed for enhancing numerical time integration to guarantee high degree of mass conservation while the CFL restriction is eliminated. The present paper reports on successful validation of the numerical model through available benchmark tests with increasing complexity, using the high quality and high spatial resolution three-dimensional data set collected from experiments.     

Reduction method for finite element dynamic analysis of submerged turbomachinery wheels

The proposed method allows the efficient computation of natural frequency mode shape and harmonic response of submerged rotationally periodic structures, such as turbine or centrifugal pump wheels. The geometry of such structures is complex and the influence of the surrounding fluid has to be taken into account. The fluid is assumed to be incompressible and at rest. Both fluid and structure domains are modelled using the finite element method. The size of the problem is appreciably reduced using the properties of wave propagation in periodic systems combined with modal techniques. The efficiency of the method is shown by an application to a submerged centrifugal pump wheel where the numerical results are compared to experimental results.     

A parallel multipolar boundary element method for internal stokes flows

A multipolar expansion technique is applied to the Boundary Element Method (direct and indirect formulation) in order to solve the two-dimensional internal Stokes Flow with first kind boundary conditions. The algorithm is based on a multipolar expansion for the far field and numerical evaluation for the inner field. Due to the nature of the algorithm, it is necessary to resort to the use of an iterative solver for the resulting algebraic linear system of equations. In comparison with the direct BEM formulation, the indirect formulation is more stable with iterative solvers, and does not need to be preconditioned to obtain a fast rate of convergence. A parallel implementation is designed to take advantage of the natural domain decomposition of fast multipolar techniques and bring further improvement. A good result in memory saving and computing time is obtained that enable us to run huge examples which are prohibitive for traditional BEM implementations.     

A space-time least-square finite element scheme for advection-diffusion equations

A space-time least-square finite element scheme is presented for the advection-diffusion problems at moderate to high Peclet numbers. This scheme is designed to eliminate spurious oscillations and can be used to define the steady-state solution as the asymptotic transient solution for large time. Numerical results, using linear elements in a 1D space and bilinear elements in a 2D space, demonstrate the accuracy and the stability of the new scheme.     

A new 3D parallel SPH scheme for free surface flows

We propose a new robust and accurate SPH scheme, able to track correctly complex three-dimensional non-hydrostatic free surface flows and, even more important, also able to compute an accurate and little oscillatory pressure field. It uses the explicit third order TVD Runge-Kutta scheme in time, following Shu and Osher [Shu C-W, Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J Comput Phys 1988;89:439-71], together with the new key idea of introducing a monotone upwind flux for the density equation, thus removing any artificial viscosity term. For the discretization of the velocity equation, the non-diffusive central flux has been used. A new flexible approach to impose the boundary conditions at solid walls is also proposed. It can handle any moving rigid body with arbitrarily irregular geometry. It does neither produce oscillations in the fluid pressure in proximity of the interfaces, nor does it have a restrictive impact on the stability condition of the explicit time stepping method, unlike the repellent boundary forces of Monaghan [Monaghan JJ. Simulating free surface flows with SPH. J Comput Phys 1994;110:399-406]. To asses the accuracy of the new SPH scheme, a 3D mesh-convergence study is performed for the strongly deforming free surface in a 3D dam-break and impact-wave test problem providing very good results.Moreover, the parallelization of the new 3D SPH scheme has been carried out using the message passing interface (MPI) standard, together with a dynamic load balancing strategy to improve the computational efficiency of the scheme. Thus, simulations involving millions of particles can be run on modern massively parallel supercomputers, obtaining a very good performance, as confirmed by a speed-up analysis. The 3D applications consist of environmental flow problems, such as dam-break flows and impact flows against a wall. The numerical solutions obtained with our new 3D SPH code have been compared with either experimental results or with other numerical reference solutions, obtaining in all cases a very satisfactory agreement.     

Mathematical aspect of the stream-function equations of compressible turbomachinery flows and their finite element approximations using optimal control

In this paper we establish the partial differential equation and properly pose conditions for the stream function on an arbitrary stream surface in a turbomachine using tensor analysis and a semi-geodesic coordinate system. We provide computational examples using the finite element method and also the error estimate of Galerkin's finite element approximation which depends on the Mach number.     

A finite element method for diffusion dominated unsteady viscous flows

A general conforming finite element scheme for computing viscous flows is presented which is of second-order accuracy in space and time. Viscous terms are treated implicitly and advection terms are treated explicitly in the time marching segment of the algorithm. A method for solving the algebraic equations at each time step is given. The method is demonstrated on two test problems, one of them being a plane vortex flow for which asymptotic methods are used to obtain suitable numerical boundary conditions at each time step.     

A control-volume finite element method for dilute gas-solid particle flows

The formulation of a co-located equal-order Control-Volume-based Finite Element Method (CVFEM) for the solution of two-fluid models of 2-D, planar or axisymmetric, incompressible, dilute gas-solid particle flows is presented. The proposed CVFEM is formulated by borrowing and extending ideas put forward in earlier CVFEMs for single-phase flows. In axisymmetric problems, the calculation domain is discretized into torus-shaped elements and control volumes: in a longitudinal cross-sectional plane, or in planar problems, these elements are three-node triangles, and the control volumes are polygons obtained by joining the centroids of the three-node triangles to the midpoints of the sides. In each element, mass-weighted skew upwind functions are used to interpolate the convected scalar dependent variables and the volume concentrations. An iterative variable adjustment algorithm is used to solve the discretized equations. The capabilities of the proposed CVFEM are illustrated by its application to two test problems and one demonstration problem, using a simple two-fluid model for dilute gas-solid particle flows. The results are quite encouraging.     

A flow-condition-based interpolation finite element procedure for incompressible fluid flows

A new finite element procedure for the solution of the incompressible Navier–Stokes equations is presented. In the Petrov–Galerkin formulation employed, the velocities are interpolated using the flow conditions over the elements and the pressure is interpolated to satisfy the inf–sup condition for incompressible analysis. Element control volumes are employed to satisfy local mass and momentum conservation (as in finite volume methods), which corresponds to using step functions as weight functions in the finite element method. An important achievement of the discretization scheme is that no artificial parameters are set in the scheme to reach stability for low and high Reynolds (and Péclet) number flows. The solutions of nontrivial test problems are presented to demonstrate the capability and potential of the scheme.     

A finite element method for 3D hydrostatic water flows

In this paper we propose a numerical method for the solution of the “Quasi-3D” hydrodynamic equations. This approach uses a combination of standard linear finite elements along the vertical direction and non-conforming Raviart-Thomas elements in the horizontal planes. We describe, also, a suitable approximation for the convective terms. Received: 17 March 1999 / Accepted: 17 June 1999     

A finite element based level set method for two-phase incompressible flows

We present a method that has been developed for the efficient numerical simulation of two-phase incompressible flows. For capturing the interface between the phases the level set technique is applied. The continuous model consists of the incompressible Navier–Stokes equations coupled with an advection equation for the level set function. The effect of surface tension is modeled by a localized force term at the interface (so-called continuum surface force approach). For spatial discretization of velocity, pressure and the level set function conforming finite elements on a hierarchy of nested tetrahedral grids are used. In the finite element setting we can apply a special technique to the localized force term, which is based on a partial integration rule for the Laplace–Beltrami operator. Due to this approach the second order derivatives coming from the curvature can be eliminated. For the time discretization we apply a variant of the fractional step θ-scheme. The discrete saddle point problems that occur in each time step are solved using an inexact Uzawa method combined with multigrid techniques. For reparametrization of the level set function a new variant of the fast marching method is introduced. A special feature of the solver is that it combines the level set method with finite element discretization, Laplace–Beltrami partial integration, multilevel local refinement and multigrid solution techniques. All these components of the solver are described. Results of numerical experiments are presented.     

Subdomain generation for non-convex parallel finite element domains

In this paper the Subdomain Generation Method (SGM), originally formulated in Khan & Topping (1993; Khan, A. I. & Topping, B. H. V., Subdomain generation for parallel finite element analysis. Comput. Syst. Engng, 1993, 4(4/6), 473–488) for convex finite element domains, is generalized for arbitrary shaped domains. Modifications to the original SGM are described which allow partitioning of non-convex domains. These modifications have been made to the formulation of the optimization module and the predictive module. The examples presented in Khan & Topping (1993) have been re-worked and two more examples have been added which demonstrate the application of the method to arbitrary shaped domains. It is shown with the aid of the examples that the method provides well-balanced subdomains very efficiently and allows parallel adaptive mesh generation. The method in its present form may be used to partition unstructured graphs in two or three dimensions. Since the computational cost for the mesh partitioning with this method depends solely upon the initial coarse mesh, hence the computational cost does not increase with the increase in the mesh density of the final mesh. The method in its present form is unsuitable for relatively coarse grained parallel computers, however the modifications which would impart a greater degree of scalability to this method are discussed.