BOUNDARY CONDITIONS FOR FINITE ELEMENT SIMULATIONS OF CONVECTIVE FLOWS WITH ARTIFICIAL BOUNDARIES

We examine the use of natural boundary conditions and conditions of the Sommerfeld type for finite element simulations of convective transport in viscous incompressible flows. We show that natural boundary conditions are superior in the sense that they always provide a correct boundary condition, as opposed to the Sommerfeld-type conditions, which can lead to a singular formulation and a great loss of accuracy. For the Navier–Stokes equations, the natural boundary conditions must be combined with a simple method to eliminate perturbations on the pressure at the open boundary, which is the source of most errors.     

A SPACE–TIME FINITE ELEMENT MODEL TO STUDY THE INFLUENCE OF INTERFACIAL KINETICS AND DIFFUSION ON CRYSTALLIZATION KINETICS

We present a space–time finite element formulation to study the cooperative growth of adjacent needle-like crystals in a two-dimensional, binary melt. It is assumed that the system is isothermal and that the compositions of the melt and the crystals are different. The growth rate of the crystals is taken to be a function of the melt composition in front of the growing crystals, and the composition of the melt as a function of space and time is determined by the diffusion equation. The positions of the growth fronts of each crystal are tracked. Good agreement is found between the numerical solution of an approximated one-dimensional problem and an analytical solution. Numerical results of the simulation of the growth of isolated and adjacent crystals are presented. © 1997 John Wiley & Sons, Ltd.     

FINITE ELEMENT SOLUTION OF INCOMPRESSIBLE FLUID–STRUCTURE VIBRATION PROBLEMS

In this paper we solve an eigenvalue problem arising from the computation of the vibrations of a coupled system, incompressible fluid – elastic structure, in absence of external forces. We use displacement variables for both the solid and the fluid but the fluid displacements are written as curls of a stream function. Classical linear triangular finite elements are used for the solid displacements and for the stream function in the fluid. The kinematic transmission conditions at the fluid–solid interface are taken into account in a weak sense by means of a Lagrange multiplier. The method does not present spurious or circulation modes for non-zero frequencies. Numerical results are given for some test cases. © 1997 by John Wiley & Sons, Ltd.     

A PARTICLE TRACKING TECHNIQUE FOR THE LAGRANGIAN–EULERIAN FINITE ELEMENT METHOD IN MULTI-DIMENSIONS

This paper presents a multi-dimensional particle tracking technique for applying the Lagrangian–Eulerian finite element method to solve transport equations in transient-state simulations. In the Lagrangian– Eulerian approach, the advection term is handled in the Lagrangian step so that the associated numerical errors can be considerably reduced. It is important to have an adequate particle tracking technique for computing advection accurately in the Lagrangian step. The particle tracking technique presented here is designed to trace fictitious particles in the real-world flow field where the flow velocity is either measured or computed at a limited number of locations. The technique, named ‘in-element’ particle tracking, traces fictitious particles on an element-by-element basis. Given a velocity field, a fictitious particle is traced one element by one element until either a boundary is encountered or the available time is completely consumed. For the tracking within an element, the element is divided into a desired number of subelements with the interpolated velocity computed at all nodes of the subelements. A fictitious particle, thus, is traced one subelement by one subelement within the element. The desired number of subelements can be determined based on the complexity of the flow field being considered. The more complicated the flow field is, the more subelements are needed to achieve accurate particle tracking results. A single-velocity approach can be used to efficiently perform particle tracking in a smooth flow field, while an average-velocity approach can be employed to increase the tracking accuracy for more complex flow fields.     

BOUNDARY ELEMENT–FINITE ELEMENT COUPLED EIGENANALYSIS OF FLUID–STRUCTURE SYSTEMS

The eigenanalysis of acoustical cavities with flexible structure boundaries, such as a fluid-filled container or an automobile cabin enclosure, is considered. An algebraic eigenvalue problem formulation for the fluid–structure problem is presented by combining the acoustic fluid boundary element eigenvalue analysis method and the structural finite elements. For many practical eigenproblems, use of finite elements to discretize the fluid domain leads to large stiffness and mass matrices. Since the acoustic boundary element discretization requires putting nodes only on the wetted surface of the structure, the size of the eigenproblem is reduced considerably, thus reducing the eigenvalue extraction effort. Futhermore, unlike in ordinary cases, the finite element discretization of pressure–displacement based fluid–structure problem gives rise to unsymmetric matrices. Therefore, the fact that the boundary element formulation produces unsymmetric matrices does not introduce additional difficulties here compared to the finite element case in the choice of an eigenvalue extraction procedure. Examples are included to demonstrate the fluid–structure eigenanalysis using boundary elements for the fluid domain and finite elements for the structure.     

SOLUTION OF THE ONE-DIMENSIONAL CONVECTION–DIFFUSION EQUATION BY A MULTILEVEL PETROV–GALERKIN METHOD

A multilevel Petrov–Galerkin (PG) finite element method to accurately solve the one-dimensional convection–diffusion equation is presented. In this method, the weight functions are different from the basis functions and they are calculated from simple algebraic recursion relations. The basis for their selection is that the given (coarse) mesh may duplicate the solutions obtained at common nodes of a finer virtual mesh. If the fine mesh is sufficiently refined, then the coarse mesh solutions converge to the exact solution. The finer mesh is virtual because its associated system of discrete equations is never solved. This multilevel PG method is extended to cases of the non-homogeneous problem with polynomial force functions. The examples considered confirm that this method is successful in accelerating the rate of convergence of the solution even when the force terms are non-polynomial. The multilevel PG method is therefore efficient and powerful for the general non-homogeneous convection–diffusion equation.     

A SPACE–TIME FINITE ELEMENT METHOD FOR THE EXTERIOR STRUCTURAL ACOUSTICS PROBLEM: TIME-DEPENDENT RADIATION BOUNDARY CONDITIONS IN TWO SPACE DIMENSIONS

A time-discontinuous Galerkin space–time finite element method is formulated for the exterior structural acoustics problem in two space dimensions. The problem is posed over a bounded computational domain with local time-dependent radiation (absorbing) boundary conditions applied to the fluid truncation boundary. Absorbing boundary conditions are incorporated as ‘natural’ boundary conditions in the space–time variational equation, i.e. they are enforced weakly in both space and time. Following Bayliss and Turkel, time-dependent radiation boundary conditions for the two-dimensional wave equation are developed from an asymptotic approximation to the exact solution in the frequency domain expressed in negative powers of a non-dimensional wavenumber. In this paper, we undertake a brief development of the time-dependent radiation boundary conditions, establishing their relationship to the exact impedance (Dirichlet-to-Neumann map) for the acoustic fluid, and characterize their accuracy when implemented in our space–time finite element formulation for transient structural acoustics. Stability estimates are reported together with an analysis of the positive form of the matrix problem emanating from the space–time variational equations for the coupled fluid-structure system. Several numerical simulations of transient radiation and scattering in two space dimensions are presented to demonstrate the effectiveness of the space–time method.     

THE s-VERSION OF FINITE ELEMENT METHOD FOR LAMINATED COMPOSITES

The s-version of the finite element method is developed for laminated plates and shells. By this technique the global domain is idealized using 2-D Equivalent Single Layer model. The regions where ESL model errs badly in capturing localized phenomena are superimposed by a stack of 3-D elements. Assumed strain formulation and selective polynomial order escalation in the two models, as well as fast iterative procedures, are employed to maintain a high level of computational efficiency.     

A SPACE-TIME COUPLED p-VERSION LEAST SQUARES FINITE ELEMENT FORMULATION FOR UNSTEADY TWO-DIMENSIONAL NAVIER–STOKES EQUATIONS

This paper presents a p-version least squares finite element formulation for two-dimensional unsteady fluid flow described by Navier–Stokes equations where the effects of space and time are coupled. The dimensionless form of the Navier–Stokes equations are first cast into a set of first-order differential equations by introducing auxiliary variables. This permits the use of C⁰ element approximation. The element properties are derived by utilizing the p-version approximation functions in both space and time and then minimizing the error functional given by the space–time integral of the sum of squares of the errors resulting from the set of first-order differential equations. This results in a true space–time coupled least squares minimization procedure. The application of least squares minimization to the set of coupled first-order partial differential equations results in finding a solution vector {δ} which makes gradient of error functional with respect to {δ} a null vector. This is accomplished by using Newton's method with a line search. A time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step. Equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance. The space–time coupled p-version approximation functions provide the ability to control truncation error which, in turn, permits very large time steps. What literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space–time coupled approach. The generality, success and superiority of the present formulation procedure is demonstrated by presenting specific numerical examples for transient couette flow and transient lid driven cavity. The results are compared with the analytical solutions and those reported in the literature. The formulation presented here is ideally suited for space–time adaptive procedures. The element error functional values provide a mechanism for adaptive h, p or hp refinements.     

二维自由振动问题的自适应有限元分析初探

自由振动反映结构动力特性,是抗震分析和结构设计的重要基础。近年来,基于单元能量投影（EEP）法的自适应有限元分析已在一系列线弹性及非线性问题中取得成功,而有限元线法（FEMOL）自适应分析在二维自由振动问题中的应用也被证实是有效的。在此基础上,该文进一步提出二维自由振动问题的自适应有限元分析方法。通过将特征值问题线性化,合理引入二维线性问题的EEP超收敛计算和自适应求解技术,该法可得到满足精度要求的自振频率和按最大模度量满足用户给定误差限的振型。该文以弹性薄膜为例,介绍了这一进展,并给出数值算例以表明该方法的有效性和可靠性。     

An alternative TLM method for steady‐state convection–diffusion

Recent papers have introduced a novel and efficient scheme, based on the transmission line modelling (TLM) method, for solving one‐dimensional steady‐state convection–diffusion problems. This paper introduces an alternative method. It presents results obtained using both techniques, which suggest that the new scheme outlined in this paper is the more accurate and efficient of the two. Copyright © 2009 John Wiley & Sons, Ltd.     

Solving the advection–diffusion equation on unstructured meshes with discontinuous/mixed finite elements and a local time stepping procedure

Explicit schemes are known to provide less numerical diffusion in solving the advection–diffusion equation, especially for advection‐dominated problems. Traditional explicit schemes use fixed time steps restricted by the global CFL condition in order to guarantee stability. This is known to slow down the computation especially for heterogeneous domains and/or unstructured meshes. To avoid this problem, local time stepping procedures where the time step is allowed to vary spatially in order to satisfy a local CFL condition have been developed. In this paper, a local time stepping approach is used with a numerical model based on discontinuous Galerkin/mixed finite element methods to solve the advection–diffusion equation. The developments are detailed for general unstructured triangular meshes. Numerical experiments are performed to show the efficiency of the numerical model for the simulation of (i) the transport of a solute on highly unstructured meshes and (ii) density‐driven flow, where the velocity field changes at each time step. The model gives stable results with significant reduction of the computational cost especially for the non‐linear problem. Moreover, numerical diffusion is also reduced for highly advective problems. Copyright © 2009 John Wiley & Sons, Ltd.     

A DISTRIBUTED FINITE ELEMENT METHOD FOR SOLVING THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS

The potential for using a network of workstations for solving the incompressible Navier–Stokes equations using a finite element formulation is investigated. A programming paradigm suitable for a heterogeneous distributed workstation environment is developed and compared to the traditional paradigm employed for distributed memory parallel computers. In particular, the issues of load balancing and fault recovery are explored. Numerical results are presented for two computer configurations: (1) a homogeneous network of workstations and (2) a heterogeneous network of workstations. The superiority of the developed paradigm over the traditional paradigm employed for distributed memory parallel computers is shown in cases where a heterogeneous network of workstations is employed or when one of the workstations of the cluster is loaded by other users.     

A VELOCITY BASED APPROACH INCLUDING ACCELERATION TO THE FINITE ELEMENT COMPUTATION OF VISCOPLASTIC PROBLEMS

We present a velocity based approach including acceleration to the finite element computation of metal forming problems, based on the viscoplastic Norton–Hoff law. In order to reduce computational cost, we suggest substituting the classical solution procedure based on standard Newton–Raphson method for solving the set of non-linear equations, with a new one which needs only one computation inside a time step and which is based on the linearization of the non-linear equations over time. The new procedure was introduced as an option in the existing computer code FORGE2©. Some examples are used for comparison between the classical procedure and the new one. They show that the new procedure is stable and accurate, and in comparison to the classical one it reduces the total number of resolutions of linear systems. Therefore, significant computer time reduction can be expected for 3-D problems.     

浸水旋转壳体振动特性的有限条—边界元分析

根据旋转壳体的结构特点,利用有限条法对其进行分析达到降维目的;对流场则提出了一种特殊的边界单元,这种边界元不仅能与结构的有限条单元协调,而且计算量小,精度高;在此基础上,通过凝聚降阶进一步减少流固耦合系统的控制方程。对典型浸水旋转壳的振动频率、模态以及响应作了全面分析,获得了理想的数值分析结果。     

COMBINATIONS OF THE RITZ–GALERKIN AND FINITE DIFFERENCE METHODS

This paper presents six combinations of the Ritz–Galerkin method and the finite difference method for solving elliptic boundary value problems. Not only optimal convergence rates of solutions but also superconvergence rates of solution derivatives can be achieved. The non-conforming combination and other five new combinations are given in algorithms, error analysis, convergence rates, outlines of proofs, numerical experiments and their comparisons.     

A NEW FORMULATION OF ISOPARAMETRIC FINITE ELEMENTS AND THE RELATIONSHIP BETWEEN HYBRID STRESS ELEMENT AND INCOMPATIBLE ELEMENT

A new method of formulating isoparametric finite element is developed, and the element strains are proposed to be resolved into two parts, constant part and higher-order one. The new method indicates two important properties of isoparametric finite element, and the equivalent relationship between hybrid stress elements and incompatible elements. © 1997 by John Wiley & Sons, Ltd.     

FREE VIBRATION ANALYSIS OF KIRCHHOFF PLATES RESTING ON ELASTIC FOUNDATION BY MIXED FINITE ELEMENT FORMULATION BASED ON GÂTEAUX DIFFERENTIAL

The main objective of the present work is to give the systematic way for derivation of Kirchhoff plate-elastic foundation interaction by mixed-type formulation using the Gâteaux differential instead of well-known variational principles of Hellinger–Reissner and Hu–Washizu. Foundation is a Pasternak foundation, and as a special case if shear layer is neglected, it converges to Winkler foundation in the formulation. Uniform variation of the thickness of the plate is also included into the mixed finite element formulation of the plate element PLTVE4 which is an isoparametric C⁰ class conforming element discretization. In the dynamic analysis, the problem reduces to solution of the standard eigenvalue problem and the mixed element is based upon a consistent mass matrix formulation. The element has four nodes and at each node transverse displacement two bending and one torsional moment is the basic unknowns. Proper geometric and dynamic boundary conditions corresponding to the plate and the foundation is given by the functional. Performance of the element for bending and free vibration analysis is verified with a good accuracy on the numerical examples and analytical solutions present in the literature. © 1997 by John Wiley & Sons, Ltd.