A new finite element technique for the analysis of steady viscous flow problems

A new finite element technique for two-dimensional viscous incompressible fluid flow problems is presented in this paper. The vorticity transport equation is integrated in a small control volume, which results in the conservation law of vorticity. The finite element technique is applied to this equation together with the continuity equation, where simple linear triangular elements with three nodes are used for the formulation. Resulting sets of algebraic equations are solved by the use of a kind of relaxation method. Numerical results for viscous flow past a cavity show good agreement with experimental results.     

Trefftz-type boundary elements for the evaluation of symmetric coefficient matrices

The classical Trefftz-method can be generalized such that different types of finite elements and boundary elements are obtained. In a Trefftz-type approach we utilize functions which a priori satisfy the governing differential equations. In this paper the systematic construction of singular Trefftz-trial functions for elasticity problems is discussed. For convenience a list of solution representations and particular solutions is given which did not appear together elsewhere. The Trefftz-trial functions with singular expressions on the boundary are constructed such that the physical components (stresses, strains, displacements) remain finite in the solution domain and on the boundary. The unknown coefficients of the linearly independent Trefftz-trial functions for the physical components can be obtained by using a variational formulation. The symmetric coefficient matrix in the discussed procedure can be obtained from the evaluation of boundary integrals. As an application of the proposed boundary element algorithm, the symmetric stiffness matrices of subdomains (finite element domains) are calculated. For the numerical example the solution domain is decomposed into triangular subdomains so that a standard finite element program could be used to assemble the system of equations. The chosen example is meant as a simple test for the proposed algorithm and should not be understood as a proposal for a new triangular finite element. Using the proposed boundary element techniques, symmetric stiffness matrices for irregular shaped subdomains (finite elements) can be derived. However, in order to use the method in a finite element package for the coupling of irregular shaped subdomains some program modifications will be necessary.     

Triangular and tetrahedral space-time finite elements in vibration analysis

Recently, several methods of time integration of the equations of motion have been proposed. Many of them result in square mass, damping and stiffness matrices. The space–time finite element method is an extension of the FEM, familiar to most engineers, over the time domain. A special approach enables the use of triangular, tetrahedral and hyper-tetrahedral elements in time and space. By special division of the space–time layer the triangular matrix of coefficients in the system of equations can be obtained. A simple algorithm enables the storage of non-zero coefficients only. Dynamic solution requires a small amount of the memory compared to other methods, and ensures considerable reduction of arithmetic operations. The method presented is also efficient in solving both linear and non-linear problems. Matrices for a beam and plane stress/strain element are derived. Exemplary problems solved by the method described have proved the effectiveness of the application of triangular and tetrahedral space–time elements in vibration analysis.     

The ALE‐method with triangular elements: direct convection of integration point values

The arbitrary Lagrangian–Eulerian (ALE) finite element method is applied to the simulation of forming processes where material is highly deformed. Here, the split formulation is used: a Lagrangian step is done with an implicit finite element formulation, followed by an explicit (purely convective) Eulerian step. The purpose of this study is to investigate the Eulerian step for quadratic triangular elements. To solve the convection equation for integration point values, a new method inspired by Van Leer is constructed. The new method is based on direct convection of integration point values without intervention of nodal point values. The Molenkamp test and a so‐called block test were executed to check the performance and stability of the convection scheme. From these tests it is concluded that the new convection scheme shows accurate results. The scheme is extended to an ALE‐algorithm. An extrusion process was simulated to test the applicability of the scheme to engineering problems. It is concluded that direct convection of integration point values with the presented algorithm leads to accurate results and that it can be applied to ALE‐simulations. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical simulation of thermogravitational energy transport of a hybrid nanoliquid within a porous triangular chamber using the two-phase mixture approach

This study is a computational investigation of transient thermogravitational energy transport in H₂O/Al₂O₃ nanoliquid and water-based copper/aluminum oxide hybrid nanofluid (water/Al₂O₃-Cu) inside a horizontal isosceles triangular enclosure with porous medium. The governing equations for two-phase mixture flow have been derived by the use of Darcy-Brinkman model for porous media without the Forchheimer term (inertia loss). The control equations have been discretized using the finite volume technique. The effects of porosity factor, Rayleigh number, and Darcy number on the liquid motion and transient energy transport have been studied. The results have shown that convective thermal transmission in the nanofluid inside the triangular cavity generally consists of three phases: initial, transient, and quasi-steady, all of which are described in detail. It has been found that a rise of the porosity factor, Rayleigh number, or Darcy number always leads to an increment of the average Nusselt number and energy transport intensity. It has been also observed that with a rise of the Darcy number and strengthening of flow motion (convection), the instability in both flow and temperature fields increases and the distribution of isotherms and streamlines becomes completely asymmetric.     

Novel quadratic Bézier triangular and tetrahedral elements using existing mesh generators: Extension to nearly incompressible implicit and explicit elastodynamics in finite strains

We present a novel unified finite element framework for performing computationally efficient large strain implicit and explicit elastodynamic simulations using triangular and tetrahedral meshes that can be generated using the existing mesh generators. For the development of a unified framework, we use Bézier triangular and tetrahedral elements that are directly amenable for explicit schemes using lumped mass matrices and employ a mixed displacement-pressure formulation for dealing with the numerical issues arising due to volumetric and shear locking. We demonstrate the accuracy of the proposed scheme by studying several challenging benchmark problems in finite strain elastostatics and nonlinear elastodynamics modelled with nearly incompressible hyperelastic and von Mises elastoplastic material models. We show that Bézier elements, in combination with the mixed formulation, help in developing a simple unified finite element formulation that is accurate, robust, and computationally very efficient for performing a wide variety of challenging nonlinear elastostatic and implicit and explicit elastodynamic simulations.     

An operator-splitting algorithm for two-dimensional convection–dispersion–reaction problems

An operator-splitting algorithm for the two-dimensional convection–dispersion–reaction equation is developed. The flow domain is discretized into triangular elements which are fixed in time. The governing equation is split into three successive initial value problems: a pure convection problem, a pure dispersion problem and a pure reaction problem. For the pure convection problem, solutions are found by the method of characteristics. The solution algorithm involves tracing the characteristic lines backwards in time from a vertex of an element to an interior point. A cubic polynomial is used to interpolate the concentration and its derivatives on an element. For the pure dispersion problem, an explicit finite element algorithm is employed. Analytical solutions are obtained for the pure reaction problem. The treatment of the boundary conditions is also discussed. Several numerical examples are presented. Numerical results agree well with analytical solutions. Because cubic polynomials are used in the interpolation, very little numerical damping and oscillation are introduced, even for the pure convection problem.     

Triangular finite elements for the generalized Bessel equation of order m

A new family of triangular finite elements is described, useful for solving the axisymmetric vector Helmholtz equation, and a variety of scalar Helmholtz equation problems which lead to generalized Bessel equations of some order m. This family is similar in principle to the scalar axisymmetric Helmholtz elements derived earlier, but requires both reformulation of its describing equations and corresponding new universal element matrices, for successful computational implementation. The necessary formulation is given in this paper. Matrix elements to the sixth-order inclusive have been calculated and extensively tested computationally.     

An efficient finite element solution using a large pre-solved regular element

In this paper a finite element algorithm is presented using a large pre-solved hyper element. Utilizing the largest rectangle/cuboid inside an arbitrary domain, a large hyper element is developed that is solved using graph product rules. This pre-solved hyper element is efficiently inserted into the finite element formulation of partial differential equations (PDE) and engineering problems to reduce the computational complexity and execution time of the solution. A general solution of the large pre-solved element for a uniform mesh of triangular and rectangular elements is formulated for second-order PDEs. The efficiency of the algorithm depends on the relative size of the large element and the domain; however, the method remains as efficient as a classic method for even relatively small sizes. The application of the method is demonstrated using different examples.     

Stability analysis of some finite difference schemes for the Navier-Stokes equations

We study the stability of two standard finite differnce schemes for the unsteady Navier-Stokes equations. We write a scheme for small values of Reynolds number, where teh diffusion is predominant, and another one for large values of Reynolds number, where convection is predominant. The purpose of this results may be extended to unsteady problems leading to mixed hyperbolic-parabolic partia differenctial equations.     

A new triangular element for finite difference solution of axisymmetric conduction problems in cylindrical co-ordinates

The triangular element is one of the most useful finite difference elements for the potential problems because of its versatility in fitting irregular boundary, in connecting one element type to another, and in changing grid fineness. Such triangular elements find many applications in conduction problems in cartesian co-ordinates. This paper presents a new triangular element for the finite' difference solution of axisymmetric conduction problems in cylindrical co-ordinates. The validity of the proposed triangular element is analyzed, and its workability is demonstrated using three selected examples. Also, an industrial application highlights the advantageous characteristics of the triangular element, and gives a comparison with known results obtained by finite element method.     

Comment on the homogeneous nanofluid models applied to convective heat transfer problems

In several recent papers, the heat transfer characteristics of nanofluids have been investigated by simply replacing the transport coefficients of the base fluid by the effective transport coefficients of the nanofluids. The present note emphasizes, however, that the governing equations of these homogeneous nanofluid models (in which the velocity-slip effects of the nanoparticles are neglected) can be reduced with the aid of elementary scaling transformations to the respective equations of the regular fluids. Thus, the corresponding nanofluid results can be recovered from the solutions of already solved regular problems by simple arithmetic operations, without any additional research effort. This feature is illustrated here by the specific examples of the classical Blasius and Sakiadis forced convection heat transfer problems.     

Modelling convection in solidification processes using stabilized finite element techniques

Solidification of dendritic alloys is modelled using stabilized finite element techniques to study convection and macrosegregation driven by buoyancy and shrinkage. The adopted governing macroscopic conservation equations of momentum, energy and species transport are derived from their microscopic counterparts using the volume‐averaging method. A single domain model is considered with a fixed numerical grid and without boundary conditions applied explicitly on the freezing front. The mushy zone is modelled here as a porous medium with either an isotropic or an anisotropic permeability. The stabilized finite‐element scheme, previously developed by authors for modelling flows with phase change, is extended here to include effects of shrinkage, density changes and anisotropic permeability during solidification. The fluid flow scheme developed includes streamline‐upwind/Petrov–Galerkin (SUPG), pressure stabilizing/Petrov–Galerkin, Darcy stabilizing/Petrov–Galerkin and other stabilizing terms arising from changes in density in the mushy zone. For the energy and species equations a classical SUPG‐based finite element method is employed with minor modifications. The developed algorithms are first tested for a reference problem involving solidification of lead–tin alloy where the mushy zone is characterized by an isotropic permeability. Convergence studies are performed to validate the simulation results. Solidification of the same alloy in the absence of shrinkage is studied to observe differences in macrosegregation. Vertical solidification of a lead–tin alloy, where the mushy zone is characterized by an anisotropic permeability, is then simulated. The main aim here is to study convection and demonstrate formation of freckles and channels due to macrosegregation. The ability of stabilized finite element methods to model a wide variety of solidification problems with varying underlying phenomena in two and three dimensions is demonstrated through these examples. Copyright © 2005 John Wiley & Sons, Ltd.     

An algorithm for adaptive refinement of triangular element meshes

A simple algorithm is developed for adaptive and automatic h refinement of two-dimensional triangular finite element meshes. The algorithm is based on an element refinement ratio that can be calculated from an a posteriori error indicator. The element subdivision algorithm is robust and recursive. Smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh. Several example problems are presented to illustrate the utility of the approach.     

Redundant structure concepts in finite element analyses

The paper examines two triangular finite elements used in plate bending analyses for which Lagrange multiplier techniques have been used to set up the necessary equations for the solution. These equations involve either equilibrium of the system or the inter-element compatibility. Starting from only the notions of equilibrium and continuity as used in the analyses of redundant structures, the paper proceeds to solutions of the same problems. It is found that end results are the same when the equations obtained from the minimization techniques are reduced by Gauss-elimination in partitioned from.     

A coupled ES-FEM/BEM method for fluid–structure interaction problems

The edge-based smoothed finite element method (ES-FEM) developed recently shows some excellent features in solving solid mechanics problems using triangular mesh. In this paper, a coupled ES-FEM/BEM method is proposed to analyze acoustic fluid–structure interaction problems, where the ES-FEM is used to model the structure, while the acoustic fluid is represented by boundary element method (BEM). Three-node triangular elements are used to discretize the structural and acoustic fluid domains for the important adaptability to complicated geometries. The smoothed Galerkin weak form is adopted to formulate the discretized equations for the structure, and the gradient smoothing operation is applied over the edge-based smoothing domains. The global equations of acoustic fluid–structure interaction problems are then established by coupling the ES-FEM for the structure and the BEM for the fluid. The gradient smoothing technique applied in the structural domain can provide the important and right amount of softening effects to the “overly-stiff” FEM model and thus improve the accuracy of the solutions of coupled system. Numerical examples of acoustic fluid–structure interaction problems have been used to assess the present formulation, and the results show that the accuracy of present method is very good and even higher than those obtained using the coupled FEM/BEM with quadrilateral mesh.     

A least-squares front-tracking finite element method analysis of phase change with natural convection

This paper focuses on the numerical modelling of phase-change processes with natural convection. In particular, two-dimensional solidification and melting problems are studied for pure metals using an energy preserving deforming finite element model. The transient Navier–Stokes equations for incompressible fluid flow are solved simultaneously with the transient heat flow equations and the Stefan condition. A least-squares variational finite element method formulation is implemented for both the heat flow and fluid flow equations. The Boussinesq approximation is used to generate the bulk fluid motion in the melt. The mesh motion and mesh generation schemes are performed dynamically using a transfinite mapping. The consistent penalty method is used for modelling incompressibility. The effect of natural convection on the solid/liquid interface motion, the solidification rate and the temperature gradients is found to be important. The proposed method does not possess some of the false diffusion problems associated with the standard Galerkin formulations and it is shown to produce accurate numerical solutions for convection dominated phase-change problems.     

A contact analysis approach based on linear complementarity formulation using smoothed finite element methods

Based on the subdomain parametric variational principle (SPVP), a contact analysis approach is formulated in the incremental form for 2D solid mechanics problems discretized using only triangular elements. The present approach is implemented for the newly developed node-based smoothed finite element method (NS-FEM), the edge-based smoothed finite element method (ES-FEM) as well as standard FEM models. In the approach, the contact interface equations are discretized by contact point-pairs using a modified Coulomb frictional contact model. For strictly imposing the contact constraints, the global discretized system equations are transformed into a standard linear complementarity problem (LCP), which can be readily solved using the Lemke method. This approach can simulate different contact behaviors including bonding/debonding, contacting/departing, and sticking/slipping. An intensive numerical study is conducted to investigate the effects of various parameters and validate the proposed method. The numerical results have demonstrated the validity and efficiency of the present contact analysis approach as well as the good performance of the ES-FEM method, which provides solutions of about 10 times better accuracy and higher convergence rate than the FEM and NS-FEM methods. The results also indicate that the NS-FEM provides upper-bound solutions in energy norm, relative to the fact that FEM provides lower-bound solutions.     

C0 finite element discretization of Kirchhoff's equations of thin plate bending

An alternative formulation of Kirchhoff's equations is given which is amenable to a standard C⁰ finite element discretization. In this formulation, the potential energy of the plate is formulated entirely in terms of rotations, whereas the deflections are the outcome of a subsidiary problem. The nature of the resulting equations is such that C⁰ interpolation can be used on both rotations and deflections. In particular, general classes of triangular and quadrilateral isoparametric elements can be used in conjunction with the method. Unlike other finite element methods which are based on three-dimensional or Mindlin formulations, the present approach deals directly with Kirchhoff's equations of thin plate bending. Excellent accuracy is observed in standard numerical tests using both distorted and undistorted mesh patterns.