Quadratic Interpolation Functions Research Articles

Finite element analysis of interfacial instability in aluminium reduction cells in a uniform, vertical magnetic field

In this paper the interfacial instability in aluminium reduction cells under a uniform, vertical magnetic field is investigated by means of numerical simulation. Shallow-water equations governing the wave motion and the electric field at the interface between two electrically conducting fluids are discretized using a finite element scheme which employs quadratic interpolation functions. For a sufficiently high magnetic field, it is observed that one global maximum and one global minimum of the interface elevation are formed in closed symmetrical and asymmetrical domains irrespective of shape. By taking account of this, an asymptotic expression for the evaluation of the growth rate and the frequency of instability is constructed. The effectiveness of the proposed model is validated through comparison with the numerical results.

Development of an adaptive Discontinuous-Galerkin finite element method for advection–reaction equations

This paper presents an adaptive Discontinuous-Galerkin finite element method to solve linear advection–reaction equations. The method was developed for the prediction of hemolysis in prosthetic devices, where the flow may include recirculation zones. To ensure the well-posedness of the problem, we solve the transient form of the PDE. Time discretization is achieved with an implicit Euler scheme in order to obtain the steady-state solution rapidly. Linear interpolation functions are used to approximate the hemolysis field. The a posteriori estimation of the error is obtained by solving an advection–reaction equation for the error, which is approximated with quadratic interpolation functions. Then, elementary estimations of the error are computed using a semi-norm developed to achieve the best possible results, even in the presence of recirculation. Code verification – to assess the convergence and conservation properties of the method – is performed with the method of manufactured solutions and with a conservation test using a sharp gaussian function as a reaction term. Excellent results are obtained in both cases. Predictions of hemolysis in a dialysis cannula are presented as an engineering application of our method.

A New High-order Time-kernel BIEM for the Burgers Equation

This paper presents a new high-order time-kernel boundary-integral-equation method (BIEM) for numerically solving transient problems governed by the Burgers equation. Instead of using high-order Lagrange polynomials such as quadratic and quartic interpolation functions, the proposed method employs integrated radial-basis-function networks (IRBFNs) to represent the unknown functions in boundary and volume integrals. Numerical implementations of ordinary and double integrals involving time in the presence of IRBFNs are discussed in detail. The proposed method is verified through the solution of diffusion and convection-diffusion problems. A comparison of the present results and those obtained by low-order BIEMs and other methods is also given.

Numerical solutions of distributed diffusion Hopfield neural network equations

A finite element method for numerical solutions of the systems governed by distributed models of Hopfield-type neural network equations is studied. When time keeps continuous and the spatial dimension is one, a semi-discrete algorithm for numerical solutions using quadratic interpolation functions is constructed, in which the Gauss–Legendre quadrature of numerical integrations of nonlinear term and Runge–Kutta method for solving ordinary differential equation are efficiently used. Finally, numerical simulations results obtained by Mathematica to show the scheme is stable.

Numerical solution of damped nonlinear Klein–Gordon equations using variational method and finite element approach

Numerical treatment for damped nonlinear Klein–Gordon equations, based on variational method and finite element approach, is studied. A semi-discrete algorithm is proposed by using quadratic interpolation functions of continuous time and spatial dimension one. The Gauss–Legendre quadrature has been utilized for numerical integrations of nonlinear terms, and Runge–Kutta method is used for solving ordinary differential equation. Finally, three dimensional graphics of numerical solutions are used to demonstrate the numerical results.

A numerical solution to Klein–Gordon equation with Dirichlet boundary condition

Klein–Gordon equation arises in relativistic quantum mechanics and field theory, so it is of a great importance for the high energy physicists. In this paper, we establish the existence and uniqueness of the solution and in the second part a numerical scheme is developed based on finite element method. For one space dimensional case, a complete numerical algorithm for the numerical solutions using the quadratic interpolation functions is constructed. The one-dimensional model equation is formulated over an arbitrary element, applying the assembly process on the elements of the domain, employing numerical scheme to integrate the nonlinear terms and solving the system of equations numerically. Finally the obtained results of simulation is visualized, which shows the overflow of the solution as expected.

On shear and extensional locking in nonlinear composite beams

A displacement beam finite element is derived based on a nonlinear kinematic model for a pretwisted composite beam. Elements based on linear, quadratic and cubic interpolation functions are developed. The reduced/selective integration technique is employed to investigate shear and membrane locking for each type of element. Various numerical integration rules are examined to determine the best integration scheme. Numerical results are presented on a wide range of beam problems to evaluate the performances of the three types of elements. None of the Gauss quadrature rules investigated, in particular, the three uniform reduced rules, have shown spurious mechanisms. Using the best scheme for each element, the performance studies carried out have shown that the cubic interpolation was the most efficient for almost all the problems studied.

Boundary element methods for transient convective diffusion. Part I: General formulation and 1D implementation

A general formulation of higher-order boundary element methods (BEM) is presented for time-dependent convective diffusion problems in one- and multi-dimensions. Free-space time-dependent convective diffusion fundamental solutions originally proposed by Carslaw and Jaeger are used to obtain the boundary integral formulation. Linear, quadratic and quartic time interpolation functions are introduced in this paper for approximate representation of time-dependent boundary temperatures and normal fluxes. Closed form time integration of the kernels is mandatory to attain both accuracy and efficiency of the numerical approach. A complete set of time integrals for the one-dimensional formulation is presented here for the first time in the literature.

An accurate integral‐based scheme for advection–diffusion equation

AbstractThis paper proposes an accurate integral‐based scheme for solving the advection–diffusion equation. In the proposed scheme the advection–diffusion equation is integrated over a computational element using the quadratic polynomial interpolation function. Then elements are connected by the continuity of first derivative at boundary points of adjacent elements. The proposed scheme is unconditionally stable and results in a tridiagonal system of equations which can be solved efficiently by the Thomas algorithm. Using the method of fractional steps, the proposed scheme can be extended straightforwardly from one‐dimensional to multi‐dimensional problems without much difficulty and complication. To investigate the computational performances of the proposed scheme five numerical examples are considered: (i) dispersion of Gaussian concentration distribution in one‐dimensional uniform flow; (ii) one‐dimensional viscous Burgers equation; (iii) pure advection of Gaussian concentration distribution in two‐dimensional uniform flow; (iv) pure advection of Gaussian concentration distribution in two‐dimensional rigid‐body rotating flow; and (v) three‐dimensional diffusion in a shear flow. In comparison not only with the QUICKEST scheme, the fully time‐centred implicit QUICK scheme and the fully time‐centred implicit TCSD scheme for one‐dimensional problem but also with the ADI‐QUICK scheme, the ADI‐TCSD scheme and the MOSQUITO scheme for two‐dimensional problems, the proposed scheme shows convincing computational performances. Copyright © 2001 John Wiley & Sons, Ltd.

Boundary element analysis of two-dimensional elastoplastic contact problems

A general boundary element formulation for contact problems, capable of dealing with local elastoplastic effects and friction, is presented. Both conforming and non-conforming problems may be analysed. The contact problem is solved by means of a direct constraint technique, in which compatibility and equilibrium conditions are directly enforced in the general system of equations. The contact areas are modelled with linear interpolation functions, and quadratic interpolation functions are used everywhere else. Elastoplasticity is solved by a BEM initial strain approach The Von Mises yield criterion with its associated flow rule is adopted. Both perfectly plastic and work hardening materials are studied in the proposed formulation. An incremental loading technique is proposed, which allows accurate development of the loading history of the problem. The non-linear nature of these problems demands the use of an iterative procedure, to determine the correct frictional conditions at every node of the contact area and the value of the plastic strains at selected points where local yielding may have occurred. Several numerical examples are presented to demonstrate the efficiency of the proposed formulation.

Application of Free Mesh Method to Fluid Analysis. A Higher-order Free Mesh Method and its Application to a Nonlinear Problem.

In this paper, Free Mesh Method (a kind of meshless method) proposed by the authors is improved for application to steady incompressible viscous flow analysis. In the finite-element analysis of incompressible viscous flow, it is better to use different-order interpolation function for velocity and pressure. In the method, quadratic interpolation function can be used by introducing medium nodes between two original nodes, and a set of advection, diffusion and gradient matrices can be assembled independently node-by-node. As a numerical example, a two-dimensional driven cavity flow is analyzed. In this example, the governing equations, i.e. a coupled system of incompressibility and Navier-Stokes equations, are solved simultaneously. To solve a set of linear equations derived from the formulation of this method, the conjugate residual (CR) method is used. It is concluded that the result obtained using the method agrees well with the numerical result obtained by Ghia et al.

A higher‐order boundary element method for three‐dimensional potential problems

AbstractA method of eliminating the singularities involved in boundary element methods for three‐dimensional potential problems is presented and the non‐singular expressions of integrals on an element on which the singular point is situated are given for linear and quadratic interpolation functions. Numerical examples are compared with analytical solutions to show that the higher‐order interpolations have better precision.

The green element method

AbstractThis paper discusses an element‐by‐element approach of implementing the Boundary Element Method (BEM) which offers substantial savings in computing resource, enables handling of a wider range of problems including non‐linear ones, and at the same time preserves the second‐order accuracy associated with the method. Essentially, by this approach, herein called the Green Element Method (GEM), the singular integral theory of BEM is retained except that its implementation is carried out in a fashion similar to that of the Finite Element Method (FEM). Whereas the solution procedure of BEM couples the information of all nodes in the computational domain so that the global coefficient matrix is dense and full and as such difficult to invert, that of GEM, on the other hand, involves only nodes that share common elements so that the global coefficient matrix is sparse and banded and as such easy to invert. Thus, GEM has the advantage of being more computationally efficient than BEM. In addition, GEM makes the singular integral theory more flexible and versatile in the sense that GEM readily accommodates spatial variability of medium and flow parameters (e.g., flow in heterogeneous media), while other known numerical features of BEM—its second‐order accuracy and ability to readily handle problems with singularities are retained by GEM.A number of schemes is incorporated into the basic Green element formulation and these schemes are examined with the goal of identifying optimum schemes of the formulation. These schemes include the use of linear and quadratic interpolation functions on triangular and rectangular elements. We found that linear elements offer acceptable accuracy and computational effort. Comparison of the modified fully implicit scheme against the generalized two‐level scheme shows that the modified fully implicit scheme with weight of about 1·25 offers a marginally better approximation of the temporal derivative. The Newton–Raphson scheme is easily incoporated into GEM and provides excellent results for the time‐dependent non‐linear Boussinesq problem. Comparison of GEM with conventional BEM is done on various numerical examples, and it is observed that, for comparable accuracy, GEM uses less computing time. In fact, from the numerical simulations carried out, GEM uses between 15 and 45 per cent of the simulation time of BEM.

2-D elastostatic analysis by a symmetric BEM/FEM scheme

An accurate and efficient BEM/FEM scheme for elastostatic structural analysis is developed. The BEM component is based on DeFigueiredo's hybrid displacement boundary element model which leads to a symmetric stiffness matrix. Linear as well as quadratic boundary elements are employed. The FEM component is a displacement based finite element model employing linear as well as quadratic interpolation functions, or the MSC/NASTRAN general purpose finite element program. The resulting BEM/FEM scheme is characterized by symmetric matrices and perfect matching of nodal forces and displacements at the interface of the BEM and FEM domains. This leads to accurate results in an efficient way. Numerical examples dealing with elastostatic stress analysis and fracture mechanics under plane stress conditions are presented in order to illustrate the proposed methodology and demonstrate its merits.

サブパネルを用いた内部ディリクレ形式のパネル法について

A panel code using subpanel technique, based on the internal Dirichlet formulation has been developed. In this study, only the case where the internal region has no flow is considered. Every panel is devided into 9 flat subpanels. Each subpanel has 4 subgrid points which are located on the biquadratic curve defined between two neibouring grid points, which effects more accurate expression of the geometry. Each subpanel is assumed to have uniform density of singularity, the value of which is expressed at the center of the subpanel using two-dimensional quadratic interpolation functions defined with the geometry of the surrounding panels. The calculated results are compared with those obtained by the panel method based on the external Neumann formulation using the same subpanel technique. Both results show good agreements in external flow examples, but in the calculation of the flow suffering strong interaction such as a flow through a duct, the panel code of the internal Dirichlet formulation gives much better results than that of the external Neumann formulation. This suggests that the internal Dirichlet panel method could be applied more effectively to the problems, such as wings with wing-tip clearance or wing-tip vortex sheet, which have severe interactions with the nearby singularity surfaces.

Treatment of numerical diffusion in strong convective flows

AbstractA three‐dimensional extension of the QUICK scheme adapted for the finite volume method and non‐uniform grids is presented to handle convection‐diffusion problems for high Peclet numbers and steep gradients. The algorithm is based on three‐dimensional quadratic interpolation functions in which the transverse curvature terms are maintained and the diagonal dominance of the coefficient matrix is preserved. All formulae are explicitly given in an appendix. Results obtained with the classical upwind (UDS), the simplified QUICK (transverse terms neglected) and the present full QUICK schemes are given for two benchmark problems, one two‐dimensional, steady state and the other three‐dimensional, unsteady state. Both QUICK schemes are shown to give superior solutions compared with the UDS in terms of accuracy and efficiency. The full QUICK scheme performs better than the simplified QUICK, giving even for coarse grids acceptable results closer to the analytical solutions, while the computational time is not affected much.

A finite element model for the simulation of the torsion and torsion-tension tests

This paper outlines a complete thermo-mechanical description of torsion and torsion-tension tests for a homogeneous, isotropic and viscoplastic material. The mechanical problem is solved by minimization of viscoplastic functional in which the incompressibility condition is enforced using a penalty method. Torsional effects are taken into account by computation of the three components of the velocity field in an axisymmetric space approximation. The domain is discretized with the Finite Element Method. Isoparametric elements in the space (r 2, z) are used for the discretization of the space coordinates and the velocity field. The stress tensor is computed using a local Finite Difference Method and the hydrostatic pressure is computed separately reconsidering the equilibrium equation. The heat transfer equation is solved by the Finite Element Method and the temperature field is computed separately on each time step. Numerical results with elements using linear and quadratic interpolation functions are compared with analytical solutions and experimental measurements. Evolutions of torque and temperature are compared with experimental measurements during a very high speed pure torsion test. Heat generation and localization of the deformation are simulated. We show that thermal effects significantly disturb the analysis of the test. Evolutions of velocity, strain and stress fields are computed in the case of a torsion-tension test.

A TRANSIENT FINITE ELEMENT ADAPTATION SCHEME FOR THERMAL PROBLEMS WITH STEEP GRADIENTS

An adaptive mesh refinement procedure that uses nodeless variables and quadratic interpolation functions is presented for analysing transient thermal problems. A temperature based finite element scheme with Crank‐Nicolson time marching is used to obtain the thermal solution. The strategies used for mesh adaptation, computing refinement indicators, and time marching are described. Examples in one and two dimensions are presented and comparisons are made with exact solutions. The effectiveness of this procedure for transient thermal analysis is reflected in good solution accuracy, reduction in number of elements used, and computational efficiency.

Finite element method versus finite difference methods : theoretical investigations of some models of potential of biological interest

In view of evaluating the specific advantages of the finite difference methods and the finite element method (FEM) in the resolution of the Schrödinger nuclear equation, we performed accurate investigations of some one-dimensional multi-well potentials. Two finite difference methods were used : an Adams method and the powerful Numerov-Cooley method. The FEM investigations were carried out with quadratic interpolation functions and the shooting method was used in place of the usual algorithms.

Wave propagation in laminated composite circular cylinders

A stiffness method is used to study the dispersion characteristics of waves propagating in laminated composite cylinders. Each lamina of the cylinder can possess distinct anisotropic properties, mass density and thickness. The objective of the study is to analyze the effects of the circumferential wave number, ply lay up configuration, number of layers, and the thickness-toradius ratio on the dispersion characteristics. A Rayleigh-Ritz type of approximation of the throughthickness variation of the displacements which maintains the continuity of displacements and tractions at the interfaces between the layers has been used. The numerical results are compared with those obtained from the method using quadratic interpolation functions and also with the analytical solutions to illustrate the accuracy and efficiency of the method. Frequency spectra for four ply [+30−30], and [+15−15]. sixteen ply [+ 15−15]. and twelve ply [O2 +45−45O2], graphite/epoxy laminated composite cylinders are also presented Numerical results show strong influence of anisotropy on the guided waves.