NMLMAP-A two dimensional finite element program for transient or static, linear or nonlinear magnetic field problems

NMLMAP is a finite element program for the solution of two dimensional magnetic field problems. The problems may be transient or static, linear or nonlinear, and planar or axisymmetric. The program has features that include a coarse-to-fine rezone technique; an adaptive acceleration method for speeding the convergence of the nonlinear iteration; and a substructuring technique for decreasing solution times in certain non-linear problems. These features, the governing and finite element equations, and example problems are discussed     

Collocation-based stochastic finite element analysis for random field problems

A stochastic response surface method (SRSM) which has been previously proposed for problems dealing only with random variables is extended in this paper for problems in which physical properties exhibit spatial random variation and may be modeled as random fields. The formalism of the extended SRSM is similar to the spectral stochastic finite element method (SSFEM) in the sense that both of them utilize Karhunen–Loeve (K–L) expansion to represent the input, and polynomial chaos expansion to represent the output. However, the coefficients in the polynomial chaos expansion are calculated using a probabilistic collocation approach in SRSM. This strategy helps us to decouple the finite element and stochastic computations, and the finite element code can be treated as a black box, as in the case of a commercial code. The collocation-based SRSM approach is compared in this paper with an existing analytical SSFEM approach, which uses a Galerkin-based weighted residual formulation, and with a black-box SSFEM approach, which uses Latin Hypercube sampling for the design of experiments. Numerical examples are used to illustrate the features of the extended SRSM and to compare its efficiency and accuracy with the existing analytical and black-box versions of SSFEM.     

An alternative high-order finite element formulation for cylindrical field problems

Earlier formulations of the finite element approach to cylindrical (rz) field problems led initially to variational expressions containing a simple term in 1/r and consequent attempts to remove it by appropriate choice of interpolation functions. The present paper uses new interpolation functions which ensure that the field behaviour near the axis is correctly modelled. High-order finite elements up to order four are derived and tested on a special cylindrical geometry to confirm, in a practical case, the theoretical claims of improved rates of convergence in solving problems of engineering significance.     

Three-dimensional elasto-plastic finite element analysis

Advances in technology and interest in limit state design have made the inclusion of non-linear effects, such as elasto-plastic behaviour, desirable in the analysis of many structures. Improvements in solution algorithms coupled with parallel developments in high speed digital computers have now made the practical solution of such problems possible. The paper presents numerical solutions to three-dimensional elasto-plastic problems illustrating the applicability of isoparametric elements and the order of computation times involved.     

A finite element weighted residual solution to one-dimensional field problems

The one-dimensional diffusion-convection equation is formulated with the finite element representation employing the Galerkin approach. A linear shape function and two-dimensional triangular and rectangular elements in space and time were used in solving the problem. The results are compared with finite difference solutions as well as the exact solution. As another example, the convective term is set equal to zero and these techniques are applied to the resulting heat equation and similar comparisons are made.     

Three-dimensional finite element contact algorithm for metal forming

This work presents a three-dimensional rigid plastic finite element formulation. The workpiece is discretized with eight node hexahedral isoparametric elements. Friction is included in the formulation by means of a shear stress depending on the relative velocity between the workpiece and the tool. Special attention is given to the contact problems, and a three-dimensional contact algorithm based on a discretization of the tool surface with triangular elements is presented. Finally, some selected examples are solved, in order to show the capabilities of the formulation.     

The B-spline finite element method applied to axi-symmetrical andnonlinear field problems

A B-spline FEM (finite-element method) using the B-spline functions for rectangular elements as shape functions is presented. It is very effective for solving two-dimensional electromagnetic field problems in regular regions. Compared with the conventional FEM, it gives more accurate potential values and due to the inherent properties of B-spline functions yields much closer field values. Both the computing time and the storage capacity are greatly reduced as well     

A finite element analysis for time-dependent problems

The finite element method is applied to dynamic linear viscoelastic analysis. The arising system of differential equations is integrated by means of the Laplace transformation. The direct numerical evaluation of the inversion integral seems to be appropriate in case of vibrational problems. The method is applied to examples of transient vibrations of elastic bodies. Initial conditions and arbitrary damping matrices can be considered.     

A finite element method for non-self-adjoint problems

This paper presents a general theory and application of the finite element method for some special class of non-self-adjoint problems. The formulation employed here is based on the Galerkin method for linear boundary value and eigenvalue problems described by the partial differential equations of elliptic type, and it can be regarded as an extension of the usual displacement method formulated by the use of the principle of minimum potential energy. In order to illustrate its validity and feasibility, the method is applied to the problems of the two-group neutron diffusion equations and of the stability of a non-conservative system.     

Galerkin finite element methods for wave problems

We compare here the accuracy, stability and wave propagation properties of a few Galerkin methods. The basic Galerkin methods with piecewise linear basis functions (called G1FEM here) and quadratic basis functions (called G2FEM) have been compared with the streamwise-upwind Petrov Galerkin (SUPG) method for their ability to solve wave problems. It is shown here that when the piecewise linear basis functions are replaced by quadratic polynomials, the stencils become much larger (involving five overlapping elements), with only a very small increase in spectral accuracy. It is also shown that all the three Galerkin methods have restricted ranges of wave numbers and circular frequencies over which the numerical dispersion relation matches with the physical dispersion relation — a central requirement for wave problems. The model one-dimensional convection equation is solved with a very fine uniform grid to show the above properties. With the help of discontinuous initial condition, we also investigate the Gibbs’ phenomenon for these methods.     

Three-dimensional magnetic field analysis method using scalarpotential formulated by boundary element method

A computational method for magnetic fields formulated by a BEM (boundary element method) has been developed. In the method, a reduced scalar potential is selected as an unknown variable to simplify the calculation of the boundary conditions. Its use requires a high numerical accuracy of the potential gradient. Conventional BEM does not provide this, because numerical element integration for the singular kernal causes a large error. To overcome this difficulty, a highly accurate numerical integration scheme is proposed based on the BEM, and it is applied to magnetic field problems. Calculation results for a spherical permeable material in a problem proposed by the Institute of Electrical Engineers of Japan (the problem of a magnetic field generated by a coil) agreed with the exact solution and the experimental data within 5%     

A frontal solution program for finite element analysis

The program given here assembles and solves symmetric positive–definite equations as met in finite element applications. The technique is more involved than the standard band–matrix algorithms, but it is more efficient in the important case when two-dimensional or three-dimensional elements have other than corner nodes. Artifices are included to improve efficiency when there are many right hand sides, as in automated design. The organization of the program is described with reference to diagrams, full notation, specimen input data and supplementary comments on the ASA FORTRAN print-out.     

A split Band-Cholesky equation solving strategy for finite element analysis of transisent field problems

An efficient strategy is outlined for out-of-core solution of the large systems of equations which specify nodal point time derivatives in finite element models of transient flow problems. The positive definiteness, symmetry, and band structure of the finite element mass matrices, as well as the nature of the equation assemblage process, is exploited by the method. Computational results are indicated for systems on the order of several thousand unknowns in size.     

A finite element solution program for large structures

A straightforward and general computer program for assembling and solving (using Gauss elimination technique) widely sparsed finite element matrix equations with very large bandwidth and capable of handling different degrees-of-freedom and variable bandwidth at different nodes, is described herein. The program assembles any type of finite elements having arbitrary number of nodes and each node may have differnt degrees-of-freedom. It requires only a small core memory in the computer, although a fast random access device is also needed. The two very important features of this program are (i) it does not store any zero submatrices within the band and (ii) during the solution of equations all operations dealing with zero submatrices within the band are automatically skipped and thus the savings of a considerable amount of disc storage space and computer time can be effected in many cases. Another feature is that many right hand sides can be handled simultaneously. Hence the program is very economical for structures having widely sparsed matrix equations. A listing of the computer program written in FORTRAN IV for CDC 6400 computer is readily available from the authors, but unfortunately could not be given here because of lack of space. The program is so general that it can be used to solve a wide class of finite element problems without actually having to understand fully the techniques behind it.     

A finite element method for stability problems in finite elasticity

In this paper a finite element method is developed to treat stability problems in finite elasticity. For this purpose the constitutive equations are formulated in principal stretches which allows a general representation of the derivatives of the strain energy function with respect to the principal stretches. These results can then be used to derive an efficient numerical scheme for the computation of singular points.     

A higher-order triangular finite element for the solution of field problems in orthotropic media

This paper presents a triangular finite element for the solution of two-dimensional field problems in orthotropic media. The element has nine degrees of freedom, these being the potential and its two derivatives at each node. The ‘stiffness’ matrix is derived analytically so that no further integration is required when computations are performed using the element. The results obtained using the element are compared with the exact mathematical solution of both a temperature distribution and a torsion problem.     

Three-dimensional non-linear fluid-structure interaction problems: Formulation by the finite and infinite element method

Using the perturbation method, the non-linear exterior fluid-structure interaction problem is separated into first- and second-order problems. With the finite element method for the structure and the finite-infinite element method for the fluid, we obtain a first-order coupled matrix system and a second-order coupled matrix system. By determining the frequencies of resonance of the structure, comparison of numerical results for a vertical cylinder and a horizontal floating cylinder shows the validity of the presented method for the resolution of three-dimensional exterior hydroelastic problems.     

Regular boundary element steady-state solutions of the traveling magnetic field problems

Regular boundary element method (R-BEM) is applied to analyze steady-state traveling magnetic field problems for which convective diffusion equation is considered as governing equation. We deal with a three-dimensional rectangular prism as a simple example in order to study stability and accuracy of regular boundary element (R-BE) solutions. It is found that R-BE solutions are unconditionally stable for a rectangular prism whose sides parallel to a traveling velocity are longer than those perpendicular to the velocity. Furthermore, we can show that R-BE solutions as well as conventional BE solutions have second-order accuracy. Finally, numerical precision is studied through the condition number of the system matrices used in the analysis for a few parameters. It is shown that the R-BEM is available for the analysis of three-dimensional steady-state convective diffusion equations.     

Comparison of finite element techniques for solidification problems

The accuracies of the computed temperatures of a liquid in a corner region under freezing conditions are compared for various fixed-grid finite element techniques using the analytical solution for this problem as a reference. In the finite element formulation of the problem different time-stepping schemes are compared: the implicit Euler-backward algorithm combined with an iterative scheme and two three-time-level methods—the Lees algorithm and a Dupont algorithm, which are both applied as non-iterative schemes. Furthermore, different methods for handling the evolution of latent heat are examined: an approximation method suggested by Lemmon and one suggested by Del Giudice, both using the enthalpy formulation as well as a fictitious heat-flow method presented by Rolph and Bathe. Results of calculations performed with the consistent heat-capacity matrix are compared with those performed with a lumped heat-capacity matrix.