Isoparametric Finite Element Method Research Articles

Fully Discrete Convergence Estimates for Vortex Methods in Bounded Domains

In this paper the authors study vortex method for 2-dimensionsal Euler equations of incompressible flow in bounded domains. To approximate the initial vortex field by a sum of vortex blobs with arbitrary high accuracy, this field is extended smoothly to a small neighborhood of the boundary. And the computation is carried out in the extended domain. To construct the velocity field from vorticity, a second-order isoparametric finite element method is applied, and to solve the characteristic equations, the explicit Euler’s scheme is considered. Optimal error bounds for this fully discrete scheme are obtained.

Numerical computation for heat transfer in crystal growth

The numerical analyses for crystallization processes are selected and summarized from the author’s recent works. The importance of heat transfer may be noted in the phase change manufacturing processes from liquid to solid. The effect of convection on the curved interface may be also noted in the floating zone crystallization process. Specifically, the processes presented are as follows. Three-dimensional cylindrical coordinate model for a Czochralski bulk flow of liquid metal in a horizontal or a vertical magnetic field is presented and numerically solved for selected conditions. Then, a floating zone crystallization method is modeled by an axisymmetric coordinate and then solved by an isoparametric finite element method for curved solid/melt interfaces and a gas/melt interface. Sample computational results are also presented.

Comparing the Time-Domain EM Response of 2-D and Elongated 3-D Conductors Excited by a Rectangular Loop Source.

We develop an algorithm for modelling the time-domain EM response of a general 2-D geometry excited by a 3-D source. A frontal solution, isoparametric finite-element method is used to solve for the y-components of the magnetic and electric fields in the frequency-wavenumber (x, ky, z, w) domain where y is the direction of strike. Step response solutions are computed for 31 frequencies over the range 1 Hz to 100 kHz and transformed into the time-domain with a digital cosine filter. The response of a 2-D simple block target in a uniform half-space is compared to 3-D models of similar cross section with strike lengths of 0.5, 1, and 2 km, computed using the compact finite-element method. Further comparisons were made with the results of a 3-D time-stepping finite-difference solution for a block on the boundary of a quarter-space. In both cases the agreement between the 2.5-D and the 3-D results is close at early delay times with the 3-D target response decaying more rapidly than the 2-D target response at later times as expected. Although the late time-amplitudes differ, the profile shapes are very similar indicating that the 2.5-D model is useful for interpreting practical exploration problems.

The use of alternating direction implicit methods to model diffusion phenomena in elliptic products

Three alternating direction implicit (ADI) methods plus the Crank-Nicolson implicit method and an 8-noded isoparametric finite element method (both included for comparison purposes), are considered to solve numerically transient thermal or mass diffusion problems in elliptic shaped bodies. Moreover, three domain discretization methodologies are proposed: polar cylindrical, unequally and equally spaced orthogonal curvilinear coordinates. In order to analyse indirectly the effectiveness of the methods in different discretizing techniques, all of them are applied to water diffusion during rice grain soaking, and the computed solutions are compared with experimental ones. The influence of various time and space discretizations upon the results is studied, and numerical stability constraints of some of the methods used, as well as performance orientations on personal computer applications are given.

Free vibrations of sandwich plates of variable thickness

The free vibration of sandwich plates of variable thickness is investigated. In this analysis, the face plates are treated as shallow shells which can withstand both membrane forces and bending moments, and the core is assumed to be inextensible in the thickness direction and to provide resistance to transverse shear stresses only. Displacement continuity conditions are used at the interfaces between face plates and the core to derive the displacement field. Energy formulations are obtained and solved by the isoparametric finite element method. Numerical results are compared with existing results for special cases, and also we presented to show the effects of taper constant, face plate thickness and boundary condition on natural frequencies.

FEM analysis of 3-D transformer leakage field and eddy current loss in the windings

The authors present a method which resolves the three-dimensional eddy current field into a 3-D time harmonic field external to the windings and n axisymmetric eddy current fields near the coils as an approximation for engineering applications. Because the isoparametric FEM (finite element method) with 3-D vector potential, is used in the time harmonic field analysis whereas, in the n axisymmetric eddy current field analysis, B-spline FEM is used, the method presented can give a solution with an accuracy quite acceptable for engineering purposes. The distributions of leakage fields, eddy current losses, and heat source in the windings obtained are compared with the test results. >

Stress analysis and the testing of Celanese and IITRI compression specimens

A quadrilateral isoparametric finite element method has been utilized to analyze the Celanese and Illinois Institute of Technology Research Institute (IITRI) compression test specimens. Formulations were based on plane strain and plane stress assumptions, and uniform displacement and uniform stress boundary conditions were both applied to solve the problem. Parametric studies were performed to show the effects on the stress distributions of the end tab material and of the thicknesses of the adhesive layer and the laminate. Experiments with the Celanese test method showed that fracture of the specimen was triggered by global buckling and was initiated from tab tip as a result of high stress concentrations. The measured apparent strength is much lower than the value measured by other testing methods. This study reveals that the Celanese and the IITRI test methods need to be modified for measuring the compressive strength of unidirectional graphite/epoxy composites and probably for other material systems as well. The results of tests on thick IITRI specimens show that appropriate interpretation of the data is necessary because of high stress concentrations and non-uniformity in the specimens.

Humidity sensing properties and electrical permittivity of new photosensitive polyimides

We present experimental and theoreitcal results for the electrical permittivity of new photosensitive as well as humidity-sensitive polyimides, of potential use in capacitive humidity microsensors. Our designs for the latter are based on interdigitated comb-like microelectrodes for which we have developed a fully three-dimensional simulation program to calculate the capacitance. We solve the Laplace equation, subject to appropriate boundary conditions, for the electric potential and field distributions outside the electrodes using the isoparametric finite-element method. The behaviour of the permittivity of these materials, as a function of relative humidity (RH), is studied at different temperatures, frequencies and thicknesses. Their response to humidity is linear for a wide range of RH. The permittivity changes are modelled using an effective-medium theory (ODEEM). Our modelling work is relevant for gas sensors in general, since gases are known to affect the permittivity of polymeric materials.

The crack resistance of a dilatation-sensitive material

The entire history of crack propagation in high-strength steel with dilatation-sensitivity is investigated in this work. Based on the experimentally determined stress-strain curves varying in the strain history according to the mean stress level, a central cracked panel made of unaged maraging steel subject to a remote tension is considered in the analysis. Due to tremendous amount of hydrostatic stress established in the vicinity of the crack tip, the effect of mean stress is accounted for in the onset for plastic flow. The J 2−I 1 non-associate flow rule is incorporated in the incremental analysis employing the isoparametric finite element method with displacement formulation. When the stage of global instability is approached, the possible formation of a secondary crack in front of the primary crack is demonstrated by the use of strain energy density criterion. Also, the size effect on the crack resistance curve is established at the end for the unaged maraging steel.

Three-dimensional modelling of capacitive humidity sensors

Abstract We present the first three-dimensional modelling of integrated capacitive humidity sensors based on interdigitated microelectrodes coated with a moisture-absorbing dielectric layer. We obtain the electric potential and field distributions outside the electrodes by solving the Laplace equation, subject to appropriate boundary conditions, using the isoparametric finite-element method. The solution also determines the surface charge induced on the electrodes, from which the capacitance of the full structure is readily obtained. We obtain excellent agreement with the experimentally measured values of the capacitance. Apart from humidity sensors, our modelling results are relevant for capacitive microdielectrometry in general.

Element by element a posteriori error estimation and improvement of stress solutions for two‐dimensional elastic problems

AbstractA quantitative method of a posteriori error estimation based on finite element solutions is developed. The proposed method can estimate the error of solutions obtained by the finite element method for two‐dimensional elastic problems with 4‐node elements. Since the error is estimated element by element, the proposed method does not require a large memory or long computing time. Not only rectangular elements but also arbitrarily shaped 4‐node elements can be used in this method for estimating the error in the finite element computation with high accuracy. The finite element solutions are improved by adding the estimated errors onto the original solutions. The proposed method can be utilized for any type of linear problem if an isoparametric finite element method is used.

Investigation of a Posteriori Error Analysis for Plane Stress Problems by the Finite Element Method and Updating Stresses

The estimation of the discretisation error in the finite element method is considered in this paper. The proposed method can estimate solution errors obtained by the finite element method for 2-dimensional elastic problems with 4-node elements. Because the quadratic term is expected to dominate the error field, an 8-node element is utilised in the estimation. Since error is estimated element by element, this does not lead to excessive computer memory or processing time usage. Not only rectangular elements but also any shape 4-node elements can be used for highly accurate error estimation in finite element computation. An error norm is computed in the paper which can indicate a relative error in the elements. The finite element solutions are also considerably improved by adding the estimated errors onto the original solutions. The proposed method can be utilised for any type of linear problem if an isoparametric finite element method is used.

Prediction of mixed-mode cracking direction in random, short-fibre composite materials

The prediction of cracking direction in composite materials is of significance to the design of composite structures. This paper presents several methods for predicting the cracking direction in the double grooved tension-shear specimen which gives mixed-mode cracking. Five different criteria are used in this analysis: two of them have been used by other investigators and the others are proposed by the present authors. The strain energy density criterion proposed by G.C. Sih is modified to take account of the influence of the anisotropy of the strength on the direction of crack. The two failure criteria of Tsai-Hill and Norris are extended to predict the crack orientation. The stress distributions in the near-notch zone are calculated by using the 8-node quadrilateral isoparametric finite element method. The predictions of all the criteria except one are in good agreement with the experimental measurement. In addition, on the basis of the FEM results, the size of the zone in which the singular term is dominant is estimated.

A STUDY ON ANALYSIS AND DESIGN OF WEB PLATE IN CURVED GIRDER BRIDGES SUBJECTED TO BENDING

This paper presents a design method of the web slenderness ratio and the corresponding rigidity of longitudinal stiffeners in the curved plate girders. The elastic non-linear behaviors of the curved web panels with/without longitudinal stiffeners subjected to pure bending are inquired by means of analysis using the up-dated lagrangian and an isoparametric finite element method. Based upon these analysis, a required slenderness ratio of web panels is investigated by setting a limit state concerning the maximum out-of-plane deflections and bending stresses. In order to evaluate the strength of the longitudinal stiffeners in the ultimate limit state of curved web panel, a curved beam-column model is proposed, then a design method for the required relative stiffness of the stiffeners is, moreover, recomended by adopting interaction curve of this beam-column model.

Some recent developments in field calculations

The paper surveys several numerical techniques, under development at the University of Manitoba, that yield improved computational efficiency in the solution of difficult magnetic field problems. The paper includes the following: a description of an isoparametric finite element method for continuously inhomogeneous and orthotropic media within curved boundaries; a description of an isoparametric boundary element method (using integral equation or source distribution techniques) for extensive or open-region problems; and coupled partial differential and integral equation techniques (i.e., picture-frame methods) which are of particular use in solving problems that involve inhomogeneous/orthotropic media within localized regions of free space. Mention is made of sparsity techniques, for the solution of large and sparse systems of equations, and of Kron's diakoptics which is a method of tearing a large system into more easily managed subsystems.

Higher-Order Approximate Solutions of Neumann Problems by Isoparametric Finite Element Method with Relevant Lumping Operator

Jn 1966, Friedriehs and Keller [1 j proposed a generalized finite difference scheme for the Neumann problem. Today, their method can be regarded as a finite element method using the piecewise linear chapeau bases. From the viewpoint of computations, however, their method is not convenient enough, except the case of a polygonal domain, since it requires to perform all the integrations which appear in obtaining Ritz coefficients and right-hand sides of determinate linear equations, in an exact domain and on an exact boundary. The purpose of this paper is to consider a higher-order approximate procedure for the Neumann problem in an approximate domain, using isoparametric finite elements. This forces us to change slightly the data functions to gtiarantee the solvability of the determinate linear system. We shall stud some kinds of errors, which are caused by the selection of the basis functions (approximation errors), by the change in domain, by the approximation of data functions and b the change of data functions for the solvability. Errors caused by the selection of die basis functions are, of course, inevitable. Tt is desirable tha t errors due to other reasons mentioned above do not destroy the order of accuracy which the basis functions could achieve. In this sense, it appears to be reasonable to expect that, when the boundary P is curved, the optimal order of accuracy can be achie ed by the use of isoparametric technique. Also, we may apply the lumping technique to the body force term/, which may simplify

A theoretical and experimental investigation into soil–structure interaction

In this Paper the three-dimensional isoparametric finite element method is used to carry out the non-linear analysis of soil, foundations and superstructures as parts of a single integral system. The method is then employed to calculate the stresses and deformations of space frames resting on a bed of sand. The stress contours σx, σy and σz throughout the depth of the foundations are presented. Details of theoretical and experimental investigations into soil-structure interaction are presented. Particular attention is given to the problems of differential settlement of the foundations and the sway deflexion of the space frames. Comparisons are made between the theoretical results and those obtained experimentally. Dans cette étude la méthode des éléments finis isoparamétriques à trois dimensions est utilisée afin de mettre en application l'analyse non-linéaire du sol, des fondations et des superstructures comme faisant partie d'un seul système intégral. La méthode est alors employée pour calculer les contraintes et les déformations d'une structure spatiale reposant sur un lit de sable. L'enveloppe des contraintes σx, σy, σz dans toute la profondeur des fondations est présentée. Les détails d'investigations théoriques et expérimentales de l'interaction sol-structure sont présentés. Une attention particulière est portée aux problèmes de tassements différentiels des fondations et la balance de déflexion des structures spatiales. Des comparaisons sont faites entre les résultats théoriques et expérimentaux.