Showing papers on "Extended finite element method published in 1994"
Book•
01 Jan 1994TL;DR: In this paper, the authors present a study on the effect of differential expressions on the performance of a computations of a continuous-time transfer of heat transfer and fluid flow.
Abstract: EQUATIONS OF HEAT TRANSFER AND FLUID MECHANICS Present Study Governing Equations of a Continuum Governing Equations in Terms of Primitive Variables Porous Flow Equations Auxiliary Transport Equations sChemically Reacting Systems Boundary Conditions sChange of Phase Enclosure Radiation Summary of Equations THE FINITE ELEMENT METHOD: AN OVERVIEW Model Differential Equation Finite Element Approximation Weighted-Integral Statements and Weak Forms Finite Element Model Interpolation Functions Assembly of Elements Time-Dependent Problems Axisymmetric Problems Convective Boundary Conditions Library of Finite Elements Numerical Integration Modeling Considerations Illustrative Examples 3D CONDUCTION HEAT TRANSFER Semidiscrete Finite Element Model Interpolation Functions Numerical Integration Computation of Surface Fluxes Semidiscrete Finite Element Model Solution of Nonlinear Equations Radiation Solution Algorithms Variable Properties sPost-Processing Operations sAdvanced Topics in Conduction sExamples of Diffusion Problems VISCOUS INCOMPRESSIBLE FLOWS Mixed Finite Element Model Penalty Finite Element Models Finite Element Models of Porous Flow Computational Considerations Solution of Nonlinear Equations Time-Approximation Schemes sStabilized Methods Post-Processing sAdvanced Topics Advanced Topics - Turbulence Numerical Examples CONVECTIVE HEAT TRANSFER Mixed Finite Element Model Penalty Finite Element Model Finite Element Models of Porous Flow Solution Methods Convection with Change of Phase Convection with Enclosure Radiation Post-Computation of Heat Flux Advanced Topics - Turbulent Heat Transfer Advanced Topics - Chemically Reacting Systems Numerical Examples sNON-NEWTONIAN FLUIDS Governing Equations of Inelastic Fluids Finite Element Models of Inelastic Fluids Solution Methods for Inelastic Fluids Governing Equations of Viscoelastic Fluids Finite Element Model of Differential Form Finite Element Model of Integral Form Unresolved Problems Numerical Examples sCOUPLED PROBLEMS Coupled Boundary Value Problems Fluid Mechanics and Heat Transfer Solid Mechanics Electromagnetics Coupled Problems in Mechanics Implementation of Coupled Algorithms Numerical Examples sADVANCED TOPICS Parallel Processing Other Topics Note: Chapters also include an Introduction, Exercises, and References APPENDIX A: COMPUTER PROGRAM--HEATFLOW Heat Transfer and Related Problems Flows of Viscous Incompressible Fluids Description of the Input Data A Source Listings of Selective Subroutines Reference sAPPENDIX B: SOLUTION OF LINEAR EQUATIONS Introduction Direct Methods Iterative Methods References for Additional Reading sAPPENDIX C: FIXED POINT METHODS AND CONTRACTION MAPPINGS Fixed Point Theorem Chord Method Newton's Method The Newton-Raphson Method Descent Methods References for Additional Reading
876 citations
Book•
01 Jan 1994
512 citations
TL;DR: The findings suggest that the finite element method is suitable for studying normal and pathological cardiac activation and has significant advantages over existing techniques.
Abstract: A new computational method was developed for modeling the effects of the geometric complexity, nonuniform muscle fiber orientation, and material inhomogeneity of the ventricular wall on cardiac impulse propagation. The method was used to solve a modification to the FitzHugh-Nagumo system of equations. The geometry, local muscle fiber orientation, and material parameters of the domain were defined using linear Lagrange or cubic Hermite finite element interpolation. Spatial variations of time-dependent excitation and recovery variables were approximated using cubic Hermite finite element interpolation, and the governing finite element equations were assembled using the collocation method. To overcome the deficiencies of conventional collocation methods on irregular domains, Galerkin equations for the no-flux boundary conditions were used instead of collocation equations for the boundary degrees-of-freedom. The resulting system was evolved using an adaptive Runge-Kutta method. Converged two-dimensional simulations of normal propagation showed that this method requires less CPU time than a traditional finite difference discretization. The model also reproduced several other physiologic phenomena known to be important in arrhythmogenesis including: Wenckebach periodicity, slowed propagation and unidirectional block due to wavefront curvature, reentry around a fixed obstacle, and spiral wave reentry. In a new result, the authors observed wavespeed variations and block due to nonuniform muscle fiber orientation. The findings suggest that the finite element method is suitable for studying normal and pathological cardiac activation and has significant advantages over existing techniques. >
402 citations
01 Jan 1994
250 citations
TL;DR: In this article, a second order linear scalar differential equation including a zero-th order term is approximated using first the standard Galerkin method enriched with bubble functions, and then the method is generalized to allow for a convection operator in the equation.
218 citations
TL;DR: In this article, the accuracy of the Darcy velocity, flux, and stream function computed from lowest-order, triangle-based, control volume and mixed finite element approximations to the two-dimensional pressure equation is considered.
Abstract: The accuracy of the Darcy velocity, flux, and stream function computed from lowest-order, triangle-based, control volume and mixed finite element approximations to the two-dimensional pressure equation is considered. The control volume finite element method, similar to integrated finite difference methods and analogous to the interpolation of Galerkin finite element results over “control volumes,” is shown to yield a conservative velocity field and smooth streamlines. The streamlines and fluxes through the system computed with the control volume finite element approach are compared to those computed from the mixed finite element method, which approximates the pressure and velocity variables separately. It is shown that for systems with only moderate degrees of heterogeneity, the control volume finite element method is the more computationally efficient alternative; i.e., it provides more accurate flow results for a given number of unknowns. For more variable or discontinuous permeability fields, by contrast, such as sand/shale systems, the mixed finite element method is shown to approximate flow variables more accurately and more realistically than the control volume method with the same number of unknowns.
206 citations
01 Dec 1994
TL;DR: In this article, the authors discuss the advantages and disadvantages of finite element method and compare and contrast the Rayleigh comment on both the methods and explain the various teps involved in finite Element method and explain them through an Example.
Abstract: 1. If a displacement field is described as follows, u = (−x+2y+6xy)10 and v = (3x + 6y Determine the strain components v xx , v yy , and v xy at the point x = 1; y = 0. 2. Explain briefly about the following: (a) Variational method. (b) Importance of Boundary cond itions. 3. Discuss the following basic principles of finite element method. (a) Derivation of element stiffness matrix. (b) Assembly of Global stiffness Matrix. 4.What are the various teps involved in finite Element method and explain them through an Example 5. Compare and contrast the “Rayleigh comment on both the methods. 6. What are the various approximate methods of anal ysis and exp 7. (a) Explain the advantages and disadvantages of Finite Element Method. (b) What is meant by total potential of elastic str uc ure? Write the expression for total potential of a cantilever beam with uniformly dist ributed 8. In a plane strain problem, we have σx = 137.90*10 6 Pa σy= -68.95*10 Determine the value of the stress
200 citations
TL;DR: In this article, a comparison between the mixed hybrid finite element and the standard finite element method is performed in the bidimensional case with a triangular space discretization, and the results of the simulations are presented in the form of streamlines.
Abstract: Selected groundwater flow scenarios are used in a two-way comparison between the mixed hybrid finite element method and the standard finite element method (also called the conforming finite element method). The simulations presented are performed in the bidimensional case with a triangular space discretization because of its practical interest for hydrogeologists. The basic idea of the mixed procedure is to approximate both the hydraulic potential and the velocity simultaneously and to satisfy an exact water balance for each element. By contrast, the conforming finite element method calculates the potential field everywhere and then calculates the velocity by differentiation of the potential. The conventional approach results in an elementwise constant velocity which can be subject to significant problems because of the normal component discontinuity of the velocity. The mixed hybrid finite element method provides velocities everywhere in the field, as well as potentials at the center of each element and each edge. Moreover, the normal component of the velocity field is continuous between adjacent elements. The results of the simulations are presented in the form of streamlines. To avoid the problem of velocity discontinuity, the method of Cordes and Kinzelbach (1992) is used; it allows the construction of a continuous velocity field from potentials obtained by the conforming finite element method. The comparison studies show that the mixed hybrid finite element is superior to the conforming method in terms of accuracy. It is also superior to the conforming method in terms of computational effort. The potential fields obtained by the mixed hybrid and the conforming finite element methods are the same.
177 citations
TL;DR: In this article, a new four-node shell element for nonlinear analysis which is useful for explicit time integration with single point quadrature is presented, and an assumed strain method is used to stabilize the zero-energy modes of the element.
TL;DR: This paper presents the first endeavour to exploit a generalized differential quadrature method as an accurate, efficient and simple numerical technique for structural analysis as well as seeking an alternate numerical method using fewer grid points to find results with acceptable accuracy.
Abstract: SUMMARY This paper presents the first endeavour to exploit a generalized differential quadrature method as an accurate, efficient and simple numerical technique for structural analysis. Firstly, drawbacks existing in the method of differential quadrature (DQ) are evaluated and discussed. Then, an improved and simpler generalized differential quadrature method (GDQ) is introduced to overcome the existing drawback and to simplify the procedure for determining the weighting coefficients. Subsequently, the generalized differential quadrature is systematically employed to solve problems in structural analysis. Numerical examples have shown the superb accuracy, efficiency, convenience and the great potential of this method. Numerical approximation methods for solving partial differential equations have been widely used in various engineering fields. Classical techniques such as finite element and finite difference methods are well developed and well known. These methods can provide very accurate results by using a large number of grid points. However, in a large number of cases, reasonably approximate solutions are desired at only a few specific points in the physical domain. In order to get results even at or around a point of interest with acceptable accuracy, conventional finite element and finite difference methods still require the use of a large number of grid points. Consequently, the requirement for computer capacity is often unnecessarily large in such cases. In seeking an alternate numerical method using fewer grid points to find results with acceptable accuracy, the method of differential quadrature (DQ) was introduced by Bellman et a!.'. The method of DQ is a global approximate method. This method is based on the ideas that the derivative of a function with respect to a co-ordinate direction can be expressed as a weighted linear sum of all the function values at all mesh points along that direction and that a continuous function can be approximated by a higher-order polynomial in the overall domain. The DQ method differs from the finite element method (FEM) in two aspects. Firstly, the FEM uses lower-order polynomials to approximate a function on a local element level, while the DQ method approximates a function on the global area using higher-order polynomials. Secondly, the DQ method directly approximates the derivatives of a function at a point, while the FEM approximates a function over a local element and the derivatives can then be derived from the approximate function. In this aspect, the DQ method is more similar to the finite difference method (FDM). However, the FDM is also a local approximation method based on lower-order polynomial approximation. In fact, it can be shown that the FDM is just a special case of the DQ method where it is applied locally on the range [.xi- xi + l]. Owing to the higher-order
TL;DR: A system of adaptive procedures for large-deformation finite element analysis of elastic and elastoplastic problems using the h -refinement approach is presented and demonstrated to be effective in solutions of two-dimensional stress analysis problems including contact conditions.
TL;DR: In this paper, the authors considered the application of least square principles to the approximate solution of the Stokes equations cast into a first-order velocity-vorticity-pressure system.
Abstract: In this paper we consider the application of least-squares principles to the approximate solution of the Stokes equations cast into a first-order velocity-vorticity-pressure system. Among the most attractive features of the resulting methods are that the choice of approximating spaces is not subject to the LBB condition and a single continuous piecewise polynomial space can be used for the approximation of all unknowns, that the resulting discretized problems involve only symmetric, positive definite systems of algebraic equations, that no artificial boundary conditions for the vorticity need be devised, and that accurate approximations are obtained for all variables, including the vorticity. Here we study two classes of least-squares methods for the velocity-vorticity-pressure equations. The first one uses norms prescribed by the a priori estimates of Agmon, Douglis, and Nirenberg and can be analyzed in a completely standard manner. However, conforming discretizations of these methods require C continuity of the finite element spaces, thus negating the advantages of the velocity-vorticity-pressure fomulation. The second class uses weighted L-norms of the residuals to circumvent this flaw. For properly choosen mesh-dependent weights, it is shown that the approximations to the solutions of the Stokes equations are of optimal order. The results of some computational experiments are also provided; these illustrate, among other things, the necessity of introducing the weights. AMS Subject Classification: 65N30, 65N12, 76M10
TL;DR: The development, validation, and application of a new finite element scheme for the solution of the compressible Euler equations on unstructured grids is described and a flow solution about a complete F-18 fighter is shown to demonstrate the accuracy and robustness of the proposed algorithm.
Abstract: We describe the development, validation, and application of a new finite element scheme for the solution of the compressible Euler equations on unstructured grids. The implementation of the numerical scheme is based on an edge-based data structure, as opposed to a more traditional element-based data structure. The use of this edge-based data structure not only improves the efficiency of the algorithm but also enables a straightforward implementation of upwind schemes in the context of finite element methods. The algorithm has been tested and validated on some well-documented configurations. A flow solution about a complete F-18 fighter is shown to demonstrate the accuracy and robustness of the proposed algorithm
TL;DR: In this article, the authors extended the standard dispersion analysis technique to include complex wavenumbers and used this complex Fourier analysis to examine the dispersion and attenuation characteristics of the p-version finite element method.
Abstract: High-order finite element discretizations of the reduced wave equation have frequency bands where the solutions are harmonic decaying waves. In these so called “stopping” bands, the solutions are not purely propagating (real wavenumbers) but are attenuated (complex wavenumbers). In this paper we extend the standard dispersion analysis technique to include complex wavenumbers. We then use this complex Fourier analysis technique to examine the dispersion and attenuation characteristics of the p-version finite element method. Practical guidelines are reported for phase and amplitude accuracy in terms of the spectral order and the number of elements per wavelength.
TL;DR: In this paper, the interior elastoacoustic problem is solved by a finite element method, which does not present spurious or circulation modes for nonzero frequencies, and consists of classical triangular lagrangian elements for the solid and lowest order triangular Raviart-Thomas elements for fluid.
TL;DR: In this article, the velocity and pressure fields, free surface shape and wave speed are computed simultaneously as functions of the Reynolds number Re and the wave number μ, and compared with predictions of long-wave, asymptotic theories and boundary-layer approximations for the form and nonlinear transitions of finite-amplitude waves that evolve from the flat film state.
Abstract: Finite‐amplitude waves propagating at constant speed down an inclined fluid layer are computed by finite element analysis of the Navier–Stokes equations written in a reference frame translating at the wave speed. The velocity and pressure fields, free‐surface shape and wave speed are computed simultaneously as functions of the Reynolds number Re and the wave number μ. The finite element results are compared with predictions of long‐wave, asymptotic theories and boundary‐layer approximations for the form and nonlinear transitions of finite‐amplitude waves that evolve from the flat film state. Comparisons between the finite element calculations and the long‐wave predictions for fixed μ and increasing Re agree well for small‐amplitude waves. However, for larger‐amplitude waves the long‐wave results diverge qualitatively from the finite element predictions; the long‐wave theories predict limit points in the solution families that do not exist in the finite element solutions. Comparisons between the finite ele...
TL;DR: A comparison between the finite element and the finite volume methods is presented in the context of elliptic, convective–diffusion and fluid flow problems and it is shown that in many cases both techniques are completely equivalent.
Abstract: In this paper a comparison between the finite element and the finite volume methods is presented in the context of elliptic, convective–diffusion and fluid flow problems. The paper shows that both procedures share a number of features, like mesh discretization and approximation. Moreover, it is shown that in many cases both techniques are completely equivalent.
TL;DR: In this paper, a number of recent developments in the finite element analysis of metal forming are discussed, with emphasis on the advantages and disadvantages of each technique for complex analyses involving contact.
TL;DR: In this article, the authors present a method for the numerical computation of the wavenumbers and associated modes of the cross-section of thin-walled waveguides based on finite element techniques, and compare results obtained by the finite element method, by Donnell's shell theory and by simple Euler-Bernoulli beam theory.
TL;DR: In this article, finite element models based on discrete-layer theories are presented for the coupled-field analysis of laminated plates containing piezoelectric layers, which allows for piecewise approximations of the variables through the thickness of each layer.
Abstract: Finite element models based on discrete-layer theories are presented for the coupled-field analysis of laminated plates containing piezoelectric layers. The three displacements and the electrostatic potential are treated as unknowns in this formulation, which allows for piece-wise approximations of the variables through the thickness of each layer. Two specific models are demonstrated in which the transverse displacement is either variable or constant, and the in-plane displacements and potential take piece-wise linear approximations through the thickness. The models are applied to example problems with applied surface tractions and specified surface potentials. Good agreement is found with exact solutions.
TL;DR: In this article, a pointwise equilibrating polynomial (PEP) element is proposed for nonlinear analysis of frames in which each member can be modeled by one element in most cases.
Abstract: A pointwise equilibrating polynomial (PEP) element is proposed for nonlinear analysis of frames in which each member can be modeled by one element in most cases. The formulation of the element is based on the imposition of compatibility conditions at end nodes as well as the satisfaction of equilibrium at midspan. The resulting expression for this new element is accurate in describing the force‐versus‐displacement relations at the element level, is simple, and it does not lead to a significant numerical truncating error in the computer analysis. Its implementation in a nonlinear analysis program is straightforward. The accuracy of the analysis results by the element was found to be considerably higher than its cubic counterpart in a number of well‐known examples. The limitation or inconvenience of the method of stability function such as the separation of the solution for tensile, compressive, and zero load cases is eliminated.
TL;DR: In this article, a parametric updating using modal test results is proposed to correct both the stiffness and the mass matrices of a given finite element model when the measures are noisy.
Abstract: Today the adjustment of structural models is an essential step in the modeling of complex structures. In this paper, we are interested in the improvement of finite element models. Our approach is a parametric updating using modal test results, which supply eigenvalues and associated eigenvectors. It is based on the computation of the error measure on the constitutive relation and allows us to correct both the stiffness and the mass matrices. In particular, this paper shows how this tuning strategy can improve a given finite element model when the measures are noisy. Several simulation examples illustrate the behavior of this method. A E e
TL;DR: In this paper, a two-dimensional eight-noded isoparametric finite element is derived for modeling the distributed coupled electromechanical behavior using higher order displacement theory, and the results obtained by the present finite element method for a given distributed mechanical load and with or without describing the electric potential on the surface of the actuator are compared with the exact solution.
TL;DR: It is proved that the natural extension of this finite element approximation to the original domain is optimal-order accurate.
Abstract: In this paper the authors consider a simple finite element method on an approximately polygonal domain using linear elements. The Dirichlet data are transferred in a natural way and the resulting linear system can be solved using multigrid techniques. Their analysis takes into account the change in domain and data transfer, and optimal-error estimates are obtained that are robust in the regularity of the boundary data provided they are at least square integrable. It is proved that the natural extension of this finite element approximation to the original domain is optimal-order accurate.
TL;DR: In this article, a hybrid vector finite element method was used for full-wave analysis of lossy dielectric waveguides, where edge elements and first-order nodal finite element basis functions were used to span the transverse and the z components of the electric field, respectively.
Abstract: This paper presents a full-wave analysis of lossy dielectric waveguides using a hybrid vector finite element method. To avoid the occurrences of spurious modes in the formulation, edge elements and first-order nodal finite element basis functions are used to span the transverse and the z components of the electric field, respectively. Furthermore, the direct matrix solution technique with minimum degree of reordering has been combined with the modified Lanczos algorithm to solve for the resultant sparse generalized eigenmatrix equation efficiently. >
TL;DR: In this paper, a triangular bending element based on an efficient higher order plate theory is developed for symmetric laminated composites, which provides an efficient and accurate tool for the analysis of symmetric multilayered composite plates.
Abstract: A triangular bending element based on an efficient higher order plate theory is developed for symmetric laminated composites. This nonconforming element has five degrees of freedom in each node. It passes proper bending and shear patch tests in arbitrary meshes in isotropic materials. Thus it converges to the exact solution. To demonstrate the element and compare with other theories, finite element solutions are obtained for a static bending problem under sinusoidal loading. The present finite element results give deflections and stresses that are in good agreement with three-dimensional elasticity solutions. Thus this element provides an efficient and accurate tool for the analysis of symmetric multilayered composite plates
TL;DR: In this paper, a 24 degree of freedom quadrilateral shell element is developed and a status report on the formulation method is provided by pointing out what works and what does not.