Showing papers on "Extended finite element method published in 1972"
Book•
01 Jan 1972
697 citations
01 Jan 1972
TL;DR: In this paper, the authors focus on variational crimes in the finite element method and discuss the convergence theory for non-conforming elements, and the modifications of the Ritz procedure which have been made to achieve an efficient finite element system.
Abstract: Publisher Summary This chapter focuses on variational crimes in the finite element method. The finite element method is nearly a special case of the Rayleigh-Ritz technique. The convenience and effectiveness of the finite element technique is regarded as conclusively established; it has brought a revolution in the calculations of structural mechanics, and other applications are rapidly developing. The chapter reviews the modifications of the Ritz procedure which have been made to achieve an efficient finite element system. On a regular mesh one could regard the system of Ritz-finite element equations KQ = F as a finite difference scheme, and then the patch test would be equivalent to the formal consistency of the difference equations with the correct differential equation. The chapter discusses the convergence theory for non-conforming elements. It is impossible for a polynomial to satisfy a condition like u = 0 on a general curved boundary. Therefore, some alteration in the boundary condition will be necessary. The most important possibility is to change the domain.
260 citations
01 Jan 1972
253 citations
TL;DR: In this article, a general theory for obtaining asymptotic estimates of the form ∥ u − Πu ∥ Hm ( K ) = O ( h k + 1− m ).
247 citations
174 citations
TL;DR: In this article, a general quadrilateral multilayer plate element is derived using the hybrid-stress method and a comparison of results with known solutions is made and excellent accuracy in predicting both displacements and stresses is observed.
Abstract: In view of the increasing interest in using composite materials for aerospace structures, the analysis of layered anisotropic plates becomes essential. The so-called classical laminated-plate theory (CPT) does not include effects of shear deformation. Recent elasticity solutions [ 1, 2, 3] of laminated plates show the importance of the transverse shear effects. While analytical solutions are usually restricted to problems with simple loading and boundary conditions, the finite-element method is found attractive in dealing with complicated problems. To include the effects of transverseshear deformation, the rotation of surface normal for different layers of a laminated plate must be considered as different. This has been found to be not feasible by the conventional finite-element displacement method [4]. The above-mentioned difficulties can be avoided if a hybrid-stress finite-element model is used [5, 6, 7]. In this paper, a general quadrilateral multilayer plate element is derived using the hybrid-stress method. The transverse shear effects, as well as the coupling effects of stretching and bending, are included. A comparison of results with known solutions is made and excellent accuracy in predicting both displacements and stresses is observed.
153 citations
TL;DR: In this article, the method of forming curved finite element shape functions from simple independent generalized strain functions is applied to a rectangular cylindrical shell element, which has 20 degrees of freedom and satisfies the conditions for rigid body displacements and constant strain (in so far as this is allowed by compatibility equations).
139 citations
TL;DR: In this paper, it was shown that if the trial space is complete through polynomials of degreek?1, then it contains a functionv h such that |u?v h | s?ch k?s|u| k.
Abstract: The rate of convergence of the finite element method depends on the order to which the solutionu can be approximated by the trial space of piecewise polynomials. We attempt to unify the many published estimates, by proving that if the trial space is complete through polynomials of degreek?1, then it contains a functionv h such that |u?v h | s ?ch k?s|u| k . The derivatives of orders andk are measured either in the maximum norm or in the mean-square norm, and the estimate can be made local: the error in a given element depends on the diameterh i of that element. The proof applies to domains Ω in any number of dimensions, and employs a uniformity assumption which avoids degenerate element shapes.
138 citations
TL;DR: In this article, a Timoshenko beam finite element based on exact differential equations of an infinitesimal element in static equilibrium is presented, and convergence tests are performed for a simply-supported beam and a cantilever.
115 citations
TL;DR: In this paper, the authors studied the convergence rate of the finite element method on polygonal domains in weighted Sobolev spaces, and showed that the use of different spaces of trial and test functions will restrict the usual low rate of convergence to a neighborhood of each vertex of the polygon.
Abstract: This paper is concerned with the rate of convergence of the finite element method on polygonal domains in weighted Sobolev spaces. It is shown that the use of different spaces of trial and test functions will restrict the usual low rate of convergence to a neighborhood of each vertex of the polygonal domain.L 2-convergence and lower bounds on the error are also studied.
107 citations
TL;DR: In this article, an estimate for the error of derivative interpolation is obtained for the finite element method, and the particular interpolation to which this estimate refers has been considered in a previous paper.
Abstract: An estimate for the error of derivative interpolation is obtained. The particular interpolation to which this estimate refers has been considered in a previous paper ; the present paper is an extension of this previous work, and is of special importance for the finite element method.
TL;DR: The chapter presents the several variational principles and the corresponding models used in the finite element formulation and discusses equilibrium problems of linear elastic solids.
Abstract: Publisher Summary The chapter presents a brief introduction to the different finite element formulations for linear elastic solids and discusses similar formulations for several other field problems. The chapter presents detailed illustrations for several typical finite element formulations. In the finite element formulation, displacement and stress fields are assumed to be continuous within each discrete element. This formulation calls for modified variational principles for which the continuity or equilibrium conditions along the interelement boundaries are introduced as conditions of constraint and appropriate boundary variables are used as the corresponding Lagrangian multipliers. The chapter presents the several variational principles and the corresponding models used in the finite element formulation. The large majority of the existing finite element formulations are based on the assumed displacement approach. The chapter discusses equilibrium problems of linear elastic solids. There are several other problems in solid mechanics, which can be formulated by means of variational principles and hence can be solved by finite element methods. The finite element methods have also been extended to nonlinear problems resulting from elastic-plastic material properties or from large deflections or finite strains.
TL;DR: In this article, a constant curvature beam finite element based on thin beam theory is shown to converge onto exact frequencies, and the element which allows shear deformation and rotary inertia is given by more accurate finite element analysis providing the correct value of shear coefficient is used.
TL;DR: In this paper, the finite element method is applied to the solution f seepage problems where both electrokinetic and hydrodynamic forces occur, and the resulting system of coupled equations is used to solve both the steady state and transient conditions in a one-dimensional system and compared with the theoretical result.
Abstract: The finite element method is applied to the solution f seepage problems where both electrokinetic and hydrodynamic forces occur. The resulting system of coupled equations is used to solve both the steady state and transient conditions in a one-dimensional system and compared with the theoretical result. An analysis is then made of a more complex two-dimensional problem where the application of electro-osmosis may be used successfully to prevent piping.
TL;DR: In this paper, a report on recent developments in the application of the finite element method in the analysis of elastohydrodynamic lubrication (e.g., H.h.l.) problems is presented.
Abstract: This paper comprises a report on recent developments in the application of the finite element method in the analysis of elastohydrodynamic lubrication (e.h.l.) problems. The basic formulation is effected, using the Galerkin approach and the domain under investigation is discretized using isoparametric elements. The techniques used to locate the inlet and outlet boundaries and those employed during successive iterations are illustrated by application to particular examples.
TL;DR: An iterative approach which combines the advantages of a finite element method and a standard finite difference technique is developed for the analysis of plates on elastic foundation subjected to general loadings and arbitrary edge support conditions as mentioned in this paper.
TL;DR: In this article, a simplified form of Cantin and Clough's cylindrical shell element, reducing the size of the element stiffness matrix from 24 × 24 to 20 × 20, is presented.
TL;DR: In this article, a finite element analysis for finite and infinite solid or hollow cylinders in axisymmetric vibration is presented, and excellent agreement is found with those from the exact Pochhammer theory and Mindlin and McNiven's three-mode theory.
TL;DR: In this article, the finite element method is applied to the modelling of magnetotelluric problems and its adaptation to geological profiles is outlined, and a novel method for obtaining surface field values, involving matrix representation of the normal derivative operator, is presented in detail.
Abstract: The finite element method, here viewed as a special case of the Galerkin projective method, is applied to the modelling of magnetotelluric problems, and its adaptation to geological profiles is outlined A novel method for obtaining surface field values, involving matrix representation of the normal derivative operator, is presented in detail
Results obtained by this method are compared with well-known infinite series solutions for the vertical fault and the outcropping dyke Two profiles containing sulphide zones are also modelled, the results being compared with field data; satisfactory agreement is obtained
TL;DR: In this paper, the authors consider matrices arising from the use of finite element techniques in least square approximation and in elliptic partial differential equations, and study their properties of numerical stability, and establish bounds for their inverses with respect to the uniform norm.
Abstract: This paper considers matrices arising from the use of finite element techniques in least-squares approximation and in elliptic partial differential equations; it studies their properties of numerical stability, and in particular, it establishes bounds for their inverses with respect to the uniform norm.
TL;DR: A gradient minimization technique for the solution of the general algebraic eigenproblem Kx = λMx arising from the application of the finite element method consists of a simultaneous linear and directional searches so as to obtain the highest rate of convergence.
TL;DR: In this paper, an alternative approach to the usual finite element treatment of temperature problems is presented, using approximations for the field of the dual variables, and the appropriate extremum principle is established and its minimization is discussed in connection with a plane triangular finite element process.
Abstract: An alternative approach to the usual finite element treatment of steady-state temperature problems is presented, using approximations for the field of the dual variables.
The appropriate extremum principle is established and its minimization is discussed in connection with a plane triangular finite element process. Original heat flow elements are derived: in conjunction with temperature elements, they enable dual analysis of a given structure and an important estimate of the convergence to the true solution by upper and lower bounds to the dissipation function, as illustrated by means of several examples.
TL;DR: In this paper, the simplified hybrid displacement method for plate bending is given systematically with some extension made to allow for the discontinuity of deflection field, and eight types of finite elements are developed utilizing this principle.
TL;DR: In this article, a displacement-type finite element method is applied to reduce the governing partial differential equations to a set of simultaneous nonlinear ordinary differential equations of motion of a lumped mass system connected by three-dimensional elements.
01 Jan 1972
TL;DR: This chapter reviews various finite element formulations for two typical partial differential equations which has its application in many continuum mechanics problems typical of which are torsion of prismatic bars, deflection of stretched membranes, heat conduction, and potential flow.
Abstract: Publisher Summary This chapter focuses on finite element formulation by variational principles with relaxed continuity requirements. The finite element method has long been recognized as an extension of the well-known Ritz procedure for constructing approximate solutions to the governing variational principle associated by a given boundary value problem. The method was originally used in the analysis of solid mechanics problems and several alternative variational principles in elasticity have been employed in the finite element formulation. As in the finite element method a continuum is subdivided by finite element mesh, it is possible to modify the variational principles by allowing discontinuous fields at the interelement boundaries and hence, to create the so-called hybrid models in finite element analyses. The chapter reviews various finite element formulations for two typical partial differential equations. The first one involves the harmonic equation which has its application in many continuum mechanics problems typical of which are torsion of prismatic bars, deflection of stretched membranes, heat conduction, and potential flow. The second one is associated with the bi-harmonic equation which is associated, for example, with the bending of thin plates.
TL;DR: In this article, a polynomial representation over each finite element of the excitation cross spectral density function is introduced, which allows the spatial integrations involved in evaluating the modal force cross spectral matrix to be carried out in closed form and this matrix can be calculated by an automatic computer process.
TL;DR: In this article, a finite element method is applied to isothermal slow channel flow of power-law fluids, where the fully developed flow is normal to the channel cross section, and the method and results are compared with a finite difference method for rectangular channels and with exact solutions for the Newtonian case.
Abstract: A finite element method is applied to isothermal slow channel flow of power-law fluids. The fully developed flow is normal to the channel cross section. The method and results are compared with a finite difference method for rectangular channels and with exact solutions for the Newtonian case. An advantage of finite element methods is the flexibility of the mesh of elements approximating the continuum, chosen to suit the particular problem. Arbitrary boundary shapes can be handled as illustrated by a rectangular channel with rounded corners.