Showing papers in "International Journal for Numerical Methods in Engineering in 1985"
TL;DR: In this article, a 4-node plate bending element for linear elastic analysis is presented, as a special case, from a general nonlinear continuum mechanics based four-node shell element formulation.
Abstract: This communication discusses a 4-node plate bending element for linear elastic analysis which is obtained, as a special case, from a general nonlinear continuum mechanics based 4-node shell element formulation. The formulation of the plate element is presented and the results of various example solutions are given that yield insight into the predictive capability of the plate (and shell) element.
1,000 citations
TL;DR: Gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed.
Abstract: Gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method. A brief review of existing quadrature rules for the triangle is given, and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed. New quadrature rules of degree 12–20 are presented, and a short FORTRAN program is included.
700 citations
TL;DR: In this article, an analysis of accuracy and stability of algorithms for the integration of elastoplastic constitutive relations is carried out in the presence of the plastic consistency condition, and the criteria derived are used to identify two second-order accurate members of the proposed algorithms.
Abstract: An analysis of accuracy and stability of algorithms for the integration of elastoplastic constitutive relations is carried out in this paper. Reference is made to a very general internal variable formulation of plasticity and to two families of algorithms that generalize the well-known trapezoidal and midpoint rules to fit the present context. Other integration schemes such as the radial return, mean normal and closest point procedures are particular cases of this general formulation. The meaning of first and second-order accuracy in the presence of the plastic consistency condition is examined in detail, and the criteria derived are used to identify two second-order accurate members of the proposed algorithms. A general methodology is also derived whereby the numerical stability properties of integration schemes can be systematically assessed. With the aid of this methodology, the generalized midpoint rule is seen to have far better stability properties than the generalized trapezoidal rule. Finally, numerical examples are presented that illustrate the performance of the algorithms.
629 citations
TL;DR: A new algorithm to generate interior nodes within any arbitrary multi-connected regions by a completely revised technique in an efficient and stable manner to generate 3-node or 6-node triangular element meshes of great variety within the most irregular heterogeneous regions.
Abstract: This paper describes a new algorithm to generate interior nodes within any arbitrary multi-connected regions. The boundary nodes and the interior nodes are then linked up to form the best possible triangular elements by a completely revised technique in an efficient and stable manner. Owing to the generality of the central generation program, the global domain is allowed to be divided into as many irregular subdomains as desired, in order to model closely the actual physical situation. Moreover, the boundaries of the sub-domains are updated from time to time when necessary to include the possibilities of progressive refinement around a sharp corner, generating radiating mesh from a prescribed node, generating mesh between two circular arcs, etc. Despite its flexibility and capabilities, data for triangulation have been kept to a minimum by a logical input module; no connectivity information between subregions is needed, and common boundaries are defined once only. All these features have contributed to a powerful method to generate 3-node or 6-node triangular element meshes of great variety within the most irregular heterogeneous regions.
487 citations
TL;DR: A comprehensive study of various mathematical programming methods for structural optimization is presented in this article, where the authors discuss the applicability of modern optimization techniques to structural design problems, and present mathematical programming method from a unified and design engineers' viewpoint.
Abstract: A comprehensive study of various mathematical programming methods for structural optimization is presented. In recent years, many modern optimization techniques and convergence results have been developed in the field of mathematical programming. The aim of this paper is twofold: (a) to discuss the applicability of modern optimization techniques to structural design problems, and (b) to present mathematical programming methods from a unified and design engineers' viewpoint. Theoretical aspects are considered here, while numerical results of test problems are discussed in a companion paper. Special features possessed by structural optimization problems, together with recent developments in mathematical programming (recursive quadratic programming methods, global convergence theory), have formed a basis for conducting the study. Some improvements of existing methods are noted and areas for future investigation are discussed.
482 citations
TL;DR: The aim in this research is the development of a solution algorithm for analysis of general contact conditions which shall include the possibilities to analyse: contact between flexible-flexible and rigid--flexible bodies; sticking or sliding conditions; large relative motions between bodies; repeated contact and separation between the bodies.
Abstract: SUMMARY A solution procedure for the analysis of planar and axisymmetric contact problems involving sticking, frictional sliding and separation under large deformations is presented. The contact conditions are imposed using the total potential of the contact forces with the geometric compatibility conditions, which leads to contact system matrices and force vectors. Some key aspects of the procedure arc the contact matrices, the use of distributed tractions on the contact segments for deciding whether a node is sticking, sliding or releasing and the evaluation of the nodal point contact forces. The solutions to various sample problems are presented to demonstrate the applicability of the algorithm. Much progress has been made during recent years in the development of computational capabilities for general analysis of certain nonlinear effects in solids and structures. In each of these developments, quite naturally, the first step was the demonstration of some ideas and possibilities for the analyses under consideration, and then the research and development for reliable and general techniques was undertaken. The second step proved in many cases much more difficult, and in the case of capabilities for analysis of contact problems has yielded few general results. Although some of the first complex contact problems have been solved using the finite element method quite some time ago,'- and much interest exists in the research and solution of contact problems (see, for example, References 4-1 5), there is still a great deaiwf effort necessary for the development of a reliable, general and cost-effective algorithm for the practical analysis of such problems. This is largely due to the fact that the analysis of contact problems is computationally extremely difficult, even for the simplest constitutive relations used. Much of the difficulty lies in that the boundary conditions of the bodies under consideration are not known prior to the analysis, but they depend on the solution variables. The aim in our research is the development of a solution algorithm for analysis of general contact conditions which shall include the possibilities to analyse: contact between flexible-flexible and rigid--flexible bodies; sticking or sliding conditions (with or without friction); large relative motions between bodies; repeated contact and separation between the bodies. Since the large deformation motion of the individual bodies can in many cases be analysed already quite effectively,'6 an algorithm of the above nature will certainly enlarge, very
474 citations
TL;DR: The purpose of this paper is to describe an interactive solid mesh generation system capable of generating valid meshes of well-proportional tetrahedral finite elements for the decomposition of multiply connected solid structures.
Abstract: Recently developed solid modelling systems for the design of complex physical solids using interactive computer graphics offer the exciting possibility of an integrated design/analysis system. Called geometric modellers, these systems build complex solids from primitive solids (cubes, cylinders, spheres, solid patches, etc.) and macro solids (combination of primitives)3, 4, 8, 16, 18, 25, 38. To provide an effective structural analysis capability for these systems, methods must be devised to ease the burden of discretizing the solid geometry into a user controlled (usually locally graded) finite element mesh. The purpose of this paper is to describe an interactive solid mesh generation system capable of generating valid meshes of well-proportional tetrahedral finite elements for the decomposition of multiply connected solid structures. The system uses a semi-automatic node insertion procedure to locate element node points within and on the surface of a structure. An independent automatic three-dimensional triangulator then accepts these nodes as input and connects them to form a valid finite element mesh oftetrahedral elements. Although this report makes use of a modeller based on a constructive solid geometry representation (a so-called CSG modeller), the mesh generation strategy elaborated herein is completely general and makes no particular use of the CSG representation.
393 citations
TL;DR: In this paper, a higher-order deformation theory is used to analyse laminated anisotropic composite plates for deflections, stresses, natural frequencies and buckling loads, and applications of the element to bending, vibration and stability of laminated plates are discussed.
Abstract: A higher-order deformation theory is used to analyse laminated anisotropic composite plates for deflections, stresses, natural frequencies and buckling loads. The theory accounts for parabolic distribution of the transverse shear stresses, and requires no shear correction coefficients. A displacement finite element model of the theory is developed, and applications of the element to bending, vibration and stability of laminated plates are discussed. The present solutions are compared with those obtained using the classical plate theory and the three-dimensional elasticity theory.
364 citations
TL;DR: In this article, a joint/interface element for three-and two-dimensional finite element analysis is presented, which can model joints/interfaces between solid finite elements and shell finite elements.
Abstract: A generally applicable and simple joint/interface element for three- and two-dimensional finite element analysis is presented. The proposed element can model joints/interfaces between solid finite elements and shell finite elements. The derivation of the joint element stiffness is presented and algorithms for the treatment of nonlinear joint behaviour discussed. The performance of the element is tested on typical problems involving shell-to-shell and shell-to-solid interfaces.
272 citations
TL;DR: A generalization of Newmark's time marching integration scheme, called the β-m method, which provides a gcneral single-step scheme applicable to any set of initial value problems and which unifies old and new methods.
Abstract: Introduced herein is a generalization of Newmark's time marching integration scheme, called the β-m method. Like the SSpj method (introduced in Parts 1 and 2 of this series), the β-m method provides a gcneral single-step scheme applicable to any set of initial value problems. The method is specialized by specifying the method order m along with rn integration parameters, β0, β1, …, βm−1. For a particular choice of m, the integration parameters provide a subfamily of methods which control accuracy and stability, as well as offering options for explicit or implicit algorithms. For the most part, attention is focused on the application to structural dynamic equations. Most well-known methods (e.g. Newmark, Wilson, Houbolt, etc.) are shown to be special cases within the β-m family. Hence, one computationally efficient and surprisingly simple algorithm unifies old and new methods. Stability and accuracy analyses are presented for method orders m = 2, 3 and 4 to determine optimal parameters for implicit and explicit schemes, along with numerical verification.
211 citations
TL;DR: Etude du rayonnement et de la diffusion d'ondes elastiques par des obstacles de forme arbitraire as discussed by the authors, a.k.a.
Abstract: Etude du rayonnement et de la diffusion d'ondes elastiques par des obstacles de forme arbitraire
TL;DR: The use of a complete and nonsingular set of Trefftz functions in the solution of quasi-harmonic equations is demonstrated and shown to be often superior to the more conventional singularity distribution in boundary-type approximation as mentioned in this paper.
Abstract: The use of a complete and nonsingular set of Trefftz functions in the solution of quasi-harmonic equations is demonstrated and shown to be often superior to the more conventional singularity distribution in boundary-type approximation. Procedures for coupling separate domains of such solution and indeed of deriving equivalent finite elements are demonstrated.
TL;DR: In this article, two different variational formulations have been used to construct special finite elements with circular and elliptic holes and internal cracks for the numerical treatment of stress concentration problems in plane elasticity.
Abstract: For the numerical treatment of stress concentration problems in plane elasticity, special finite elements with circular and elliptic holes and internal cracks have been developed. Two different variational formulations have been used to construct elements, which may be combined with conventional displacement elements. Using complex functions and conformal mapping techniques the systematic construction of trial functions is shown which not only satisfy a priori the governing differential equations but also the boundary conditions on such influential boundary portions as hole or crack surfaces. For the evaluation of the stiffness matrices of the special elements, only boundary integral computations arc necessary. The numerical results of various examples are very accurate for both functionals.
TL;DR: In this article, a new and efficient method for the evaluation of singular integrals in stress analysis of elastic and elasto-plastic solids, respectively, by the direct boundary element method (BEM), is presented.
Abstract: The purpose of this paper is to report on a new and efficient method for the evaluation of singular integrals in stress analysis of elastic and elasto-plastic solids, respectively, by the direct boundary element method (BEM). Triangle polar co-ordinates are used to reduce the order of singularity of the boundary integrals by one degree and to carry out the integration over mappings of the boundary elements onto plane squares. The method was subsequently extended to the cubature of singular integrals over three-dimensional internal cells as occur in applications of the BEM to three-dimensional elasto-plasticity. For this purpose so-called tetrahedron polar co-ordinates were introduced. Singular boundary integrals stretching over either linear, triangular, or quadratic quadilateral, isoparametric boundry elements and singular volume integrals extending over either linear, tetrahedral, or quadratic, hexahedral, isoparametric internal cells are treated. In case of higher order isoparametric boundary elements and internal cells, division into a number of subelements and subcells, respectively, is necessary. The analytical investigation is followed by a numerical study restricted to the use of quadratic, quadrilateral, isoparametric boundary elements. This is justified by the fact that such elements, as opposed to linear elements, yield singular boundary integrals which cannot be integrated analytically. The results of the numerical investigation demonstrate the potential of the developed concept.
TL;DR: In this article, Bernoulli-Euler theory and Bessel functions are used to obtain explicit expressions for the exact dynamic stiffnesses for axial, torsional and flexural vibrations of any beam which is tapered such that A varies as yn and GJ and I both vary as y(n + 2), where y = (cx/L) + 1; c is a constant such that c > − 1; L is the length of the beam; and x is the distance from one end of a beam.
Abstract: Bernoulli-Euler theory and Bessel functions are used to obtain explicit expressions for the exact dynamic stiffnesses for axial, torsional and flexural vibrations of any beam which is tapered such that A varies as yn and GJ and I both vary as y(n + 2), where A, GJ and I have their usual meanings; y = (cx/L) + 1; c is a constant such that c > − 1; L is the length of the beam; and x is the distance from one end of the beam. Numerical checks give better than seven-figure agreement with the stiffnesses obtained by extrapolation from stepped beams with 400 and 500 uniform elements. A procedure is given for calculating the number of natural frequencies exceeded by any trial frequency when the ends of the member are clamped. This enables an existing algorithm to be used to obtain the natural frequencies of structures which contain tapered members.
TL;DR: In this article, a new type of infinite elements with r−1/2 decay was proposed, which does not require any artificial movement of the origin and can be applied to exterior wave problems.
Abstract: Recently, a new type of infinite elements which uses r−1 decay was proposed. They were applied to exterior wave problems and good results were obtained. In two-dimensional problems, however, it was necessary to move the origin of the r−1 decay in order to model the outgoing wave more accurately, because it decays roughly as r−1/2. In this paper, the mapped infinite elements with r−1/2 decay and the necessary numerical integration procedure are presented. These elements do not require any artificial movement of the origin. Several example problems are solved. The results show that the infinite elements with r−1/2 decay here give much more accurate values than the infinite elements with exponential decay and any damper elements.
TL;DR: In this article, a Taylor expansion in the local parametric co-ordinates is developed for the singular integrand, so allowing singular terms to be integrated in closed form, even for curved surface elements.
Abstract: The problem treated is the integration of singular functions which arise in three-dimensional isoparametric formulations of boundary integral equations. A Taylor expansion in the local parametric co-ordinates is developed for the singular integrand, so allowing singular terms to be integrated in closed form, even for curved surface elements. The remainder integral obtained by subtracting out the worst singularities is integrated by repeated Gaussian quadrature.
Two groups of tests are presented. First, the accuracy of the integrations has been checked for plane parallelograms (for which exact solutions have been developed) and for curved elements on a sphere. Secondly, results from complete boundary element calculations based on point collocation have been compared with known analytical solutions to two problems; zonal surface harmonics on a sphere and the capacitance of an ellipsoid. The agreement obtained with few degrees-of-freedom suggests that errors which have previously been attributed to point collocation might have arisen in the numerical integration.
TL;DR: In this paper, a uniform beam element of open thin-walled cross-section is studied under stationary harmonic end excitation and an exact dynamic (transcendentally frequency-dependent) 14 × 14 element stiffness matrix is derived from Vlasov's coupled differential equations.
Abstract: A uniform beam element of open thin-walled cross-section is studied under stationary harmonic end excitation. An exact dynamic (transcendentally frequency-dependent) 14 × 14 element stiffness matrix is derived from Vlasov's coupled differential equations. Special attention is paid to the computational problems arising when coefficients vanish in these equations because of symmetric cross-section, zero warping stiffness, etc. The dynamic element stiffness matrix is established via a generalized linear eigenvalue problem and a system of linear algebraic equations with complex matrices. A static stiffness matrix is also derived and the associated consistent mass and geometric stiffness matrices are given. Modal masses are evaluated. A FORTRAN program and a numerical example are included.
TL;DR: In this paper, the methode de programmation dynamique for the resolution of problemes inverses de conduction thermique is described, impliquant the determination of 2 evolutions dans le temps inconnues de flux thermique sur les 2 faces d'une plaque.
Abstract: Description de la methode de programmation dynamique pour la resolution de problemes inverses de conduction thermique. Application a un probleme impliquant la determination de 2 evolutions dans le temps inconnues de flux thermique sur les 2 faces d'une plaque. Execution de differentes experiences numeriques pour etudier les effets du bruit et des parametres de ponderation
TL;DR: Algorithms are presented for the accurate and efficient treatment of singular kernels frequently encountered in the boundary element method (BEM) based upon the use of appropriately weighted Gaussian quadrature formulae, together with numerical geometrical transformations of the region of integration.
Abstract: Use of the Green function, for the solution of boundary-value problems, frequently results in singular integral equations. Algorithms are presented for the accurate and efficient treatment of singular kernels frequently encountered in the boundary element method (BEM). They are based upon the use of appropriately weighted Gaussian quadrature formulae, together with numerical geometrical transformations of the region of integration. The use of high-order subdomain expansion functions, for interpolation over nonplanar elements, allows boundary curvature to be accommodated. In particular, the handling of Green functions with logarithmic and r−1 behaviour are detailed. Volume integrals, with r−2 singularity, are outlined. Operations are performed on a simplex, thus resulting in generality and ease of automation. This scheme has been incorporated into boundary element method software and successfully applied to a variety of problems.
TL;DR: It seems possible to determine optimal integration rules that will allow the extensional deformation of a curved beam/deep arch/finite ring element to be modelled by independently chosen low order polynomial functions and which will recover the inextensional case in the penalty limit of extreme thinness without spurious locking constraints.
Abstract: An attempt is made to understand the errors arising in curved finite elements which undergo both flexural and membrane deformation. It is shown that with elements of finite size (i. e. a practical level of discretization at which reasonably accurate results can be expected), there can be errors of a special nature that arise because the membrane strain fields are not consistently interpolated with terms from the two independent field functions that characterize such a problem. These lead to errors, described here as of the 'second kind' and a physical phenomenon called 'membrane locking'. It seems possible to determine optimal integration rules that will allow the extensional deformation of a curved beam/deep arch/finite ring element to be modelled by independently chosen low order polynomial functions and which will recover the inextensional case in the penalty limit of extreme thinness without spurious locking constraints. What is emphasized is that the choice of shape functions, or subsequent operations to determine the discretized functionals, must consistently model the physical requirements the problem imposes on the field variables.
TL;DR: It turns out that globally convergent algorithms (multiplier methods, in particular) are very reliable but not efficient, and Primal algorithms (like CONMIN), which are not proved to be globally Convergent, are efficient but not reliable.
Abstract: Various mathematical programming methods for structural optimization are studied. In a companion paper, these methods have been studied based on certain theoretical considerations. In this paper, the methods are studied based on solving a set of test problems. The methods that are studied include recursive QP, feasible directions, gradient projection, SUMT and multiplier methods. Various computer codes have been developed, and are studied together with some existing programs such as CONMIN and OPTDYN. The test problems considered have 3–47 design variables and 3–252 constraints. The evaluation criteria consist of studying the accuracy, reliability and efficiency of a code. It turns out that globally convergent algorithms (multiplier methods, in particular) are very reliable but not efficient. Primal algorithms (like CONMIN), which are not proved to be globally convergent, are efficient but not reliable.
TL;DR: In this article, a finite element formulation is developed with emphasis primarily focused on providing stress predictions for thin to moderately thick plate (shell) type structures, and a selective reduced integration technique is utilized in computing element stiffness matrices.
Abstract: A finite element formulation is developed with emphasis primarily focused on providing stress predictions for thin to moderately thick plate (shell) type structures. Plate element behaviour is specified by prescribing independently the neutral surface displacements and rotations, thus relaxing the Kirchhoff hypothesis. Numerical efficiency is achieved due to the simplicity of the element formulation, i.e. the approach yields a displacement dependent multi-layer model. In-plane layer stresses are determined via the constitutive equations, while the transverse shear and short-transverse normal stresses are determined via the equilibrium equations. Accurate transverse stress variations are obtained by appropriately selecting the displacement field for the element. A selective reduced integration technique is utilized in computing element stiffness matrices. Static and spectral (eigenvalue) tests are performed to demonstrate the element modelling capability.
TL;DR: In this article, explicit expressions for second variations are derived for the case when the material stiffness or compliance variations depend on a set of parameters, by using the solutions of two additional sets of adjoint systems.
Abstract: For a linear elastic structure, the first variations of arbitrary stress, strain and displacement functionals were explicitly expressed in terms of variations of material stiffness parameters within specified domain, in References 1 and 2, by using the solutions for primary and adjoint systems. In this paper, explicit expressions for second variations are derived for the case when the material stiffness or compliance variations depend on a set of parameters. These expressions arc obtained in terms of variations of stiffness or compliance parameters by using the solutions of two additional sets of adjoint systems. Furthermore, the variation of local stress, strain or displacement is considered. Several illustrative examples are provided for the first- and second-order sensitivity analysis of beams.
TL;DR: The importance of a good preconditioning is emphasized, various methods including ICGG(n) and incomplete block factorization are looked at, and some practical recommendations are made.
Abstract: We present the results of a numerical study of the preconditioned conjugate gradient algorithm, the minimal residual algorithm, the biconjugate gradient algorithm and the bi-minimal residual algorithm using both simple test matrices and more realistic test matrices derived from physical problems. The application of the methods to unsymmetric matrices is considered. We emphasize the importance of a good preconditioning, look at various methods including ICGG(n) and incomplete block factorization, and make some practical recommendations. Some of the folk-lore surrounding the semi-iterative methods is dispelled.
TL;DR: In this article, a procedure to discretize explicit enthalpy formulations for one-dimensional planar phase change problems is derived, which removes numerical oscillations in temperature and phase front position.
Abstract: A procedure to discretize explicit enthalpy formulations for one-dimensional planar phase change problems is derived, which removes numerical oscillations in temperature and phase front position. The technique is based on an enthalpy balance for the control volume containing the phase front in combination with linearized temperature profiles near the phase front. A continuous casting solidification problem and a problem with known analytical solution are applied to demonstrate the effect of the scheme.
TL;DR: In this article, an explicit expression for energy changes due to virtual crack extensions is formulated based on a variation of isoparametric element mappings, which is shown to produce very accurate solutions even with fairly coarse element meshes.
Abstract: A new finite element technique for calculating energy release rates is presented. An explicit expression for energy changes due to virtual crack extensions is formulated based on a variation of isoparametric element mappings. Energy release rates are calculated directly from integral expressions evaluated over singular quarter-point isoparametric elements surrounding the crack tip. Since the energy release rates are expressed in variational form, there is no need for the analyst to select a small finite crack extension to simulate a virtual crack extension. The method is shown to produce very accurate solutions even with fairly coarse element meshes. A similar technique for mixed-mode fracture based on mutual potential energy release rates is described.
TL;DR: In this paper, the stiffness matrix for the DKT plate-bending element is formulated explicitly in a global co-ordinate system, which avoids transformations of stiffness, and elasticity properties for anisotropic materials, from local to global coordinates.
Abstract: The stiffness matrix for the DKT plate-bending element is formulated explicitly in a global co-ordinate system. This approach avoids transformations of stiffness, and elasticity properties for anisotropic materials, from local to global co-ordinates, which were required in previous formulations. A FORTRAN listing of the algorithm is appended for potential users.
TL;DR: In this paper, a modified finite difference procedure is presented which improves the accuracy of the calculated derivatives of a simple algebraic example as well as an element-by-element preconditioned conjugate gradient iterative solution technique applied to truss examples.
Abstract: The calculation of sensitivity derivatives of solutions of iteratively solved systems of algebraic equations is investigated. A modified finite difference procedure is presented which improves the accuracy of the calculated derivatives. The procedure is demonstrated for a simple algebraic example as well as an element-by-element preconditioned conjugate gradient iterative solution technique applied to truss examples.
TL;DR: In this article, a new explicit and conditionally stable finite difference equation for heat conduction was reported, which predicts results with an accuracy comparable with or better than that obtainable by other methods.
Abstract: This paper reports a new explicit and conditionally stable finite difference equation for heat conduction. It predicts results with an accuracy comparable with or better than that obtainable by other methods. Stability of operation can be extended to any desired degree by subdividing the basic time step and increasing the number of nodes. Some existing difference equations are special cases of the new equation reported in this paper. The new solution has been tested by calculating the response of a slab to transient and progressive waves whose analytical solutions are known.