Showing papers in "International Journal for Numerical Methods in Engineering in 1989"
TL;DR: In this paper, a numerical scheme for crack modelling by means of continuous displacement fields is presented, where a crack is modelled as a limiting case of two singular lines (with continuous displacements, but discontinuous displacement gradients across them) which tend to coincide with each other.
Abstract: A numerical scheme for crack modelling by means of continuous displacement fields is presented. In two-dimensional problems a crack is modelled as a limiting case of two singular lines (with continuous displacements, but discontinuous displacement gradients across them) which tend to coincide with each other. An analysis of the energy dissipated inside the band bounded by both lines allows one to obtain an expression for the characteristic length as the ratio between the energy dissipated per unit surface area (fracture energy) and the energy dissipated per unit volume (specific energy) at a point. The application of these mathematical expressions to the finite element discretized medium allow one to obtain a general spatial and directional expression for the characteristic length which guarantees the objectivity of the results with respect to the size of the finite element mesh. The numerical results presented show the reliability of the proposed expressions.
449 citations
TL;DR: In this paper, the classical and shear deformation theories up to the third-order are presented in a single theory, and results of linear and non-linear bending, natural vibration and stability of composite laminates are presented for various boundary conditions and lamination schemes.
Abstract: Finite element models of the continuum-based theories and two-dimensional plate/shell theories used in the analysis of composite laminates are reviewed. The classical and shear deformation theories up to the third-order are presented in a single theory. Results of linear and non-linear bending, natural vibration and stability of composite laminates are presented for various boundary conditions and lamination schemes. Computational modelling issues related to composite laminates, such as locking, symmetry considerations, boundary conditions, and geometric non-linearity effects on displacements, buckling loads and frequencies are discussed. It is shown that the use of quarter plate models can introduce significant errors into the solution of certain laminates, the non-linear effects are important even at small ratio of the transverse deflection to the thickness of antisymmetric laminates with pinned edges, and that the conventional eigenvalue approach for the determination of buckling loads of composite laminates can be overly conservative.
282 citations
TL;DR: In this article, the numerical technique of differential quadrature for the solution of linear and non-linear partial differential equations, first introduced by Bellman and his associates, is applied to the equations governing the deflection and buckling behaviour of one-and two-dimensional structural components.
Abstract: The numerical technique of differential quadrature for the solution of linear and non-linear partial differential equations, first introduced by Bellman and his associates, is applied to the equations governing the deflection and buckling behaviour of one- and two-dimensional structural components. Separate transformations are used for higher-order derivatives, as suggested by Mingle, thus extending the method to treat fourth-order equations and to include multiple, boundary conditions in the respective co-ordinate directions. Results are obtained for various boundary and loading conditions and are compared with existing exact and numerical solutions by other methods. The application of differential quadrature to this class of problems is seen to lead to accurate results with relatively small computational effort.
258 citations
TL;DR: In this article, the authors present a procedure to establish Reissner-Mindlin plate bending elements based on the idea to combine known resuits on the approximation of Stokes problems with known results on approximation of elliptic problems.
Abstract: DCpartement de Mathtmatiyue, L'nit,ersitt 1aral Quthec, Canado SUMMARY We present in this paper a procedure to establish Reissner-Mindlin plate bending elements The procedure is based on the idea to combine known resuits on the approximation of Stokes problems with known results on the approximation of elliptic problems The proposed elements satisfy the mathematical conditions of stability and convergence, and some of them promise to provide efficient elements for practical solutions
250 citations
TL;DR: In this article, a new three node, nine d.o.f. triangular plate bending element called DST (Discrete Shear Triangle) was proposed for the analysis of thin and thick plates.
Abstract: This paper deals with the formulation and the evaluation of a new three node, nine d.o.f. triangular plate bending element valid for the analysis of thick to thin plates. The formulation is based on a generalization of the discrete Kirchhoff technique to include the transverse shear effects. The element, called DST (Discrete Shear Triangle), has a proper rank and is free of shear locking. It coincides with the DKT (Discrete Kirchhoff Triangle) element if the transverse shear effects are not significant. However, an incompatibility of the rotation of the normal appears due to shear effects. A detailed numerical evaluation of the characteristics and of the behaviour of the element has been performed including patch tests for thin and thick plates, convergence tests for clamped and simply supported plates under uniform loading and evaluation of stress resultants. The overall performance of the DST element is found to be very satisfactory.
194 citations
TL;DR: A plate bending element based on the generalized laminate plate theory (GLPT) developed by the senior author is described and its accuracy is investigated by comparison with the exact solutions ofthe generalized plate theory and the 3D-elasticity theory as discussed by the authors.
Abstract: .( SUMMARY A plate bending element based on the generalized laminate plate theory (GLPT) developed by the senior author is described and its accuracy is investigated by comparison with the exact solutions ofthe generalized plate theory and the 3D-elasticity theory. The element accounts for transverse shear deformation and layer wise description of the inplane displacements of the laminate. The element has improved description of the inplane as well as the transverse deformation response. A method for the computation of interlaminar (transverse) stresses is also presented. 1. BACKGROUND Laminated composite plates are often modelled using the classical laminate plate theory (CLPT) or the first-order shear deformation plate theory (FSDT). In both cases the laminate is treated as a single-layer plate with equivalent stiffnesses, and the displacements are assumed to vary through the thickness according to a single expression (see Reddy 1 ), not allowing for possible discontinuities in strains at an interface of dissimilar material layers. Recently, Reddy2 presented a general laminate plate theory that allows layer-wise representation of inplane displacements, and an improved response of inplane and transverse shear deformations is predicted. Similar but different theories have appeared in the literature. 3-6 In the generalized laminate plate theory (0LPT) the equations of three-dimensional elasticity are reduced to differential equations in terms of unknown functions in two dimensions by assuming layer-wise approximation of the displacements through the thickness. Consequently, the strains are different in different layers. Exact analytical solutions of the theory were developed by the authors 7 ,8 to evaluate the accuracy ofthe theory compared to the 3D-elasticity theory. The results indicated that the generalized laminate plate theory allows accurate determination ofinterlaminar stresses. The present study deals with the finite-element formulation of the theory and its application to laminated composite plates. In the interest of brevity only the main equations of the theory are reviewed and the major steps of the formulation are presented. The accuracy of the numerical
185 citations
TL;DR: In this article, an assumed strain interpolation method is used to construct a stabilization matrix for the 9-node shell element, which can be justified based on the Hellinger-Reissner variational method.
Abstract: An assumed strain (strain interpolation) method is used to construct a stabilization matrix for the 9-node shell element. The stabilization procedure can be justified based on the Hellinger–Reissner variational method. It involves a projection vector which is orthogonal to both linear and quadratic fields in the local co-ordinate system of each quadrature point. All terms in the development involve 2 × 2 quadrature in the 9-node element. Example problems show good accuracy and an almost optimal rate of convergence.
165 citations
TL;DR: In this article, an a-posteriori error estimator for finite element analysis proposed by Zienkiewicz and Zhu is analyzed and shown to be effective and convergent.
Abstract: An a-posteriori error estimator for finite element analysis proposed by Zienkiewicz and Zhu is analysed and shown to be effective and convergent. In addition we analyse wider classes of estimators of which the Zienkiewicz–Zhu estimator is a special case. It is shown that some of these estimators will be asymptotically exact. Numerical evidence is presented supporting the analysis.
164 citations
TL;DR: Two practical and effective, h–p-type, finite element adaptive procedures are presented, which allow not only the final global energy norm error to be well estimated using hierarchic p-refinement, but in addition give a nearly optimal mesh.
Abstract: Two practical and effective, h–p-type, finite element adaptive procedures are presented. The procedures allow not only the final global energy norm error to be well estimated using hierarchic p-refinement, but in addition give a nearly optimal mesh. The design of this is guided by the local information computed on the previous mesh. The desired accuracy can always be obtained within one or at most two h–p-refinements.
The rate of convergence of the adaptive h–p-version analysis procedures has been tested for some examples and found to be very strong.
The presented procedures can easily be incorporated into existing p- or h-type code structures.
154 citations
TL;DR: In this paper, an explicit expression for the coupled bending and torsional dynamic stiffness matrix of a uniform beam element is derived by solving the governing differential equation of the beam element.
Abstract: Explicit expressions for the coupled bending–torsional dynamic stiffness matrix of a uniform beam element are derived in an exact sense by solving the governing differential equation of the beam. Implementation of the derived dynamic stiffness matrix in a space frame computer program is discussed with particular reference to an established algorithm to enable vibration analysis of coupled systems to be made. The application of the theory is demonstrated by an illustrative example wherein the results for a cantilever beam with a substantial amount of coupling between bending and torsion are highlighted. The correctness of the theory is confirmed to a high degree of accuracy by computed results and numerical checks.
151 citations
TL;DR: In this paper, the finite element method is used to simulate freezing and melting including convection in the melt and the phase boundaries are recovered from the solution as the isotherms corresponding to the beginning and end of the transformation.
Abstract: SUMMARY Methods are described for modelling of phase change processes using the finite element method to simulate freezing and melting including convection in the melt. Evaluation of several enthalpy/specific heat methods and time marching schemes is also included. Suppression of velocities in the solid region is described, and example problems are given. Comparison is made to simulations performed by other researchers using finite difference methods. Substantially different results were found for one of these problems, and this result is shown to be caused by numerical problems in the earlier work. strong effect on the resulting microstructure. A number of researchers have shown reorientation of columnar grains,' alteration of the size and location of equiaxed zones' and macro~egregation,~. all due to melt convection. Mathematical models have been used in attempts to better understand the processes and thus control them. Although the most convenient mathematical models would use analytical solutions to the coupled heat and momentum transport equations, very few such solutions exist for these problems, and none would extend to the realistic problems where complicated geometries and temperature dependent material properties are included. For this reason, nearly all of the efforts in this area have been numerical. There are different types of numerical methods which are appropriate to phase change problems, depending on the kind of material involved. In pure materials, eutectics or congruent melting phases, the liquid-solid interface is sharp and corresponds to an isotherm. For these kinds of problems it may be appropriate to have part of the mesh coincide with the solidification front at all times, and distort the mesh in both phases as the boundary moves. A number of these front- tracking methods have been de~eloped,~, but none exists for three-dimensional problems. For alloys which freeze over a range of temperatures, front-tracking methods are no longer applicable. Instead, what is normally done is to specify the evolution of latent heat over a freezing range as part of the material properties. The phase boundaries are then recovered from the solution as the isotherms corresponding to the beginning and end of the transformation. The two
TL;DR: In this article, a consistent tangent stiffness matrix for the analysis of non-linear contact problems is presented, which accounts for the nonlinear kinematics of large deformation analysis and guarantees a quadratic convergence rate.
Abstract: A consistent tangent stiffness matrix for the analysis of non-linear contact problems is presented. The associated element has three or four nodes and establishes contact between three-dimensional structures like solids and shells. It accounts for the non-linear kinematics of large deformation analysis and guarantees a quadratic convergence rate. Two formulations, the penalty method and the Lagrange multiplier method, are investigated.
TL;DR: In this article, the main results of the homogenization theory for periodic media are presented, and two applications related to damage evaluation and simulation for composite materials are discussed. But neither of them are related to our work.
Abstract: After presenting shortly the main results of the homogenization theory for periodic media, we give two applications related to damage evaluation and simulation for composite materials: (i) simulation of the evolution of damage by fibre rupture in a unidirectional composite, involving parameters defined on the microscale; and (ii) prediction of debonding near an unloaded boundary in a stratified structure, using boundary layer asymptotic expansions.
TL;DR: It is shown that the quality of all the triangulations iteratively generated depends only on the geometric characteristics of the initial grid.
Abstract: The construction of changing sequences of irregular and nested triangulations, based on the use of conforming refinement/derefinement algorithms for triangulations, is presented and discussed. This strategy is particularly appropriate to combine adaptivity and full multigrid algorithms for dealing with moving fronts or fluid dynamics problems. It is shown that the quality of all the triangulations iteratively generated depends only on the geometric characteristics of the initial grid. A data structure suitable to create, manage and modify series of nested triangulations as well as the main features of the DEREF prototype package are described, and numerical examples are given.
TL;DR: In this paper, the Zienkiewicz-Zhu error estimator is used in combination with an adaptive mesh generator allowing a prescribed size of elements to be developed, very fast adaptive convergence for results of specified accuracy is achieved.
Abstract: The Zienkiewicz–Zhu error estimator is shown to be effective in problems of plate flexure. When used in conjunction with triangular elements and an adaptive mesh generator allowing a prescribed size of elements to be developed, very fast adaptive convergence for results of specified accuracy is achieved.
TL;DR: The numerical strategies employed in the evaluation of singular integrals existing in the Cauchy principal value (CPV) sense are one of the key aspects which remarkably affect the performance and accuracy of the boundary element method (BEM).
Abstract: The numerical strategies employed in the evaluation of singular integrals existing in the Cauchy principal value (CPV) sense are, undoubtedly, one of the key aspects which remarkably affect the performance and accuracy of the boundary element method (BEM).
Thus, a new procedure, based upon a bi-cubic co-ordinate transformation and oriented towards the numerical evaluation of both the CPV integrals and some others which contain different types of singularity is developed.
Both the ideas and some details involved in the proposed formulae are presented, obtaining rather simple and-attractive expressions for the numerical quadrature which are also easily embodied into existing BEM codes.
Some illustrative examples which assess the stability and accuracy of the new formulae are included.
TL;DR: This work characterize the conditions under which an elastic–plastic stress update algorithm preserves the symmetries inherent to the material response and shows that holonomic methods and the fully implicit algorithm may indeed be brought into correspondence.
Abstract: In this work we seek to characterize the conditions under which an elastic–plastic stress update algorithm preserves the symmetries inherent to the material response. From a numerical standpoint, the aim is to determine under what conditions a stress update algorithm produces symmetric consistent tangents when applied to materials obeying normality. For the ideally plastic solid we show that only the fully implicit or closest point return mapping algorithm is symmetry preserving. For hardening plasticity, symmetry cannot be preserved in general unless suitable restrictions are imposed on the constitutive equations. We show that these restrictions amount to the existence of a pseudo-internal energy function acting as a joint potential for both the direction of plastic flow and the hardening moduli. In view of the fact that holonomic methods based on incrementally extremal paths also result in update rules possessing a potential structure and, hence, in symmetric tangents, we address the question of whether any connections exist between the two approaches. We show that holonomic methods and the fully implicit algorithm may indeed be brought into correspondence.
TL;DR: In this paper, the propagation of non-planar hydraulic fractures is modelled using a three-dimensional numerical simulator, and different components of the model (stress/displacement analysis, fluid-flow analysis, propagation criterion) with an emphasis on the numerical techniques used.
Abstract: The propagation of non-planar hydraulic fractures is modelled using a three-dimensional numerical simulator. This paper describes the different components of the model (stress/displacement analysis, fluid-flow analysis, propagation criterion) with an emphasis on the numerical techniques used. A few examples of out-of-plane fracture geometries are provided.
TL;DR: In this paper, a domain method of shape design sensitivity analysis that uses the material derivative concept of continuum mechanics is employed for numerical implementation of shape optimal design, parameterization of the boundary shape of mechanical components is defined and illustrated using a Bezier surface.
Abstract: A unified method for continuum shape design sensitivity analysis and optimal design of mechanical components is developed. A domain method of shape design sensitivity analysis that uses the material derivative concept of continuum mechanics is employed. For numerical implementation of shape optimal design, parameterization of the boundary shape of mechanical components is defined and illustrated using a Bezier surface. In shape design problems, nodal points of the finite element model move as the shape changes. A method of automatic regridding to account for shape change has been developed using a design velocity field in the physical domain that obeys the governing equilibrium equations of the elastic solid. For numerical implementation of the continuum shape design sensitivity analysis and automatic regridding, an established finite element analysis code is used. To demonstrate the feasibility of the method developed, shape design optimization of a main engine bearing cap is carried out as an example.
TL;DR: In this paper, the hp-finite element method is applied to a boundary collocation method, and it is shown in numerical examples that even in the presence of singularities in the exact solution exponential convergence is obtained.
Abstract: Recently very promising results in a so-called hp-version of the finite element method have been obtained. The basic idea is a balanced combination of mesh refinement and increase of the polynomial degree of the shape functions. This idea is applied to a boundary collocation method in this paper. The new method is compared with adaptive h- and p-versions and it is shown in numerical examples that even in the presence of singularities in the exact solution exponential convergence is obtained.
TL;DR: Through the study of numerous examples of various characteristics, it is found that high-quality triangular element meshes are obtained by the proposed algorithm, and the mesh generation time bears a linear relationship with the number of elements/nodes of the triangulation.
Abstract: This paper investigates the possibility of integrating the two currently most popular mesh generation techniques, namely the method of advancing front and the Delaunay triangulation algorithm. The merits of the resulting scheme are its simplicity, efficiency and versatility. With the introduction of ‘non-Delaunay’ line segments, the concept of using Delaunay triangulation as a means of mesh generation is clarified.
An efficient algorithm is proposed for the construction of Delaunay triangulations over non-convex planar domains. Interior nodes are first generated within the planar domain. These interior nodes and the boundary nodes are then linked up together to produce a valid triangulation. In the mesh generation process, the Delaunay property of each triangle is ensured by selecting a node having the smallest associated circumcircle. In contrast to convex domains, intersection between the proposed triangle and the domain boundary has to be checked; this can be simply done by considering only the ‘non-Delaunay’ segments on the generation front.
Through the study of numerous examples of various characteristics, it is found that high-quality triangular element meshes are obtained by the proposed algorithm, and the mesh generation time bears a linear relationship with the number of elements/nodes of the triangulation.
TL;DR: Errors in the strain-displacement relationship and numerical integration of the strains are used to estimate the total response error and to rationalize the connection between optimal stress points and reduced integration.
Abstract: The presented work addresses the relationships between optimal sampling points, reduced integration and geometric distortion with the objective of estimating errors in terms of those considerations. Isoparametric quadratic plane and solid elements are used as a vehicle for the study. Geometric distortion measures and evaluation conditions, based on convergence requirements, are first defined in terms of the polynomial orders of the geometry and applied strain. Using these, the concept of optimal stress sampling, already established for undistorted elements, is extended to distorted geometry and shown to be effective over a range of geometries and strains. Errors in the strain-displacement relationship and numerical integration of the strains are used to estimate the total response error and to rationalize the connection between optimal stress points and reduced integration. Enhanced convergence, by extension to the representation of linear strains in elements with quadratic geometry, is identified as the main advantage of reduced integration. The applicability of the proposed, and other, distortion parameters to vetting of element geometry and error prediction is discussed.
TL;DR: Special types of boundary elements are discussed which can be used for the modelling of surfaces which extend to infinity and it is shown that the elements perform extremely well even for cases in which they are located close to the area of interest.
Abstract: Special types of boundary elements are discussed which can be used for the modelling of surfaces which extend to infinity. The theoretical background and details of implementation are discussed. On test examples it is shown that the elements perform extremely well even for cases in which they are located close to the area of interest. A practical application of the use of the elements for the modelling of mining excavations is given.
TL;DR: In this paper, a cohesive crack model is proposed aiming at describing the size effects of fracture mechanics, i.e. the transition from ductile to brittle structure behavior by increasing the size scale and keeping the geometrical shape unchanged.
Abstract: The nature of the crack and the structure behaviour can range from ductile to brittle, depending on material properties, structure geometry, loading condition and external constraints. The influence of variation in fracture toughness, tensile strength and geometrical size scale is investigated on the basis of the π-theorem of dimensional analysis. Strength and toughness present in fact different physical dimensions and any consistent fracture criterion must describe energy dissipation per unit of volume and per unit of crack area respectively. A cohesive crack model is proposed aiming at describing the size effects of fracture mechanics, i.e. the transition from ductile to brittle structure behaviour by increasing the size scale and keeping the geometrical shape unchanged.
For extremely brittle cases (e.g. initially uncracked specimens, large and/or slender structures, low fracture toughness, high tensile strength, etc.) a snap-back instability in the equilibrium path occurs and the load–deflection softening branch assumes a positive slope. Both load and deflection must decrease to obtain a slow and controlled crack propagation (whereas in normal softening only the load must decrease). If the loading process is deflection-controlled, the loading capacity presents a discontinuity with a negative jump. It is proved that such a catastrophic event tends to reproduce the classical LEFM-instability (KI = KIC) for small fracture toughnesses and/or for large structure sizes. In these cases, neither the plastic zone develops nor slow crack growth occurs before unstable crack propagation.
IBM1
TL;DR: The results of the numerical experiments show that preconditioned iterative methods are a practical alternative to direct methods in the solution of large, sparse systems of equations, and can offer significant savings in storage and CPU time.
Abstract: In this paper we compare direct and preconditioned iterative methods for the solution of nonsymmetric, sparse systems of linear algebraic equations. These problems occur in finite difference and finite element simulations of semiconductor devices, and fluid flow problems.
We consider five iterative methods that appear to be the most promising for this class of problems: the biconjugate gradient method, the conjugate gradient squared method, the generalized minimal residual method, the generalized conjugate residual method and the method of orthogonal minimization. Each of these methods was tested using similar preconditioning (incomplete LU factorization) on a set of large, sparse matrices arising from finite element simulation of semiconductor devices. Results are shown where we compare the computation time and memory requirements for each of these methods against one another, as well as against a direct method that uses LU factorization to solve these problems.
The results of our numerical experiments show that preconditioned iterative methods are a practical alternative to direct methods in the solution of large, sparse systems of equations, and can offer significant savings in storage and CPU time.
TL;DR: In this article, a set of three new hybrid elements with rotational degrees of freedom (d.o.f.) is introduced for solving three-dimensional elasticity problems, and the stiffness matrix is derived based on the Hellinger-Reissner variational principle.
Abstract: A set of three new hybrid elements with rotational degrees-of-freedom (d.o.f.'s) is introduced. The solid, 8-node, hexahedron element is developed for solving three-dimensional elasticity problems. This element has three translational and three rotational d.o.f.'s at each node and is based on a 42-parameter. three-dimensional stress field in the natural convected co-ordinate system. For two-dimensional, plane elasticity problems, an improved triangular hybrid element and a quadrilateral hybrid element are presented. These elements use two translational and one rotational d.o.f. at each node. Three different sets of five-parameter stress fields defined in a natural convected co-ordinate system for the entire element are used for the mixed triangular element. The mixed quadrilateral element is based on a nine-parameter complete linear stress field in natural space. The midside translational d.o.f.'s are expressed in terms of the corner nodal translations and rotations using appropriate transformations. The stiffness matrix is derived based on the Hellinger–Reissner variational principle. The elements pass the patch test and demonstrate an improved performance over the existing elements for prescribed test examples.
TL;DR: The primary goal of this paper is to show how second derivative information can be used in an effective way in structural optimization problems, and a primal–dual approach is employed, that can be interpreted as a sequential quadratic programming method.
Abstract: The primary goal of this paper is to show how second derivative information can be used in an effective way in structural optimization problems. The basic idea is to generate such an information at the expense of only one more ‘virtual load case’ in the sensitivity analysis part of the finite element code. To achieve this goal a primal–dual approach is employed, that can also be interpreted as a sequential quadratic programming method.
Another objective is to relate the proposed method to the well known family of approximation concepts techniques, where the primary optimization problem is transformed into a sequence of non-linear explicit subproblems. When restricted to diagonal second derivatives, the new approach can be viewed as a recursive convex programming method, similar to the ‘Convex Linearization’ method (CONLIN), and to its recent generalization, the ‘Method of Moving Asymptotes’ (MMA).
This new method has been successfully tested on simple problems that can be solved in closed form, as well as on sizing optimization of trusses. In all cases the method converges faster than CONLIN, MMA or other approximation techniques based on reciprocal variables.