Natural description of large inelastic deformations for shells of arbitrary shape-application of trump element

The paper presents an extension of the range of application of the simple triangular shell element evolved in [1] to the domain of large inelastic deformations. The element is partly based on ideas of physical lumping which contain a simple mechanical interpretation. None of the virtual work or variational principles are applicable to the present model of a finite element thus implying its approximate character. Nevertheless it is hoped that the proposed element, denoted as in [1] by TRUMP, will prove useful for the study of geometrically complicated and materially nonlinear problems of shell systems in which transverse shear may have to be included. In such cases the application of the high order “formal” displacement elements may prove difficult and expensive. Some examples illustrate the technique.     

Shape design sensitivity analysis of eigenvalues using “exact” numerical differentiation of finite element matrices

As has been shown in recent years, the approximate numerical differentiation of element stiffness matrices which is inherent in the semi-analytical method of finite element based design sensitivity analysis, may give rise to severely erroneous shape design sensitivities in static problems involving linearly elastic bending of beam, plate and shell structures.This paper demonstrates that the error problem also manifests itself in semi-analytical sensitivity analyses of eigenvalues of such structures and presents a method for complete elimination of the error problem. The method, which yields exact numerical sensitivities on the basis of simple first-order numerical differentiation, is computationally inexpensive and easy to implement as an integral part of the finite element analysis.The method is presented in terms of semi-analytical shape design sensitivity analysis of eigenvalues in the form of frequencies of free transverse vibrations of plates modelled by isoparametric Mindlin finite elements. Finally, the development is illustrated via two examples of occurrence of the error phenomenon when the traditional method is used and it is shown that the problem is completely eliminated by the application of the new method.     

Optimization of acoustic response — a numerical and experimental comparison

Acoustic optimization within structural dynamics involves automatic changes of structural design variables such as geometric dimensions, shell thickness, material parameters, fiber density and orientation angles, and others to obtain minimum noise or a specified sound quality in specified regions inside or outside the structure. The objective of the present paper is to compare numerical optimization results with experimental ones. The analysed structure is geometrically simple; a closed cylinder. The objective function is the sound intensity at specified points outside the structure. The variable used is the shell thickness. The structural dynamic behaviour is analysed with the finite element method and the acoustic analysis is performed with the boundary element method. Received February 10, 1999     

Dual BEM for crack growth analysis on distributed-memory multiprocessors

The Dual Boundary Element Method (DBEM) has been presented as an effective numerical technique for the analysis of linear elastic crack problems [Portela A, Aliabadi MH. The dual boundary element method: effective implementation for crack problems. Int J Num Meth Engng 1992;33:1269–1287]. Analysis of large structural integrity problems may need the use of large computational resources, both in terms of CPU time and memory requirements. This paper reports a message-passing implementation of the DBEM formulation dealing with the analysis of crack growth in structures. We have analyzed the construction of the system and its resolution. Different data distribution techniques have been studied with several problems. Results in terms of scalability and load balance for these two stages are presented in this paper.     

Nonlinear finite element analysis of elastic systems under nonconservative loading-natural formulation. part I. Quasistatic problems

The paper presents a nonlinear finite element analysis of elastic structures subject to nonconservative forces. Attention is focused on the stability behaviour of such systems. This leads mathematically to non-self-adjoint boundary-value problems which are of great theoretical and practical interest, in particular in connection with alternative modes of instability like divergence of flutter. Only quasistatic effects are however considered in the present part.The methodology of our theory is general, but the specific thrust of the present research is directed towards the analysis of structures acted upon by displacement-dependent nonconservative (follower) forces. In a finite element formulation the analysis of geometrically nonlinear elastic systems subject to such forces gives, in general, rise to a contributory nonsymmetric stiffness matrix known as the load correction matrix. As a result, the total tangent stiffness matrix becomes unsymmetric - an indication of the non-self-adjoint character of the problem. Our theory is based on the natural mode technique [1, 2, 3]and permits i.a. a simple but elegant derivation of the load correction matrix. The application of the general theory as evolved in this paper is demonstrated on the beam element in space. A number of numerical examples are considered including divergence and flutter types of instability, for which exact analytic solutions are known. The problems demonstrate the efficiency of the present finite element formulation.The paper furnishes also a novel and concise formulation of finite rotations in space which may be considered as a conceptual generalization of the theory presented in [2, 3].     

A conical shell finite element

In the analysis of rocket and missiles structures one frequently encounters cylindrical and cornica' shells. A simple finite element which fits the above configuration is obviously a conical shell finite element. In this paper stiffness matrix for a conical shell finite element is derived using Novozhilov's strain-displacement relations for a conical shell. Numerical integration is carried out to ge. the stiffness matrix. The element has 28 degrees-of-freedom and is nonconforming. An eigenvalue analysis of the stiffness matrix showed that it contains all the rigid body modes (six in this case) adequately, which is one of the convergence criteria. An advantage of this element is that a cylindrical shell, an annular segment flat plate, a rectangular flat plate elements can easily be obtained as degenerate cases. The effectiveness of this element is shown through a variety of numerical examples pertaining to annular plate, cylindrical shell and conical shell problems. Comparison of the present solution is made with the existing ones wherever possible. The comparison shows that the present element is superior in some respects to the existing elements     

Resultant-stress degenerated-shell element

In this paper, a resultant-stress degenerated-shell element is described and a variety of numerical examples, including the post-buckling analysis of an axially loaded perfect cylinder, are presented. The general degenerated nonlinear shell theory of Hughes and Liu is employed in deriving this resultant-stress degenerated-shell element.Contrary to the traditional integration through the thickness approach, which assumes no coupling between the in-plane and transverse material and structural response matrices, the present approach can permit use of arbitrary, three-dimensional (3-D) nonlinear constitutive equations. Furthermore, explicit expressions of the element matrices for a 4-node shell element are developed. This rank-sufficient 4-node shell element, termed the resultant-stress degenerated-shell (RSDS) element, avoids the need for the costly numerical quadrature function evaluations of the element matrices and force vectors. And thus there are large increases in computational efficiency with this method. The comparisons of this RSDS element with six other shell elements are also given in this paper.     

The alternating schwarz method applied to some biharmonic variational inequalities

Recently there has been a renewed interest in the alternating Schwarz method and its numerical applications; this interest has also regarded developments in parallel computing (see [19] for an ample bibligraphy). In this paper we deal with the problems known as the free and clamped plate with two obstacles problems. We study the theoretical and numerical aspects of such problems with response to the Schwarz method, applying the mixed finite element method. Finally, we construct a simple algorithm permitting an estimate of the m.th step Schwarz error in the continuous and discrete case.     

Interval Finite Elements as a Basis for Generalized Models of Uncertainty in Engineering Mechanics

Latest scientific and engineering advances have started to recognize the need for defining multiple types of uncertainty. Probabilistic modeling cannot handle situations with incomplete or little information on which to evaluate a probability, or when that information is nonspecific, ambiguous, or conflicting [12], [47], [50]. Many interval-based uncertainty models have been developed to treat such situations. This paper presents an interval approach for the treatment of parameter uncertainty for linear static structural mechanics problems. Uncertain parameters are introduced in the form of unknown but bounded quantities (intervals). Interval analysis is applied to the Finite Element Method (FEM) to analyze the system response due to uncertain stiffness and loading. To avoid overestimation, the formulation is based on an element-by-element (EBE) technique. Element matrices are formulated, based on the physics of materials, and the Lagrange multiplier method is applied to impose the necessary constraints for compatibility and equilibrium. Earlier EBE formulation provided sharp bounds only on displacements [32]. Based on the developed formulation, the bounds on the system’s displacements and element forces are obtained simultaneously and have the same level of accuracy. Very sharp enclosures for the exact system responses are obtained. A number of numerical examples are introduced, and scalability is illustrated.     

Application of a new stress finite element to analysis of shell structures

A new stress finite element for analysis of shell structures of arbitrary geometry and loading has been introduced in Ref. [1]. The purpose of the present paper is to demonstrate the versatility of the proposed element with respect to all kinds of shell structures.     

On theories and applications of mixed finite element methods for linear boundary-value problems

This paper is devoted to a study of mathematical properties of certain mixed finite element approximations of linear boundary-value problems, and the application of such methods to simple representative problems which are designed to test the validity of the theory. In particular, the Oden—Reddy theory[1,2] is studied in some depth. An alternate approach to convergence questions, suggested by certain theorems of Babu ka[3], is also devised, and predictions of the two theories are briefly compared. As a result of this investigation, a number of criteria for using mixed methods in practical problems are identified, and it is shown that these criteria are supported by both theoretical arguments and by numerical experiments.     

Efficient domain decomposition methods for elliptic problems arising from flows in heterogeneous porous media

In this paper we study domain decomposition methods for solving some elliptic problem arising from flows in heterogeneous porous media. Due to the multiple scale nature of the elliptic coefficients arising from the heterogeneous formations, the construction of efficient domain decomposition methods for these problems requires a coarse solver which is adaptive to the fine scale features, [4]. We propose the use of a multiscale coarse solver based on a finite volume – finite element formulation. The resulting domain decomposition methods seem to induce a convergence rate nearly independent of the aspect ratio of the extreme permeability values within the substructures. A rigorous convergence analysis based on the Schwarz framework is carried out, and we demonstrate the efficiency and robustness of the preconditioner through numerical experiments which include problems with multiple scale coefficients, as well as problems with continuous scales. Communicated by: G. Wittum     

Finite element analysis of viscous fluid flow

A finite element model for the analysis of two dimensional viscous flows is formulated using the virtual work method. The model is in part based on a finite element shell model, using the same reduced integration of quadratic interpolations for all variables[1]. Differences from preceding formulations are that integration by parts is applied to the continuity equation, yielding different loading terms which are more easily defined in some problems, and a new approach is used for the convective inertia terms, giving a clearer interpretation of their effects which are distributed to both sides of the nonlinear recurrence relation. In the case of compressible flow, for which comparatively few formulations have been proposed to date, the thermal energy equation is used to form a two stage solution and here this seems the most natural and economical approach.     

Deterministic and stochastic approximating problems with applications to production control

Improper mathematical programming problems are analyzed and deterministic and stochastic approaches to correcting these problems are suggested. Numerical experiments with test examples are presented. The paper focuses on numerical analysis of improper linear programming problems [1–4], which arise in the context of scarce resources in economics [5]. Parametrization is applied to examine one of the possible approaches to approximation of improper LP problems under deterministic and stochastic conditions. Although the main focus is on improper problems of the 1st kind, we also touch upon some issues connected with improper problems of 2nd and 3rd kind [1]. The analysis of improper LP problems is based on duality theory [2]. Some results specialize the ideas previously presented in [1, 6]. The present paper is a continuation of [4].Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 114–125, May–June, 1992.     

Efficient sensitivity analysis and optimization of shell structures by the ABAQUS code

In this paper, a new efficient sensitivity analysis procedure is presented for the optimization of shell structures without access to the finite element source code. It is devised as a general interface tool to extend existing finite element systems from pure structural analysis to design capability. The implementation is performed based on the ABAQUS code. Kirchhoff flat shell elements are taken into account in the study with the element thickness as design variables. To ensure the performance and the validity of the proposed procedure, satisfactory sensitivity and optimization results are illustrated for numerical examples.     

Blending curves for landing problems by numerical differential equations II. Numerical methods

A landing curve of airplane is a blending curve that smoothly joins the two given boundary points, which is described by the parametric functions x(s), y(s), and z(s) governed by a system of ordinary differential equations (ODEs) with certain boundary conditions. In Part I, Mathematical Modelling [1], existence and uniqueness of the ODE system are explored to produce the optimal landing curves in minimum energy. In this paper, numerical techniques are provided by the finite element method (FEM) using piecewise cubic Hermite polynomials, to give the optimal solutions. An important issue is how to deal with infinite solutions occurring in the landing problems reported in [1]. Moreover, error analysis is made, and numerical examples are carried to verify the theoretical results made. This paper displays again the effectiveness and flexibility of the ODE approach to complicated blending curves. Besides, the numerical techniques in this paper can be applied directly to other landing and trajectory problems given in [1], as well as other kinds of blending curves and surfaces of airplane, ships, grand building, and astronautic shuttle-station.     

A combined scheme of edge-based and node-based smoothed finite element methods for Reissner–Mindlin flat shells

In this paper, a combined scheme of edge-based smoothed finite element method (ES-FEM) and node-based smoothed finite element method (NS-FEM) for triangular Reissner–Mindlin flat shells is developed to improve the accuracy of numerical results. The present method, named edge/node-based S-FEM (ENS-FEM), uses a gradient smoothing technique over smoothing domains based on a combination of ES-FEM and NS-FEM. A discrete shear gap technique is incorporated into ENS-FEM to avoid shear-locking phenomenon in Reissner–Mindlin flat shell elements. For all practical purpose, we propose an average combination (aENS-FEM) of ES-FEM and NS-FEM for shell structural problems. We compare numerical results obtained using aENS-FEM with other existing methods in the literature to show the effectiveness of the present method.     

Convergence and accuracy of the path integral approach for elastostatics

This paper addresses convergence rate and accuracy of a numerical technique for linear elastostatics based on a path integral formulation [Int. J. Numer. Math. Eng. 47 (2000) 1463]. The computational implementation combines a simple polynomial approximation of the displacement field with an approximate statement of the exact evolution equations, which is designated as functional integral method. A convergence analysis is performed for some simple nodal arrays. This is followed by two different estimations of the optimum parameter ζ: one is based on statistical arguments and the other on inspection of third order residuals. When the eight closest neighbors to a node are used for polynomial approximation the optimum parameter is found to depend on Poisson's ratio and lie in the range 0.5<ζ<1.5. Two well established numerical methods are then recovered as specific instances of the FIM. The strong formulation––point collocation––corresponds to the limit ζ=0 while bilinear finite elements corresponds exactly to the choice ζ=0.5. The use of the optimum parameter provides better precision than the other two methods with similar computational cost. Other nodal arrays are also studied both in two and three dimensions and the performance of the FIM compared with the corresponding finite element and collocation schemes. Finally, the implementation of FIM on unstructured meshes is discussed, and a numerical example solving Laplace equation is analyzed. It is shown that FIM compares favorably with FEM and offers a number of advantages.     

Knowledge-based control for finite element analysis

This paper shows that control logic may be separated from analysis software and that a knowledge-based expert system can use this logic to perform interactive computation. Heuristics that control a simple interactive finite element analysis program are represented using a rule-based format and are used by a goal-driven logic processor to invoke analysis activity.Traditional algorithm-oriented control and the proposed knowledge-based control are compared in a simple displacement computation scenario to identify the advantages/disadvantages of the two approaches. General activities and constraints, practical methods of reasoning and representation, and knowledge-based expert systems are discussed with emphasis on applications to interactive finite element analysis.An analysis control expert system has been developed for use in the numerical analysis of two-dimensional linear problems in solid and structural mechanics. An example problem is used to clarify the methods used to direct activity and to identify the problems associated with conditional task processing for interactive analysis.The main difference between the analysis program described in this paper and conventional analysis programs is related to the control architecture. The general conclusion of this paper is that knowledge-based control is more effective and flexible than algorithm-oriented control.