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.     

Sensitivity analysis of shape memory alloy shells

This paper presents procedures for efficient design sensitivity analysis for shape memory alloy (SMA) structures modeled with shell elements. Availability of sensitivity information at low computational cost can dramatically improve the efficiency of the optimization process, as it enables use of efficient gradient-based optimization algorithms. The formulation and computation of design sensitivities of SMA shell structures using the direct differentiation method is considered, in a steady state electro-thermo-mechanical finite element context. Finite difference, semi-analytical and refined semi-analytical sensitivity analysis approaches are considered and compared in terms of efficiency, accuracy and implementation effort, based on a representative finite element model of a miniature SMA gripper.     

Developments of sizing sensitivity analysis with the ABAQUS code

A practical tool is developed to deal with sizing sensitivity analysis of linear and geometrically nonlinear problems. Design variables are selected as cross-sectional areas of bar elements or the thickness of the membrane, beam, plate and shell elements. This tool is considered as a general open interface implemented outside the finite element package. It can extend existing finite element systems from pure structural analysis to engineering design capabilities. The formulation is based on the direct differentiation method. Numerical results are typically provided with the application of the ABAQUS code.     

Free vibration analysis and shape optimization of variable thickness plates and shells—II. Sensitivity analysis and shape optimization

In this paper an automated approach is used to carry out sensitivity analysis and to obtain optimum shapes for plates and shells in which the natural frequencies are maximized. The free vibration analysis is carried out with the nine-noded, degenerated, Huang-Hinton shell element implemented and tested in Part I of this paper. Design variables that specify either the shape or thickness distribution of the structures are considered. Special attention is focused on the sensitivity calculations and problems connected with their accuracy and performance are highlighted when the semi-analytical and finite difference methods are used. Advantages and disadvantages of each method are discussed. The optimal solution is found by the use of a structural optimization algorithm which integrates the finite element module (Part I), sensitivity analysis and a mathematical programming method: sequential quadratic programming (SQP). Optimal forms are then obtained for a set of benchmark examples using the two sensitivity analysis techniques and their results are compared. The results obtained for optimum solutions in the present paper justify the usage of the semi-analytical method for sensitivities calculations for structural shape optimization purposes.     

Mixed elements in the optimal design of plates

The theory of design sensitivity analysis of structures, based on mixed finite element models, is developed for static, dynamic and stability constraints. The theory is applied to the optimal design of plates with minimum weight, subject to displacement, stress, natural frequencies and buckling stresses constraints. The finite element model is based on an eight node mixed isoparametric quadratic plate element, whose degrees of freedom are the transversal displacement and three moments per node. The corresponding nonlinear programming problem is solved using the commercially available ADS (Automated Design Synthesis) program. The sensitivities are calculated by analytical, semi-analytical and finite difference techniques. The advantages and disadvantages of mixed elements in design optimization of plates are discussed with reference to applications.     

‘Integral’ finite elements for designing the strong frames of highly maneuverable airplanes

The experience of using classical finite element programs in the computer-aided design of airframes is discussed. A set of features in the classical finite element method is mentioned which hinder the implementation of programs for automated structural optimization with respect to strength requirements. Principles simplifying implementation of such programs are proposed, including the principle of two-dimensional elements being identical to natural structural elements. The approximating functions are introduced on the basis of exact solutions for a beam and an anisotropic plate subjected to concentrated and distributed loads, these solutions being subsequently used to describe displacements of these elements. This approach is exemplified by test problems and is demonstrated in application to the highly maneuverable aircraft strong frame design.     

Frequency response of shell-plate combinations

Natural frequencies of cylindrical shells with a circular plate attached at arbitrary locations are determined for various boundary conditions and L/D ratios. The semi-analytical finite element method is used for the analysis. A conical shell element with four degrees of freedom per node and two nodes per element is used. For clamped-clamped and simply-supported boundary conditions the plate is attached at the center of the shell. For a clamped-free boundary condition the plate is at the free end of the shell. The effects of plate thickness and L/D ratio of the shell on the frequencies of the shell-plate combination are investigated.     

An improved formulation of space stiffeners

This paper presents a displacement based finite element model for predicting the constraint torsion effect of stiffeners. In structural modelling, the plate/shell and the stiffeners are treated as separate elements where the displacement compatibility transformation between these two types of elements takes into account the constraint torsional warping effect in the stiffeners. The development is based on a general beam theory which includes flexural-torsion coupling, constrained torsion warping, and shear-centre location. The virtual work principle includes the second order terms of finite beam rotations. For finite element analysis, cubic Hermitian polynomials are used as shape functions of the straight space frame element with two nodes. Elastic stiffness and geometric stiffness matrices for an arbitrary cross-section are evaluated in a closed form, and load correction stiffness for eccentric stiffener loads are considered. To demonstrate the importance of torsion warping constraints and to illustrate the accuracy of this formulation, finite element solutions are presented and compared with available solutions.     

Geometrically nonlinear analysis of shell structures using a flat triangular shell finite element

Summary This paper presents a state of the art review on geometrically nonlinear analysis of shell structures that is limited to the co-rotational approach and to flat triangular shell finite elements. These shell elements are built up from flat triangular membranes and plates. We propose an element comprised of the constant strain triangle (CST) membrane element and the discrete Kirchhoff (DKT) plate element and describe its formulation while stressing two main issues: the derivation of the geometric stiffness matrix and the isolation of the rigid body motion from the total deformations. We further use it to solve a broad class of problems from the literature to validate its use.     

Finite element analysis of shell structures

Summary A survey of effective finite element formulations for the analysis of shell structures is presented. First, the basic requirements for shell elements are discussed, in which it is emphasized that generality and reliability are most important items. A general displacement-based formulation is then briefly reviewed. This formulation is not effective, but it is used as a starting point for developing a general and effective approach using the mixed interpolation of the tensorial components. The formulation of various MITC elements (that is, elements based on Mixed Interpolation of Tensorial Components) are presented. Theoretical results (applicable to plate analysis) and various numerical results of analyses of plates and shells are summarized. These illustrate some current capabilities and the potential for further finite element developments.     

Study of inaccuracy in semi-analytical sensitivity analysis — a model problem

The semi-analytical method of sensitivity analysis (Zienkiewicz and Campbell 1973; Esping 1983; Cheng and Liu 1987) of finite element discretized structures is attractive due to the balance between computational cost and ease of implementation (Cheng and Liu 1987; Haftka and Adelman 1989), but unfortunately the method may exhibit serious inaccuracies when applied in shape optimization of structures modelled by beam, plate, shell and Hermite elements (Cheng and Liu 1987; Haftka and Adelman 1989; Barthelemyet al. 1988; Barthelemy and Haftka 1988; Choi and Twu 1991, Pedersenet al. 1989; Chenget al. 1989).In the present paper, we perform an exact analysis of the error of sensitivity for a simple model problem which has earlier been considered by Barthelemyet al. (1988), Barthelemy and Haftka (1988), Pedersenet al. (1989). The analysis gives a deep insight into the nature of the general inaccuracy problem and enables us to devise methods by which the severe error of the sensitivity can be substantially reduced or removed for the model problem. The results of the paper are illustrated via an example.A method of error elimination for an extended class of semianalytical analysis problems is developed and presented in a companion paper (Olhoff and Rasmussen 1991).     

A general three-dimensional L-section beam finite element for elastoplastic large deformation analysis

In this paper, the development of a general three-dimensional L-section beam finite element for elastoplastic large deformation analysis is presented. We propose the generalized interpolation scheme for the isoparametric formulation of three-dimensional beam finite elements and the numerical procedure is developed for elastoplastic large deformation analysis. The formulation is general and effective for other thin-walled section beam finite elements. To show the validity of the formulation proposed, a 2-node three-dimensional L-section beam finite element is implemented in an analysis code. As numerical examples, we first perform elastic small and large deformation analyses of a cantilever beam structure subjected to various tip loadings, and elastoplastic large deformation analysis of the same structure under reversed cyclic tip loading. We then analyze the failures of simply supported beam structures of different lengths and slenderness ratios under elastoplastic large deformation. The same problems are solved using refined shell finite element models of the structures. The numerical results of the L-section beam finite element developed here are compared with the solutions obtained using shell finite element analyses. We also discuss the numerical solutions in detail.     

Vibration analysis of laminated anisotropic shells of revolution

An efficient computational procedure is presented for the free vibration analysis of laminated anisotropic shells of revolution, and for assessing the sensitivity of their response to anisotropic (nonorthotropic) material coefficients. The analytical formulation is based on a form of the Sanders-Budiansky shell theory including the effects of both the transverse shear deformation and the laminated anisotropic material response. The fundamental unknowns consist of the eight stress resultants, the eight strain components, and the five generalized displacements of the shell. Each of the shell variables is expressed in terms of trigonometric functions in the circumferential coordinate and a three-field mixed finite element model is used for the discretization in the meridional direction.The three key elements of the procedure are: (a) use of three-field mixed finite element models in the meridional direction with discontinuous stress resultants and strain components at the element interfaces, thereby allowing the elimination of the stress resultants and strain components on the element level; (b) operator splitting, or decomposition of the material stiffness matrix of the shell into the sum of an orthotropic and nonorthotropic (anisotropic) parts, thereby uncoupling the governing finite element equations corresponding to the symmetric and antisymmetric vibrations for each Fourier harmonic; and (c) application of a reduction method through the successive use of the finite element method and the classical Bubnov-Galerkin technique.The potential of the proposed procedure is discussed and numerical results are presented to demonstrate its effectiveness.     

Finite element modal analysis of an HDD considering the flexibility of spinning disk–spindle, head–suspension–actuator and supporting structure

This paper presents a finite element method to analyze the free vibration of a flexible HDD (hard disk drive) composed of the spinning disk–spindle system with fluid dynamic bearings (FDBs), the head–suspension–actuator with pivot bearings, and the base plate with complicated geometry. Finite element equations of each component of an HDD are consistently derived with the satisfaction of the geometric compatibility in the internal boundary between each component. The spinning disk, hub and FDBs are modeled by annular sector elements, beam elements and stiffness and damping elements, respectively. It develops a 2-D quadrilateral 4-node shell element with rotational degrees of freedom to model the thin suspension efficiently as well as to satisfy the geometric compatibility between the 3-D tetrahedral element and the 2-D shell element. Base plate, arm, E-block and fantail are modeled by tetrahedral elements. Pivot bearing of an actuator and air bearing between spinning disk and head are modeled by stiffness elements. The restarted Arnoldi iteration method is applied to solve the large asymmetric eigenvalue problem to determine the natural frequencies and mode shapes of the finite element model. Experimental modal testing shows that the proposed method well predicts the vibration characteristics of an HDD. This research also shows that even the vibration motion of the spinning disk corresponding to half-speed whirl and the pure disk mode are transferred to a head–suspension–actuator and base plate through the air bearing and the pivot bearing consecutively. The proposed method can be effectively extended to investigate the forced vibration of an HDD and to design a robust HDD against shock.     

基于等几何分析的柔性多体系统建模方法研究

近三十年来,柔性多体系统动力学取得长足进步,尤其是以绝对节点坐标方法(Absolute Nodal Coordinate Formulation, ANCF)为代表的非线性有限元已被用来处理复杂的柔性多体系统动力学问题.但绝对节点坐标方法采用斜率矢量作为广义坐标,导致系统自由度多,计算效率低.针对柔性多体系统,基于非均匀有理B样条(Non-Uniform Rational B-Splines, NURBS)曲线和曲面分别提出了Euler-Bernoulli细长梁单元和Kirchhoff-Love薄壳单元,在完全拉格朗日格式下,根据Green应变张量对单元变形进行描述,结合第二类Piola-Kirchhoff应力张量给出单元应变能公式,推导了单元的弹性力和弹性力雅可比矩阵表达式,最后通过静力学及动力学数值算例对提出的两类单元的性能进行对比和验证,为柔性多体系统建模提供了一种精确高效的新单元.     

On shape sensitivity approaches in the numerical analysis of structures

The semi-analytical, analytical and direct methods for numerical structural shape sensitivity analysis are discussed for a beam model and the general three-dimensional case. While the two first methods are applied directly to the finite element model of a structure, the direct approach follows from a continuous formulation and only the final results can be discretized.     

Finite element analysis of stiffened plates

A finite element method is presented in which the constraint between stiffener and member is imposed by means of Lagrange multipliers. This is performed on the functional level, forming augmented variational principles. In order to simplify the initial development and implementation of the proposed method, two-dimensional stiffened beam finite elements are developed. Several such elements are formulated, each showing monotonic convergence in numerical tests. In the development of stiffened plate finite elements, the bending and membrane behaviors are treated seperately. For each, the stiffness matrix of a standard plate element is modified to account for an added beam element (representing the stiffener) and additional terms imposing the constraint between the two. The resulting stiffened plate element was implemented in the SAPIV finite element code. Exact solutions are not known for rib-reinforced plated structures, but results of numerical tests converge monotonically to a value in the vicinity of an approximate “smeared” series solution.     

Application of the compound strip method for the analysis of slab-girder bridges

The compound strip method is illustrated for the analysis of slab-girder bridges modeled as a linear elastic plate continuous over deflecting supports. This approach incorporates the effects of support elements in a direct stiffness methodology by creating a substructure composed of plate. beam. and column elements which is termed a “compound strip.” The theory and application of the compound strip method is presented. The finite element and compound strip methods are compared in an illustrative analysis for a slab-girder bridge. The results of the compound strip analysis compare well with the finite element method. The methodology presented herein can be used to efficiently model any slab-girder bridge configuration. Typically, the compound strip method requires significantly less computational resources than does the finite element method and is well suited for use on today's microcomputers.     

Axisymmetric Finite Element Method for Analysis of Plate on Layered Soil

To obtain the fundamental solution of soil has become the key problem for the semi-analytical and semi-numerical (SASN) method in analyzing plate on layered soil. By applying axisymmetric finite element method (FEM),an expression relating the surface settlement and the reaction of the layered soil can be obtained. Such a reaction can be treated as load acting on the applied external load. Having the plate modelled by four-node elements,the governing equation of the plate can be formed and solved. In this ca...