Abstract: Recent developments are reviewed in two major areas of structural sensitivity analysis: sensitivity of static and transient response; and sensitivity of vibration and buckling eigenproblems. Recent developments from the standpoint of computational cost, accuracy, and ease of implementation are presented. In the area of static response, current interest is focused on sensitivity to shape variation and sensitivity of nonlinear response. Two general approaches are used for computing sensitivities: differentiation of the continuum equations followed by discretization, and the reverse approach of discretization followed by differentiation. It is shown that the choice of methods has important accuracy and implementation implications. In the area of eigenproblem sensitivity, there is a great deal of interest and significant progress in sensitivity of problems with repeated eigenvalues. In addition to reviewing recent contributions in this area, the paper raises the issue of differentiability and continuity associated with the occurrence of repeated eigenvalues.

Abstract: In order to use an anisotropic material effectively it should be oriented optimally with respect to the actual strain condition. Orientations with extreme energy density are obtained for orthotropic materials. It is found that the optimal orientation depends on one non-dimensional material parameter only, plus the ratio of the two principal strains. Complete analytical results are derived, including local as well as global maxima and minima.

Abstract: The Convex Linearization method (CONLIN) exhibits many interesting features and it is applicable to a broad class of structural optimization problems. The method employs mixed design variables (either direct or reciprocal) in order to get first order, conservative approximations to the objective function and to the constraints. The primary optimization problem is therefore replaced with a sequence of explicit approximate problems having a simple algebraic structure. The explicit subproblems are convex and separable, and they can be solved efficiently by using a dual method approach. In this paper, a special purpose dual optimizer is proposed to solve the explicit subproblem generated by the CONLIN strategy. The maximum of the dual function is sought in a sequence of dual subspaces of variable dimensionality. The primary dual problem is itself replaced with a sequence of approximate quadratic subproblems with non-negativity constraints on the dual variables. Because each quadratic subproblem is restricted to the current subspace of non zero dual variables, its dimensionality is usually reasonably small. Clearly, the Hessian matrix does not need to be inverted (it can in fact be singular), and no line search process is necessary. An important advantage of the proposed maximization method lies in the fact that most of the computational effort in the iterative process is performed with reduced sets of primal variables and dual variables. Furthermore, an appropriate active set strategy has been devised, that yields a highly reliable dual optimizer.

Abstract: Multicriterion structural optimization problems pertaining to minimization of the maximum (or maximization of the minimum) of a set of weighted criteria are considered. In order to alleviate the inherent difficulty of non-differentiability of min-max problems, we adopt a so-called “bound formulation” and show that this approach even provides us with a very simple means of performing a switch from a prescribed-resource to a cost-minimization formulation of a given type of problem. The bound approach was found very useful in admitting the treatment of min-max problems by usual variational analysis; we demonstrate in this paper that the technique is also extremely well-suited to mathematical programming. Illustrative examples are presented at the end of the paper.

Abstract: A unified approach to various problems of structural optimization is presented. It is based on a combination of mathematical models of different complexity. The models describe the behaviour of a designed structure. From the computational point of view, it is connected with the sequential approximation of design problem constraints and/or an objective function. In each step, a subregion of the initial search region in the space of design variables is chosen. In this subregion, various points (designs) are selected, for which response analyses are carried out using a numerical method (mostly FEM). Using the least-squares method, analytical expressions are formulated, which then replace the initial problem functions. They are used as functions of a particular mathematical programming problem. The size and location of sequential subregions may be changed according to the result of the search. The choice of one particular form of the analytical expressions is described. The application of the approach is shown by means of test examples and comparison with other optimization techniques is presented.

Abstract: In this paper, various methods based on convex approximation schemes are discussed, that have demonstrated strong potential for efficient solution of structural optimization problems. First, theconvex linearization method (CONLIN) is briefly described, as well as one of its recent generalizations, themethod of moving asymptotes (MMA). Both CONLIN and MMA can be interpreted as first order convex approximation methods, that attempt to estimate the curvature of the problem functions on the basis of semi-empirical rules. Attention is next directed toward methods that use diagonal second derivatives in order to provide a sound basis for building up high quality explicit approximations of the behaviour constraints. In particular, it is shown how second order information can be effectively used without demanding a prohibitive computational cost. Various first and second order approaches are compared by applying them to simple problems that have a closed form solution.

Abstract: Continuum-type optimality criteria for the iterative optimization of “large” finite element systems (i.e. systems with over ten thousand finite elements) are discussed. By investigating optimization problems with up to one million elements and one million variables, it is shown that for a single displacement constraint the proposed method results in a rapid and almost uniform convergence, the rate of which, even for relatively ill-conditioned problems, does not depend significantly on the number of elements. Additional refinements, including upper and lower limits on the cross-sectional dimensions, segmentation, allowance for selfweight and the cost of supports, non-linear and non-separable objective functions, inclusion of shear deformations, built-up cross-sections as well as additional stress constraints and two-dimensional (plane stress) problems, will be considered in Part II of this contribution. The current development constitutes an extension and generalization of pioneering work by Berke, Khot, Venkayya and their associates, whose methods are also reviewed herein. In addition to an elementary truss example and a more advanced beam example, some simple layout optimization problems are considered in this Part. A special feature of the paper is that all numerical results presented are confirmed by closed form analytical solutions.

Abstract: Optimum design of structures with path dependent response is studied in this paper. The direct differentiation and the adjoint structure methods of design sensitivity analysis are summarized. The reference volume concept is used to unify the conventional and shape design problems. It is concluded that the direct differentiation method is more effective for this class of problems. The design sensitivity analysis — developed with continuum formulation — is discretized using the finite element method. Two cases for an example problem are optimized using a sequential quadratic programming algorithm to demonstrate how the developed procedures work and to study the optimization process for the problems with path dependent response.

Abstract: In this paper, the static aeroplastic characteristics, divergence velocity, control effectiveness and lift effectiveness are considered in obtaining an optimum weight structure Swept wing structures are used with upper and lower skins, spar and rib thicknesses, and spar cap and vertical post cross-sectional areas as the design parameters The aerodynamic strip theory is used to derive the constraint formulations and aerodynamic load matrices A Sequential Unconstrained Minimization Technique (SUMT) algorithm is used to optimize the wing structure to meet the desired aeroelastic constraints

Abstract: This paper describes a non-gradient formulation for solving shape optimal design problems involving structures in plane stress or having an axially symmetric geometry. The minimization of the maximum von Mises stress value at a traction free boundary poses a non-linear optimization problem in which the design variables do not appear explicitly in the formulation. The most commonly used approach is to apply a standard non-linear programming technique. There exists in this field no universally accepted solution method. The major difficulty of shape optimization in connection with FEM is to perform an accurate and efficient sensitivity analysis. The perturbation analysis introduced here takes advantage of the character of the problem. It is based on methods from the theory of notches. The results are applied to an FE-model of the structural component. The iterative method with such a direction of search works efficiently even for a large number of design variables as shown by Schnack (1977b, 1978, 1979, 1980, 1983 and 1985). Using a dynamic programming formulation (see also Schnack and Sporl 1986), the existence of a solution for the shape optimal problem will be discussed. Examples of applications to structural components from mechanical engineering are presented to demonstrate the power of this approach.

Abstract: The aim of structural design is to determine the dimensions and the shape of a system that fulfils certain requirements in an optimal way. Although this problem is not at all new, the application of mathematical algorithms together with multicriteria and shape optimization techniques as strategies for achieving an optimum are still very rarely used in practice. The efficiency of the optimization procedure SAPOP is demonstrated through the shape optimization of ultra light shell structures (e.g. satellite tanks) for which bending effects as well as the influence of large deformations in the shell theory are taken into account. After establishing a corresponding transfer matrix method for analysis, the meridional shape can be determined by using special shape functions (modified ellipsoids) and a “direct” shape optimization strategy. In addition, simultaneous optimization of shape and wall thickness distribution is introduced.

Abstract: Natural Structural Shapes are derived for axisymmetrically loaded shells of revolution within the membrane theory of shells. The concept of natural structural shapes is based on the simultaneous minimization of the mass and the strain energy of the loaded structure, a multicriteria optimization problem with Edgeworth-Pareto optimality as the basic optimality concept. The problem is formulated as a multicriteria control problem and necessary conditions for arbitrary loading and boundary are derived. Exact and numerical results are obtained for both the case of uniform pressure and that of a ring load with zero surface loads.

Abstract: Geometrical optimization of trusses, i.e. optimization of the cross-sectional areas of the members and the coordinates of the joints, is solved by atwo-level approximation concept (TLAC). Displacements and element forces are approximated by first order Taylor series expansions in terms of generalized variables (or their reciprocal) which define the geometrical properties of the elements. This approximation leads to a high-qualityfirst level approximate problem (FA) which is solved by considering a sequence ofsecond level approximate problems (SA) in the design variable space. The method presented here represents a new approach to the solution of geometrical optimization problems. Numerical examples are given to demonstrate the high efficiency of the proposed method.

Abstract: 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.

Abstract: Optimal structural design of a noncircular cylindrical shell under overall bending and axial force is considered. The material is assumed to be governed by the Norton nonlinear steady creep law. Minimal cross-sectional area is the design objective, the middle line of the profile and the wall-thickness are the design variables and the constraints refer to local stability of the wall according to Gerard's criterion and to brittle creep rupture as described by the Kachanov-Robinson hypothesis. In view of bending, optimal design requires some concentrated areas (longitudinal ribs) located at the outer fibres of the cross-section.

Abstract: In nonlinear optimization, the dual problem is in general not easier to solve than the primal problem. Convex separable optimization problems, frequently arising in electrical and mechanical engineering, constitute a notable exception to the above rule. The dual problem is to optimize the dual objective functionl over a non-negative orthant, and the evaluation ofl reduces to the execution of independentlinear searches only. To generalize the idea, we also consider partially-separable problems with objective and constraint functions such that the Hessian matrix of the Lagrange function is a block-diagonal matrix with 2*2 blocks. The evaluation of the dual objective function is accordingly reduced to a number of independentplanar searches. Obviously, 3*3 blocks would lead tospatial searches, etc. We compare the performance of a primal and a dual method on a graded set of artificial test problems with increasing size, increasing degree of degeneracy, and increasing ill-conditioning. The observed speed-up by the dual approach varies between 2 and 30. Finally, we consider the potential of the dual approach for execution on parallel computers.

Abstract: A computer-based structural design methodology is presented for the least-weight design of planar frameworks subjected to multiple dynamic loads. The method makes use of dynamic finite element analysis, sensitivity analysis, first-order Taylor series approximations and the identification of response extrema to convert the time-parametric design problem into an explicit non-parametric form, which is solved iteratively using a dual optimization routine. The focus of the study is not to develop a general-purpose design capability but, rather, to examine the computational aspects of accounting for multiple dynamic loads in the design process. Two example truss designs under multiple regular sinusoidal wave loadings and multiple irregular earthquake loadings are presented.

Abstract: Confidence of designers in the theory of structural optimization is of fundamental importance for practical applications. The so-called “erosion of optimal designs” is discussed from this point of view, in the context of ring and plate examples for which the limit and post-limit behaviors are obtained either experimentally or by numerical simulations of experiments. Tentative conclusions are suggested.

Abstract: The paper deals with the optimal design of circular arches against creep buckling in the Rabotnov-Shesterikov sense. The design vairable, i.e. the cross-sectional area function, is determined so as to minimize the total volume of an arch under given external load (radial pressure) and the critical time. Multimodal formulation of the problem is introduced, i.e. both symmetric and antisymmetric instability modes are considered for in-plane and out-of-plane buckling. The effect of the behaviour of loading in the course of buckling on optimal shapes is analysed. The problem is solved by the use of Pontryagin's maximum principle.

Abstract: A unified numerical approach, based on a control parameterization technique, for solving structural crosssectional optimization problems is presented. The key factor to the unified formulation lies in the framing of the objective functional and the constraints into the same unified canonical form. Consequently, the different types of objective functionals, geometrical and performance constraints can be treated in the same way, thus paving the path for the problems to be solved under a single approach using a general purpose software. To demonstrate this versatile approach, several illustrative examples of cross-sectional shape optimization of structural members under a variety of constraints were examined.

Abstract: Nonlinearly elastic discrete systems under nonconservative, compressive loading [1,2,3] of follower type, that may lose their stability through divergence, are considered. Using a general mathematical formulation a thorough parametric discussion of the critical, prebuckling and postbuckling, large displacement response, is comprehensively presented. The predominant effects on the nonlinear divergence instability of the material nonlinearity as well as of the loading parameters defining the degree of nonconservativeness, are completely revealed. Necessary and sufficient conditions for the existence of regions of devergence instability, are properly established. By means of these conditions the boundary between divergence (static) and flutter (dynamic) instability is found. The case of existence of a double critical point (coincidence of the first and second static buckling eigenmodes), obtained as a result of the linear stability analysis [4–6], is also discussed. At the aforementioned boundary, the (critical) buckling load corresponds to the maximum load-carrying capacity that can be determined by means of a nonlinear (static) stability analysis. Thus, a further insight into the role of certain parameters of paramount importance for the change of mechanism of instability from divergence to flutter, and vice-versa, is also gained.

Abstract: Various methods for performing the sensitivity analysis in solving optimal shape design problems are outlined. The methods are illustrated in detail in the finite setting of a unilateral boundary value problem of the Dirichlet-Signorini type. The methods are compared in several numerical examples.

Abstract: Absolute structural optimization problems such as leastweight and layout problems become both more realistic and more complex when their parameters are described probabilistically. The theoretical framework for absolute optimal design of sandwich beams, grillages and reinforced slabs is extended herein to deal with probabilistic parameters. It is shown that using a simplified probabilistic framework (socalled First Order Second Moment), some classical solutions for optimal layout remain valid. Remarks about the more general problem and the difficulty of its solution close the paper.

Abstract: Optimization of transversely vibrating shafts with respect to eigenfrequencies, with constraints on the design variables, has been almost fully investigated in recent years - as far as the Bernoulli-Euler equation of motion is concerned. In the present paper, Timoshenko's equations are applied, i.e. the effects of shear and rotational energy are taken into account in the calculation of the transverse vibration modes and frequencies of the shaft. This is done via a finite element formulation. The sensitivity analysis of the transverse vibration frequencies is based on analytical differentation of the stiffness- and mass matrices. The computer program developed provides the user with an option to suppress the Timoshenko effects, such that the analysis can be carried out within the Bernoulli-Euler theory as well and then comparisons can be made. Torsional vibrations of shafts are also considered, computing the vibration modes and natural frequencies by means of a finite difference approach. Again, the sensitivity analysis is carried out analytically. Objective functions and behavioral constraints are selected from fundamental frequencies, higher order frequencies and differences between adjacent frequencies of the vibrational types considered. The cross-sectional area function of the shaft is used as the design variable and the total volume of structural material is assumed to be given, along with some sizing constraints. The shafts considered may be equipped with non-structural disks and flexible supports. Several examples are presented and some notable differences between optimized Timoshenko and Bernoulli-Euler shafts derived.