Niels Olhoff

An investigation concerning optimal design of solid elastic plates

Abstract We consider the problem of maximizing the integral stiffness of solid elastic plates described by thin plate theory. Assuming the material volume and plate domain to be given, we use the plate thickness function as the design variable and take both maximum and minimum allowable thickness values into account. On the basis of a convenient tensorial formulation of the problem, where the governing equations are derived by variational analysis and constitute necessary conditions for stationarity, we develop an efficient and quite general numerical algorithm by means of which a number of stationary solutions for rectangular and axisymmetric annular plates with various boundary conditions are obtained. These numerical results enable us to investigate the optimization problem itself in terms of its major parameters, particularly the maximum and minimum values specified for the plate thickness. For problems associated with large ratios between these constraint values, plate designs with significant integral stiffeners are obtained. We find however, that these designs are only local optima and that a global optimal plate thickness function does generally neither exist within the class of smooth functions nor within the class of smooth functions with a finite number of discontinuities. In order to determine a global optimal solution associated with given thickness constraint values, it is therefore necessary to change the optimal design formulation. With implications for a number of similar two-dimensional optimization problems, our results offer valuable indications of the lines along which such changes should be performed.

Multiple eigenvalues in structural optimization problems

This paper discusses characteristic features and inherent difficulties pertaining to the lack of usual differentiability properties in problems of sensitivity analysis and optimum structural design with respect to multiple eigenvalues. Computational aspects are illustrated via a number of examples.Based on a mathematical perturbation technique, a general multiparameter framework is developed for computation of design sensitivities of simple as well as multiple eigenvalues of complex structures. The method is exemplified by computation of changes of simple and multiple natural transverse vibration frequencies subject to changes of different design parameters of finite element modelled, stiffener reinforced thin elastic plates.Problems of optimization are formulated as the maximization of the smallest (simple or multiple) eigenvalue subject to a global constraint of e.g. given total volume of material of the structure, and necessary optimality conditions are derived for an arbitrary degree of multiplicity of the smallest eigenvalue. The necessary optimality conditions express (i) linear dependence of a set of generalized gradient vectors of the multiple eigenvalue and the gradient vector of the constraint, and (ii) positive semi-definiteness of a matrix of the coefficients of the linear combination.It is shown in the paper that the optimality condition (i) can be directly applied for the development of an efficient, iterative numerical method for the optimization of structural eigenvalues of arbitrary multiplicity, and that the satisfaction of the necessary optimality condition (ii) can be readily checked when the method has converged. Application of the method is illustrated by simple, multiparameter examples of optimizing single and bimodal buckling loads of columns on elastic foundations.

On single and bimodal optimum buckling loads of clamped columns

Abstract We study the problem of determining the optimum shape of a thin, elastic, clamped column of given length and volume, such that the fundamental buckling load is a maximum. The column cross-sections are assumed to be geometrically similar, and a minimum allowable value is specified for the cross-sectional area. Investigating the optimization problem parametrically in terms of this minimum constraint, we reveal a significant feature. There exists a threshold value for the constraint, beyond which the optimum columns are all associated with single mode optimum buckling loads, whereas, for any value of the constraint less than the threshold value, the optimum columns are associated with bimodal fundamental buckling loads. This bimodal behaviour necessitates an extension and a mathematical reformulation of the current optimization problem, which is outlined and solved in the paper. In particular, we revise the result hitherto considered to be the optimum solution for an unconstrained column with clamped ends.

Multicriterion structural optimization via bound formulation and mathematical programming

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.

Optimum topology and reinforcement design of disk and plate structures with multiple stiffness and eigenfrequency objectives

Abstract This paper deals with topology optimization of statically loaded or freely vibrating disk and plate structures, and with layout optimization of different types of integral rib-reinforcement of plates. The design parametrization is based on different configurations of layered microstructures, and the effective stiffness properties of the microstructures are derived in a unified way by a homogenization approach. The effects of using microstructures of second rank and of arbitrary rank (based on moment variables) are studied and compared using examples. All structures are analyzed applying anisotropic Mindlin finite elements with membrane capability. For the disk and plate structures under consideration, special emphasis is devoted to problems of stiffness optimization subject to several load cases and problems of optimization of eigenfrequencies of free vibration that belong to a given spectrum of eigenfrequencies, i.e. optimization problems that imply multiobjective formulations are considered. The multiobjective problems are treated via a max–min formulation based on a variable lower bound technique, and this approach is shown to yield much more significant results than the usual weighted sum formulation of multiobjective problems. Several illustrative numerical examples of multiobjective topology and layout optimization problems of statically loaded and vibrating disks and plates are presented in the paper.

A Variational Formulation for Multicriteria Structural Optimization

ABSTRACT Multicriterion structural optimization problems stated as minimization of the maximum of a set of weighted criteria are considered. Interpretation of the min-max problem as a simple minimization problem for an upper bound on the criteria leads to a formulation that is convenient for analysis of problems with global as well as local objectives. The method is demonstrated on multipurpose design of a vibrating cantilever, on design of a

A Method of “Exact” Numerical Differentiation for Error Elimination in Finite-Element-Based Semi-Analytical Shape Sensitivity Analyses

ABSTRACT The traditional, simple numerical differentiation of finite-element stiffness matrices by a forward difference scheme is the source of severe error problems that have been reported recently for certain problems of finite-element-based, semi-analytical shape design sensitivity analysis. In order to develop a method for elimination of such errors, without a sacrifice of the simple numerical differentiation and other main advantages of the semi-analytical method, the common mathematical structure of a broad range of finite-element stiffness matrices is studied in this paper. This study leads to the result that element stiffness matrices can generally be expressed in terms of a class of special scalar functions and a class of matrix functions of shape design variables that are defined such that the members of the classes admit “exact” numerical differentiation (exact up to round-off error) by means of very simple correction factors to upgrade standard computationally inexpensive first-order finite di...

Regularized formulation for optimal design of axisymmetric plates

Abstract This paper presents a regularized mathematical formulation, necessary conditions of optimality and solutions of problems of optimal design of solid, elastic, axisymmetric plates of prescribed total material volume. Single-purpose design criteria of minimum static compliance (maximum stiffness) and maximum frequency of free, transverse vibrations, are considered. The regularization, which alleviates some anomalies and difficulties encountered earlier in plate optimization problems, is based on a new compound plate model with two simultaneous design variables, namely, variable thickness of a solid part of the plate and variable concentration of a dense system of thin, integral stiffeners attached to the solid plate part. A numerical, optimality-criterion-based method of solution is developed for the problem, and several optimal designs are presented. The results are compared with results obtained from optimal design formulations applied heretofore, and substantiate the superiority of the new, regularized formulation.

On the Solid Plate Paradox in Structural Optimization

ABSTRACT This paper discusses the minimum weight design of solid plastic plates subject to constraints on the highest and lowest allowable values of the plate thickness. It is shown that the maximum thickness constraint alone does not ensure a smooth global minimum weight solution because the least weight is furnished, in the limit, by a grillage-like continuum consisting of a dense system of ribs of infinitesimal spacing and uniform depth. The optimal layout of such continua has already been determined for most loading and boundary conditions by the first author and Prager and is found to be the same for (a) plastic limit design, (b) elastic stress design, (c) design for given compliance, and (d) design for given fundamental frequency. Two refinements of the above layout theory are also considered in the current paper. One formulation takes into consideration the weight savings due to rib intersections in high density grillages. The other development deals with minimum as well as maximum thickness constr...

Optimal design of vibrating circular plates

Abstract Related to a given volume, the shape of a rotationally symmetric plate is determined so that its first natural frequency of transverse vibrations becomes optimal. Three different cases of boundary conditions are investigated. Assuming that the lowest mode is rotationally symmetric, the corresponding mathematical problem is shown to be an ordinary, fourth order, non-linear and singular, but homogeneous eigenvalue problem. The differential equation is derived by variational analysis, and is solved numerically by successive iterations.