Keywords: structural optimisation, geometrical imperfections, random fields, convex model, distributed components, CORBA.

Optimisation of slender structures which are subjected to compressive stresses requires careful attention to geometrical imperfections. In particular, it is crucial to pay attention to the fact that the influence of a given imperfection shape highly depends on the actual structural design. Hence, it is not possible to prescribe an imperfection shape at the beginning and to keep it unchanged during the optimisation process. In this contribution, an optimisation model for complex engineering structures is presented which comprises geometrical imperfections as part of the structural model such that the design optimised, using this model, performs well for various probable imperfection shapes.

Within the optimisation model, the structural design is described by means of a set of optimisation variables collected in the design vector. For a given design vector, the structural performance is computed by the model and expressed in terms of an objective function and a set of constraint functions. By that, the structural performance has to reflect the possible variability of the imperfection shape which is introduced by the stochastic nature of geometrical imperfections.

Within the optimisation model, the structural model is described by means of a geometry model in which the design is constructed hierarchically upon a base geometric element representing the overall shape of the structure. The use of NURBS allows an efficient geometrical modelling with only a few, but substantial key points.

Based upon the ideal geometry, the imperfect structural system is described by an imperfection model. In order to represent the probabilistic nature of geometrical imperfections realistically, the mathematical concept of random fields [1] is used. Hereby, the probabilistic correlation of the deviations (as described by the random field) are evaluated at discrete points (typically FE-nodes) and collected in the a correlation matrix. Applying a spectral decomposition of the correlation matrix, the imperfection shape is represented in terms of a linear combination of deterministic base vectors (the eigenvectors of the correlation matrix) and random amplitudes which have the important property of being uncorrelated.

A finite-element model is used to perform the structural analysis. In order to include the stability behaviour of the system adequately, a geometrically nonlinear analysis has to be carried out.

Despite the fact, that some system parameters are uncertain, "sharp" statements with respect to the structural performance are needed for optimisation. For this purpose, an uncertainty model is introduced in which the uncertainties are represented by a convex model [2]. A probabilistic approach does not seem to be applicable because sufficient knowledge about the stochastic properties of geometrical imperfections for complex systems are hardly available. As a consequence of the convex model, a bound for the set of imperfection shapes can be prescribed by requiring the norm of the random vector to be smaller than a given quantity. This quantity is chosen such that the imperfection shape is contained within a reasonable fractile (e.g. 95%) of all possible imperfection shapes. The worst imperfection shape can then be conveniently found by employing optimisation methods ("anti-optimisation").

The software system developed is based on a component-oriented design and mainly comprises three server sided software components. (i) The optimisation component provides a unified object-oriented interface to various optimisation algorithms including several gradient based methods as well as evolutionary strategies. (ii) In the optimisation model component, the optimisation problem is established by means of so called process objects. A process object performs a clearly defined computation task which has to be carried out inside the optimisation model. The optimisation problem is thus established by adequately connecting various process objects in a network. (iii) For the structural analysis component, an object-oriented interface to the public domain FEA package FElt has been developed. In addition, a further software component has been implemented, which interacts with the commercial FEA system ANSYS. With this component, ANSYS can be integrated in the above problem component as a process object. Each of the three software component is a CORBA server which exhibits a interface defined in CORBA IDL and is equipped with a graphical client application.

A practical example from the field of civil engineering illustrates both the necessity to include imperfections in the optimisation as well as the capability of the approach suggested in designing structures that are sufficiently robust against geometrical imperfections.

1

E. Vanmarcke, "Random Fields: Analysis and Synthesis". The MIT Press, 1983.

2

Y. Ben-Haim, I. Elishakoff, "Convex Models of Uncertainty in Applied Mechanics". Studies in Applied Mechanics, Amsterdam: Elsevier, 1990.

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