Colby C. Swan and Salam F. Rahmatalla

Center for Computer-Aided Design, University of Iowa, Iowa City, Iowa 52242, USA E-mail: colby-swan@uiowa. edu, srahmata@engineering. uiowa. edu


A methodology of enhanced computational efficiency is presented for continuum topology optimization of sparse structural systems. Such systems are characterized by the structural material occupying only a small fraction of the structure’s envelope volume. When modeled within a continuum mechanics and topology optimization framework such structures require models of very high refinement which is computationally very expensive. The methodology presented herein to deal with this issue is based on the idea of starting with a relatively coarse mesh of low refinement and employing a sequence of meshes featuring progressively greater degrees of uniform refinement. One starts by solving for an initial approximation to the final material layout on the coarse mesh. This design is then projected onto the next finer mesh in the sequence, and the material layout optimization process is continued. The material layout design from the second mesh can then be projected onto the third mesh for additional refinement, and so forth. The process terminates when an optimal design of sufficient sparsity, and sufficient mesh resolution is achieved. Within the proposed methodology, additional computational efficiency is realized by using a design-dependent analysis problem reduction technique. As one proceeds toward sparse optimal designs, very large regions of the structural model will be devoid of any structural material and hence can be excluded from the structural analysis problem resulting in great computational efficiency. The validity and performance characteristics of the proposed methodology are demonstrated on three different problems, two involving design of sparse structures for buckling stability, and the third involving design of a hinge-free gripper compliant mechanism.


For nearly twenty years, beginning approximately with the work of Bendsoe and Kikuchi (1988), continuum structural topology optimization has been used to investigate optimal forms of structures and mechanical systems. During this time, myriad gains have been realized in understanding how different continuum topology formulations work in relation to an ever-widening circle of applications. Although there are many exceptions, continuum structural topology optimization methods have for the most part been applied to the design of structures and mechanical systems in two-dimensions. One would hope that the methods can eventually be applied to design of structures and mechanical systems in three-dimensions with the same ease that they currently enjoy in two dimensions. If this vision is to become reality, the potentially huge computational costs must be reduced to the extent possible while preserving the inherent generality and flexibility of continuum structural topology optimization. In many design applications, the structural will be very sparse in that the volume of structural material will occupy only a small fraction of the structural system’s envelope volume. In such cases, continuum structural topology optimization can require highly refined finite element meshes to achieve convergent and interpretable material layout solutions and models that realistically estimate system performances (i. e. overall compliance; buckling stability; vibrational eigenvalues; etc).

One approach to achieving and dealing with fine meshes in continuum topology optimization [Maute et al. 1998] is to use adaptive mesh refinement to decrease the number of design variables and to seek smooth final topological forms. A similar approach is now extensively involved in finding optimal design forms using the evolutionary structural optimization method (ESO)(3). In another approach, researchers enforced design symmetry during the optimization process by reducing the design space(4) by, or to remove the void nonstructural elements temporarily from the structural analysis, but reintroduce them if they needed (5,6). The latter approach has been shown to be very effective in dealing with problems involving geometrical nonlinearity.


M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 673-683. © 2006 Springer. Printed in the Netherlands.

The material usage constraint is another important factor and plays a considerable role in achieving low weight and certain performances when utilizing continuum topology optimization method. For example, most existing optimal large civil structures such as long span bridges are sparse in nature, where the real structural material occupies only a small percentage (less than 1%) of the structure’s envelope volume. Therefore, in utilizing continuum structural topology optimization to obtain optimal design forms for such systems, it is crucial to impose stringent material usage constraint and implement very fine meshes in order to capture a realistic performance for such systems. The importance of such an approach has been demonstrated in a previous work (6) where it proved to be very effective when designing structural systems for buckling instability. Similarly, it has been shown in designing hinge-free compliant mechanisms to achieve considerable flexibility (7), the amount of structural material comprising the mechanism can be progressively reduced until the desired flexibility of the mechanism is achieved. It is also crucial toward this end to use stringent material usage constraint with very fine finite element meshes

This article presents a methodology for solving large-size sparse systems in continuum structural topology design framework based on sequential refinement and size reduction strategy in a new way that is conceptually simple and theoretically sound. In sequential refinement, the proposed methodology solved a preliminary problem involving relatively coarse meshes and moderate material usage constraint. The resulting optimal form from this stage, which comprises the solid structural material, is then mapped onto a finer mesh and with realistic material usage constraint. The new problem is then solved where a new topological form is obtained. The mesh refinement process is repeated until the final design converges to a realistic shape and performance with minimum error.

A size reduction strategy is implemented within each structural analysis, where the void nonstructural elements are removed temporarily from structural analysis but can come back quite easily and naturally if needed. The proposed size reduction technique has been tested on many linear and nonlinear systems involving geometrical nonlinearity and buckling instability and shown to be a very effective and powerful tool for reducing the computational costs, especially when dealing with sparse systems and very fine meshes.

It should be noted here that the current methodology is based on interpolation of nodal design variables using nodal basis functions1-8-1 as opposed to element-based design variables. Although node­based design variables feature C0 continuity, they must generally be used with perimeter constraints to achieve design convergence with mesh refinement.