Keywords: topology, optimization, Lagrangian, sensitivities, optimality criterion, design.

Topology optimization methods have shown great promise in designing highly efficient lightweight structures for a variety of applications. Many different numerical optimization methods have been applied to topology optimization problems. This work examines the usefulness of the optimality criterion method in the topology optimization of structures.

Given a finite element mesh with a defined set of static load cases and boundary conditions, the optimality criterion method seeks to find the most optimal structural topology by treating the element thicknesses as a set of design variables that can be optimized. An optimal solution is one that minimizes a given objective function while meeting specific design constraints. For this work, the objective function to be minimized is the total weight of the structure or a combination of weight and pseudo-surface area. A set of maximum deflections at a selected points are set as the design constraints.

After introducing the optimality criterion methodology, four examples of its application to two-dimensional cantilever problems are presented. The first example demonstrates the method on a simple six-element cantilever illustrating the process through which the optimality criterion method is applied to topology optimization problems. Using the method described in Terlaje et al. [1], side constraints are introduced as a way to keep design variables from exceeding thickness limitations without the addition of design constraints. The next example uses the same weight based algorithm but on a more refined 1200 element cantilever. In order to create optimal solutions that contain material voids, the third example introduces a pseudo-surface area function into the objective function. A mesh connectivity filter developed by Sigmund [2] is used in conjunction with the new objective function to eliminate the occurrences of checkerboard and spider web like solutions. In the final example, the method is applied to a cantilever with multiple load cases and deflection limits to illustrate the method's versatility. To handle multiple inequality constraints, a method of active and passive constraint selection is discussed. In all examples, an optimal solution was reached while controlling specific deflection constraints. Computational effort was considered by examining the number of iterations necessary to reach an optimal solution. The number of design variables has little affect on the number of iterations required to reach a solution, but altering the objective function to include pseudo-surface area, a nonlinear objective function, did increase the number of iterations necessary for convergence.

1

A.S. Terlaje, K.Z. Truman, "A Static Response Based Parameter Identification Algorithm using Optimality Criterion Optimization", in Proceedings of the Eleventh International Conference on Civil, Structural and Environmental Engineering Computing, B.H.V. Topping, (Editor), Civil-Comp Press, Stirlingshire, United Kingdom, paper 88, 2007. doi:10.4203/ccp.86.88

2

O. Sigmund, "A 99 Line Topology Optimization Code Written in MATLAB", Structural and Multidisciplinary Optimization, 21(2), 120-127, 2001. doi:10.1007/s001580050176

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