Topology optimization (pages from Bendsoe and Sigmund and Section 6.5) Looks for the connectivity of the structure. How many holes. Optimum design of bar in tension, loaded on right side

Structural Optimization categories Fig. 1.1

Problem optimization classification Provide examples of sizing, shape, and topology optimization in the design of a car structure.

History Microstructure based approach by various mathematicians in the 1970s and early 1980s Engineers caught on after landmark paper of Martin Bendsoe of the Technical University of Denmark and Noboru Kikuchi of the University of Michigan in 1988 Method dominated by Danes Alternative based on simpler mathematics called Evolutionary Structural Optimization developed by Australians Mike Xie and Grant Steven in mid 1990s.

Basic elements Loads, boundaries, full and empty regions

Example Rectangular domain, 50% volume fraction, 3200 finite elements

Design freedom Goal is to specify the region where there is material Simplifications: The same material everywhere, and it is isotropic

Challenge and answer We will divide domain into large numbers of elements (pixels or voxels) and will have a binary decision for each. With 10,000 elements, there are 2 10,000 possible designs! Answer 1: Find trick to convert to continuous design (so can use derivatives) Answer 2: Find objective function with cheap derivatives.

9 Optimal shapes of bike frames Least weight Least deflection

Solid Isotropic Material with Penalization (SIMP) Micro structure leads to power-law where elastic moduli vary like power of density Later it turned out that microstructure is not necessary, just SIMP First ingredient: Density can take any value in [0,1]. Second ingredient: Power law for Young modulus favors 0-1 solution. Why?

Problem SIMP Assume E is proportional to the square of the density. Compare the compliance of a bar in tension for a volume fraction of 0.5 between uniform density of 0.5 and half of the area at full density and half empty.

Compliance minimization Compliance is the opposite of stiffness Inexpensive derivatives

Density design variables Recall For density variables Want to increase density of elements with high strain energy and vice versa To minimize compliance for given weight can use an optimality criterion method.

Ole Sigmund’s Site Good summary and many examples Minimize compliance for given volume Provides also a 99-line computer code that we will analyze. Can get also a mobile phone ap that would do for you topology optimization.

Problem top Use the top ap or the web site to design a bar in tension with aspect ratio of 3, with the tensile loads applied at two corners of the rectangle.