This document provides an overview of topology optimization. It discusses how topology optimization looks for the optimal connectivity of a structure's design. It also provides a brief history, noting pioneering work done in the 1970s-1980s and a landmark 1988 paper. The document outlines the basic elements of topology optimization problems, including loads, boundaries, material regions, and examples of applying it to optimize bike frame designs. It also describes the challenges of discrete designs and how continuous approximations like the Solid Isotropic Material with Penalization (SIMP) method were developed to allow for optimization using derivatives.

1. 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

2. Structural Optimization
categories
• Fig. 1.1

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

4. 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.

5. Basic elements
• Loads, boundaries, full and empty regions

6. Example
• Rectangular domain, 50% volume fraction,
3200 finite elements

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

8. 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 210,000
possible designs!
• Answer 1: Find trick to convert to
continuous design (so can use derivatives)
• Answer 2: Find objective function with
cheap derivatives.

9. tusharg@ufl.edu 9
Optimal shapes of bike frames
Least weight
Least deflection

10. 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?
0 1
p
E E p

 

11. 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.

12. Compliance minimization
• Compliance is the opposite of stiffness
• Inexpensive derivatives
T T
C K
 
f u u u
2
But since if does not depend on x
T T
T
dC d dK
K
dx dx dx
K
d dK
K
dx dx
dC dK
dx dx
 

 
 
u
u u u
u f f
u
u
u u

13. 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.
T
dC dK
dx dx
 u u
1
T p e
e
dC
K
d



 u u

14. Ole Sigmund’s Site
• http://www.topopt.dtu.dk/
• 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.

15. 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.