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  mat 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 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.
Optimal shapes of bike frames
Least weight
Least deflection
tusharg@ufl.edu
9
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
E = E0 
p
p 1
• 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
C = f u = u Ku
T
T
• Inexpensive derivatives
dC
du
dK
= 2uT K
+ uT
u
dx
dx
dx
But since Ku = f if f does not depend on x
du
dK
K
=−
u
dx
dx
dC
T dK
= −u
u
dx
dx
Density design variables
• Recall
dC
T dK
= −u
u
dx
dx
• For density variables
dC
T
p −1 e

−u

Ku
e
d
• 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
• 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.
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.