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topology.pptx
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
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.
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.
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.
Editor's Notes
Topology optimization is the most fundamental inquiry into what form of a structure will carry the loads most efficiently. The material in this lectures is mostly taken from Chapter 1 of Bendsoe and Sigmund’s Topology Optimizaiton, and some from Section 6.5 of Haftka and Gurdal’s Elements of Structural Optimization.
The figure shows what kind of structure we will get from a topology optimization routine that gets as input a concentrated horizontal load acting to the right and the fact that it has to be transmitted to a wall that is the boundary on the left.
The difference between the two solutions is the region specified that the structure can occupy.
The example is based on a famous problem of the design of a floor beam of an MBB civil transport. A simply supported beam is loaded by uniformly distributed load down.
The top figure shows what happens if we decide to use a truss beam and do sizing optimization of the truss cross sectional areas. In the middle we decide to design it as a plate with holes, selecting six holes and designing their shape, which is an example of shape optimization. The bottom is topology optimization, where we do not make any assumptions on holes, and let the optimization design them. We get a truss-like continuum structure.
The objective of this lecture is to discuss how we perform such optimization.
Topology optimization started as a highly mathematical approach assuming that the structure possesses micro-structure and optimizing the microstructure. The method took off when a simpler formulation was developed by Bendsoe and Kikuchi in 1988.
The method is dominated by Danish authors, and the best textbook is Topology Optimization by Bendsoe and Sigmund. There is however an alternative that is based on a simpler concept developed by Australians Mike Xie and Grant Steven in the mid 1990s called Evolutionary Structural Optimzation based on the idea of removing elements with low stresses..
The formulation that we will study defines a region where we can have material which is shown by the outer contour. Inside we can have regions (shown in black, where we must have given amount of material, and there are regions, shown in white that must be empty. So the real design domain is shown in gray. Loads are also specified as well as the support region, shown in shading at the top left.
This is an example that includes all the elements except for the forces on the inside. We have a rectangular region with a given black region and a specified hole in white. The objective of the design problem is to have the stiffest structure that will transmit the loads to the fixed boundary with a specified amount of material. The specification is usually done as a percentage of the area of the domain. On the right we see the solution we could obtain with the top program if we model the domain with 3200 rectangular finite element and specify that we will use 50% of the area.
The design problem, is therefore to decide where we will have material and where we will not have material. We also make the simplifications that the material is the same everywhere, and it is isotropic. For 2D problems we assume that it is of constant depth or thickness.
This is shown in the equations (Eq. 1.3 in Bendsoe and Sigmund). The equation also includes a volume constraint that tells us how much material we can use.
We solve the problem by dividing the domain into square or cubic elements, view them as pixels or voxels, and look for them to take only the values 0 or 1. The challenge is that one needs a very large number of elements to get a good resolution of the material distribution. For 2D problems, possibly 100x100=10,000 is reasonable, but for 3D problem, we may need a million elements. With 10,000 elements we will have 2^10,000 possible designs and it is a very challenging integer programming problems.
We use two tricks in order to make it manageable. We first convert it to continuous design variables so that we can use gradient based local optimization algorithms. Second we find an objective function that has very inexpensive derivatives.
The first answer that allows us to convert the problem to continuous design variables is called SIMP for solid isotropic material with penalization. Instead of a black and white solution, we allow gray by having in each element a density design variable that can vary in [0,1]. We need though a trick that will induce this design variable to prefer values of 0 or 1 over intermediate values.
This is done by the power law linking the density with Young’s modulus. With the modulus proportional to the density to a power greater than one, we create an incentive for it to be zero or one.
To see that consider the fact that in tension, for example, the stiffness is proportional to EA, where A is the cross sectional area. If we have a volume constraint that the volume be half of the total volume, we could possibly have a density of 0.5 and have material everywhere. For p=2, for example, that would mean that Young’s modulus will be 0.25 of it full value everywhere. So we will use the full area, but will get only 0.25EA. If we used only half of the area with full density, we will get 0.5EA.
The second trick to make the topology optimization affordable is replacing stress and displacement constraints by a stiffness objective function. The compliance of the structure is defined as the product of the loads times the displacements, which is twice the work done by the forces or twice the elastic energy in the structure.
Consider a design variable x that for topology optimization will be the density in one element. The derivation in the slide shows that the derivative of the compliance can be obtained by pre- and post-multiplying the derivative of the stiffness matrix by the displacement vector.
Since the design variable will be controlling the stiffness of one element, we can replace K by the element stiffness matrix and replace the global displacement vecvtor by the element nodal displacements.
The equation gives us the directive that we want to increase the density of elements with high strain energy (or high stresses) and decrease it for elements with low strain energy (low stresses). This is indeed the basis of another topology optimization method that we will study, called evolutionary structural optimization.
Because we have an optimization method with an objective and a single constraint (volume) it can be done with a simple specialized optimization method, called the optimality criterion method that we will study later.
Professor Sigmund’s site is a good source of examples, interactive use of topology optimization as well as an ap that one can use on a mobile device.
