Topology Optimization

P Venkat Vijay KumarMechanical branch

KLUniversity

What is Topology???

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing

Then… Optimization???

In mathematics, computational science, or management science, mathematical optimization (alternatively, optimization or mathematical programming) refers to the selection of a best element from some set of available alternatives.

Finally… Topology Optimization

Topology optimization is a mathematical approach that optimizes material layout within a given design space, for a given set of loads and boundary conditions such that the resulting layout meets a prescribed set of performance targets.

How???

• Any optimization technique can be used as tool of optimization in Topology Optimization.

• An objective function has to be formulated which has to be optimized / maximized / minimized.

• Then using the optimization techniques number of probabilities are produced and the result that suits best is choosen.

Detailed how????

• Lets start with little more detail.• Consider a structure or component, say a

cantilever beam. Let our objective be minimizing the deflection if possible, minimizing the mass

Cantilever beam

Load F

Original deflections and stresses in the beam in the initial conditions of load and structure

Divide the cantilever into number of checks i.e. rows and columns.

Now, considering the presence of material as 1 and void as 0, number the checks one by one

Cantilever beam

1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1

• Now would be the task of optimization.• The numbers considered are taken into a row

matrix[1 1 1 .. .. .. 40 1’s]

• From the original analysis its clear that some portion of the beam is not taking the load, therefore lets remove the portion with less stress.

[1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

Cantilever beam

1 1 1 1 1 1 1 1 1 10 0 0 1 1 1 1 1 1 10 0 0 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1

- A parent

[1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

Cantilever beam

1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 1 1 10 0 0 0 0 0 1 1 1 11 1 1 1 1 1 1 1 1 1

- One more parent

• For each parent we analyze the amount of mass and deflection that occurs

• And keep doing until our objective is met.• A topology optimized cantilever beam is as

follows…

Software… soft procedure

• Modeling• Analysis• Optimizing• Remodeling• Producing manufacturable design

Applications

• Now Topology optimization is being spread to composite structures

• Topology optimization is the major tool for optimizing aero structures

• There are lot of things around us that have to be optimized… ..

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Than

k yo

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