Generating Optimal Topologies in Structural Design Using a Homogenization Method

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Generating Optimal Topologies in Structural
Design Using a Homogenization Method

• A paper by
• Martin Philip Bendsoe
• and
• Noboru Kikuchi
• AME 60662
• Topology Optimization

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Basic Concept

• The method consists of getting an optimal
  distribution for anisotropic space which is
  constructed by introducing small periodically
  distributed holes in a homogeneous, isotropic
  material such that it satisfies all the required
  constarints.

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Basic Approach

• The structural element is taken as being defined
  only by
• The loads it is supposed to carry
• Its volume (cost)
• Stress and strain limitations
• Note Restriction on allowable shapes is that the
  element should connect to the given surface
  tractions.

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Optimal Design of Linearly Elastic Structures

• Seeking minimum compliance is equivalent to
  finding optimal elasticity tensor Eijkl for a
  given set of admissible elasticity tensors Uad
  where Uad (L8(O)).
• The minimum compliance takes the form

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Cont

• Where

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1. For Optimal Shape design

• Indicator function
• 2. For Optimal Sizing Problems
• Sizing Function

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Stepwise Method

• Suitable reference for defining surface tractions
  and fixed boundaries.
• Choosing a composite having periodic repetitive
  unit cells with one or more holes
• Using Homogenization Theory, get a functional
  relationship between the density of material and
  effective material properties.
• Compute the optimal distribution of material in
  the reference domain treating density as the
  sizing variable.
• Interpret the optimal distribution of material as
  defining shape.

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Homogenization Method
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• The Finite Element Analysis is used on cell
  element for getting homogenized elasticity tensor
  which is applied for shape optimization.

Question??
Sensitivity of results with respect to FEM meshing
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Conclusions from examples

• Its better not to replace holes by soft
  materials when the goal is to find the
  homogenized elasticity tensor.
• The present study takes a cell structure with
  rectangular hole.
• Continuously varying size of rectangular hole is
  to be taken to make density a design variable.

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The Optimization Method
Homogenization method

• The problem of minimum compliance becomes

The design variables µ and ? are discretized. An
optimization scheme based on Optimality criteria
is constructed
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Optimality Criteria
Where ? and ?i are Lagrange multipliers for
volume and density constraints.
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Iteration used over µ and ? is
Where D is
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Computational Results

• Two types of basic square cells were used
• One with square holes
• One with rectangular holes
• Note For both the cases optimization was
  performed including and excluding cell
  rotation

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Comparing results obtained with and without the
cell rotation as a design variable. Case A with
voids as rectangular holes. Left-hand rotation of
voids, right-hand column not. Volumes are 91,
64 and 36
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Results of the lumping process. Case A with voids
as square holes that can be rotated. Left-hand
column shows results prior to lumping, right-hand
column the resulting lumped designs. Volumes are
91, 64 and 36.
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Conclusions

• The proposed optimization method can provide the
  optimal shape as well as the topology of the
  element.
• The method is a material distribution method,
  based on the use of artificial composite material
  with microscopic voids.