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An Introduction to Structural Optimization

Umer Khan, Professor W. L. Cleghorn

University of Toronto

1. Terminology

Design Space: The volume enclosed by the topology of a part.

Design Variable: A parameter which can be varied to optimize the operation of a part.

Response: A function of the design variable (e.g. mass, stress, displacement and etc.)

used to measure the performance of a part.

Objective Function: Responses which are being optimized.

Constraint Function: Restrictions on the response. They have to be fulfilled in order for

the design to be considered feasible.

A feasible design is one which satisfies all constraints and an infeasible design is one

which does not satisfy one or more of the constraints. The optimum design is one which

minimizes/maximizes the objective function while satisfying all constraints.

2. Structural Optimization

Structural optimization yields a result which exhibits “optimal” structural performance of

a given design space under a set of loads and boundary conditions. There are four types

of structural optimization, each serving a different function: topography, size, shape, and

topology. These types of optimization consist of algorithms which are fed a variety of

constraints and objectives to converge onto a solution. The only other alternative to these

methods is trial and error which can be tedious and time consuming.

2.1 Size Optimization

This type of optimization can determine the optimal dimensions of a cross-section while

keeping the shape and topology constant. An example would be the dimensions of an I-

beam cross section under some loading conditions as shown in figure 1.

Figure 1: Size Optimization

2.2 Shape Optimization

Like the shape optimization method, the topology is also kept constant but the shape is

varied. Sizing optimization is a byproduct of this method. Figure 2 illustrates shape

optimization where an abstract cross-section has been optimized into an I-beam.

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Figure 2: Shape optimization of abstract shape into I-beam.

2.3 Topography Optimization

This type of optimization is only applicable to 2-D shell elements. Protrusions are

introduced onto the surface of a part to increase stiffness. Examples include placement of

bosses and ribs on plastic parts which provide excellent structural support at minimal

material usage. Figure 3 illustrates the process of topography optimization on a flat plate.

The topography of the part has been changed to increase the stiffness.

Figure 3: Topography Optimization of a plate under torsion.

2.4 Topology Optimization

Topology optimization aims at providing the best possible arrangement of material over a

given design space, or spatial domain, to minimize/maximize an objective. It is

essentially a material distribution problem, whereby the content of material for each

element in the spatial domain is determined by introducing voids. For example, starting

with an element of 1 x 1 units, as shown in Figure 4, a rectangular void with parameters a

and b is introduced. The amount of material and void space depends on specific

parameters and their values. Setting a and b equal to 1 would yield a complete void,

whereas if a and b were set to 0.5, the element would be “porous.” Therefore, for any

element defined in a structure, there exists either a complete void, or an intermediary void

resembling porous material, or no void at all.

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Figure 4: A unit cell showing an element with a void.

The software used for performing topology optimization, Altair’s HyperWorks 10, uses

the SIMP (Solid Isotropic Material with Penalization) method, also called the density

method. Each element in the spatial domain is assigned a fictitious stiffness tensor, E,

which is related to the element density (ratio of volume of material in element to total

volume of element) in the following manner:

E = ρpE0

E0 – Stiffness tensor of the bulk material

E – Stiffness tensor of element

ρ – Element density (0 ≤ ρ ≤ 1)

p – Penalization factor

The software is programmed to determine the density value for each element. The

exponent p is a penalty factor which is used to reach discrete solutions by penalizing

intermediate densities. In other words, it is used to move solutions of element densities

closer to either 1 (no void) or 0 (void). Therefore, if the element density tends towards 0,

then the stiffness tensor also tends to zero signifying a hole, and vice versa if the element

density tends towards 1. The new stiffness tensor (E) calculated from the density signifies

the importance of that element to the part. Stiffness defined as the material’s ability to

resist deformation. A high stiffness tensor value for an element thus implies that it plays a

crucial role in resisting deformation.

For example, take a simply supported beam with the a load of 50 N, applied at the center

as shown at the left of figure 5. The maximum deflection occurs at the nodes where the

force is applied, as shown at the right of figure 5. Using these results, we now apply

topology optimization. The maximum deflection is the constraint, and the objective is to

minimize compliance (maximize stiffness). The solution, in terms of density distribution,

is shown in figure 6. The black elements indicate a density of 1, which implies that those

elements are crucial to the structure, thus possessing nominal stiffness properties. The

gray elements indicate a density of close to 0, and signify wasted material. Therefore, the

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structure should be build around the red elements. The results obtained are intuitive and

outline the capabilities of topology optimization.

Figure 5: Simply supported beam divided into elements for analysis.

Figure 6: Results of topology optimization

3. Problem Statement

Structural optimization problems are structured in the following manner:

Minimize the mass of the cantilever beam with a point force applied at the end:

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OBJECTIVE FUNCTION

I want to minimize mass. min m(b,h)

DESIGN VARIABLES I can change the width and

the height to minimize the

mass.

bL ≤ b ≤ b

U (20< b< 40)

hL ≤ h ≤ h

U (30 < h< 90)

DESIGN CONTRAINTS The stresses have to be

under a certain value.

σ(b,h) ≤ 150 MPa

τ(b,h) ≤ 50 MPa

h ≤ 2b

DESIGN REGION All beam elements

The objective of this optimization problem is to minimize the mass. This can only be

achieved through varying the design variables, height and width, within a specified

intervals. Constraints are set on the maximum allowable stress. And finally, this process

is to be performed only in the design space of the beam. Note that in this case, the length

(l) is kept constant and so the objective function is only a function of the base and height.

4. Interpretation of the Results

Interpretation of the results can be done by asking the following questions:

1. Were any constraints violated?

2. Did we reach our objective?

3. How much did the objective improve?

4. What are the values of variables for the improved design?

Remember that optimization is an iterative process and that the finite element model

might require refinement in terms of mesh or boundary conditions in order to obtain a

better solution. If the software does not provide a good solution, there could be a variety

of problems with the finite element model. One of the most common mistakes is over

constraining the model which would not provide a solution at all. To overcome this

problem, try increasing the constraints.