# Topology optimization

Topology optimization (TO) is a mathematical method that optimizes material layout within a given design space, for a given set of loads, boundary conditions and constraints with the goal of maximising the performance of the system. TO is different from shape optimization in the sense that the design can attain any shape within the design space, instead of dealing with predefined configurations.

The conventional TO formulation uses a finite element method [FEM] to evaluate the design performance. The design is optimised using nonlinear programming techniques such as the optimality criteria algorithm, the method of moving asymptotes and genetic algorithms.

Currently, engineers mostly use TO at the concept level of a design process. Due to the free forms that naturally occur, the result is often difficult to manufacture. For that reason the result emerging from TO is often fine-tuned for manufacturability. Adding constraints to the formulation in order to increase the manufacturability is an active field of research. In some cases results from TO can be directly manufactured using additive manufacturing; TO is thus a key part of design for additive manufacturing.

## Problem statement

A topology optimisation problem can be written in the general form of an optimization problem as

${\displaystyle \min _{\rho }\;F=F(\mathbf {u(\rho ),\rho } )=\int _{\Omega }f(\mathbf {u(\rho ),\rho } )\mathrm {d} V}$

subject to

• ${\displaystyle \scriptstyle \rho \,\in \,\{0,\,1\}}$
• ${\displaystyle G_{0}(\rho )=\int _{\Omega }\rho (\mathbf {u} )\mathrm {d} V-V_{0}\leq 0}$
• ${\displaystyle G_{j}(\mathbf {u} (\rho ),\rho )\leq 0{\text{ with }}j=1,...,m}$

The problem statement includes the following:

• An objective function ${\displaystyle \scriptstyle \left(\int _{\Omega }f(\mathbf {u(\rho ),\rho } )\mathrm {d} V\right)}$. This function indicates the quality of the design and is to be minimised. A classic example is the goal of generating a stiff structure by minimisation of compliance.
• The material distribution as a problem variable. This is described by the density of the material at each location ${\displaystyle \scriptstyle \rho (\mathbf {u} )}$. Material is either present, indicated by a 1, or absent, indicated by a 0.
• The design space ${\displaystyle \scriptstyle (\Omega )}$. This indicates the allowable volume within which the design can exist. Assembly and packaging requirements, human and tool accessibility are some of the factors that need to be considered in identifying this space . With the definition of the design space, regions or components in the model that cannot be modified during the course of the optimization are considered as non-design regions.
• ${\displaystyle \scriptstyle m}$ constraints ${\displaystyle \scriptstyle G_{j}(\mathbf {u} (\rho ),\rho )\leq 0}$ a characteristic that the solution must satisfy. Examples are the maximum amount of material to be distributed (volume constraint) or maximum stress values.

Evaluating ${\displaystyle \scriptstyle \mathbf {u(\rho )} }$ often includes solving a differential equation. This is most commonly done using the finite element method since these equations do not have a, known, analytical solution.

## Implementation methodologies

There are various implementation methodologies that have been used to solve TO problems.

### Discrete

Solving TO problems in a discrete sense is done by discretizing the design domain into finite elements. The material densities inside these elements are then treated as the problem variables. In this case material density of one indicates the presence of material, while zero indicates an absence of material. Due to the attainable topological complexity of the design being dependent of the amount of elements, a large amount is preferred. Large amount of finite elements increase the attainable topological complexity, but come at a cost. Firstly, solving the FEM system becomes more expensive. Secondly, algorithms that can handle a large amount (several thousands of elements is not uncommon) of discrete variables with multiple constraints are unavailable. Moreover, they are unpractically sensitive to parameter variations.[1] In literature problems with up to 30000 variables have been reported [2]

### Solving the problem with continuous variables

The earlier stated complexities with solving TO problems using binary variables has caused the community to search for other options. One is the modelling of the densities with continuous variables. The material densities can now also attain values between zero and one. Gradient based algorithms that handle large amounts of continuous variables and multiple constraints are available. But the material properties have to be modelled in a continuous setting. This is done through interpolation. One of the most implemented interpolation methodologies is the SIMP method (Solid Isotropic Material with Penalisation [3]).[4] This interpolation is essentially a power law ${\displaystyle \scriptstyle E\;=\;E_{0}\,+\,\rho ^{p}(E_{1}-E_{0})}$. It interpolates the Young's modulus of the material to the scalar selection field. The value of the penalisation parameter ${\displaystyle p}$ is generally taken between ${\displaystyle \scriptstyle [1,\,3]}$. This has been shown to confirm to micro-structure of the materials.[5] In the SIMP method a lower bound on the Young's modulus is added, ${\displaystyle \scriptstyle E_{0}}$, to make sure the derivatives of the objective function are non-zero when the density becomes zero. The higher the penalisation factor, the more SIMP penalises the algorithm in the use of non-binary densities. Unfortunately, the penalisation parameter also introduces non-convexities [6]).

## Examples

Checker Board Patterns are shown in this result.
Topology Optimization result when filtering is used.
This is an example of a compliance problem done by the program called ToPy.

### Topology Optimization for stiff structures

A stiff structure is one that has the least possible displacement when given certain set of boundary conditions. A global measure of the displacements is the strain energy (also called compliance) of the structure under the prescribed boundary conditions. The lower the strain energy the higher the stiffness of the structure. So, the problem statement involves the objective functional of the strain energy which has to be minimized.

On a broad level, one can visualize that more the material, lesser will be the deflection as there is more material to resist the loads. So, the optimization requires an opposing constraint, the volume constraint . This is in reality a cost factor, as we would not want to spend a lot of money on the material. To obtain the total material utilized, an integration of the selection field over the volume can be done.

Finally the elasticity governing differential equations are plugged in so as to get the final problem statement.

${\displaystyle \min _{\rho }\;\int _{\Omega }{\frac {1}{2}}\mathbf {\sigma } :\mathbf {\varepsilon } \,\mathrm {d} \Omega }$

subject to:

• ${\displaystyle \scriptstyle \rho \,\in \,[0,\,1]}$
• ${\displaystyle \scriptstyle \int _{\Omega }\rho \,\mathrm {d} \Omega \;\leq \;V^{*}}$
• ${\displaystyle \scriptstyle \mathbf {\nabla } \cdot \mathbf {\sigma } \,+\,\mathbf {F} \;=\;{\mathbf {0} }}$
• ${\displaystyle \scriptstyle \mathbf {\sigma } \;=\;{\mathsf {C}}:\mathbf {\varepsilon } }$

But, a straightforward implementation in the Finite Element Framework of such a problem is still infeasible owing to issues such as:

1. Mesh dependency—Mesh Dependency means that the design obtained on one mesh is not the one that will be obtained on another mesh. The features of the design become more intricate as the mesh gets refined.
2. Numerical instabilities—The selection of region in the form of a chess board.

Some techniques such as Filtering based on Image Processing are currently being used to alleviate some of these issues.

### 3F3D Form Follows Force 3D Printing

The current proliferation of 3D Printer technology has allowed designers and engineers to take advantage of topology optimization techniques when designing new products.

## References

1. ^ Sigmund, O., Maute, K., Topology optimization approaches A comparative review. Structural and Multidisciplinary Optimization, 2013, p. 1031-1055
2. ^ Beckers, M. Topology optimization using a duel method with discrete variables. Structural Optimization, p. 14-24
3. ^ Bendsøe, MP. Optimal shape design as a material distribution problem.. Structural Optimization, 1989, p. 193-202
4. ^ [1], a monograph of the subject.
5. ^ [2], A reference that proved the validity of the interpolation scheme.
6. ^ van Dijk, NP. Langelaar, M. van Keulen, F. Critical study of design parameterization in topology optimization; The influence of design parameterization on local minima.. 2nd International Conference on Engineering Optimization, 2010