Topology optimization

Solving topology optimization 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. Owing to the attainable topological complexity of the design being dependent on the number of elements, a large number is preferred. Large numbers of finite elements increases the attainable topological complexity, but come at a cost. Firstly, solving the FEM system becomes more expensive. Secondly, algorithms that can handle a large number (several thousands of elements is not uncommon) of discrete variables with multiple constraints are unavailable. Moreover, they are impractically sensitive to parameter variations.[1] In literature problems with up to 30000 variables have been reported.[2]

The earlier stated complexities with solving topology optimization 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 Solid Isotropic Material with Penalisation method (SIMP).[3][4] This interpolation is essentially a power law ${\displaystyle E\;=\;E_{0}\,+\,\rho ^{p}(E_{1}-E_{0})}$. 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 ${\displaystyle [1,\,3]}$. This has been shown to confirm the micro-structure of the materials.[5] In the SIMP method a lower bound on the Young's modulus is added, ${\displaystyle E_{0}}$, 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]

There are several commercial topology optimization software on the market. Most of them use topology optimization as a hint how the optimal design should look like, and manual geometry re-construction is required. There are a few solutions which produce optimal designs ready for Additive Manufacturing.

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 objective function of the problem is to minimize the strain energy.

On a broad level, one can visualize that the more the material, the less the deflection as there will be 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:

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

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.[7]
2. Numerical instabilities—The selection of region in the form of a chess board.[8]

Some techniques such as filtering based on image processing[9] are currently being used to alleviate some of these issues. Although it seemed like this was purely a heuristic approach for a long time, theoretical connections to nonlocal elasticity have been made to support the physical sense of these methods.[10]

Fluid-structure-interaction is a strongly coupled phenomenon and concerns the interaction between a stationary or moving fluid and an elastic structure. Many engineering applications and natural phenomena are subject to fluid-structure-interaction and to take such effects into consideration is therefore critical in the design of many engineering applications. Topology optimisation for fluid structure interaction problems has been studied in e.g. references[11][12][13] and.[14] Design solutions solved for different Reynolds numbers are shown below. The design solutions depend on the fluid flow with indicate that the coupling between the fluid and the structure is resolved in the design problems.

Design solution and velocity field for Re=1

Design solution and velocity field for Re=5

Design solution and pressure field for Re=10

Design solution and pressure field for Re=40

Design solutions for different Reynolds number for a wall inserted in a channel with a moving fluid.

Sketch of the well-known wall problem. The objective of the design problem is to minimize the structural compliance.

Design evolution for a fluid-structure-interaction problem from reference.[14] The objective of the design problem is to minimize the structural compliance. The fluid-structure-interaction problem is modelled with Navier-Cauchy and Navier-Stokes equations.

A sketch of the design problem. The aim of the design problem is to spatially distribute two materials, Material A and Material B, to maximise a performance measure such as cooling power or electric power output

Design evolution for an off-diagonal thermoelectric generator. The design solution of an optimisation problem solved for electric power output. The performance of the device has been optimised by distributing Skutterudite (yellow) and bismuth telluride (blue) with a density-based topology optimisation methodology. The aim of the optimisation problem is to maximise the electric power output of the thermoelectric generator.

Design evolution for a thermoelectric cooler. The aim of the design problem is to maximise the cooling power of the thermoelectric cooler.

Thermoelectricity is a multi-physic problem which concerns the interaction and coupling between electric and thermal energy in semi conducting materials. Thermoelectric energy conversion can be described by two separately identified effects: The Seebeck effect and the Peltier effect. The Seebeck effect concerns the conversion of thermal energy into electric energy and the Peltier effect concerns the conversion of electric energy into thermal energy.[15] By spatially distributing two thermoelectric materials in a two dimensional design space with a topology optimisation methodology,[16] it is possible to exceed performance of the constitutive thermoelectric materials for thermoelectric coolers and thermoelectric generators.[17]

The current proliferation of 3D printer technology has allowed designers and engineers to use topology optimization techniques when designing new products. Topology optimization combined with 3D printing can result in less weight, improved structural performance and shortened design-to-manufacturing cycle. As the designs, while efficient, might not be realisable with more traditional manufacturing techniques.

The direction, magnitude, and location of a design-dependent load alter with topology optimization iterations. Therefore, dealing with such loads in a TO setting is a challenging task. One can find novel methods to deal with such loads (e.g. pressure load, self-weight, etc) in Refs.[18] [19] [20]

Internal contact can be included in topology optimization by applying the third medium contact method.[21][22][23] The third medium contact (TMC) method is an implicit contact formulation that is continuous and differentiable. This makes TMC suitable for use with gradient-based approaches to topology optimization.

Design development and deformation of self-engaging hooks resulting from topology optimization of a contact problem using the TMC method [21].