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Mathematical method for optimizing material layout under given conditions

Topology optimization 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 maximizing the performance of the system. Topology optimization is different from shape optimization and sizing optimization in the sense that the design can attain any shape within the design space, instead of dealing with predefined configurations. The conventional topology optimization formulation uses a finite element method (FEM) to evaluate the design performance. The design is optimized using either gradient based mathematical programming techniques such as the optimality criteria algorithm and the method of moving asymptotes or non gradient based algorithms such as genetic algorithms. Topology optimization has a wide range of applications in aerospace, mechanical, bio chemical and civil engineering. Currently, engineers mostly use topology optimization 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 topology optimization 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 topology optimization can be directly manufactured using additive manufacturing; topology optimization is thus a key part of design for additive manufacturing.

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

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. In literature problems with up to 30000 variables have been reported.

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). This interpolation is essentially a power law It interpolates the Young's modulus of the material to the scalar selection field. The value of the penalisation parameter is generally taken between This has been shown to confirm the micro structure of the materials. In the SIMP method a lower bound on the Young's modulus is added, , 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.

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.

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 and. 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. [[File:Fluid structure interaction design evolution. gif|thumb|Design evolution for a fluid structure interaction problem from reference. By spatially distributing two thermoelectric materials in a two dimensional design space with a topology optimisation methodology, it is possible to exceed performance of the constitutive thermoelectric materials for thermoelectric coolers and thermoelectric generators.

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.

Internal contact can be included in topology optimization by applying the third medium contact method. 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. thumb|Design development and deformation of self engaging hooks resulting from topology optimization of a contact problem using the TMC method .