material to be distributed (volume constraint) or maximum stress values. Evaluating 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 topology optimization problems. Discrete 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. Solving the problem with continuous variables 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. Shape derivatives Topology optimization can be achieved by using shape derivatives. Topological derivatives Level set Phase