Structural optimization has over the past decades qualified as an important tool in the design process. The method can be grouped into topology, size and shape optimization. The objective of the optimization can be to minimize the stresses weight or compliance for a given amount of material and boundary conditions. The method can be utilized to design engineering structures but it can also be used to tailor mircostructures.
The most widely used numerical scheme for topology optimization is the Solid Isotropic Material with Penalization (SIMP) scheme where the density is approximated as constant within each element. The objective of the optimization is in the present work to find a design that maximizes the stiffness for a given amount of material. The advantage of using the stiffness as the objective, or rather its complement the compliance, is that it is a global measure and thus can be represented by a scalar value. Moreover, the constraint on the volume is also particular simple since it is linear and monotone which in most cases gives rise to a robust numerical algorithm. The SIMP procedure is based on a sequence of convex approximations and the algorithm is simple to implement and at the same time numerically efficient. A distinct black/white solution is obtained by penalizing gray designs via a scaling of the elastic constitutive relation. This formulation has been shown to be ill-posed due to the length scale missing in the formulation. The remedy to this problem regularize the problem via e.g. a filter.
Phase-field based topology optimization
Several procedures for regularization of the topology optimization problem has been proposed, these can be local such that the smallest size of a segment is limited, or it can restrict the total length of the perimeter. Such methods are known as filtering techniques, perimeter control, gradient control and recently also phase-field based approaches. All these methods has in common that they introduce a length scale in the formulation and thus restricting the smallest characteristic size either locally or globally. In  the Cahn-Hilliard approach was used to solve the topology optimization problem. This procedure has the advantage of being able to inherently preserve the material. The method is, however, computational expensive since a fourth order PDE is governing the optimality.
In Fig.1 one example of an optimal design derived in  is shown. Initial a rectangular domain if filled with material, in a random manner, the sequence of snap-shots then shown how the material is redistributed during the iteration process. The final figure then shows the optimal structure.
To to improve the computational efficiency use can be made of the Allen-Cahn approach. In , the Allen-Cahn approach was combined with the Howard's policy iteration schemer. The advantage of the Howard policy iteration scheme is that the optimality is defined by a second order PDE which is significantly more efficient than the Cahn-Hilliard approach. In  the possibility by including boundary control directly into the objective functional was investigated.
Most topology optimization is performed for the assumptions that the strains are small. If finite strains are considered the optimal design depends on the load level, i.e. different optimal designs will be found for different load levels. In , the phase-field approach was extended to take finite strains into account.
An example of a phase-field based topology optimization for large deformations can be seen in Fig. 2. Starting with an initial homogenous material distribution the sequence clearly shows that the structure becomes stiffer when the material is redistributed.
A straight-forward generalization of the results shown in Fig. 2 is to allow the design to consist of N materials with different where each phase has specific properties. With this approach the design can be tailored to e.g. place expensive high strength materials in critical regions. An example of this is shown in Fig. 3.
Topology optimizatation of elasto-plastic materials and structures
Most topology optimization is performed for linear elastic isotopic materials. This is in many cases adequate, however, it is not a good approximation for structures that are designed to absorb energy during for instance an impact. One of the major challenges for this problem is to the sensitivities accurately and efficiency. In Fig. 4, an example where a simple structure has been designed to absorb maximum energy is shown. The optimization is performed for large plastic deformations and the objective is to maximize the plastic during during the deformation process.
Wallin, M., Ristinmaa, M., & Askfelt, H. (2012). Optimal topologies derived from a phase-field method. Structural and Multidisciplinary Optimization, 45(2), 171–183.
Wallin, M., & Ristinmaa, M. (2013). Howard's algorithm in a phase-field topology optimization approach. International journal for numerical methods in engineering, 94(1), 43–59.
Wallin, M., & Ristinmaa, M. (2014). Boundary effects in a phase-field approach to topology optimization. Computer Methods in Applied Mechanics and Engineering, 278, 145–159.
Wallin, M., & Ristinmaa, M. (2015). Finite strain topology optimization based on phase-field regularization. Structural and Multidisciplinary Optimization, 51(2), 305–317.
Wallin, M., & Ristinmaa, M. (2015). Topology optimization utilizing inverse motion based form finding. Computer Methods in Applied Mechanics and Engineering, 289, 316–331.
Wallin, M., Ivarsson, N., & Ristinmaa, M. (2015). Large strain phase-field-based multi-material topology optimization. International Journal for Numerical Methods in Engineering, 104(9), 887–904.
Wallin, M., Jönsson, V., & Wingren, E. (2016). Topology optimization based on finite strain plasticity. Structural and multidisciplinary optimization, 54(4), 783–793.
Wallin, M., Ivarsson, N., & Tortorelli, D. (2018). Stiffness optimization of non-linear elastic structures. Computer Methods in Applied Mechanics and Engineering, 330, 292–307.
Ivarsson, N., Wallin, M., & Tortorelli, D. (2018). Topology optimization of finite strain viscoplastic systems under transient loads. International Journal for Numerical Methods in Engineering, 114(13), 1351–1367.
Fan, Z., Yan, J., Wallin, M., Ristinmaa, M., Niu, B., & Zhao, G. (2020). Multiscale eigenfrequency optimization of multimaterial lattice structures based on the asymptotic homogenization method. Structural and Multidisciplinary Optimization, 61(3), 983–998.
Wallin, M., & Tortorelli, D. (2020). Nonlinear homogenization for topology optimization. Mechanics of Materials, 145, 103324.
Wallin, M., Ivarsson, N., Amir, O., & Tortorelli, D. (2020). Consistent boundary conditions for PDE filter regularization in topology optimization. Structural and Multidisciplinary Optimization, 62(3), 1299–1311.
Ivarsson, N., Wallin, M., & Tortorelli, D. (2020). Topology optimization for designing periodic microstructures based on finite strain viscoplasticity. Structural and Multidisciplinary Optimization, 61, 2501–2521.
Dalklint, A., Wallin, M., & Tortorelli, D. (2020). Eigenfrequency constrained topology optimization of finite strain hyperelastic structures. Structural and Multidisciplinary Optimization.
Niu, B., Liu, X., Wallin, M., & Wadbro, E. (2020). Topology optimization of compliant mechanisms considering strain variance. Structural and Multidisciplinary Optimization, 62(3), 1457–1471.