Topology and material optimization are used to optimize the resource usage for loaded structures. For example, the structural stiffness should be maximized for restricted resources, i.e. the maximum amount of material or financial cost, for a given mechanical boundary value problem defined by loads and supports. To optimize the structural performance, design variables are introduced, which are controllable production parameters and define the local and or global structure properties. The objective of the optimization is to find those values for the design variables which minimize or maximize the objective function, i.e. maximum performance or minimal cost. From a material modeling perspective, design variables are similar to internal variables, which are usually used to describe (microstructural) evolution of the material based on thermodynamic principles. Within the thermodynamic topology optimization, the design variables are treated as internal variables and thermodynamic principles for dissipative processes are used to derive differential equations for the evolution of the design variables towards the optimal material configuration.
To improve numerical efficiency and speed up computation times, the mechanical problem, i.e. finding the displacements for a given design, and the solution of the evolution equations, i.e. the design update, are solved nonmonolithically. The numerical optimization process contains an alternating process of finite element solutions and differential equation solutions: the finite element method (FEM) is used to find the displacements, which are the only unknowns within the FEM, for a given design. Driving forces (also called sensitivities) are calculated with the resulting displacements and are used to solve the evolution equations. The evolution equations are solved with fixed displacements and independently from the finite element formulation. In combination an backward Euler scheme (explicit time discretization) for the evolution equations, the computation times become significantly shorter compared with classical monolithical solution schemes within material modeling.
Besides the topology of the structure, also local and global material properties, as for example different material phases and orientation of anisotropic materials, can be defined as design variable and subject to optimization. Not only the application of different materials but also the local optimization of those properties have a significant influence on the optimal topology. Thus, it is reasonable to optimize the topology and the material simultaneously. The thermodynamic optimization approach can be applied to derive evolution equations and therefore update schemes for the optimal design for various types of design variables describing different material properties.
Topology and material optimization

Topology optimizationFor the optimization of the topology, the local material density is defined as design variable within a given design space. The design space describes the geometrical bounds of the structure and to which the (mechanical) boundary value problem is applied. In each point of the design space, the density indicates whether material should be applied in that region or not. For mathematical relaxation, the density variable is continuous allowing intermediate densities during the optimization process, i.e. porous material. Intermediate densities are penalized so that the final topology contains approximately only full and void material (SIMPapproach). The underlying mathematical problem is illposed and according regularization techniques have to be applied. A gradientenhanced regularization is added for the density field and the evolution equation is formulated in its strong form. With the backward Euler scheme and an internal loop for numerical stability, no additional equation systems besides the FEM have to be solved within the optimization process. The second spacial derivatives in the strong form are computed via the neighbored element method. Herein, only the minimum number of neighboring points are used to calculate the required second spatial derivatives to reduce the calculation effort even further. The formulation is independent of the spacial discretization of the design variable: only data on the close neighborhood between points is required. Therefore the method is suitable for meshbased as well as for meshfree methods. The minimum member size, i.e. the minimum cross section width of a structure feature, can be directly controlled by a usergiven parameter. Furthermore, the regularization technique can also be applied to regularization in other material models, as for example damage, wherein the width of the damaged zone can be controlled directly.Leitung: D. R. Jantos, P. JunkerJahr: 2021

Large deformationsWith only minor modifications to the optimization, the model is also capable to optimize structures under finite (large) deformations including buckling phenomena. The optimized topology is rather different than for small deformations and with large deformations, the direction of the applied forces influence the final topology (changing the sign of the forces considering small deformations does not change the result). The regularization and therefore the minimum member size are applied from perspective of production, i.e. the undeformed state, and do not require any recalculation of the operator matrices. The calculation effort of the optimization compared to the mechanical analysis is negligible.Leitung: P. JunkerJahr: 2021

PlasticityPlastic deformation or plastic zones can weaken the structure drastically or are also planned into the design of structure. Usual approaches for optimization with plastic material require the calculation of a full plasticity analysis with multiple load steps until convergence for each design optimization step, which results in a lot of mechanical analysis steps and therefore large calculation efforts. In the novel approach, a dissipationfree plasticity model is applied, whose evolution is pathindependent, so that only one mechanical analysis step is required for each optimization step. In combination with the operator split, the calculation effort for the optimization with plastic material is negligible higher than for an optimization with pure elastic material.Leitung: P. Junker, D. R. JantosTeam:Jahr: 2021

Anisotropic materialsHigh performance materials as for example carbon fiber reinforced polymers but also structures produced with additive manufacturing inhere anisotropic material properties, which can be be influenced during the production process, i.e. the applied direction of fibers or print path within 3D printing. Since the material orientation has a major influence on the structure performance, the local material orientation should also be considered as design variable for the optimization process. With the thermodynamic optimization approach, evolution equations for the optimal material direction described by Euler angles can be found and are combined with a simultaneous topology optimization, which results in significantly different varying optimal typologies in comparison to a topology optimization with isotropic material. For some production processes, as for example reinforcement with long fibers, or simply for a smoother fiber path design, the maximum fiber curvature can be constrained via a filtering technique with the filter radius R given by the user.Leitung: D. R. Jantos, P. JunkerJahr: 2021

Tension and compression affine materialsConcrete is economical but rather weak under tension load, whereas steel may bear tension and compression very well, but is much less economical. Therefore, an simplified approach for economical steelconcrete structures is to apply concrete only in regions predominant to compression loading and steel under tension loading. By introducing an energetic penalization, this approach can be implemented into an topology optimization with two elastic materials, in which one material is affine to compression (e.g. concrete) and one is affine to tension (e.g. steel). Due to different elastic properties of the both materials, i.e. Young's modulus an Poisson's ratio, the resulting optimization depends strongly on the load direction.Leitung: D. R. Jantos, P. JunkerJahr: 2021

Optimization and additve manufacturingThe results from the topology optimization are usually very difficult or even impossible to manufacture with conventional methods. But with additive manufacturing, as for example 3D printing, the production becomes not only feasible but most optimized structures can be directly produced without modification. However, the material characteristics and also bounds of the additive manufacturing processes, as for example material anisotropy, print directions, overhangs, thermomechanical properties, and so on, should be considered as constraints for the optimization. Those effects strongly depend on the chosen additive manufacturing process and are considered in future projects.Leitung: D. R. Jantos, P. JunkerJahr: 2021