Thermodynamic topology optimization

Publications:

https://doi.org/10.1002/nme.5988
https://doi.org/10.1007/s00158-020-02667-4

For 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 (SIMP-approach). The underlying mathematical problem is ill-posed and according regularization techniques have to be applied. A gradient-enhanced 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 mesh-based as well as for mesh-free methods. The minimum member size, i.e. the minimum cross section width of a structure feature, can be directly controlled by a user-given 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.