Budapest University of Technology and EconomicsFaculty of Civil Engineering - Since 1782

Topology optimization is a design tool that sketches lightweight components searching the design domain for the distribution of a limited amount of material that minimizes a prescribed objective function (generally compliance of the structure), given a set of constraints (such as equilibrium and volume).

Techniques such as additive manufacturing (AM) are well-suited to bring optimal layouts from concept to reality, since they considerably reduce restrictions imposed by traditional manufacturing techniques, see e.g. [1]. 

 

Adopting the conventional Solid Isotropic Model with Penalization (SIMP) to interpolate the elastic properties of the material with respect to its point-wise density, i.e. the minimization unknown, optimal layouts made of void and solid material are found. However, an improved overall performance can be achieved adopting alternative interpolation schemes with reduced penalization of the intermediate densities [2]. Extended grey regions arise in the optimal layouts, which can be handled as infill zones of porous material to be further optimized, see e.g. [3]. Alternatively, multi-scale topology optimization can be adopted to design both the macrostructure and the microstructure using a consistent material interpolation scheme, see e.g. [4]. A wide range of lattices is achieved, and it is not straightforward to effectively combine their different shapes.

 

In this proposal, topology optimization approaches will be investigated that distribute a limited amount of material by grading throughout the domain the main geometrical features of a selected set of microstructures of assigned shape. Reference is made in particular to [5] for an optimization problem that provides graded cells, but in case of a fixed macroscale. 

Numerical homogenization will be employed to derive the relevant material schemes that interpolate the elastic properties of isotropic and orthotropic lattices/microstructures both in 2D problems and 3D problems. The Cauchy model and the Cosserat model will be used at the macroscale depending on the reference size of the microstructure with respect to that of the macrostructure. 

Optimal shapes will be investigated considering manufacturability when using AM techniques. Continuity among adjacent microstructures with different material density will be sought, as well as shapes that avoid undesired overhangs. The latter point is crucial for printers that use Fused Deposition Modelling (FDM) to melt a plastic filament while positioning it layer-by-layer. At 45 degree, the newly printed layer is supported only by half of the previous layer, often still enough to build upon. For angles in excess of 45 degrees, some uneconomic support is generally required, at least for non-negligible “bridges”. 

The material laws derived from homogenization of the selected graded microstructures will be used, at first, to replace SIMP in the conventional volume-constrained minimum compliance problem. This is equivalent to a minimum volume problem with constraints on the displacement fields in case of identical boundary conditions. Hence, additional types of enforcement will be included in the formulation, such as stress-based constraints and eigenvalue-based constraints. The former topic addresses strength and fatigue requirement, whereas the second allows considering dynamic properties and buckling failure. Both topics have been widely investigated in the literature within the framework of the SIMP [6], whereas a lack of studies is reported for multiscale design based on porous microstructures.

The above problem will be investigated at first in the case of deterministic single loading. It will be investigated later the alternative loading and the multiple loading cases. The effect of probabilistic loads will be considered using first order (FORM), second order (SORM, )Monte Carlo approaches and ad hoc formulations [7,8,9].