Optimal material design based on variational sensitivity analysis

Abstract

Today's state of the art within industrial applications requires the usage of efficient and high-performance materials, which are optimally designed in terms of production costs, material savings, fuel consumption or their mechanical behaviour. In many cases, a useful choice of materials of components contributes to the overall performance significantly. Modern materials can not be classified as homogeneous, but are composed of various ingredients. Often, considerations at the structural level are no longer sufficient and investigations on a lower length scale, the so-called material- or microscale, become necessary.

Based on experimental data and results, practical experience and available methods for computer-aided simulation and analysis, the responsible design engineer has to decide which ingredients are suitable for a goal-oriented composition of materials. The evaluation of the choice made can be performed based on so-called multiscale methods for structural analysis. These also include the FE2 method, which allows a combined analysis of the macroscopic structural behaviour and the microscopic material behaviour by a numerical homogenisation scheme applied to the microscopic material structure.

The work at hand provides an enhancement of methods for the numerical multiscale simulation of the physical behaviour and enables improvements of characteristic properties. This extension leads to the research field of structural optimisation. Apart from sizing, it deals with the optimal design of components in the sense of optimal topological distribution of material as well as shape optimisation. The classical sequence of steps within a structural optimisation process contains the definition of goals, restrictions and design variables as well as the choice of algorithms for mathematical optimisation. The proposed method extends the formulation of a single-scale optimisation task and allows choices of objective functions, constraints and design parameters on multiple scales. Design parameters are for instance characteristic material properties, number and location of holes and inclusions, and geometrical parameters in general. Since the numerical effort within methods for multiscale structural analysis raises with the complexity of referred problems, it is useful to apply efficient methods in the context of structural optimisation. This class of methods includes the so-called gradient based optimisation methods. Within the present work, the required gradient and sensitivity information are derived and provided based on the variational approach for sensitivity analysis. By means of suitable examples and numerical investigations, it is shown that this approach can be classified as performant by nature.