Topology optimisation with the discrete element method

Abstract

The drive towards sustainability in engineering system design has renewed interest in topology optimisation as a method to maximise performance while minimizing waste. Typically these methods employ continuum-based analyses using numerical techniques, such as finite elements, to quantify performance. Whilst this approach is efficient for linear elastic systems, non-linearity adds complexity and discontinuous behaviours including rigid body motion cannot be included due to singularity in the stiffness matrix. As a result inherently discontinuous processes such as material fragmentation, powder-based 3D printing, and granular mechanics in general have not benefited from the development of topology optimisation. This thesis proposes an original approach of coupling penalisation-based topology optimisation with computational simulations using the discrete element method. In the penalisation-based approach the stiffness of individual finite elements are scaled based on a penalised element density variable. Here the proposed adaptation is derived from a scaling of interaction forces and potentials between interacting particles. This formulation is developed into a complete topology optimisation framework including analytical and numerical definitions for sensitivity and the formulation of a filtering technique. This new methodology is first implemented in a simple, proof-of-concept, 2D implementation, for validation against well-known cases from the continuum regime, such as simply supported beams, and columnar systems. These systems are discretised as lattices of particles connected by harmonic springs; at this validation stage bond breakage is not allowed, but some cases involve geometric and material non-linearlity, which the new method captures already in its basic formulation are shown. The method is then implemented in combination with a stateof- the-art, 3D simulator for particle based mechanics. This generalised implementation provides flexibility to define complex objectives for the optimisation and enables the incorporation of fully discontinuous behaviours and rigid motion. Examples are presented showing the incorporation of discontinuous processes such as the maximisation of fragmentation energy under impact in a beam and the optimal design of granular systems.