CONFIGURATIONAL FORCES IN STRUCTURAL AND CONTINUUM OPTIMISATION

Abstract

This thesis deals with optimisation using the principles of continuum mechanics. Both shape and mesh optimisation will be covered. A unified approach will be introduced to obtain shape and mesh optimisation for hyperelastic, hyperelastodynamic and hyperelastoplastic settings. The approach makes use of the generated material force method in mesh optimisation and the so-called imposed material force method in shape optimisation. To this end, the appropriate spatial and material continuum mechanic Equations will be developed in hyperelastic, hyperelastodynamic and hyperelastoplastic settings. A summary of the four main parts is as follows. The first part begins with structural optimisation in hyperelastic setting. After introducing the necessary Equations, the effectiveness of the material force method to obtain global optimised solutions for truss structures will be demonstrated. The implementation produces the global optimised undeformed configuration and the global optimised deformed configuration. The shape and mesh optimisation will be tested for two and three dimensional truss structures under small and large deformations. In addition, these formulations will be extended to obtain constrained optimised solutions. The penalty method is used to realise optimised truss structures within certain design criteria. The second part develops a new Arbitrary Lagrangian Eulerian (ALE) hyperelastic setting in rate form. It will deal with two systems of partial differential Equations, namely the spatial and the material momentum Equation. Both are discretised with the finite element method. The spatial Equation will then be linearised by taking the material time derivative while the material Equation will be linearised by taking the spatial time derivative. The solution defines the optimal spatial and material configuration in the context of energy minimisation in hyperelastic setting. The implemented examples will illustrate shape optimisation under the effect of mesh refinement The third part provides the formulation and implementation details of ALE hyperelastodynamic problem classes. This ALE formulation is based on the dual balance of momentum in terms of both spatial and material forces. The balance of spatial momentum results in the usual Equation of motion, whereas the balance of the material momentum indicates deficiencies in the nodal positions, hence providing an objective criterion to optimise the finite element mesh. The main difference with traditional ALE approaches is that the combination of the Lagrangian and Eulerian description is no longer arbitrary. In other words the mesh motion is no longer user defined but completely embedded within the formulation. This presents a discretisation and linearisation for a recently developed variational arbitrary Lagrangian Eulerian framework in hyperelastodynamics setting. The spatial and material variational Equations will be discretised to obtain the weak form of the momentum and continuity Equations. The discretised ALE Hamiltonian Equations of the spatial motion problem introduces the balance of the discretised spatial momentum and the discretised spatial continuity Equation while the corresponding material motion problem defines the balance of the discretised material (or configurational) momentum and the discretised material continuity Equation. We will deal with two systems of partial differential Equations: the scalar continuity Equation and the vector balance of momentum Equation. The momentum and continuity Equations will then be linearised. The time integration of both the spatial and the material Equations is performed with Newmark scheme. A monolithic solution strategy solving both the spatial and the material momentum Equations has been carried out while updating of the spatial and the material densities were attained through solving the spatial and material continuity Equations (mass conservation). The concept of generated material force has been implemented to optimise the mesh and consequently the wave propagation. The solution defines the optimal spatial and material configuration in the context of energy minimisation. The fourth part provides the framework and implementational details of ALE hyperelastoplasticity problem classes. This ALE formulation is based on the dual balance of momentum in terms of spatial forces (the well-known Newtonian forces) as well as material forces (also known as configurational forces). The balance of spatial momentum results in the usual Equation of motion, whereas the balance of the material momentum indicates deficiencies in the nodal positions, hence providing an objective criterion to optimise the shape or the finite element mesh. The earlier developed ALE hyperelastic setting will provide the platform to extend the formulation to include plasticity. The new ALE hyperelastoplasticity setting will be developed at finite strain. In ALE hyperelastoplastic formulation additional Equations are required to update the stresses. The principle of maximum plastic dissipation as well as the consistency conditions in spatial and material setting will introduce the spatial and material plastic parameters and rate form of the stress-strain relations. The solution defines the optimal spatial and material configuration in the context of energy minimisation in hyperelastoplasticity setting. The concepts of imposed and generated material force are implemented to provide improvements over Lagrangian solutions