Structural shape optimisation is a field that has been studied since early on in the development of finite element methods. The sub-fields of shape and topology optimisation are continuously growing in industry and aim to leverage the benefits of technologies such as 3D printing and additive manufacturing. These fields are also being used to optimise designs to improve quality and reduce cost.
Gradient-based optimisation is well understood as an efficient method of obtaining solutions. In order to implement gradient-based optimisation methods in the context of structural shape optimisation, sensitivities describing the change of the domain stiffness are required. To obtain the stiffness sensitivities, mesh deformation sensitivities are required. In this study, a mesh generating method is developed that provides mesh deformation sensitivities.
For shape optimisation it is advantageous to employ an optimisation algorithm that allows for the manipulation of CAD geometry. This means that the CAD geometry is finalised upon completion of the optimisation process. This, however, necessitates the calculation of accurate sensitivities associated with non-linear geometries, such as NURBS (those present in CAD), by the mesher.
The meshing method developed in this study is analogous to a linear truss system. The system is solved for static equilibrium through a geometrically non-linear finite element analysis using Newton’s method. Sensitivities are made available by Newton’s method for use in generating mesh sensitivities for the system.
It is important for the mesher to be able to accurately describe the geometrical domain which approximates the geometry being modelled. To do so, nodes on the boundary may not depart from the boundary. Instead of prescribing all boundary nodes, this mesher frees the boundary nodes to move
University of Pretoria ii
Department of Mechanical and Aeronautical Engineering
along, but not away from the boundary. This is achieved using multipoint constraints since they allow for an analytical relationship between boundary node movement and the boundary.
Two multipoint constraint (MPC) methods are investigated for boundary discretisation, namely, the Lagrangian and master-slave elimination methods (MSEM). The MSEM presents several difficulties in obtaining convergence on non-linear boundaries in general when compared to the Lagrangian method.
The MSEM has reduced computational requirements for a single Newton step, especially when direct solvers are used. However, when indirect solvers are implemented the time difference between the two MPC methods reduces significantly. For a “medium” curvature geometry the Lagrangian implementation has only a 6% time penalty.
The Lagrangian method is selected as the preferred MPC method for implementation in the mesher to avoid the convergence problems associated with the MSEM. This is justified on the basis of reliability outweighing the 6% time penalty for what is intended to be a tool in the shape optimisation process.
Analytical sensitivities are obtained for the truss system in order to account for the MPC boundaries. The analytical mesh sensitivities are proven to be accurate through comparison with numerical sensitivities. The method is demonstrated to be able to accurately described the mesh deformation throughout the domain for both uniform and non-uniform meshes in the presence of non-linear boundaries.
Dissertation (MEng)--University of Pretoria, 2020.