Topology Optimization Using the Lattice Boltzmann Method

Abstract

Human design is limited by our ideas of geometries. Topology optimization is a tool to cross that barrier and improve design especially for applications that are irrevocable once they are in use. This thesis presents a Python implementation of the Lattice Boltzmann Method (LBM) with a porosity model and a separate Python implementation of the adjoint method using the Lagrange multiplier method. The implementation is intended as a basis for topology optimization and is created with the deliberate application of blood flow related topology optimization. The two-dimensional LBM is implemented with a no-slip fullway bounce-back boundary for the closed boundary and solid nodes. For the open boundaries a non-equilibrium density boundary condition and an equilibrium velocity Boundary Condition (BC) are provided. The porosity model constitutes a hybrid method of the Stokes and Brinkman equations. The implemented test case represents a quadratic domain with a velocity boundary building a flow inlet and a pressure boundary that is the flow outlet. New topologies are created by adjusting the porosity of the nodes, so that the ideal shape of the tube under the specified conditions based on a scalar objective function is obtained, when the LBM is combined with an optimization algorithm as the adjoint method. The adjoint method is applied to a one-dimensional problem, where the geometry of a tube is optimized based on the radius along the axis of the tube. The theory and the implementation procedure are documented in a detailed manner to facilitate the understanding of the methods and provide a foundation for own implementations.