Topology optimization is an important design tool for high performance components and devices. In this thesis, we explored numerical methods for topology optimization and applications.

The minimum compliance problem is one of the most important problems in structural optimization. We reviewed the energy principles in linearized elasticity theory and reformulated the minimum compliance problem into a minimum-minimum problem, which is suitable for a threshold dynamics method to be developed. We proved energy decaying property and verified it numerically. Two effective continuation methods have been proposed to get a better local minimum. The same thresholding algorithm has also been applied to the heat transfer problem, which performs well when volume fraction is small.

Phase field method...[ Read more ]

Topology optimization is an important design tool for high performance components and devices. In this thesis, we explored numerical methods for topology optimization and applications.

The minimum compliance problem is one of the most important problems in structural optimization. We reviewed the energy principles in linearized elasticity theory and reformulated the minimum compliance problem into a minimum-minimum problem, which is suitable for a threshold dynamics method to be developed. We proved energy decaying property and verified it numerically. Two effective continuation methods have been proposed to get a better local minimum. The same thresholding algorithm has also been applied to the heat transfer problem, which performs well when volume fraction is small.

Phase field method models the motion of interface. With the help of sensitivity analysis, we formulated the gradient flow of the total energy which consists of compliance and interface energy. To increase the time step, a convex splitting scheme to the Cahn-Hilliard equation and truncation to the phase field function have been applied. We also discussed the application of Allen-Cahn formulation and pointed out some further directions.