Study On Level Set Methods And Its Applications In Topology Optimization

Level Set method is a new method to describe the fronts propagating of curveand curved surface, developed by curve evolvement with curvature. It can be used tocalculate and analyze interface motion, which depends on time, position, geometryproperty, and the exterior physical characteristic of the interface. It has been used inmultiphase motion successfully, such as image segmentation, combustion, crystalgrowth, numerical simulation of computational fluid dynamics, computer visiontechnique, semi-conductor process etc. And now, it is one of the dominant methodsto calculate and to track the evolution of interfaces.In this paper, based on the viscosity solution of Hamilton-Jacobi equations, andcombined with Godunov's monotony differential scheme, essentially non-oscillatorydifferential scheme (ENO), weighted essentially non-oscillatory differential scheme(WENO), integral of TVD Runge-Kutta scheme, a high fidelity algorithm for levelset equations is proposed and deduced. With this algorithm, shock in hyperbolicconservation laws for multidimensional scalar equation(s), the non-reacting shockproblems and detonation discontinuities in multimaterial flows are trackedsuccessfully. Based on the high fidelity algorithm, a narrow band level set method isconstructed. Through modifying the level set function values near the local boundary,pseudo oscillation problem of local boundary condition is eliminated availably, butthe evolvement trend of interface will not be infected. When the level set function issolved in local area, regularized function c(φ) is introduced to avoid considering oflocal boundary condition in computing. In this thesis, the narrow band level setmethod is applied to track the evolvement of interfaces in multi-material flow, and iscompared to global level set method in computing time. Numerical examples showthat the narrow band level set method is fast, the physical and geometrical essencesof every step strategy for tracking fronts are revealed.Then, a new structural topology optimization method is proposed based on thelevel set method. The structural topology configurations are concentrated to handle the structural boundary. When level set function is introduced, the method is toembed the structural boundary as its zero level set of a higher dimensional function.This method can handle topology change, describe the shape of structural boundary,easily, and can deal with the hole fuse naturally in structure topology. So, it is anevolutionary structural optimization method. In the optimization iteration, the levelset function values of nodes are considered as design variables. To computedifferences normally, the Heaviside function is handled for smoothing, theregularized Heaviside function is selected to avoid the oscillation (checkerboard) inhomogenizations method, and the results can converge smoothly. A local structuraltopology optimization method is developed, using the level set method, the narrowband is introduced in sensitivity to eliminate the computing time. Numericalexamples are analyzed and compared. The limitation of the local topologyoptimization method in eliminating computing time is analyzed, and a new revealedstructural topology optimization method is developed to eliminate computing time inlarge scale.In this paper, the main research work is concentrated to develop a local level setmethod, related to the numerical techniques of partial differential equations. Usedthis method, evolving process of interface motion is tracked, the sensitivity functionwhich contain information of stress and strain is constructed, thus, structuraltopology optimization can be carried on. Numerical experiments indicate that themethod is fast, robust and accurate enough.