Study On The New Methods In Shape And Topology Optimization Within Fixed Grid

Shape optimization and topology optimization are two important methods in the field of structural optimization,which can help engineers to design innovative structures with stringent demands,high quality and high performance.However,there are still a series of problems in the traditional structural optimization methods based on the finite element method.For example,the optimization process needs the body-fitted mesh to be refreshed unavoidably,and the optimization result appears gray elements and mesh-dependency,which objectively restricts the development of structural optimization.This thesis is dedicated to the new method of shape/topology optimization under the framework of fixed grid.By combining the advanced B-spline finite cell method and the level set method that can describe the structural boundary accurately,this work develops a new framework of shape and topology optimization which improves the convenience and effectiveness of the structure optimization technology.The main contents are as follows:(1)A weighted B-spline finite cell method is proposed to accurately apply Dirichlet boundary conditions.In the classic fixed grid-based finite cell method,the structural boundary hardly coincides with the mesh nodes,so it cannot impose the boundary conditions as easily as the finite element method.In this thesis,two implicit level set functions(weighting function and boundary value function)are exploited to modify the traditional approximated function to exactly satisfy the prescribed homogeneous or inhomogeneous Dirichlet boundary conditions.The new equilibrium equations are deduced with the use of variational principle.The modeling and construction method of two implicit functions are discussed in detail.The feasibility and effectiveness of the method are verified by typical examples in statics and thermoelastic coupling.(2)The shape/topology optimization method with designable Dirichlet boundary conditions is investigated.Based on the weighted B-spline finite cell method,the design variables are included in the weighting function which can describes the shape of the Dirichlet boundary.Since the weighting function can force the approximated function to be zero at the Dirichlet boundary,changing the geometric design variables of the weighting function can achieve a flexible design of the structural fixation.Based on the gradient optimization algorithm,shape/topology optimization problems of several typical examples are carried out involving simultaneously structure and its supports.(3)A novel topology optimization method based on closed B-spline features is proposed.In this thesis,closed B-spline curve is used to represent the free-form holes in the structure,and the topology of the structure can be changed by its movement,deformation,fusion,shrinkage and expansion.The parametric form can be used to define the design variables,while implicit form can be used for the precise geometric modeling of level set method.Numerical examples show that the proposed closed B-spline-based topology optimization method can be carried out easily in the way of shape optimization and can obtain optimal designs with small number of design variables and smooth structural boundaries as well as free of intermediate density regions and jugged boundaries.