Efficient Numerical Algorithms For Structural Topology Optimization

With the rapid development of computer technology and numerical methods,topology optimization is widely applied in aerospace,structural design,architecture,water conservancy,etc.This thesis studies the application of variable density method and level set method in structural topology optimization.We make some improvements to the two popular topology optimization methods.The variable density method considers the densities of elements in the design domain as the design variables.The number of variables increases as the mesh scale becomes larger.Materialfield expansion is a method proposed recently that can effectively reduce the number of variables.Firstly,we combine it with the density method to realize numerical examples of two classical topology optimization problems: minimum compliance and compliant mechanism design.Secondly,we propose a parameterized level set method based on polygonal mesh partition in complex domains.The level set method uses a level set function as the design variable to implicitly represent the boundary of the structure.In the parameterized level set method,the level set function is expressed as a linear combination of radial basis functions and works well in rectangular design domains.However,the square elements that previous topology optimization methods used lead to large geometric errors for general irregular domains.This thesis adopts a polygonsl mesh generation method proposed in literature,and uses the Wachspress shape function on polygonal elements.In complex design domains,the parameterized level set method is applied to numerically solve the compliance minimization and compliant mechanism problem.Finally,we propose to apply the meshless method of radial basis functionfinite difference to update the level set function in the level set method.We present some numerical examples of compliance and compliant mechanism problem under the polygonal meshes in complex design domains.Compared with the parameterized level set method,it further reduces computation.