Moving Mesh Method And Its Applications In Topology Optimization

There are a lot of nonlinear problems in modern science and technology,most of which have the characteristic of large gradient or even discontinuity.Due to the complexity of the problem and the requirement of the accuracy of the solution,we often need to improve the accuracy of the numerical solution as far as possible.Adaptive finite element method emerges as the times require.In this paper,the adaptive moving mesh method is studied,and the structural topology optimization problem based on the adaptive moving mesh technique is studied.The whole paper has six chapters.The background of the adaptive finite element,the background and significance of the structure optimization and the current state of the structure optimization are given in the introduction.The second chapter introduces the basic knowledge of Sobolev space,basic inequalities,grid management strategy and other preparatory knowledge.In chapter 3,the moving mesh method is introduced,and the equal distribution principle and De boor algorithm are introduced in detail.The principle of equal distribution is the basis ofmoving mesh method.Based on the principle of equal distribution,we study the knowledge of partial differential equation of moving mesh and monitor function.In one-dimensional case.we often select the arc length control function to solve the problem.The choice of control function is relatively complex in two dimensional cases.We construct the control function by using a posteriori error estimate.We mainly introduce two methods,semi-a posterior method and hierarchical basis method.In chapter 4,numerical examples of solving partial differential equations by moving mesh method are introduced,including one-dimensional Burgers equation,two-dimensional Burgers equation and two-dimensional Navier-Stokes equation.Although the selection of monitor function is different,the mesh moving strategy is different.The numerical example shows that the adaptive moving mesh method can obtain higher precision and better effect than uniform mesh.Chapter 5 mainly focuses on the application of moving mesh method in topology optimization of continuous structure.Taking the two-dimensional cantilever beam as an example,the SIMP method and density filtering technique are used to optimize the topology of the cantilever beam with moving mesh method.The numerical results show that our method is effective.