General And High-performenced Discrete Variable Topology Optimization Method Via Sequential Approximate Integer Programming(SAIP)

Structural topology optimization has developed rapidly in recent years,due to the strong demand for super-lightweight and multi-function structures in many high-tech fields.The classic pixel-based topology optimization approaches take the element density as the design variable,where 0 and 1 denote the void and material,respectively.Hence,the mathematical essence of pixel-based structural topology optimization is large-scale nonlinear integer programming.To bypass combinatory complexity and dimensional curse,the traditional method is to relax the discrete variables into continuous variables through material interpolation scheme.The complex post-processing of this method is evitable to achieve the manufacturable design because it allows the intermediate(gray)densities..Besides,it is hard to extract and control the geometrical or topological information during the whole iterative process.Further,the intermediate(grey)densities may destruct the physical meaning of the optimization problems,which may lead to the analysis difficulties.In fact,how to eliminate the intermediate densities has been a core and challenging subject in the topology optimization field.Directly tackling this nonlinear integer programming problem can validly break this difficulty.However,this route has been considered impossible for a long time.This dissertation aims to break this bottleneck and develops a novel general and high-performance discrete variable topology optimization method.This thesis covers the theoretical and numerical studies on the following four aspects.(1)By using discrete variable sensitivity,we generalize the classic Sequential Approximate Programming(SAP)into discrete variable topology optimization to formulate the Sequential Approximate Integer Programming(SAIP).And then,the Canonical relaxation algorithm is developed to solve the SAIP subproblems.This algorithm is as efficient as the gradient based continuous variable sequential approximate programming.Besides,this algorithm owns strong asymptotic dual property that can guarantees the excellent property of the approximate solution.Two move limit strategies that are suitable for discrete variables are proposed.This method successfully solves four different kinds of topology optimization problems with the compliance objective function but with different constraints and physical problems.These numerical results demonstrate that the new method can efficiently solve the discrete variable structural topology optimization problems with multiple nonlinear constraints or many local linear constraints in a unified and mathematical way.(2)How to solve the non-monotonical objective function that owns indefinite signs of the sensitivity is the major obstacle to generalize the discrete variable method into the complicate non-compliance problems.Hence,we develop the Sequential Approximate Integer Programming with Trust-Region(SAIP-TR)framework.We prove that the trust-region constraint in discrete variables is linear and propose the strategy to dynamically adjust the trust region radius based on a merit function.Besides,the geometric constraint that can control the minimum length scale for the material phase and void phase under the discrete design variables is proposed.This geometric constraint is solved by SAIP-TR method to suppress de facto hinges and reduce stress concentration in the optimized design.Numerical examples demonstrate that the SAIP-TR method can solve the non-monotonical objective/constraint functions represented by the compliant mechanism design and minimum length scale control problem.Besides,since the trust region is a rational move limit strategy that can directly control the variation range of discrete variables,SAIP-TR method can handle the topology optimization problems without material usage constraint.SAIP-TR method greatly expands the application scope of the discrete variable topology optimization method.(3)For the distinct black-white topology configuration obtained by the discrete variable method,we propose a programmable Euler-Poincaré formula that only relates with the nodal density and nodal neighborhood information to efficiently calculate the topological invariants of the topology optimization results,i.e.,Euler and Betti numbers.We further demonstrate that the Betti number can directly represent the structural complexity(the number of holes).By combining the programmable Euler-Poincaré formula and three strategies that are relaxed linear structural complexity constraint,skeleton-based constraints and material usage decrement,the solvable optimization formulations to control the structural complexity are established.These formulations can be efficiently solved by Sequential Approximate Integer Programming and Canonical relaxation algorithm for various 2D and even complicated 3D problems with different initial and targeted number of holes.We further believe that this study bridges the gap between structural topology optimization and computational topology analysis,which is much expected in the structural optimization community.(4)To extend the discrete variable method into the large-scale 3D problems,we combine the physical multiscale method with the mathematical multigrid method to develop the multiresolution discrete variable topology optimization method.We firstly utilize the Extended Multiscale Finite Element Method to calculate the numerical shape functions.Secondly,the numerical shape functions are formulated into the projection matrix between the coarse and fine mesh.By applying the projection matrix with the mathematical multigrid method from the numerical algebra,EMs FEM＿MGPCG solver is achieved.This solver can avoid the island structure(QR pattern)and obtain high-multiresolution optimal stiffness design with rich details.