Level Set Topology Optimization Method Considering Multiple Constraints And Geometric Nonlinearity

Topological optimization is a component of structural optimization design.By considering loads and various constraints,it achieves the optimal material density distribution of the entire structure,fully improving the structural mechanical performance and minimizing unnecessary material loss,thereby achieving the best mechanical and economic performance in structural design.As a level set method that has emerged in the past decade,it has become an important branch of topological optimization,with smooth boundaries and no sawtooth conditions in the case of volume constraints.However,classical level set methods only consider elastic materials and volume constraints,while actual engineering often involves nonlinear problems such as large deformations,thermal coupling,or multiple constraints,so classical level set methods have some limitations when applied to such engineering problems.This article expands on classical level set methods by introducing new types of constraints,such as displacement,frequency,and stress constraints,while also considering geometric nonlinearity and thermal coupling effects.It proposes a multi-constraint parameterized level set topological optimization method and a multi-constraint geometric nonlinear level set topological optimization method based on thermal coupling.To address the issue of numerical instability in geometric nonlinearity under large deformations,it also presents a Lagrange multiplier optimization method.By improving upon traditional level set methods and considering multiple constraint scenarios and thermal coupling geometric nonlinearity scenarios,this article aims to enhance convergence and computational efficiency.The main contributions and innovations of this article are as follows:1.The traditional level set method is a mathematical model that minimizes flexibility with volume rate as the constraint.However,in practical engineering scenarios,there are often multiple constraints.In this article,we add displacement,frequency,and global stress constraints to the existing model and reconstruct the element iteration format to balance the Lagrange multipliers among the three constraints.We verify the feasibility of this method through examples.2.In actual engineering practice,factors such as temperature and load may cause changes in the displacement of the structure,leading to significant changes in the overall stress of the structure.Based on this,this paper adds geometric nonlinearity considering temperature effects,and the research results of the numerical examples show good convergence stability.3.The original Lagrange multiplier search method can handle multi-constraint optimization problems with constraint overshoot ratios(the ratio of overshoot to constraint limits)within 1.However,under the condition of large displacement in geometric nonlinearity,the overshoot ratio can reach 8-10,which makes the original Lagrange multiplier search method inapplicable.Therefore,this paper proposes a Lagrange multiplier optimization method.Through numerical examples of geometric nonlinearity,its applicability and stability are verified.4.Combining the parameterized level set method with three commonly used composite material properties in engineering,the elastic matrix D and the element stiffness equation are reshaped,successfully extending the parameterized level set method to the field of composite materials.