Multi-resolution Geometric Topology Optimization Based On Volume Subdivision

Isogeometric analysis is a new method of physical simulation using the spline representation of CAD(Computer Aided Design)models.This method provides a new idea for the seamless integration of CAD and CAE(Computer Aided Engineering),and also injects new vitality into the mature geometric modeling field.It has become a new focus of attention in the field of geometric design and computing.Generative product design is a hot research topic in the field of computer-aided design and computer graphics.The shape and topology of the product can be determined by numerical simulation of physical problems.Through the Generative design concept,the output CAD product can meet some specific performance requirements and achieving some optimization goals,such as the lowest material cost and the lightest weight.With the rapid development of 3D printing technology,the use of topology optimization methods to design product models has become increasingly popular and plays a key role in generative design.However,the traditional topology optimization method requires some post-processing operations to make the final topology optimization results suitable for CAD systems,and the integration of seamless data in complex shape design,simulation and optimization stages is also a challenging problem.In order to tackle these problems,this paper propose a novel multi-resolution 3D isogeometric topology optimization framework with an unstructured trivariate spline model.Firstly,explicit limit point formula of Catmull-Clark subdivision solids is given with mathematical proof.Secondly,based on the proposed limit point formula,the approximation representation of Catmull-Clark subdivision solid with a minimal set of tricubic volumes is given.That is,for an input hex-mesh,could construct an unstructured tricubic spline representation withC~2/C~0-continuity.The geometry volumes approximate the shape and silhouette of the Catmull-Clark solids and are smooth everywhere except along volume faces containing an extraordinary vertex where the volumes areC~0.Compared with ordinary approximate subdivision.This method can generate Catmull-Clark volume with arbitrary subdivision accuracy in one step,which is efficient and simple.This method is similar to dynamic programming,which divides the entire model into many small models,processes the small model first,and then integrates all the processing results.So there is a significant efficiency advantage when dealing with large models.Finally,a multi-resolution 3D isogeometric topology optimization framework is proposed with the proposed unstructured spline solid representation.In our framework,consistent spline language is used for geometry representation,isogeometric analysis and topology optimization,which leads to an optimal shape represented by smooth spline patches that can be imported for CAD applications directly.In particular,for generative design problem with shape constraints,the interface geometry between different components can be exactly preserved during topology optimization.Moreover,the proposed topology optimization framework naturally has a multi-resolution property,that is,a volume parameterization with fixed resolution is used to perform the simulation,and a volume parameterization with high-resolution is used for design optimization,which can achieve computationally efficient and high-resolution designs.Several topology optimization examples for minimum compliance(minimum strain energy)problems are presented to show the effectiveness and efficiency of the proposed framework.