Topology Optimization Design Of Structures Based On Topological Derivative And Isogeometric Analysis Utilizing Manifold Basis Function

The structural isogeometric analysis method(IGA)is a promising numerical method in the field of computational solid mechanics,dedicating to the integration of geometric modeling(CAD)and numerical analysis(CAE)systems.Owing to the basis function with high-order continuity,the NURBS(Non-Uniform Rational B-Spline)elements is connected continually at the boundary of elements,which can improve the computational accuracy of structural response and smooth the field of stress function.Structural topology optimization is a generative design method which can provide a design with optimal mechanical properties.It is of great importance to take the stress concentration phenomenon into account,because there are plenty of structures with curved beam in the engineering.In view of this,the author proposed a topology optimization method based on topological derivative and isogeometric analysis utilizing manifold basis function.Firstly,the stress-related topological optimization of the structure is carried out by combining topological derivative concept and the NURBS-based IGA method.The finite element method(FEM)is commonly used for structural response and sensitivity analysis in stress related topology optimization methods.On one hand,the piecewise polynomial approximation is used in the FEA,and the geometric exactness cannot be guaranteed even for some common objects.On the other hand,there is no enough stress prediction accuracy for the C~0-continuous finite elements,which is important for the stress-related topology optimization problems.Although the extremely refined mesh may improve the analysis accuracy to some extent,it will significantly increase the computational cost.In view of the above facts,this paper will carry out the stress-related topology optimization under the framework of NURBS-based isogeometric analysis.The topological derivative concept is utilized to provide analytical topological sensitivity information and the P-norm stress function is used to deal with numerous local stress constraints.Optimal results indicate that the proposed method can effectively control the local stress level while reducing the material of the structure.Secondly,IGA methods are closely related to geometric modeling techniques.The smoothness and continuity of the physics field to be solved in analysis model depend on the basis functions in the geometric modeling techniques.So far,most of the researches on isogeometric analysis has been focusing on NUBRS.Due to the tensor product nature of NURBS,it is easy to construct a surface which is a homeomorphism of the rectangle.However,multiple NURBS patches are required to construct surfaces with complex topology.In addition,some extra smoothing techniques are needed to achieve the smoothness and continuity at the concatenation point.On the other hand,manifold surface construction technique based on discrete mesh is an emerging surface modeling method in the field of computational geometry.It utilizes the concept of manifold covering in differential geometry to directly construct a smooth surface with complex topological structure,which can fundamentally avoid the patch stitching techniques.This paper adopts the manifold basis function in the manifold surface construction technique to interpolate the physical field to be solved in analysis model.High-order continuity is obtained at the extraordinary points of unstructured meshes duel to the high-order continuous blending function.Several planar stress problems are analyzed in the paper,and the accuracy of this method is verified compared with NURBS IGA and Q4-FEM.Finally,the manifold basis function IGA is employed to the structural topology optimization design for the compliance minimization problems and stress minimization problems.The topological derivative is utilized to form the level-set function directly and it is worth noting that the proposed method does not have to calculate the complex Jacobian equation like common level-set methods do.Several optimization examples indicate that the stiffness of structure will be weakened while reducing the volume of material.The stress of the structure will be distributed more equally under the stress minimization formulation.Both optimal configurations have clear boundaries which could provide references for practical engineering design.