Implicit Heaviside filter with high continuity based on suitably graded THB splines

The variable density topology optimization(TO)method has been applied to various engineering fields because it can effectively and efficiently generate the conceptual design for engineering structures.However,it suffers from the problem of low continuity resulting from the discreteness of both design variables and explicit Heaviside filter.In this paper,an implicit Heaviside filter with high continuity is introduced to generate black and white designs for TO where the design space is parameterized by suitably graded truncated hierarchical B-splines(THB).In this approach,the fixed analysis mesh of isogeometric analysis is decoupled from the design mesh,whose adaptivity is implemented by truncated hierarchical B-spline subjected to an admissible requirement.Through the intrinsic local support and high continuity of THB basis,an implicit adaptively adjusted Heaviside filter is obtained to remove the checkboard patterns and generate black and white designs.Threefold advantages are attained in the proposed filter:a)The connection between analysis mesh and adaptive design mesh is easily established compared with the traditional adaptive TO method using nodal density;b)the efficiency in updating design variables is remarkably improved than the traditional implicit sensitivity filter based on B-splines under successive global refinement;and c)the generated black and white designs are preliminarily compatible with current commercial computer aided design system.Several numerical examples are used to verify the effectiveness of the proposed implicit Heaviside filter in compliance and compliant mechanism as well as heat conduction TO problems.