Explicit Isogeometric Topology Optimization Method with Suitably Graded Truncated Hierarchical B-Spline

This work puts forward an explicit isogeometric topology optimization(ITO)method using moving morphable components(MMC),which takes the suitably graded truncated hierarchical B-Spline based isogeometric analysis as the solver of physical unknown(SGTHB-ITO-MMC).By applying properly basis graded constraints to the hierarchical mesh of truncated hierarchical B-splines(THB),the convergence and robustness of the SGTHB-ITOMMC are simultaneously improved and the tiny holes occurred in optimized structure are eliminated,due to the improved accuracy around the explicit structural boundaries.Moreover,an efficient computational method is developed for the topological description functions(TDF)ofMMC under the admissible hierarchicalmesh,which consists of reducing the dimensionality strategy for design space and the locally computing strategy for hierarchical mesh.We apply the above SGTHB-ITO-MMC with improved efficiency to a series of 2D and 3Dcompliance design problems.The numerical results show that the proposed SGTHB-ITO-MMC method outperforms the traditional THB-ITO-MMCmethod in terms of convergence rate and efficiency.Therefore,the proposed SGTHB-ITO-MMC is an effective way of solving topology optimization(TO)problems.

Isogeometric topology optimization based on energy penalization for symmetric structure

We present an energy penalization method for isogeometric topology optimization using moving morphable components(ITO–MMC),propose an ITO–MMC with an additional bilateral or periodic symmetric constraint for symmetric structures,and then extend the proposed energy penalization method to an ITO–MMC with a symmetric constraint.The energy penalization method can solve the problems of numerical instability and convergence for the ITO–MMC and the ITO–MMC subjected to the structural symmetric constraint with asymmetric loads.Topology optimization problems of asymmetric,bilateral symmetric,and periodic symmetric structures are discussed to validate the effectiveness of the proposed energy penalization approach.Compared with the conventional ITO–MMC,the energy penalization method for the ITO–MMC can improve the convergence rate from 18.6%to 44.5%for the optimization of the asymmetric structure.For the ITO–MMC under a bilateral symmetric constraint,the proposed method can reduce the objective value by 5.6%and obtain a final optimized topology that has a clear boundary with decreased iterations.For the ITO–MMC under a periodic symmetric constraint,the proposed energy penalization method can dramatically reduce the number of iterations and obtain a speedup of more than 2.

Level set-based isogeometric topology optimization for maximizing fundamental eigenfrequency

Maximizing the fundamental eigenfrequency is an efficient means for vibrating structures to avoid resonance and noises.In this study,we develop an isogeometric analysis(IGA)-based level set model for the fonnulation and solution of topology optimization in cases with maximum eigenfrequency.The proposed method is based on a combination of level set method and IGA technique,which uses the non-uniform rational B-spline(NURBS),description of geometry,to perfonn analysis.The same NURBS is used for geometry representation,but also for IGA-based dynamic analysis and parameterization of the level set surface,that is,the level set function.The method is applied to topology optimization problems of maximizing the fundamental eigenfrequency for a given amount of material.A modal track method,that monitors a single target eigenmode is employed to prevent the exchange of eigenmode order number in eigenfrequency optimization.The validity and efficiency of the proposed method are illustrated by benchmark examples.

Massively efficient filter for topology optimization based on the splitting of tensor product structure

In this work,we put forward a massively efficient filter for topology optimization(TO)utilizing the splitting of tensor product structure.With the aid of splitting technique,the traditional weight matrices of B-splines and non-uniform rational B-spline implicit filters are decomposed equivalently into two or three submatrices,by which the sensitivity analysis is reformulated for the nodal design variables without altering the optimization process.Afterwards,an explicit sensitivity filter,which is decomposed by the splitting pipeline as that applied to implicit filter,is established in terms of the tensor product of the axial distances between adjacent element centroids,and the corresponding sensitivity analysis is derived for elemental design variables.According to the numerical results,the average updating time for the design variables is accelerated by two-order-of-magnitude for the decomposed filter compared with the traditional filter.In addition,the memory burden and computing time of the weight matrix are decreased by six-and three-order-of-magnitude for the decomposed filter.Therefore,the proposed filter is massively improved by the splitting of tensor product structure and delivers a much more efficient way of solving TO problems in the frameworks of isogeometric analysis and finite element analysis.

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.