Study On Structural Optimization Design Based On NURBS Isogeometric Analysis

Structural isogeometric analysis is a type of emerging numerical method in the field of computational solid mechanics,and aims to integrate the CAD(computer-aided design)and CAE(computer-aided engineering)in a unified mathematical expression framework.Isogeometric analysis(IGA)is closely related to the geometry information,which can combine the geometric modelling,structural analysis and design process,and provides a new choice and opportunity for structural optimization design.Owing to the geometry exactness,the isogeometric analysis method is very suitable for structures with curved boundary,such as curved beam and plate and shell structures,which are widely applied in the engineering.Furthermore,the IGA enable to maintain high-order continuity between elements,which guarantees the continuity of stress function,and improves the computational accuracy of structural response analysis and sensitivity calculation and the solution quality of design optimization.It is of great importance to predict the natural frequency and optimize the dynamic performance of curved beam for structural design.However,there may exist numerical locking problems when the Timoshenko beam model is adopted for slenderly curved beams,and the Euler-Bernoulli model without considering the shear deformation is not fit for the analysis of relatively thick curved beam.Moreover,the FEM(finite element method)-based size design optimization of curved beam cannot achieve smooth size distribution.On the other hand,structural topology optimization design considering stress constraints can avoid obvious stress concentration in the optimized result,which is more suitable for the practical engineering application.Generally,the large computational cost and iterative oscillation in optimization caused by local nature and high nonlinearity of stress constraints is the prominent problem.Meanwhile,the C0-continuous FEM cannot ensure the accuracy and continuity of stress function,especially for the Kirchhoff thin plates with the requirement of high-order continuity,by which the stress sensitivity analysis presents computational difficulty.Based on the NURBS(non-uniform rational B-spline)isogeometric analysis,this dissertation develops a free vibration analysis method for general models of slender/thick beams,and smooth size design framework for optimizing the natural frequencies of curved beam.The stress constraint stabilization schemes are proposed,and the IGA-based stress constraint topology optimization method of plane continuum and thin plate is established.The research contents of this dissertation are presented as follows:(1)Isogeometric method based in-plane and out-of-plane free vibration analysis of curved Timoshenko beams.To improve the computational accuracy and efficiency,this dissertation presents an exact geometry-based method for in-plane and out-of-plane free vibration analysis of curved Timoshenko beams.Based on the Timoshenko beam theory,the numerical locking can be avoided by using the selective reduced integration in conjunction with the strain projection method for slender beam models.Comparison with the FEA solution demonstrates the higher accuracy of IGA for moderately thick beam models.The proposed method is suitable for curved beams with different slenderness,namely the general models of slender/relatively thick beams.(2)Isogeometric analysis based smooth size optimization of curved Timoshenko beams.This dissertation develops a NURBS IGA-based size design approach to optimize the natural frequencies and amount of material of curved Timoshenko beams,which can achieve optimum designs with smoothly variable cross-section size.It is shown that there exists the parameterization dependency of size design for the optimized results.Specifically,a small number of NURBS-based size variables may limit the optimized results,while a large number of size variables using refined NURBS allow abrupt change of size distribution.To overcome this difficulty,a STM-based K-S correction formulation is proposed to regularize the size design and limit the size changing rate of beam.Numerical examples indicate that the present approach can achieve smooth optimum design of curved beams,and provide a precise and effective control of the maximum size changing rate.(3)Constraint aggregation approaches for stress-constrained topology optimization of continuum structures.By considering the maximum stress in the global and local forms,respectively,this dissertation proposes two effective constraint approaches to address the stress-constrained topology optimization of continuum structures.One is stress correction with the stabilization scheme,another is the violated stress set based constraint approach.The optimization problems include the stress-constrained volume minimization design and the compliance minimization with the constraints of both volume and stress.It is found that the numerical performance of the stress constraint approach is closely related to the problem formulation of topology optimization.The constraint approach does not show enough effectiveness for the stress-constrained volume minimization design when only some selected local stresses are considered in the stress constraint function.The STM(stable transformation method)-based stabilization scheme can avoid iterative oscillation and achieve stable convergence of the volume minimization design with stress constraint,while the violated stress set-based approach without any aggregation parameters is more suitable for the compliance minimization design with the constraints of stress and volume.(4)Isogeometric analysis based stress-constrained topology optimization of continuum structures.This dissertation establishes an isogeometric analysis(IGA)-SIMP method for the stress-constrained topology optimization of continuum structures,which realizes the integration of geometric modeling,structural analysis and optimization design.By virtue of high-order continuous NURBS basis functions,the isogeometric analysis enhances the computational accuracy of stress and its sensitivities,and thus enhances the credibility of optimization results.Moreover,IGA can naturally meet the requirement of C1-continuity for formulations of Kirchhoff thin plate,which facilitates the calculation of stress and stress sensitivity for thin plate,without stress smoothing procedures.To deal with the numerous local stress constraints and improve the iterative stability,the combination of P-norm correction stress constraint and the STM is employed.Compared with the STM-based stress correction,there are more stable iterations and faster convergence when the STM is applied to the design variables.Representative examples of topology optimization for plane stress and bending of thin plate show the effectiveness and accuracy of the proposed method,and the local stress level of structure can be controlled very effectively.