Research On The Topological Optimization Of Three-dimensional Continuum Structure With Frequency Constraints

The topology optimization design of the continuum structures is one of the mostchallenging research topics in the field of the structural optimization and full ofinnovation. As an advanced level of structural optimization, topology optimization ismore efficient and more difficult to be solved than cross-sectional optimization andshape optimization. Throughout the works available on line, it is not difficult torealize that, most of current research works on topology optimization focus on staticstructures, but limit effort was put into dynamic problems where there exists lots ofchallenging numerical algorithms to be developed. Although some attention has beenpaid to dealing with topology optimization of dynamics for continuum structures,most of these papers largely focus on some simple-type structures such as beam, shelland thin plate, etc. Unfortunately, studies on3D continuum structures are still less. Itis well known that the purpose of the topology optimization design is to find theoptimal lay-out of a structure within a specified region, and the dynamic and staticperformance of a structure can be improved via such optimization, which is propitiousto the realistic engineering. It should be noticed that3D continuum structures is usedmainly. Hence, not only the investigation on the topological design of dynamics for3D continuum structures with eigenfrquency constraints is of great importance ofrealistic engineering, but also of significances in theoretical and practical value.In our present study, we mainly focus on the structural optimal design of dynamicsfor3D continuum structures, and aim at constructing the topological optimalformulation by using of the ICM method, proposed by Prof.SUI, which is consideringweight as object function and the fundamental eigenfrequency as constraints. Inaddition, some efficient approaches will be produced to handle some numericalproblems such as checkerboard patterns, localized eigenmodes and switch of modes.The main contents of this paper can be summarized as follows:(1) At the beginning, we study the connotations of ‘independent topology variable’intensively, which is one of the basic concepts of ICM method. And then, we discussand the characteristics of filter function in optimization modeling and solution.(2) A topological optimal formulation is constructed by using of the ICM method,which is considering weight as object function and the fundamental eigenfrequency asconstraints. An explicit expression of the frequency-constraint(s) with respect to thetopological variables is obtained by some technique. And two types of model withdifferent filter functions (power-type filter function and exponential filter function)are standardized. As a result, the topology optimization problem is solved by the dualquadratic programming. Thus, an unified optimal formulation and solving scheme is developed(3) In order to handle the localized eigenmodes, we employ Pedersen’s suggestion‘the ratio of the mass matrix to stiffness matrix should be about20~100’ to twomodels with different filter function mentioned above. The numerical examplesdemonstrate the feasibility and efficiency of the proposed method.(4) To avoid the switch of modes—another common problem in dynamic topologyoptimization, an efficient approach called moving constraint is used. The numericalexamples also show the efficiency of the method.(5) The optimal model and its algorithm in this paper have been implemented byusing the development language of MSC.Patran and MSC.Nastran software. Andfinally the topology optimization software integration of three-dimensionallightweight continuum structure with frequency constraint is formed. Some numericalexamples demonstrate that the present optimal model is considerably robust and itsalgorithm is feasible and reliable.