Expanded Simp Method For Structural Topology Optimization By Transplanting ICM Method

The Solid Isotropic Material with Penalization (SIMP) method and IndependentContinuous and Mapping (ICM) method are both very efficient numerical methods forstructural topology optimization. They are widely used in various fields. Although theSIMP method has been given more attention internationally because of simpleformula and easy implementation, its development is stuck in the theory system withonly one penalty function (Young’s modulus penalty function). However, the ICMmethod has made significant progress over the decades. Its theory foundation istamped and its modeling and solutions approaches are tempered. Several numericallaws have been concluded with this method. In order to furthering the development ofthe SIMP method, some progress in the ICM method is used for reference andtransplanted into the SIMP method for implementation.The reason why it is able to do so, is there are analogies between the ICM andSIMP methods. Firstly, the artificial relative density variables are defined in [0,1],just as the independent continuous topology variables do in the ICM method. This isthe analogy of design variables. Secondly, although the penalty function in the SIMPmethod has the concept of penalization, while the filter functions in the ICM methodhas the concept of approximation, their mathematical formulations are similar.Therefore, the progress of the ICM method can be transplanted into the SIMP methodby analogy.The conception of transplantation has been elaborated concretely by sevenchapters in this paper. By analogy with and transplanting the progress of the ICMmethod, several studies have been done in the SIMP method: to discuss the reasonablemodeling method for structural topology optimization, to introduce the elementweight penalty and material allowable stress penalty functions, to verify the invariantrelationship between the element weight penalty and Young’s modulus penaltyfunctions, to trace the utilization of the expanded SIMP method in structural topologyoptimization problems for plates with three constraints of displacement, stress and combined displacement and stress. The usage of nonlinear element weight penaltyfunction can improve the convergence speed in topology optimization for plates withdisplacement constraints. Transplanting the stress constraint globalization approach inthe ICM method into the SIMP method is effective to address the issues of largenumber of stress constraints and sensitivity computation difficulty.The main innovation of this research can be summarized into three aspects asbelow:(1) A basic theory is proposed better to choose the economic indicators asobjective functions and the performance indicators as constraints when constructing astructural topology optimization model. A basic approach, the strategy of adaptivedynamical adjusting constraint bounds, is utilized to address constraint breach issuesin optimization problems.(2) A basic concept is presented by extending the number of penalizationfunctions in the SIMP method from one to three. The invariant relationship betweenthe power exponents of the element weight penalty and material allowable stresspenalty functions, and the stress constraints globalization method, are introduced.(3) The expanded SIMP method is used in structural topology optimization forplates. The unified derivations and solutions for the optimization models areproceeded to address three situations with displacement constraints, stress constraints,and combined displacement and stress constraints, respectively. They are allimplemented on the platform of Abaqus software based on Python.A large number of numerical examples for plates have been calculated and gotsatisfactory results. This research indicates that, the SIMP method can indeedreference from the ICM method. Its theory and numerical approaches both canachieve great progress and success.