Two-Point Piecewise Adaptive Approximation for Structural Optimization

本研究提出兩點分段適應近似法應用於結構最佳化上。為使數學最佳化理論能與結構設計結合，必須透過近似法將結構之行為諸如應力、位移、頻率等轉換成以設計變數表示的顯函數。最佳解便能透過解決數個由近似函數構成的最佳化問題得到。為確保近似品質，近似函數會考量函數的單調性來建立。由於許多結構行為對設計變數的變化近乎單調函數，兩點分段適應近似法確保建立單調的近似函數以確保近似品質，並在兩點微分值異號時亦能建立非單調函數以符合兩點靈敏度值。並且此近似法採用分段函數解決過往近似法中不當近似的產生。此研究亦整合最佳化程式、CAD軟體與有限元素分析軟體進行自動化結構最佳設計，並以多個結構最佳化的問題驗證本近似法於結構最佳化的實用性，並另實際應用於電路板等效有限元素模型建立與精密檢測平台的設計之中。This study proposes a new two-point approximation method called two-point piecewise adaptive approximation (TPPAA) for structural optimization. For applying the mathematical optimization to structural design, several kinds of structural behavior, including stress, displacement and natural frequency, are represented as explicit functions of design variables by approximation technique. The optimum design can be found with sequential sub-problems solved, which is known as sequential approximate optimization (SAO). To ensure the approximation quality, structural behavior is approximated with considering the monotonicity. Monotonic functions are available in TPPAA when the first order derivatives of two successive design points have the same signs since many kinds of structural behavior vary quasi-monotonically with respect to design variables. Non-monotonic form can also be obtained when the two derivatives of two successive design points have different signs. TPPAA adopts the piecewise approximate functions to avoid inappropriate approximation that existing approximation schemes would encounter. In this study, a program integrating ANSYS, AutoCAD and Microsoft Visual C++ is developed for automated structural optimization. The practicability of TPPAA is examined in several structural optimization problems and the comparison of several approximation methods are also presented. Furthermore, TPPAA is applied to optimum design of large structures, such as effective FE model construction of PCB and design of high-accuracy measuring stage structure