Engineering Analysis of Super Elliptic Plates on Nonlinear Elastic Foundation

超級橢圓板在非線性彈性基礎上之結構力學分析

賴新一(Hsin-Yi Lai) ； 陳詩璇(Shih-Hsuan Chen) ； 湯祥雯(Hsiang-Wen Tang) ； 陳朝光(Cha'o Kuang Chen)

《中國機械工程學刊》 37卷5期 (2016/10) Pp. 397-405

摘要

A new combined numerical algorithm is introduced in this paper to analyze the structural behavior of super elliptic plates on nonlinear foundation. The objectives are two folds, one is to investigate the behavior of elliptic plates of different sizes residing on nonlinear foundation in terms of different power orders, and the other is to obtain approximate solutions by using genetic algorithms for comparison. Detailed description of nonlinear structural behaviors of elliptical plates of different geometric shapes are elaborated. Especially, as the power becomes large enough to have the shape turned into a rectangular one, or on the other hand, as the power equals to one, the shape of the ellipse goes to the other extremity of a circle. In either case, differential equations are derived based upon the maximum principle, and the residual solutions of monotony are presented with rigor validation. The problem solving of differential equations is further converted into a mathematical programming problem with practical constraints. By combining with optimization algorithms the solutions are obtained from bilateral sides and the solutions of minimal upper and maximal lower bounds are obtained. The weighted residual and collocation methods are also used to solve for the differential equations in combination with genetic algorithms (GA) to obtain satisfactory bounds that comply with specific optimization rules.

關鍵字

double side approach ； weighted residual method ； mathematical programming ； genetic algorithms ； super ellipse

並列摘要

本文主要是呈現超級橢圓平板在非線性基礎上的結構分析。本文內容中分成兩個部分作探討。第一部分探討不同次方尺寸之超級橢圓板其正解之印證與比對。第二部分應用基因演算法求取超級橢圓板之快速電腦化近似解，超級橢圓的形狀則決定於所選取的次方數大小。當次方數越大，其所呈現之板面即近似於方形。當次方數為一時，其所呈現的板面即近似於圓形。在此基礎上，本文詳細地探討了不同形狀的超級橢圓板在非線性之支撐結構上所展現的力學行為與特性。文中首先對微分方程之最大值原理與殘差解之單調性提出驗證，接著將雙側逼近法求得的解與單調性殘差方程數值解之差異提出相互比較。為了解決上述所提問題，首先採用傳統加權殘值法作雙側逼近，將一個原本是微分方程求解之問題，轉換為一個具有限制條件之數學規劃問題。透過與最佳化法則結合，從兩側來逼近正解，逼近的條件即是分別找出正解的最小偏大值與最大偏小值。文中並利用加權殘值法之配點轉換求解微分方程，且進而採用基因演算法作為求取上下界解之最佳化法則。本文將雙側逼近法應用於超級橢圓板以深入探討其結構行為。因過去文獻，所探討之問題多為矩形、圓形與橢圓形等一般規則形狀，對於超級橢圓板結構，尤其是在非線性基礎上之力學分析的相關資訊是欠缺的，因此本文首先使用Galerkin法求其正解，再將基因演算法所得結果與解析正解進行比對，其結果相當令人滿意，也驗證了此法在邊界值問題應用上之可行性與正確性，值得進一步推廣