本文主要是呈現非線性彈性超級橢圓平板的力學分析，在本文內容中分成兩個部分做討論。第一部分探討改變次方的不同尺寸超級橢圓平板力學分析情形，第二部分應用基因演算法在超級橢圓平板上求得近似解。超級橢圓的形狀決定於次方大小。當次方越大，則近似方形，次方為一，則為圓形。在本文中詳細的探討具非線性彈性超級橢圓平板在不同形狀下的力學行為特性。
文中首先對微分方程式最大值原理，殘差關於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的數值解相互比較其中差異。為了解決上述提及的問題，雙側逼近法需先採用傳統之加權殘值法，將一求解微分方程之問題，轉換為一具有限制條件之數學規劃問題。透過結合最佳化法則從兩側逼近正確解，逼近的條件為分別找出正確解之最小偏大近似解與最大偏小近似解。於本文中利用加權殘值法中之配點法轉換求解之微分方程，再進而採用基因演算法做為我們求解上下界之最佳化法則。
本文將雙側逼近法應用超級橢圓平板之靜態行為分析，於過去之文獻中，多為探討矩形、圓形、橢圓形等一般形狀之問題，鑑於對於超級橢圓形相關資訊之缺乏，因此先使用Galerkin’s Method求得近似解，再將基因演算法所求解之結果與解析解比對，均相當地令人滿意，驗證了此方法在邊界值問題上應用之可行性與正確性。

This thesis presents the engineering analysis of a super elliptical plate on a nonlinear foundation. The content is divided into two parts including (1) the behavior of the structure under different size of the elliptical plates for various power of orders, and (2) the application of genetic algorithms to several special cases of elliptical plates to obtain approximate solutions. The shapes of super ellipse are characterized by different power of the orders. As the power becomes large enough, the shape of the ellipse is shaped to be rectangle-like. And as the power equals to one, the shape of the ellipse is shaped to be circle-like. In this thesis, detailed description of nonlinear engineering structural behavior of elliptical plates under different geometric shapes thoroughly detailed.
Differential equations are derived first based on the maximum principle, and then the monotony of the residual solutions are presented with rigor validation. Followed by that the results obtained by a double side approach method and the monotony of the residual equation of the numerical solution are presented and compared with each other to show the difference. In order to solve the above-mentioned problem, the traditional method of weighted residuals by double side approach is employed, and the problem for solving the differential equations is further converted into a mathematical programming problem with constraints. By combining the optimization algorithms to obtain the accurate solutions from bilateral sides, the conditions to find minimal upper approximate solution and maximal lower approximate solution are formulated and incorporated into the solution process. The weighted residual method and collocation method for solving the differential equations are programming into the code with the genetic algorithms (GA) to solve for the bounds to comply with optimization rules given in the context.
This thesis elaborates the details of the application of algorithm to analyze of the static structural behavior of thin super ellipse. In the past, many researchers work studied the linear behavior of the circular, ellipse and other general shape of the plate separately. The employment of super-ellipse to accommodate the structural behavior of various shapes in general, not available, applies the Galerkin's Method for analytical solutions. This paper, therefore, as compared with the results of genetic algorithms with other analytical solution, the results of genetic algorithms are found much satisfactory for dealing with boundary value problems, somehow superior to the solutions of others obtained by traditional analytical approaches.

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