本文同時運用了廣義微分數值法及漸近展開理論，推導出功能性材料環狀球殼靜定問題之三維漸近解析解，其中之材料性質假設為具等向性、非均質，沿厚度方向呈冪級數變化。藉由代入適當之無因次化參數，利用漸近展開法將各場量變數展開成微小參數之冪級數型式，吾人得以使原先的三維方程式分離成一系列不同階數之微分方程組。再沿著厚度方向，將所得之各階方程式透過連續積分程序，進一步推得由低階至高階完整之遞迴型式的控制方程式。
相應於不同階數的邊界條件，則經由能量原理推衍，導得其合力之型式。且各階的控制方程式與其所對應之邊界條件，將構成一合宜的邊界值問題。為求解此各階相應的邊界值問題，吾人乃採用廣義微分數值法。由於在各階問題之控制方程式中，其微分運算子保持不變，且高階問題中的非齊性項可由低階解求得。因此，首階問題的求解程序可重複應用於高階問題中，經由此逐階循環修正過程，即可得一收斂解。
本文所推衍的功能性材料環狀球殼之三維漸近理論，應用於簡支承受均佈側向載重作用之數值範例，其微分數值解經驗證，不論在收斂性與精確度上均顯示理想之結果。

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