Abstract
分片檢驗長久以來被用來檢驗有限單元的相容性，已有許多文獻對於分片檢驗做為收歛的充分條件，其有效性加以探討。對於混合應力單元及非協調單元，則已發展出許多用以通過分片檢驗的方法。因此對分片檢驗完整的瞭解及其用於收歛性的評估，遂成為一重要課題。本文乃企圖以更清晰的觀點來說明分片檢驗所應用的原理。本文引用變分法中，「發散表示式」會使歐依勒表式自動為零的概念來探討三種單元的相容條件，分別為場問題中的非協調單元、平面彈力單元及板彎單元。在變分法中，我們可以在積分式中任意加上邊界項或邊界積分項來修改邊界條件而不會影響歐依勒方程式，應用此概念，場問題中的非協調單元通過分片檢驗的相容條件為：「非協調單元模態所產生的應變，其積分必需為零」。在平面彈力單元中的應力不變量及板彎單元中的高斯曲率亦應用相同的概念，它們都是「發散表示式」的形式，其歐依勒表示式乃自動為零，因此不會影響歐依勒方程式。本文所探討的單元限於非扭曲的情形，扭曲單元的相容條件則需另行討論。The consistency of finite elements traditionally is evaluatedby using the patch test. Although numerous papers have beenwritten on this subject, the current understanding of the patchtest can not be considered completely satisfactory. In thisstudy, a new explanation of the patch test is proposed based onthe concept of identical vanishing of the Euler expression incalculus of variation. The proposed theory is shown to beconsistent with the Strang's interpretation of the test, butit is more fundamental and physically more meaningful. Forelements in field problems, plane elasticity problems, andKirchhoff plate bending problems, the proposed explanation isalso shown to have led to definite conclusion on the use ofpatch test in assessing finite element convergence. As theexplanation is based on a general mathematical principle, theextension of which to other types of problems is feasible. Thediscussions in this work are limited to problems with constantcoefficients. The problem of the effects of geometricdistortion on consistency is recognized but is not specificallytreated in this study.