臺灣博碩士論文加值系統

無網格法近年來已成為一個數值分析中被廣泛使用的方法，不同於傳統有限元素法(Finite Element Method，FEM)，無網格法能夠被應用的問題更加全面，因此許多不同目的而產生的無網格法也接連被提出。本研究中主要是利用再生核質點法(Reproducing Kernel particle method，RKPM)作為數值近似的核心，用以分析船舶結構中的船殼(Hull)之非線性變形模擬，但由於船隻的船殼本身為一個三維空間中的複雜曲面形狀，因此只要當船殼的曲面有和再生核形函數中的修正方程共面時，將會導致再生核形函數產生奇異性的問題。針對這個問題，本文中利用「等參座標尺度單元(Isoparametric element)」之概念，建立一個在二維母空間(parent domain)中的再生核質點法近似方程，利用此二維近似方程投影到三維真實空間中，並引進形狀重建函數以重建船殼之複雜幾何形狀。此方法避免了再生核形函數奇異性的發生，也因為維度的降低而減少了計算形函數的時間花費。
另外在利用數值方法分析板殼問題時，當數值近似的階數不夠，會產生位移閉鎖的現象，此現象造成了偽內能的發生，進而影響了解的精確度。一般面對閉鎖問題時，能利用降階積分(例如一點高斯積分)求解，在無網格法中，使用點積分同樣可以消除閉鎖現象的產生，但是這些方法卻都會導致秩缺(rank deficiency)的問題；因此在本研究中，提出利用高階點積分法(High Order Nodal Integration，HONI)建構剛度矩陣，本方法藉由在母空間中建立積分網格以計算積分點上之權重，並在母空間中進行積分，避免處理複雜船殼形狀。高階點積分法可以經由調整積分網格的大小，改變積分點上的權重，藉以達到能提高積分近似的階數，獲得較高的收斂率。高階點積分法不但具有點積分的性質，卻不會有秩缺的問題，並且可以獲得較高的收斂率以及精度。本文中利用線性與非線性分析問題，驗證所發展之等參座標尺度下的再生核質點法以及高階點積分法之可靠性與適用性。

Meshfree methods (MMs) recently have become a popular research area in numerical analysis. By improving some drawback of general Finite Element Methods (FEM), MMs have a wide range of applications and several meshfree methods have been proposed. In this study, we applied Reproducing Kernel Particle Method (RKPM) for analysis of Hull deformation in ship structure. RKPM has improved the consistency condition in the Smoothed Particle Hydrodynamics (SPH). That makes RKPM more suitable for various engineering problems. The Hull has complex geometry in the three dimensional space. When the RK nodes are co-plain on the surface of Hull, it will lead to a singularity of shape function. This study applied the Isoparametric finite element concept to develop the two-dimensional RK shape function in a parent domain, which is called the IsoRK approximation. For arbitrary shape of plate, we mapped the IsoRK shape function into the original three-dimensional global domain to reconstruct the complex geometry. The IsoRK shape functions avoid the singularity and have more efficient discretization and less computational cost.
In the numerical analysis of Mindlin plate problems, if the inappropriate order of approximation is applied, it will lead to the locking phenomenon. The locking phenomenon comes from the spurious shear energy, which will greatly suppress the displacement solution. Generally, the reduced integration, including one point Gauss quadrature and nodal integration, is applied to avoid the locking phenomenon. However, it may lead to rank deficiency. To overcome these difficulties, we proposed a High Order Nodal Integration (HONI) for developing stiffness matrix in the weak form. Through the HONI, we can increase the accuracy by adjusting the quadrature weight in the parent domain. Several numerical analyses are performed to show the reliability of the proposed methods.

致謝 I
摘要 II
Abstract III
目次 IV
圖次 VI
符號表 VIII
第一章 緒論 1
1.1研究動機與目的 1
1.2文獻回顧 1
1.3 研究方法與架構 4
第二章 基本理論 5
2.1 再生核質點法 ( RKPM ) 5
2.2等參座標尺度下的再生核質點法近似任意幾何形狀 15
2.2.1 Full Transformation 17
2.3 Mindlin板假說 24
2.4等參座標尺度下的再生核質點法近似板方程 29
2.4.1局部座標系統 29
2.4.2 Mindlin板之離散方程 31
2.5板的閉鎖現象 34
第三章 數值積分 37
3.1 牛頓-柯特斯積分法 38
3.2 高階點積分法 41
3.3 高階點積分收斂分析 41
3.3.1 多項式函數積分近似 43
3.3.2 求解Poisson方程問題 45
3.4 模態分析 48
第四章 數值算例 51
4.1 方形板問題 52
4.2 Scordelis-Lo屋頂問題 53
4.3 閉鎖現象測試 56
4.4 兩端為自由端之圓筒問題 56
4.5 兩端為剛性隔板之圓筒問題 57
4.6 頂端開洞之半圓問題 59
第五章 結論 61
參考文獻 63

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