對稱葛勒金邊界元素法與有限元素∼邊界元素連結模式應用於三維靜力問題之分析

Symmetric Galerkin BEM and FE~BE Coupling for 3D Elastostatic Problems

柯永彥(Yung-Yen Ko)

指導教授 ： 陳正興

國立臺灣大學/工學院/土木工程學研究所/博士(2006年)

https://doi.org/10.6342/NTU.2006.01031

摘要

對稱葛勒金邊界元素法(SGBEM)的最大優點為：能建立具對稱形式之系統矩陣，同時由於加權剩餘法之應用，使得奇異積分之處理較容易，且在數值精確度與收斂行為上，較傳統點配置邊界元素法(CBEM)表現為佳，因此在近年來，被廣泛應用在各領域之邊界元素法分析。本研究將針對均質等向材料之三維靜力問題，利用無限域基本解(Kevin’s fundamental solution)，組成對偶邊界積分方程(DBIE)，建立一採四邊形線性元素之SGBEM分析模式。對於SGBEM之雙重邊界積分中，所遭遇之奇異積分問題，包含弱奇異、強奇異與超奇異積分，將利用座標轉換與積分域投影技巧，配合基本解核函數之對稱特性與基本性質處理之。經由範例分析驗證可知，本研究所建立之SGBEM模式，應用在具有良好支承條件之三維靜力問題，能得到相當正確的分析結果。

在數值分析方法中，結合有限元素法(FEM)與邊界元素法(BEM)共同使用，能兼採兩者之優點，有利於分析幾何配置與材料性質複雜的問題。當採用SGBEM與FEM連結使用時，經由適當的處理技巧，能夠得到具對稱形式之整體系統矩陣，而不破壞原本有限元素部分之系統對稱性。在本研究所採用之FE~BE連結模式中，將藉由SGBEM推導式中與自由項係數相關之部分，建立FE與BE間節點力與曳引力之轉換關係。

然而，SGBEM應用於分析力邊界值問題時，不論其屬內域或外域問題，所得位移解均會存在有一剛體運動項，而有解非唯一之現象。本研究將討論此現象之內涵與成因，並彙整一些能去除剛體運動影響的方法，藉由範例分析，討論這些方法運用在SGBEM與FE~BE連結時之適用性。結果顯示，對於外力條件之合力與合力矩為零的一般力邊界值問題，這些方法多能有效的去除剛體運動，得到唯一且正確的解；而在應用自由表面截切技巧所從事之半無限域問題分析上，由於其外力條件之合力或合力矩不為零，且因自由表面之截切造成誤差，因此雖然能夠得到唯一解，但各種方法所得之結果，仍然會存在一些不合理的現象。在所檢驗的方法中，以基於Fredholm理論所建立之修正邊界積分方程，所得之結果相對上較佳。

關鍵字

葛勒金對稱邊界元素法 ； 三維靜力分析 ； 無限域靜力基本解 ； 奇異積分 ； 有限元素∼邊界元素連結 ； 解非唯一 ； 半無限域問題

並列摘要

As compared to the conventional collocation boundary element method (CBEM), the primary advantage of the symmetric Galerkin boundary element method (SGBEM) is the ability to produce a symmetric system matrix. Besides, through the application of the weighted residual method in SGBEM, it becomes easier to regularize the singular integrals and shows better numerical behaviors than the conventional CBEM. In this study, a SGBEM procedure to form linear quadrilateral elements for 3-D elastostatic problems is established. The formulation adopts the form of dual boundary integral equations (DBIEs) based on the Kevin’s fundamental solutions. In order to evaluate the singular double integrals in the SGBEM, including the types of weak, strong or hyper-singularity, techniques of coordinate transformation and mappings of integral domains are utilized in company with the basic properties of kernel functions. Through some numerical examples, the validity of this SGBEM procedure in application to well-supported 3-D elastostatic problems is verified.

The coupling of FEM and BEM is a profitable result that exploits the advantages of each. When SGBEM is combined with FEM through appropriate techniques, a symmetric global system matrix can be obtained without ruining the symmetric virtue of the FE part. The FE~BE coupling strategy adopted in this study is to use the free-term components in the SGBEM, so that the equilibrium conditions between the nodal forces on the FE part and the nodal tractions on the BE part are reserved.

Nevertheless, when SGBEM is applied to interior or exterior Neumann problems, the displacement solution cannot be uniquely determined because of the inevitable rigid body motion term involved. In this study, discussions on this phenomenon and methods that had been used to remove the rigid-body-motion terms are investigated for the problems modeled with the SGBEM and the FE~BE coupling formulations. For general equilibrated Neumann problems, the rigid body motions can be effectively removed by using these approaches. However, for half-space problems in which the free surface are modeled by limited number of elements, the solutions obtained are still not satisfactory because of the errors introduced from the truncation of the free surface. Among the methods investigated, the one using the modified boundary integral equations based on the Fredholm theory is relatively preferable.

並列關鍵字

symmetric Galerkin BEM ； 3D elastostatics ； singular integrals ； FE~BE coupling ； non-uniqueness ； half-space problems

參考文獻

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