本文提出一套自動配置背景網格之四叉樹無網格伽遼金法（Quadtree Element Free Galerkin Method），並探討壓電超級橢圓管受管內壓及熱負載之破壞分析。本文內容主要分為三個部分進行探討。
　　第一部分以本文提出之四叉樹無網格伽遼金法與傳統無網格伽遼金法（Element Free Galerkin Method）計算應力強度因子（Stress Intensity Factor）。針對彈性平板與文獻比較之誤差分別為0.42%及1.60%，相比傳統方法，QEFGM之精度提高73.75%。再者針對壓電平板，在裂紋長度極長時與文獻比較之誤差分別為1.28%及5.17%，QEFGM之精度更是提高75.24%，由此證明本文所提之四叉樹無網格伽遼金法之正確性及優良性。
　　第二部分詳細探討溫度變化導致應力強度因子之影響，發現溫度升高時材料明顯有裂紋密合之現象，並以四種不同溫差分布進行破壞分析，發現管內緣之溫差為影響系統物理場及應力強度因子之重要因素。
　　第三部分詳細探討幾何外型對應力強度因子之影響，以四種不同長短軸比之橢圓，及三種不同冪次之超級橢圓進行破壞分析，發現長短軸比及冪次越高，都會導致應力強度因子越大，導致裂紋尖端應力發散之趨勢越明顯。
　　本文針對不同工作環境下，對壓電材料進行了多種破壞分析，繪製其物理響應及計算應力強度因子，希望在工件使用壽命之評估及材料破損之預防，能為業界提供有效的資訊。

In this article, the quadtree element-free Galerkin method (QEFGM) is proposed for automatically configuring the background cells. The thesis explores fracture analysis of piezoelectric elliptical tube subject to internal pressure and thermal loadings. The content of this paper is mainly divided into three parts to discuss.
First of all, the stress intensity factors (SIF) is calculated by the QEFGM and the traditional element-free Galerkin method (EFGM). The errors compared with the literature are 1.28% and 5.17%, respectively. Compared with the traditional method, the accuracy of the QEFGM is increased by 75.24%. Therefore, the correctness and superiority of the QEFGM proposed in this paper is proved.
Secondly, this paper discusses the effects of temperature changes on the SIF. It is found that the crack of the material is closed when the temperature rises. Further, the fracture analysis is carried out with four different temperature distributions. It is found that the temperature difference of the inner edge of the tube is an important factor affecting the physical field and the SIF.
Finally, the thesis explores the influence of geometric shape on the SIF. The fracture analysis is carried out with four ellipses with different axial ratios and three super ellipses of different powers. It is found that the SIF increases when the axial ratio and the powers increase.
In this paper, various fracture analysis of piezoelectric material is carried out under different working conditions. It is hoped that the research results can provide effective information for the industry in extending material life cycle.

[1] G. R. Liu and Y. T. Gu, An introduction to meshfree methods and their programming. Springer Science & Business Media, 2005.
[2] A. Khosravifard, M. Hematiyan, T. Bui, and T. Do, "Accurate and efficient analysis of stationary and propagating crack problems by meshless methods," Theoretical and Applied Fracture Mechanics, vol. 87, pp. 21-34, 2017.
[3] J. Curie and P. Curie, "Développement par compression de l'électricité polaire dans les cristaux hémièdres à faces inclinées," Bulletin de minéralogie, vol. 3, no. 4, pp. 90-93, 1880.
[4] G. Lippmann, "Principe de la conservation de l'électricité, ou second principe de la théorie des phénomènes électriques," Journal de Physique Théorique et Appliquée, vol. 10, no. 1, pp. 381-394, 1881.
[5] J. Curie and P. Curie, "Contractions et dilatations produites par des tensions électriques dans les cristaux hémièdres à faces inclinées," Compt. Rend, vol. 93, pp. 1137-1140, 1881.
[6] W. Voigt, Lehrbuch der kristallphysik. Teubner Leipzig, 1928.
[7] P. Liu, T. Yu, T. Q. Bui, C. Zhang, Y. Xu, and C. W. Lim, "Transient thermal shock fracture analysis of functionally graded piezoelectric materials by the extended finite element method," International journal of solids and structures, vol. 51, no. 11-12, pp. 2167-2182, 2014.
[8] J. Sladek, V. Sladek, P. Solek, and C. Zhang, "Fracture analysis in continuously nonhomogeneous magneto-electro-elastic solids under a thermal load by the MLPG," International Journal of Solids and Structures, vol. 47, no. 10, pp. 1381-1391, 2010.
[9] A. J. Chorin, "Numerical study of slightly viscous flow," Journal of fluid mechanics, vol. 57, no. 4, pp. 785-796, 1973.
[10] R. A. Gingold and J. J. Monaghan, "Smoothed particle hydrodynamics: theory and application to non-spherical stars," Monthly notices of the royal astronomical society, vol. 181, no. 3, pp. 375-389, 1977.
[11] B. Nayroles, G. Touzot, and P. Villon, "Generalizing the finite element method: diffuse approximation and diffuse elements," Computational mechanics, vol. 10, no. 5, pp. 307-318, 1992.
[12] T. Belytschko, Y. Y. Lu, and L. Gu, "Element‐free Galerkin methods," International journal for numerical methods in engineering, vol. 37, no. 2, pp. 229-256, 1994.
[13] S. N. Atluri and T. Zhu, "A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics," Computational mechanics, vol. 22, no. 2, pp. 117-127, 1998.
[14] G.-R. Liu and Y. Gu, "A point interpolation method for two‐dimensional solids," International Journal for Numerical Methods in Engineering, vol. 50, no. 4, pp. 937-951, 2001.
[15] L. Yuqiu and Z. Yiqiang, "Calculation of stress intensity factors in plane problems by the sub-region mixed finite element method," Advances in Engineering Software, vol. 7, no. 1, pp. 32-35, 1985.
[16] A. Naderi and G. Baradaran, "Element free Galerkin method for static analysis of thin micro/nanoscale plates based on the nonlocal plate theory," International Journal of Engineering, vol. 26, no. 7, pp. 795-806, 2013.
[17] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, "Meshless methods: an overview and recent developments," Computer methods in applied mechanics and engineering, vol. 139, no. 1-4, pp. 3-47, 1996.
[18] M. Fleming, Y. Chu, B. Moran, and T. Belytschko, "Enriched element‐free Galerkin methods for crack tip fields," International journal for numerical methods in engineering, vol. 40, no. 8, pp. 1483-1504, 1997.
[19] B. Rao and S. Rahman, "An efficient meshless method for fracture analysis of cracks," Computational mechanics, vol. 26, no. 4, pp. 398-408, 2000.
[20] J. Yau, S. Wang, and H. Corten, "A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity," Journal of applied mechanics, vol. 47, no. 2, pp. 335-341, 1980.
[21] R. Talebitooti, K. Daneshjoo, and S. Jafari, "Optimal control of laminated plate integrated with piezoelectric sensor and actuator considering TSDT and meshfree method," European Journal of Mechanics-A/Solids, vol. 55, pp. 199-211, 2016.
[22] X. L. Li and L. M. Zhou, "An element-free Galerkin method for electromechanical coupled analysis in piezoelectric materials with cracks," Advances in Mechanical Engineering, vol. 9, no. 2, pp. 1-11, 2017.
[23] S. Park and C. Sun, "Effect of electric field on fracture of piezoelectric ceramics," International Journal of Fracture, vol. 70, no. 3, pp. 203-216, 1993.
[24] B. Rao and M. Kuna, "Interaction integrals for fracture analysis of functionally graded piezoelectric materials," International Journal of Solids and Structures, vol. 45, no. 20, pp. 5237-5257, 2008.
[25] T. L. Anderson, Fracture mechanics: fundamentals and applications. CRC press, 2017.
[26] H. Yu, L. Wu, L. Guo, J. Ma, and H. Li, "A domain-independent interaction integral for fracture analysis of nonhomogeneous piezoelectric materials," International Journal of Solids and Structures, vol. 49, no. 23-24, pp. 3301-3315, 2012.
[27] F. Shang and M. Kuna, "Thermal stress around a penny-shaped crack in a thermopiezoelectric solid," Computational Materials Science, vol. 26, pp. 197-201, 2003.