自然界中許多材料是離散體，例如砂土和岩石，擁有許多不同的形狀。對於數值模擬分析上，以分離元素法來模擬較為合適。前人研究混合型剛體顆粒系統之數值模擬與卵形顆粒介質運動之數值模擬，本研究將兩者研究之成果結合並修改。並且加上可以排列顆粒在空中之前處理程式，使顆粒在模擬開始前即可以隨機排列且不會重疊，後處理則修改前人使用MATLAB寫出之動畫程式，使其可以畫出多面形塊體。
本研究主要將卵形顆粒和多面塊體接觸判斷完成，卵形顆粒與橢球顆粒一樣係由四圓弧逼近法所形成，然而橢球顆粒只是卵形顆粒的特例形狀。將卵形顆粒加入到混合型剛體顆粒系統可以讓其更為廣義。卵形顆粒的球面與弧面分別可能會和多面形塊體的(1) 頂點(2) 稜線(3) 面接觸，然而卵形顆粒和多面形塊體稜線或面接觸時，程式可能會誤判在範圍之外接觸，於是必須加入界內外判斷。在程式完成後，以不同的算例來驗證程式的正確性，包含: (1) 卵形顆粒拋體運動驗證 (2) 多面形剛體與卵形顆粒碰撞的線動量守恆 (3) 摩擦力驗證。在確認程式無誤之後，使用程式模擬巴西堅果效應，在尺寸不同的混合顆粒中，經過震動後，體積較大的顆粒會往上運動，在自然界也會發生這種狀況，例如土石流發生時，大的石頭會在上層，小的石頭會在下層。

Many materials in the world are discrete material, such as sand and rocks which always get irregular shape. It is proper to discrete element method for numerical simulation. The main idea of this thesis is combination the achievement of “Numerical Simulation for Granular Assemblies Egg-shaped Particle” and ” Numerical Simulation for Granular Assemblies with Mixed 3D Rigid Body ”. In addition, add the pre-processor code to arrange the position of the particles and prevent the particle overlap when the code assigns the position of the particle. In the post-processor part, the animate code is expanded the polyhedron block.
Actually, the main idea of this research is accomplishment of the contact detection between the egg-shaped particle and polyhedron block. The egg-shaped particle is formed by four-circle approximation method so as to form the ellipsoid. The ellipsoid only is the special case of the egg-shaped particle. Thus the “Numerical Simulation for Granular Assemblies with Mixed 3D Rigid Body” is more generalized. The egg-shaped particle would contact with the polyhedral block in these three ways: (1) ball- vertex (2) curve-vertex (3) ball-edge (4) curve-edge (5) ball-plane (6) curve-plane. When the egg-shaped particle contact with the edge of the polyhedron block or the plane of the polyhedron block, it could detect the out range of the polyhedron block as contact. So it must be to add the mechanism to detect the contact point inside or outside. After finishing the code, there are three examples to test the correction of the code. (1) projectile of the egg-shaped particle (2) the linear momentum equilibrium of the collision of polyhedron block and the egg-shaped particle (3) friction test. This research simulates the Brazil nut effect. After shaking the particle assemblies , the bigger particle would motion to the upper of the particle assemblies. Many examples occur the Brazil nut effect in the world. For example, the mud slide is. The bigger rocks would appear upper portion of the rocks.

參考文獻
1. 王泉博，「二維圓盤顆粒之堆積模擬與應用」，碩士論文，國立中央大學土木工程研究所，中壢(1996)。
2. 李宗龍，「橢圓顆粒介質體運動之數值模擬」，碩士論文，國立中央大學土木工程研究所，中壢(1997)。
3. 李承恩，「三維球形顆介質體變形分析」，碩士論文，國立中央大學土木工程研究所，中壢(1997)。
4. 劉建宏，「橢球顆粒介質體運動之數值模擬」，碩士論文，國立中央大學土木工程研究所，中壢(1998)。
5. 陳俊全，「混合型橢球顆粒介質體運動之數據模擬」，碩士論文，國立中央大學土木工程研究所，中壢(1999)。
6. 許秀真，「剛性多面體系統之數值模擬」，碩士論文，國立中央大學土木工程研究所，中壢(2000) 。
7. 張義隆，「以複合圓形顆粒模式探討粒狀材料之力學行為」，博士論文，國立中興大學土木工程研究所，台中(2003) 。
8. 林佳宏，「混合型剛體顆粒系統之數值模擬」，碩士論文，國立中央大學土木工程研究所，中壢(2006) 。
9. 林昇鴻，「蛋形顆粒介質運動之數值模擬」，碩士論文，國立中央大學土木工程研究所，中壢(2006) 。
10. 陳筱葳，「二維圓盤柔性約束結構之模擬」，碩士論文，中原大學土木工程研究所，中壢(2006) 。
11. 吳柏翰，「三維圓球顆粒運動之數值模擬」，碩士論文，中原大學土木工程研究所，中壢(2008) 。
12. Cundall, P. A., “A Computer Model of Simulating Progressive Large Scale Movement in Blocky Rock System,” Symposium of International Society of Rock Mechanics, Nancy, France, pp. 11-19(1971).
13. Cundall, P. A., and O. D. L. Strack, “A Discrete Numerical Model for Granular Assemblies,” Geotechnique, Vol. 29, No. 1, pp. 47-65 (1979).
14. Cundall, P. A., and O. D. L. Strack , “Modeling of Microscopic Mechanisms in Granular Material.” Mechanics of Granular Materials: Mew Models and Constitutive Relations, Amsterdam, Netherlands, pp. 137-149 (1983).
15. Lin, J.-S., and S. Chen, “Discrete Element Modeling Of Direct Shear and Simple Shear Tests,” Proceedings of the 2nd Int. Conference on DEM, IESL Publications, pp. 245-252 (1993).
16. Rothenburg, L., and R. J. Bathurst, “Numerical Simulation of Idealized Granular Assemblies With Plane Elliptical Particles,” Computers and Geotechnics, Vol. 11, pp. 315-329 (1991).
17. Shi, G.-H., “Discontinuous Deformation Analysis, A new Numerical Model for the Static and Dynamics of Block systems,” Ph. D. Thesis, Department of Civil Engineering, University of California at Berkeley(1988).
18. Wang, C.-Y., and V.-C. Liang, “A Packing Generation Scheme for the Granular Assemblies With Planar Elliptical Particles,” International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 29, pp. 347-358 (1997).
19. Ng, T.-T., and X. Lin, “ Numerical Simulations of Naturally Deposited Granular Soil with Ellipsoidal Elements,” Proceedings of the 2nd Int. Conference on DEM, IESL Publications, pp. 557-567 (1993).
20. Tran, T. X., and R. B. Nelson, “Analysis of Particle Disjoint Two-Dimensional Assemblies.” Journal of Engineering Mechanics, ASCE, Vol. 122, No. 12, pp. 1139-1148 (1996).
21. Cundall, P. A., “Formulation of a Three-dimensional Distinct Element Model-Part I.A Scheme to Detect and Represent Contacts in a System Composed of Many Polyhedral Blocks.”International J. Rock Mechanics Sci. & Geomech. Abstr., Vol. 25, No. 3, pp.107-116(1988).
22. Cundall, P. A., “Formulation of a Three-dimensional Distinct Element Model-Part II. Mechanical Calculations for Motion and Interaction of a System Composed of Many Polyhedral Blocks. ”International J. Rock Mechanics Sci. & Geomech. Abstr., Vol. 25, No. 3, pp.117-125(1988).
23. Nezami, E. G., Hashash, Y. M.A., Zhao, Dawei ,and Ghaboussi, J., “A fast contact detection algorithm for 3-D discrete element method” Computers and Geotechnics, Vol.31, PP.578-587(2004).
24. Nezami, E. G., Hashash, Y. M.A., Zhao, Dawei ,and Ghaboussi, J., “Short link method for contact detection in discrete element method” International Journal for Numerical and Analytical Methods in Geomechanics, Vol.30, PP.783-801(2006).
25. Oldenburg, M., and L. Nilsson, “The Position Code Algorithm for Contact Searching,” Int. J. Numer. Meth. Engng., Vol. 37, pp. 359-368 (1994).
26. Wang, S.P., and E. Nakamachi, “The Inside-Outside Contact Search Algorithm,” Int. J. Numer. Meth. Engng., Vol. 40, pp. 3665-685(1997)
27. Gallas, J. A. C., Herrmann, H. J., and Sokolowski, S., “Convection Cells in vibrating Granular Media,” Physical review letters, Vol.69, pp.1371-1374(1992)
28. Ellenberger, J., Vandu C.O., and Krishna, R., “Vibration-induced granular segregation in a pseudo-2D column: The (reverse) Brazil nut effect,” Powder technology, Vol.164, pp.168-173(2006)