A PACKING GENERATION SCHEME FOR THE GRANULAR.pdfTh

文件名称: A PACKING GENERATION SCHEME FOR THE GRANULAR.pdf

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详细说明：This paper introduces a new generator algorithm and computer program for 3-D numerical simulation of packing con"guration in a granular assemblies composed of ellipsoidal particles of di!erent a/b aspect ratios. Each ellipsoidal particle is approximated by the revolution of an ellipse, formed by fouPACKING GENERATION SCHEME FOR GRANULAR ASSEMBLIES 817 ellipsoids, based on the four-arc approximation theory of ellipse developed by the first author One may argue this 3-D work is a natural extension of the previous work 4 by the first author But, it has to point out that the derivation of the contact detection scheme in 3-D is not straightforward at all. a three-dimensional contact detection theory of any two ellipsoids in space is developed as the core of any 3-D computational method in the particulate mechanics. This algorithm requires no solving of high-order polynomial equation but with more applications of the mathematical theory of geometry. Some simple rules for contact detection are derived and summarized for user to develop ones own numerical simulation code It is believed that the contact algorithm proposed in this paper can compete with lin and ngs algorithms both in the efficiency and the accuracy of numerical simulation 2. APPROXIMATION OF AN ELLIPSOID The first author derived a four-arc approximation method of an ellipse with major and minor axes of length 2a and 2b.4 This method described each ellipse as four connected arcs with unique outward normal and no singularity at any point on the particle boundary As shown in the Figure 1, point C is the centroid of the ellipse, while points l,J, G, f are the centre of the radius of arcs KAL, LEM, MBH and hDK, respectively. The associate formulae for this approximation method are summarized as follows(see Figure 1) ac=cB= a DC=ce=b 2 CG(a2-b2)c+(a+b)d (3) ac CF (a2-b2)c+(a2+b2)d ) Figure 1. Approximation of an ellipse by four connected arcs Copyright C 1999 John Wiley Sons, Ltd IntJ Numer. Anal. Meth. Geomech, 23, 815-828(1999) 818 Figure 2. An ellipsoid formed by revolving the four-ac approximated ellipse where b (6) CF d= tan tan CG b GB=GH=GM=AI=AK=AL=R1=a-CG 8) FK=FD=FH=JL=JE=M=R,=b+cF Based on these fundamental equations, the co-ordinates of the end points and centre of radius of each arc for any ellipse in plane translation from the origin with a rotation respect to horizontal axis, can be easily obtained by superimposing the associated rigid-body mode A special type of ellipsoid with minor axes of equal length (i.e. b=c) can be generated by revolving the major axis of the four-arc approximated ellipse as shown in Figure 2. Though, this is a special type of ellipsoid, it can be used to characterize the main geometrical features of many natural and artificial particles. An ellipsoid, then, is formed by spherical surfaces I and Ill, and a curved surface II. Surfaces I and Ill are formed by the revolving of arcs KaL and MBH respectively, while curved surface II is formed by revolving arcs Kh and lm about the major axis. A surrounding ring IV with central axis matching the major axis is formed as the trace of the radius centers J and f during the revolving of the four-arc ellipse. The radius of circle Iv is equal to the length of Cf as expressed by equation (4). For an ellipsoid in space as shown by Figure 3 major axis directed in n=(nx, ny, n=), the co-ordinates of a point located on the surrounding ring IV can be derived as X=Cx 0/n2nx r sin (10) R R Copyright C 1999 John Wiley Sons, Ltd IntJ Numer. Anal. Meth. Geomech, 23, 815-828(1999) PACKING GENERATION SCHEME FOR GRANULAR ASSEMBLIES 819 Surrounding ring AZ Figure 3. Cartesian co-ordinates of a point p located on the surrounding ring of an ellipsoid in spa y + rcos 0 + o/nx (11) R R R (12) where the 0 is measured counter clock wise from the point with the minimum z co-ordinate and R (13) b2)a2+b2+(a2+b2)(a-b) (14) 2b Since any point on the ellipsoidal surface has an arc passing through it, the contact detection of any two ellipsoids in the packing space can be easily conducted by checking the distance between the corresponding radius centres. This is the main feature of our contact detection scheme which is different from the other proposed methods of solving complicated geometric equations of 3-D ellipsoids to detect contacts 3. CONTACT DETECTION 3 The contact conditions of ellipsoidal particles during motion or packing inside a container can e identified as the following five conditions (1) Spherical surface( or III) of a particle contacts a plane(Figure 4(a) (2)Curved surface (II)of a particle contacts a plane(Figure 4(b) (3)Spherical surface(I or Ill)of a particle contacts the spherical surface(I or III)of another particle(Figure 4c) (4)Spherical surface (I or Ill) of a particle contacts the curved surface(Il) of another particle (Figure 4d) (5)Curved surface(II)of a particle contacts the curved surface(II)of another particle(Figure 4e) The contact detection between the two spherical surfaces can be made by comparing the distance between centres with the sum of the radius of the two spheres The contact detection Copyright C 1999 John Wiley Sons, Ltd IntJ Numer. Anal. Meth. Geomech, 23, 815-828(1999) 820 C-Y. WANG. ET AL between a sphere and a plane can be conducted by checking the distance of the spherical centre to the plane with the radius of the sphere. Any three-dimensional contact detection related to the curved surface II is somewhat complicated. The curved surface is composed of many arcs with different radius centers If it can be determined which arc on this surface will contact with the other arc or plane, then the contact detection can be made by comparing the distance between the centre of this arc and the centre of other arc or plane with the impenetrability condition. The contact detection methods between surfaces relating to a curved surface are discussed in the ollowing sections 3.1. Contact detection between two curved surfaces Figure 5 shows two curved surfaces from the separate ellipsoids in the packing space contact at point P. Only one tangent plane passes through point P for both curved surfaces. Since both vector AP and vector BP are normal to this tangent plane, one can conclude that the points, A Pand B are on a straight line. Points g and h are the intersection points of line ab with major axis of the ellipsoid, respectively. Points A and b are the centres of the arc passing point P on each curved surface. According to equations(10)-(14), the location of point A (or point B)on the surrounding ring can be determined by only one parameter OA (or 0B). The length of the line AC1 is equal to the radius of the surrounding arc, so one can determine that the vectors ACl, n, and AB are on a plane. The same co-plane condition for vectors BC2, n2 and aB can also be obtained But, it has to point out that the planes aGCl and bHC2 are not necessary to be on the same plane. In general, there are only one edge of each plane, like AG and bh, located on the line ab, while the edges C,G and C2H are inclined to each other in the space. From these two properties of co-plane condition, the following two vector equations can be used to find the co-ordinates of points a and B (AC1×n1)·AB=0 (15) (BC2×n2)·AB=0 (16) Figure 4. Contact conditions for particles inside a container,(a) sphere-plane(b)curved surface-plane, (c)sphere-sphere (d)sphere-curved surface,(e)curved surface-curved surface Copyright C 1999 John Wiley Sons, Ltd IntJ Numer. Anal. Meth. Geomech, 23, 815-828(1999) PACKING GENERATION SCHEME FOR GRANULAR ASSEMBLIES 821 Figure 4. Continued fter applying equations (10)-(14) into equations (15) -(16), a set of non-linear simultaneous equations of the following form can be determined a1 CoS 0A +a2 sin 0A +a3 sin BA Sin 0B +a4 sin 0A COS OB + as COS 0A Sin 0B +a6 COS 0A COS OB=0 (17) b, cos BB+b2 sin 0B+b3 sin 0A Sin Ob+b4 sin 0A COS 0B +bs cos 0a sin 0B+b6 COS 0A coS OB=0 (18) where the coefficients ai, bi(i=1, 6)are constants determined by the directional cosines of vectors n, and n 2 of the major axes, co-ordinates of the centroid of both ellipsoids and the radii of the surrounding rings Instead of solving these non-linear equations for Oa and ], the following iteration method was used to find the positions of a and b on each surrounding ring. As shown by Figure 6, it can be Copyright C 1999 John Wiley Sons, Ltd IntJ Numer. Anal. Meth. Geomech, 23, 815-828(1999) 822 C-Y. WANG. ET AL B Figure 5. Co-plane conditions of vector set(AB, AC1, n,) and vector set(AB, BC2, n2), respectively Figure 6. Point on the surrounding ring having largest distance with the contact point P seen that if point a is the center of the arc passing through the contact point P, then point a is the farthest one to the point P compared with other points located on the surrounding ring. The same conclusion can be drawn for point b on the other surrounding ring. Since points A, P, and b are in a line, then it can be proved that the distance between points a and b in Figure 5 is the largest one among any pair of points selected separately from those two surrounding rings. An iterative method is used to find the pair of points with the longest distance between them. First we choose three points on each ring with 120 divided angle. Then, nine combinations must be evaluated to find the pair(a) and B() with the longest distance. In the second iteration step, another two Copyright C 1999 John Wiley Sons, Ltd IntJ Numer. Anal. Meth. Geomech, 23, 815-828(1999) PACKING GENERATION SCHEME FOR GRANULAR ASSEMBLIES 823 points on each ring are found by adding and subtracting 60 from the co-ordinates 0a ( or Ob D) just obtained. Again, the pair of points with the longest distance should be selected from these nine new combinations. After a few iterations, it will converge to two points with the absolute longest distance. For n iteration steps taken, the error is equal to +60/(2n-1). The value of the longest distance will compare with the sum of the two rings'radii to check whether these two ellipsoids contact. The iteratively determined OAm) and OB m values have been checked with equations(17)and(18) to verify the iteration scheme accuracy 3.2. Contact detection of a curved surface with a spherical surface Figure 7 shows the contact between a spherical surface of ellipsoid Cl and the curved surface of the ellipsoid c2. The point z is the centre of the sphere and p the contact point of these two ellipsoids. The discussion in the previous section shows that points B, P and Z are on the same line and that point b has the longest distance with point Z Using the iterative method mentioned in the previous section, the location of point b can be found. The contact condition can be determined by comparing the distance between points Z and b with the sum of the sphere radius and the curved surface arc radius 3. 3. Contact detection of a curved surface with a plane Figure 8 shows that the ellipsoid contacts a plane at point p. Point c is the ellipsoid centroid and the 0 is the angle between the plane normal vector n and the direction vector n of the ellipsoid major axis. Figure shows that when the curved surface of an ellipsoid has contact with a plane, the distance between the centroid and the plane is equal to Ch=b+ (a2-b2)a2+b2+(an2+b2)(a-b) SIn (19) 2b Figure 7. Contact detection between spherical surface and curved surface Copyright C 1999 John Wiley Sons, Ltd IntJ Numer. Anal. Meth. Geomech, 23, 815-828(1999) 824 C-Y. WANG. ET AL N 111 Figure 8. Contact detection between a curved surface and a plane B E Figure 9. Angles used for the determination of the contact detection type Then, the contact detection criterion for the curved surface with a plane is the distance between the centroid and the plane with the value obtained from equation(19) 3. 4. Determination of the contact detection type Since there are five types of contact condition that an ellipsoid may admit during packing or motion, some rules will help analysts determine which contact detection type accelerates the simulation speed. Followings are the summarized rules for the determination of contact detection Copyright C 1999 John Wiley Sons, Ltd IntJ Numer. Anal. Meth. Geomech, 23, 815-828(1999)

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