N. Abolfathi et al/ Computational Materials Science 43(2008)1193-1206 formly distributed load at infinity For load case 6 two forces at the u=-uf, uf=-u1, 5=-u's(i=1, 2, 3) (1 center of faces 3 and 5 are necessary to produce a required tor- sional load Periodicity constraints: Periodicity requires that opposite faces 2. 2. Node pairs on faces of the unit cell deform identically. This requires certain constraint For all nodes on face 2 except the center node and aints requires the number and distribution of nodes on oppo- edges e24 and ez6 and the corner node n246. One ha site faces to be identical. it is also convenient to have a node located at the geometric center of each face. To show how these achieved, consider again the solid model shown in Applying Eq (8)one gets Fig 4. On this geometry therefore six faces, 12 edges, six center- uf2-uf1-2uf1=0 face nodes and eight corner nodes. The displacement degrees of freedom for the nodes on half of the faces(2, 4 and 6) edges Ing constraInt r 4, 26, 46,.)and corners(246,. . must be written in terms of the degrees of freedom of the nodes on the other half. These u 24=u m-2(uf1+uf) algebraic relations are such that they force opposite faces to de- u 26=u ts-2(u 1+ form to the same shape though they may have a rigid body trans- lation between them At corner node n246, one has To enforce the repeating behavior, the following constraint u 2=u 35-2(u1+u2+u5) (5 relations are enforced. In the following, u(i=1, 2, 3)represents the displacement in the ith-direction, c(i=1, 2,..., 6)represents he center face nodes, ny and n represent the node pair on oppo 2.3. Node pairs ing faces i and j, ey stands for the edge i, sharing the faces i and j. sents the corner node sharing faces i,j and k. The constraint equa- 246 and the nodes on edges ez4 and e46, one has e ny stands for the nodes located on edge e, and finally nik repre- For all nodes on face 4 except the center node, the corner node ions are defined such that displacement components of ea node on faces 2, 4 and 6 are removed in terms of the respective components for the pair node on faces 1, 3 and 5. To enforce For the nodes on edge eas the following relation n is enforced: deformation compatibility between opposite faces yet still allow rigid body motion between the two faces, the displacements for the nodes on each face are expressed relative to the center node on that face. Because edge and corner nodes are shared be- 2. 4. Node pairs on faces 5 and tween multiple faces, care must be taken to avoid redundant (over)constraints For all nodes on face 6 except the center node, the corner node Additional constraints on the center nodes of the opposite faces n246 and the nodes on edges e26 and eas one has are applied. The slave nodes on the center of faces 2, 4 and 6 are related to the active nodes on the faces 1.3 and 5 as shown below: u=ul5-2ur5 Face 2: Opposite to Face 1 Face 4: Opposite to Face 3 Face 6: Opposite to Face 5 Face 3 Edge 23 Edge 36 Comer 136 er235 Face 6 Comer 14 Edge 14 Face 5 Face 4 Comer 245 Edge 15 Fig. 4. The unit cell faces, edges, and comers designations related to periodic constraint descriptions.formly distributed load at infinity. For load case 6 two forces at the center of faces 3 and 5 are necessary to produce a required tor￾sional load. Periodicity constraints: Periodicity requires that opposite faces of the unit cell deform identically. This requires certain constraint relations between the nodes on the faces. Invoking these con￾straints requires the number and distribution of nodes on oppo￾site faces to be identical. It is also convenient to have a node located at the geometric center of each face. To show how these constraints are achieved, consider again the solid model shown in Fig. 4. On this geometry therefore six faces, 12 edges, six center￾face nodes and eight corner nodes. The displacement degrees of freedom for the nodes on half of the faces (2, 4 and 6), edges (24, 26, 46, ...) and corners (246, ...) must be written in terms of the degrees of freedom of the nodes on the other half. These algebraic relations are such that they force opposite faces to de￾form to the same shape though they may have a rigid body trans￾lation between them. To enforce the repeating behavior, the following constraint relations are enforced. In the following, ui (i = 1, 2, 3) represents the displacement in the ith-direction, ci (i = 1, 2, ... , 6) represents the center face nodes, ni and nj represent the node pair on oppos￾ing faces i and j, eij stands for the edgeij, sharing the faces i and j, nij stands for the nodes located on edge eij, and finally nijk repre￾sents the corner node sharing faces i, j and k. The constraint equa￾tions are defined such that displacement components of each node on faces 2, 4 and 6 are removed in terms of the respective components for the pair node on faces 1, 3 and 5. To enforce deformation compatibility between opposite faces yet still allow a rigid body motion between the two faces, the displacements for the nodes on each face are expressed relative to the center node on that face. Because edge and corner nodes are shared be￾tween multiple faces, care must be taken to avoid redundant (over) constraints. Additional constraints on the center nodes of the opposite faces are applied. The slave nodes on the center of faces 2, 4 and 6 are related to the active nodes on the faces 1, 3 and 5 as shown below: uc2 i ¼ uc1 i ; uc4 i ¼ uc3 i ; uc6 i ¼ uc5 i ði ¼ 1; 2; 3Þ ð1Þ 2.2. Node pairs on faces 1 and 2 For all nodes on face 2 except the center node and the nodes on edges e24 and e26 and the corner node n246, one has, un2 i ¼ un1 i uc1 i þ uc2 i ð2Þ Applying Eq. (8) one gets, un2 i un1 i 2uc1 i ¼ 0 ð3Þ On the edges e24 and e26, the following constraint relations apply, respectively, un24 i ¼ un13 i 2ðuc1 i þ uc3 i Þ un26 i ¼ un15 i 2ðuc1 i þ uc5 i Þ ð4Þ At corner node n246, one has un246 i ¼ un135 i 2ðuc1 i þ uc3 i þ uc5 i Þ ð5Þ 2.3. Node pairs on faces 3 and 4 For all nodes on face 4 except the center node, the corner node n246 and the nodes on edges e24 and e46, one has un4 i ¼ un3 i 2uc3 i ð6Þ For the nodes on edge e46 the following relation is enforced: un46 i ¼ un35 i 2ðuc3 i þ uc5 i Þ ð7Þ 2.4. Node pairs on faces 5 and 6 For all nodes on face 6 except the center node, the corner node n246, and the nodes on edges e26 and e46 one has un6 i ¼ un5 i 2uc5 i ð8Þ Fig. 4. The unit cell faces, edges, and corners designations related to periodic constraint descriptions. N. Abolfathi et al. / Computational Materials Science 43 (2008) 1193–1206 1197