tailieunhanh - dohrmann Episode 1 Part 3

Tham khảo tài liệu 'dohrmann episode 1 part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | As with the least squares formulation the alternative formulation can be implemented efficiently. The derivatives in Eq. 87 can be calculated using dVijki dxi ỉ fc i 6 2 z fc z 6 88 dVijki dyi zfc - Zi xj - Xi - Zj - zi xk -xz 6 89 dVijki dzi xfc - Xi yj - yỉ - Xj - xi yk - yi Q 90 In addition the alternative formulation allows one to ignore specified mid-edge or mid-face nodes. For example a seven-node tetrahedral element without mid-face node 8 is obtained simply by neglecting the volume V1328 in Eq. 71 . The least squares formulation can also be modified to ignore certain nodes but the approach is not as straightforward. The midedge nodes of the six-node triangle and mid-face nodes of the eight-node tetrahedron can be constrained to possess only a normal degree of freedom by simple modifications of the expressions for area and volume in Eqs. 68-69 . Finally the equivalent nodal loads given in Eqs. 26-27 29-30 32-33 can also be determined by calculating the virtual work done by a uniform distributed force on the edges or faces of the triangular and tetrahedral elements. By making use of Eqs. 74-86 one arrives at the same expressions for the equivalent loads provided the mid-edge and mid-face nodes are centered. 15 References 1. D. p. Flanagan and T. Belytschko A Uniform Strain Hexahedron and Quadrilateral with Orthogonal Hourglass Control International Journal for Numerical Methods in Engineering 17 679-706 1981 . 2. 0. c. Zienkiewicz and R. L. Taylor The Finite Element Method Vol. 1 4th Ed. McGraw-Hill New York New York 1989. 3. J. c. Simo and T. J. R. Hughes On the Variational Foundations of Assumed Strain Methods Journal of Applied Mechanics 53 51-54 1986 . 4. 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 139 3-47 1996 . 5. G. H. Golub and c. F. Van Loan Matrix Computations 2nd Ed. John Hopkins Baltimore Maryland 1989. 6. s. w. Key M. w. .