Numerical Methods and Computational Algorithms
- PDF / 293,498 Bytes
- 13 Pages / 441 x 666 pts Page_size
- 74 Downloads / 248 Views
Numerical Methods and Computational Algorithms
9.1 Introduction In this chapter use discuss the numerical techniques and computational algorithms used in our work. Together the equations derived, form a procedure for a comprehensive analysis of thick plates and shells. Given the complexity of the constitutive equations, analytical solutions would be possible only for specific problems. Recent developments in computer technology, as well as numerical methods, provide a powerful tool that allows approximations of the sometimes very complicated systems of equations, used to describe engineering problems. To take advantage of this tool, a set of instructions must be given to the computer, which defines in suitable form the sequence of operations required to solve a given problem (Ketter and Prawel, 1969). We gradually built a computer program, Voyiadjis-Woelke-SHELLS, based on the shell constitutive equations presented in the preceding chapters, as well as on the computer program published by Voyiadjis and Shi (1990). The programming language was Fortran 95, along with the Compaq Visual Fortran compiler, version 6.6.C. We conducted our analyses on a personal computer. Some of the reference solutions were computed using the commercial finite element program ABAQUS installed on a Silicon Graphics Onyx 3200 workstation. Below, we discuss computational issues and numerical techniques of the computer program. First, we present a method for solving a system of linear algebraic equations. We then discuss the solution scheme of the non-linear equations followed by the overall structure of the program.
9.2 Linear Elastic Analysis – System of Linear Algebraic Equations In Chapter 3 we formulated a shell finite element, based on the shell constitutive equations derived in Chap. 2. The stiffness matrix of the element K was determined using a quasi-conforming technique. This method allows for the explicit determination of K without a need for numerical integration. Once the stiffness
G.Z. Voyiadjis, P. Woelke, Elasto-Plastic and Damage Analysis of Plates and Shells, C Springer-Verlag Berlin Heidelberg 2008
185
186
9 Numerical Methods and Computational Algorithms
matrix is calculated, the analysis involves a solution of the system of linear algebraic equations given by: Kq = R
(9.1)
where K is the stiffness matrix of the structure given by equation (3.83), q is a vector of unknown nodal displacements of the structure, and R is an external load vector. In a linear problem, the coefficients of the stiffness matrix do not depend on the unknowns. There are many different methods of successive elimination of the unknowns, which is a direct way of solving simultaneous linear equations. One of the most popular elimination method the Gauss method, is used here. It is demonstrated through consideration of a system of equations of the form: a11 x1 + a12 x2 + a13 x3 + . . . + a1n xn = c1 a21 x1 + a22 x2 + a23 x3 + . . . + a2n xn = c2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. an1 x1 + an2 x2 + an3 x3
Data Loading...
