Use of the computer is essential for dynamic structural analyses; therefore, it is important to have some understanding of how the computer is actually used to accomplish these analyses. This chapter introduces the basic computer methods and algorithms used in dynamic analyses; these are schemes for solving systems of equations, time integration of simultaneous equations, and eigenvalue problems. References 5, 27, and 88 consider many other aspects of using computers for dynamic structural analysis.
It is difficult to describe computer algorithms without describing the complete programming context. Therefore, most of the discussions presented here refer to the algorithms implemented in the finite-element programs SDframe/SDsolid, which are available on the associated website www.cambridge.org/doyle_structures_FEM and are reduced from the QED package . The performance of the algorithms is estimated based on a hypothetical benchmark machine and problem. The machine operates at 1 Gflops (one-thousand million floating-point operations per second); small, medium, and large problems refer to systems of equations with 1,000, 10,000, and 100,000 unknowns, respectively. Algorithms appropriate for structural analysis should be suitably scalable across this range of problems.
Solving Large Systems of Equations
The primary computational task associated with matrix analysis of structures consists of solving a set of N simultaneous linear algebraic equations for N unknowns. For small-scale problems, a wide variety of schemes can be used, but for large systems where robustness and scalability are important attributes, the choice is rather limited.