This chapter presents the theoretical basis of the various analysis procedures. The derivation of the individual element matrices and load vectors is discussed in Derivation of Structural Matrices, Derivation of Electromagnetic Matrices, Derivation of Heat Flow Matrices, and Derivation of Acoustic Matrices.

In the matrix displacement method of analysis based upon finite element idealization, the structure being analyzed must be approximated as an assembly of discrete regions (called elements) connected at a finite number of points (called nodes). If the “force-displacement” relationship for each of these discrete structural elements is known (the element “stiffness” matrix) then the “force-displacement relationship” for the entire “structure” can be assembled using standard matrix methods. These methods are well documented (see, for example, Zienkiewicz([39])) and are also discussed in Analysis Tools. Thermal, fluid flow, and electromagnetic analyses are done on an analogous basis by replacing the above words in quotes with the appropriate terms. However, the terms displacement, force, and stiffness are used frequently throughout this chapter, even though it is understood that the concepts apply to all valid effects also.

All analysis types for iterative or transient problems automatically reuse the element matrices or the overall structural matrix whenever it is applicable. See Reuse of Matrices for more details.