CAL-EL—An extension of CAL for coding element formulations from any variational principle

The computer analysis language CAL was designed to open the door to the black box of structural analysis general purpose programs. The present paper removes the shroud surrounding the generation of element matrices. Eight new CAL operations are presented which can be combined with general matrix operations to construct element matrices from any conceivable variational principle. Any structural, fluid, or thermal system, covering one and two dimensions, can be treated. The extension to three dimensions is straightforward. The matrices appearing in the integral formulation of the variational principle may be symmetric or nonsymmetric and may contain differential operators of various orders. Differentiation is treated as a linear transformation by matrix multiplication of a complete one- or two-dimensional polynomial vector basis. Differentiation is performed after the matrices from integration and evaluation of the vector basis have been created. Examples are shown of beam, plane stress membrane and plate bending element formulations approximated by various types of polynomials. Elastic and geometric stiffness matrices, mass, load, and stress evaluation matrices etc. can now be programmed within minutes with the aid of the new CAL operations.