Parametric study of finite element formulations

A finite element is the product of many ingredients: structural theory, variational principle, element configuration, interpolation functions and integration technique. Any change in these ingredients will definitely produce some effects on element performance. It is difficult, if not impossible, to systematically study the effects without a symbolic computational tool. In mixed field elements, the selection of assumed stresses or strains is crucial for element performance. It is believed that for a certain setting, there is a set of optimal assumed stresses or assumed strains for an element. But at this moment there is no criterion to judge the optimality. Therefore, a trial-test method have to be used in determining the 'optimal' assumed stresses or assumed strains. That is, a number of sets of assumed stresses or strains are preliminary selected, element stiffness matrices are derived, which is quite tedious and error-prone without using symbolic computer algebra, and a group of benchmark tests are used to compare their performance. In [12,13] with the aid of Maple, some mixed field elements are developed, based on the alternative assumed stress or assumed strain method. Among them, the best elements behave quite satisfactorily in the selected benchmarks, but the assumed stresses or assumed strains are still not necessarily the optimal. Nevertheless, such parametric studies might provide information for establishing optimality criteria.