From the individual element test to finite element templates: Evolution of the patch test

This paper starts a sequence of three articles that follow an unconventional approach in finite element research. The ultimate objective is to construct high-performance elements and element-level error estimators for those elements. The approach takes off from our previous work in high-performance elements and culminates with the development of finite element templates. The present paper concentrates on the patch test and evolved versions of the test that have played a key role in this research. Following a brief review of the historical roots, we present the Individual Element Test (IET) of Bergan and Hanssen in an expanded context that encompasses several important classes of new elements. The relationship of the IET to the multielement forms A, B and C of the patch test and to the single-element test are investigated. An important consequence of the IET application is that the element stiffness equations decompose naturally into basic and higher-order parts. The application of this decomposition to the “sanitization” of the non-convergent BCIZ element is described and verified with numerical experiments. Two sequel papers in preparation are subtitled ‘the algebraic approach’ and ‘element-level error estimation’. These apply the fundamental decomposition to the derivation of templates for specific mechanical elements and to the construction of element-level error estimators, respectively.