### Finite Element Templates: A Summary of Recent Developments

#### Abstract

This lecture highlights recent accomplishments of a line of research in basic technology of lhe finite element method (FEM) pursued by the author at the University of Colorado over lhe past six years. The research began in 1987 with modest goals: to develop high-performance plale and shell finile clements with unconventional techniques, in particular the Free Formulation and the Assumed Natural Strain method.

These approaches. although successful. did not fit the standard variational framework of canonical functionals. Efforts to "varationalize" success led to a number of discoveries discussed below. As of this writing the next logical step of this research is the orderly production of finile clement templates (detined below) as well as further development of a theoretical foundation that explains and justities their existence.

The application of templates to element-level error estimation also appears promising but will also require

a sounder theoretical basis.

A finite element template, or simply template. is a parametrized algebraic form that yields a continuum of convergent finite elements of fixed type. Here by "type" is meant an element selected for a specific application and with a given degree-of-freedom (dol) configuration; for example a 3-node. 9-dof Kirchhoff plate bending triangle. A form that yields all convergent elements of a given type is called an universal template. A from that yields a practically useful subset is called a generic template, where "generic" is used in the biological sense of "pertaining to a genus".

Obtaining an explicit universallemplate may be viewed as "closing the book" on an element type once and for all. For multidimensional elements. however, universal templates can become too complex or bak physical transparency. In such cases a generic template may represent a viable compromise.

These approaches. although successful. did not fit the standard variational framework of canonical functionals. Efforts to "varationalize" success led to a number of discoveries discussed below. As of this writing the next logical step of this research is the orderly production of finile clement templates (detined below) as well as further development of a theoretical foundation that explains and justities their existence.

The application of templates to element-level error estimation also appears promising but will also require

a sounder theoretical basis.

A finite element template, or simply template. is a parametrized algebraic form that yields a continuum of convergent finite elements of fixed type. Here by "type" is meant an element selected for a specific application and with a given degree-of-freedom (dol) configuration; for example a 3-node. 9-dof Kirchhoff plate bending triangle. A form that yields all convergent elements of a given type is called an universal template. A from that yields a practically useful subset is called a generic template, where "generic" is used in the biological sense of "pertaining to a genus".

Obtaining an explicit universallemplate may be viewed as "closing the book" on an element type once and for all. For multidimensional elements. however, universal templates can become too complex or bak physical transparency. In such cases a generic template may represent a viable compromise.

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Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**