Group-theoretic exploitations of symmetry in computational solid and structural mechanics

• Published in: International journal for numerical methods in engineering Vol. 79; no. 3; pp. 253 - 289
• Main Author: Zingoni, A.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 16.07.2009; Wiley
• Subjects: bifurcation analysis; Computation; Computational techniques; Exact sciences and technology; finite element formulation; finite element methods; Fundamental areas of phenomenology (including applications); Group theory; Mathematical analysis; Mathematical methods in physics; Mathematical models; Physics; representation theory; Simplification; Solid mechanics; Strategy; Structural and continuum mechanics; structural mechanics; structures; Symmetry; symmetry group; Vibration analysis; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); vibrations; Solid mechanics; Branching; Simplification; Vibration; structural mechanics; Group theory; Bifurcation; finite element formulation; structures; bifurcation analysis; Modeling; Finite element method; Symmetry group; Vector space; finite element methods; symmetry; Representation theory; vibration analysis; vibrations; Structural analysis
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.2576

The use of group theory in simplifying the study of problems involving symmetry is a well‐established approach in various branches of physics and chemistry, and major applications in these areas date back more than 70 years. Within the engineering disciplines, the search for more systematic and more efficient strategies for exploiting symmetry in the computational problems of solid and structural mechanics has led to the development of group‐theoretic methods over the past 40 years. This paper reviews the advances made in the application of group theory in areas such as bifurcation analysis, vibration analysis and finite element analysis, and summarizes the various implementation procedures currently available. Illustrative examples of typical solution procedures are drawn from recent work of the author. It is shown how the group‐theoretic approach, through the characteristic vector‐space decomposition, enables considerable simplifications and reductions in computational effort to be achieved. In many cases, group‐theoretic considerations also allow valuable insights on the behaviour or properties of a system to be gained, before any actual calculations are carried out. Copyright © 2009 John Wiley & Sons, Ltd.