Efficient finite element solution of regular and near-regular systems using graph products

In this paper, an efficient finite element (FE) solution of regular and near-regular systems is presented using graph product rules. Such systems can be observed in an FE solution of differential equations or direct construction of the stiffness matrix in a structural/mechanical problem. Graph product rules are developed to decompose a regular pattern to its preliminary generating components, solving the problem using efficient divide and conquer approaches. Regularity is regarded as a specific condition mathematically; however, due to natural repetition of elements in FE models, regular/near-regular patterns are very common in these models. In this paper, graph product mathematical formulations for FE models are developed, and the efficiency of the method is demonstrated using computational complexity. Comprehensiveness of the method is shown using diverse complementary examples.

[1]  Thomas J. R. Hughes,et al.  A simple and efficient finite element for plate bending , 1977 .

[2]  T. Healey.  A group-theoretic approach to computational bifurcation problems with symmetry , 1988 .

[3]  R.J. Wenzel,et al.  Fast analysis and optimization of combline filters using FEM , 2001, 2001 IEEE MTT-S International Microwave Sympsoium Digest (Cat. No.01CH37157).

[4]  J. Reddy.  An introduction to the finite element method , 1989 .

[5]  Alphose Zingoni,et al.  Group‐theoretic exploitations of symmetry in computational solid and structural mechanics , 2009 .

[6]  D. Logan.  A First Course in the Finite Element Method , 2001 .

[7]  Claudio De Persis,et al.  Proceedings of the 41st IEEE Conference on Decision and Control , 2002 .

[8]  S. Pellegrino,et al.  AN INTRODUCTION TO THE ANALYSIS OF SYMMETRIC STRUCTURES , 1999 .

[9]  A. Kaveh,et al.  Analysis of structures convertible to repeated structures using graph products , 2013 .

[10]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[11]  Đorđe Zloković,et al.  Group supermatrices in finite element analysis , 1992 .

[12]  Ali Kaveh,et al.  Analysis of Irregular Structures Composed of Regular and Irregular Parts Using Graph Products , 2014, J. Comput. Civ. Eng..

[13]  Kazuo Murota,et al.  Bifurcation analysis of symmetric structures using block-diagonalization , 1991 .

[14]  A. Zingoni.  A group-theoretic finite-difference formulation for plate eigenvalue problems , 2012 .

[15]  Kazuo Murota,et al.  Computational Use of Group Theory in Bifurcation Analysis of Symmetric Structures , 1991, SIAM J. Sci. Comput..

[16]  T. Healey,et al.  Exact block diagonalization of large eigenvalue problems for structures with symmetry , 1991 .

[17]  Alphose Zingoni,et al.  Symmetry recognition in group-theoretic computational schemes for complex structural systems , 2012 .

[18]  Y. K. Cheung,et al.  Exact Analysis of Structures With Periodicity Using U-Transformation , 1998 .

[19]  A. Kaveh,et al.  EFFICIENT ANALYSIS OF STRUCTURES HOLDING TRIDIAGONAL AND BLOCK TRI-DIAGONAL STIFFNESS MATRICES, GENERALIZING THE METHOD TO OTHER STRUCTURES USING HOUSEHOLDER AND BLOCK HOUSEHOLDER TRANSFORMATION , 2014 .

[20]  M. Solal,et al.  A new efficient combined FEM and periodic Green's function formalism for the analysis of periodic SAW structures , 1995, 1995 IEEE Ultrasonics Symposium. Proceedings. An International Symposium.

[21]  Alphose Zingoni,et al.  A group-theoretic formulation for symmetric finite elements , 2005 .

[22]  D. L. Thomas.  Dynamics of rotationally periodic structures , 1979 .

[23]  A. Kaveh,et al.  New developments in the optimal analysis of regular and near-regular structures: decomposition, graph products, force method , 2015 .

[24]  S. Dinkevich.  Finite symmetric systems and their analysis , 1991 .

[25]  Ali Kaveh,et al.  Optimal Analysis of Structures by Concepts of Symmetry and Regularity , 2013 .

[26]  M.A. Hasan,et al.  Block eigenvalue decomposition using nth roots of the identity matrix , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[27]  Luca Lucchese,et al.  Frequency domain classification of cyclic and dihedral symmetries of finite 2-D patterns , 2004, Pattern Recognit..

[28]  Y. Kato,et al.  Biomechanical analysis of the spinal cord in Brown-Séquard syndrome , 2013 .

[29]  H. L. Zou,et al.  Skewed rotational symmetry detection from a 2D line drawing of a 3D polyhedral object , 2006, Comput. Aided Des..

[30]  Alain Combescure,et al.  Efficient FEM calculation with predefined precision through automatic grid refinement , 2005 .

[31]  I. Ario,et al.  Block-diagonalization analysis of symmetric plates , 1992 .