A matrix function solution for the scaled boundary finite-element equation in statics
By Chongmin Song. 2004
189 citations recorded by the source corpus.
The scaled boundary finite-element method is a fundamental-solution-less boundary element method based on finite elements. It leads to semi-analytical solutions for displacement and stress fields, which permits problems with singularities or in infinite domains to be handled conveniently and accurately. However, the present eigenvalue method for solving the scaled boundary finite-element equation requires additional treatments for multiple eigenvalues with parallel eigenvectors, which results from logarithmic terms in the solutions. A matrix function solution for the scaled boundary finite-element equation in statics is presented in this paper. It is numerically stable for multiple or near-multiple eigenvalues as it is based on real Schur decomposition. Power functions, logarithmic functions and their transitions as occurring in fracture mechanics, composites and two-dimensional unbounded domains, are represented semi-analytically. No a priori knowledge on the types and orders of singularity are required when simulating stress singularities.
Subfields: Civil and Structural Engineering, Mechanics of Materials
View Paper: https://www.semanticscholar.org/paper/b0564830b3b0ca15c5fed6eb782cc273e51ee2ef
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In Civil and Structural Engineering
- E. Ooi
- Sundararajan Natarajan
- Francis Tin-Loi
- Chongmin Song
- A. Deeks
- I. Chiong
- T. Vu
- Stéphane P.A. Bordas
In Mechanics of Materials
- E. Ooi
- Chongmin Song
- Francis Tin-Loi
- Sundararajan Natarajan
- I. Chiong
- T. Vu
- A. Deeks
- Stéphane P.A. Bordas
Citations in the corpus
- A review of the scaled boundary finite element method for two-dimensional linear elastic fracture mechanics (2017, methodology): “Equation (7) is decoupled by the block-diagonal Schur decomposition [88] of the matrix [Z] [Z][Ψ] = [Ψ][S] (9) where [Ψ] is a transformation matrix with independent column vectors.”
- A p‐hierarchical adaptive procedure for the scaled boundary finite element method (2002, methodology): “Song [2] addresses the efficiency issue by reformulating the problem in terms of orthogonal deformation modes, then computing and employing a reduced set of these modes, those corresponding to the smoothest deformation of the boundary.”
- A scaled boundary polygon formulation for elasto-plastic analyses (2014, background): “Φ(ξ, η) can reproduce all rigid body modes and constant strain terms for uncracked polygons [69, 65].”
- Scaled boundary polygons with application to fracture analysis of functionally graded materials (2014, methodology): “E0, E1, and E2 are coefficient matrices that are related to the geometry and material properties in each polygon [33]”
- Crack propagation modelling in functionally graded materials using scaled boundary polygons (2015, methodology): “Sn and Ψ u are the block diagonal Schur matrix and the corresponding transformation matrix related to the displacement modes, respectively, and are obtained from a block-diagonal Schur decomposition of the Hamiltonian matrix Z (Song 2004)”