### Abstract

The conventional finite element method is difficult to converge for a non-positive definite stiffness matrix, which usually occurs when the material displays softening behavior or when the system is near the state of bifurcation. We have developed two stable algorithms for a non-positive definite stiffness matrix, one for the direct linear equation solver and the other for the iterative solver in the finite element method for minimization problems. For a direct solver with non-positive definite stiffness matrix, energy minimization of a system with multiple degrees of freedom (DOF) is decomposed to the minimization of many 1-DOF systems, and for the latter an efficient and robust minimization method is developed to ensure that the system energy decreases in every incremental step, regardless of the positive definiteness of the stiffness matrix. For an iterative solver, the stiffness matrix is modified to ensure the convergence, and the modified stiffness matrix indeed leads to the correct solution. An example of a single wall carbon nanotube under compression is studied via the proposed algorithms.

Original language | English (US) |
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Pages (from-to) | 1251-1254 |

Number of pages | 4 |

Journal | Journal of Computational and Theoretical Nanoscience |

Volume | 5 |

Issue number | 7 |

State | Published - Jul 2008 |

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### Keywords

- Atomic-scale finite element method
- Direct solver
- Iterative solver
- Minimization

### ASJC Scopus subject areas

- Chemistry(all)
- Materials Science(all)
- Condensed Matter Physics
- Computational Mathematics
- Electrical and Electronic Engineering

### Cite this

*Journal of Computational and Theoretical Nanoscience*,

*5*(7), 1251-1254.