## Abstract

A well-posed problem has three main criteria; that a solution exists, is unique, and any small deviation in the input will not result in any arbitrarily large deviation in the original solution. If the criteria are not met, the problem would be considered ill-posed, with a numerical differential being a classic example. When using a numerical differential, large deviations are a result of the numerical instability within the solution, which can mostly be avoided by choosing a superior algorithm. However, mathematical instability is a result of having an ill-posed problem, which requires changing the parameters of the problem to meet the well-posed criteria. Damage detection using the modal-curvature algorithm demonstrates the well- and ill-posed conditions when comparing the methods of measuring curvature through in-plane (bending) strain and out-of-plane deflection, respectfully.

The mathematical instability was examined through a finite element analysis (FEA). The model consisted of a two-dimensional cantilever beam-like structure that was modelled with and without damage. The curvature profiles were then extracted from the beam using the node values of both the out-of-plane deflection and in-plane strain. The damage detection was then computed by performing a double numerical differential of the deflection to obtain the beam curvature profile and then subtracting the difference from the damaged and undamaged state. These values will be compared against an alternative method of using the difference of in-plane (blending) strain, which is assumed to have significantly less numerical noise within the damage detection algorithm.

For various damage sizes, the ill-pose of the modal-curvature algorithm was investigated. The main finding shows that the modal-curvature damage detection method is mathematically unstable when using a numerical differential. Alternatively, using a well-posed parameter such as in-plane strain has resulted in the stability of the algorithm.

The mathematical instability was examined through a finite element analysis (FEA). The model consisted of a two-dimensional cantilever beam-like structure that was modelled with and without damage. The curvature profiles were then extracted from the beam using the node values of both the out-of-plane deflection and in-plane strain. The damage detection was then computed by performing a double numerical differential of the deflection to obtain the beam curvature profile and then subtracting the difference from the damaged and undamaged state. These values will be compared against an alternative method of using the difference of in-plane (blending) strain, which is assumed to have significantly less numerical noise within the damage detection algorithm.

For various damage sizes, the ill-pose of the modal-curvature algorithm was investigated. The main finding shows that the modal-curvature damage detection method is mathematically unstable when using a numerical differential. Alternatively, using a well-posed parameter such as in-plane strain has resulted in the stability of the algorithm.

Original language | English |
---|---|

Number of pages | 12 |

Publication status | Published - 2021 |

Event | 10th Australasian Congress on Applied Mechanics - Held online Duration: 29 Nov 2021 → 1 Dec 2021 Conference number: 10th https://acamconference.com.au/ |

### Conference

Conference | 10th Australasian Congress on Applied Mechanics |
---|---|

Abbreviated title | ACAM10 |

Period | 29/11/21 → 1/12/21 |

Other | ACAM provides an international forum for delegates to share their experiences, present their research on the wide-ranging topics in applied mechanics. |

Internet address |

## Keywords

- damage detection
- finite element analysis
- finite difference method
- modal curvature
- structural health monitoring