There is no best method. Damage detection completly depends on your problem. If you monitor load paths at joints other methods are suitable than finding impact damages in an aircraft structure. The method you use defines your algorithm as well.

It depends on your problem. you need to specify your problem.

you can see this

https://www.kaggle.com/c/cee-498-project9-structural-damage-detection/code

It depends on your problem type. You need to do research for your specific problem.

Depends your view.

For high-rise buildings as well as beam-like system:

1. Eigen-problem based inverse solution.

Article Combination of an inverse solution and an ANN for damage ide...

2. Artificial neural networks.