Graph-Theoretical Based Algorithms for Structural Optimization

TL;DR

This paper introduces five new algorithms aimed at optimizing the conditioning of structural matrices to reduce analytical errors and improve computational efficiency in skeletal structure analysis.

Contribution

The study presents novel algorithms for improving matrix conditioning in structural analysis, focusing on minimizing errors and enhancing analysis accuracy.

Findings

Successfully optimized flexibility matrices with better conditioning.

Reduced analytical errors in skeletal structural analysis.

Improved computational efficiency in structural matrix analysis.

Abstract

Five new algorithms were proposed in order to optimize well conditioning of structural matrices. Along with decreasing the size and duration of analyses, minimizing analytical errors is a critical factor in the optimal computer analysis of skeletal structures. Appropriate matrices with a greater number of zeros (sparse), a well structure, and a well condition are advantageous for this objective. As a result, a problem of optimization with various goals will be addressed. This study seeks to minimize analytical errors such as rounding errors in skeletal structural flexibility matrixes via the use of more consistent and appropriate mathematical methods. These errors become more pronounced in particular designs with ill-suited flexibility matrixes; structures with varying stiffness are a frequent example of this. Due to the usage of weak elements, the flexibility matrix has a large number of non-diagonal terms, resulting in analytical errors. In numerical analysis, the ill-condition of a matrix may be resolved by moving or substituting rows; this study examined the definition and execution of these modifications prior to creating the flexibility matrix. Simple topological and algebraic features have been mostly utilized in this study to find fundamental cycle bases with particular characteristics. In conclusion, appropriately conditioned flexibility matrices are obtained, and analytical errors are reduced accordingly.