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An eigen-based theory for developing numerical methods for structural dynamics
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DOI：10.1007/s11803-025-2313-9.png)
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Abstract
En 中文
The feasibility of using a problem-dependent method to solve systems of second order ODEs is corroborated by an eigen-based theory and a methodology to develop such a numerical method is constructed. The key steps of this methodology are to decouple a system of ODEs of second order into a set of uncoupled ODEs of second order; next, an eigen-dependent method is proposed to approximate the solution of each uncoupled ODE of second order. It is vital to transform all eigen-dependent methods to a problem-dependent method to bypass an Eigen analysis. The development of an eigen-dependent method plays a key role in this methodology so that slow eigenmodes can be accurately integrated while there is no instability or excessive amplitude growth in fast eigenmodes. This can explain why a problem-dependent method can simultaneously combine the explicitness of each step and A-stability. Consequently, huge computational efforts can be saved for solving nonlinear stiff problems. A new family of problem-dependent methods is developed in this work so that the feasibility of the proposed methodology can be affirmed. It has almost the same performance as that of the HHT-alpha method. However, it can save more than 99.5% of CPU demand in approximating a solution for a system of 1000 nonlinear second order ODEs.
Keywords:
an eigen-based theory
unconditional stability
accuracy
eigen-dependent method
structure-dependent method
nonlinear dynamics
Journal
IF:
3.3
Papers:
115
Citations:
3.0K