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In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
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    Area of Science:

    • Computational mechanics
    • Numerical analysis
    • Finite element methods

    Background:

    • Maintaining geometric validity in 3D high-order finite elements is crucial for simulation accuracy.
    • Existing methods for calculating maximum allowable step sizes can be computationally intensive and prone to numerical issues.

    Purpose of the Study:

    • To develop a robust and efficient algorithm for computing maximum allowable step sizes for 3D high-order finite elements.
    • To ensure continuous geometric validity of elements during motion along specified directions.

    Main Methods:

    • Transforming the problem of finding maximum step sizes into solving cubic polynomial roots.
    • Employing interval arithmetic to circumvent numerical challenges in cubic equation solving.
    • Applying the interval version of the Newton-Raphson iteration after completely enumerating cubic polynomial roots.

    Main Results:

    • The proposed algorithm effectively computes maximum allowable step sizes, preserving geometric validity.
    • Extensive testing demonstrates the algorithm's robustness and effectiveness.
    • The novel method achieves higher computational efficiency compared to the current state-of-the-art.

    Conclusions:

    • The new method provides an efficient and reliable solution for determining maximum step sizes in 3D finite element analysis.
    • The use of interval arithmetic and cubic polynomial root finding enhances numerical stability and performance.
    • This advancement contributes to more accurate and faster simulations in fields utilizing high-order finite elements.