Robust and Efficient Preservation of High-order Continuous Geometric Validity

We propose a novel method to robustly and efficiently compute the maximum allowable step sizes so that the 3D high-order finite elements continuously preserve geometric validity when moving along the given directions with positive step sizes smaller than the computed ones. We transform the problem of finding the maximum allowable step sizes to one of solving roots of cubic polynomials. To use interval arithmetic to avoid numerical issues in cubic equation solving, we completely enumerate the roots of cubic polynomials and apply the interval version of the Newton-Raphson iteration. The effectiveness of our algorithm is demonstrated through extensive testing. Compared to the state-of-the-art method, our algorithm achieves higher efficiency.