Gröbner bases plugged into graphical skills to solve a set of multiple bifurcation equations in structural compound stability problems

• Published in: International journal for numerical methods in engineering Vol. 123; no. 23; pp. 5779 - 5800
• Main Authors: Tanaka, Masato, Matsubara, Seishiro, Schröder, Jörg, Fujii, Fumio
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 15.12.2022; Wiley Subscription Services, Inc
• Subjects: bifurcation equations; Bifurcations; compound stability; Eigenvalues; graphical solution; Gröbner bases; multiple bifurcation; Polynomials; rank deficiency two; Simultaneous equations; Skills; Stiffness matrix; Structural stability
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.7088

A core issue in structural multiple bifurcations (MB) in computational engineering is to identify all existing branching paths emanating from the MB point in compound stability problems. The governing MB equations (MBEs) will commonly result in a set of three (or occasionally two) polynomial equations in asymptotic stability theory when the singular stiffness matrix is subject to a rank deficiency of two (i.e., two null eigenvalues). However, no general solution strategy has been established to solve MBEs so far. This study proposes innovative graphical solution ideas to intuitively visualize multiple path branching in 2D‐ and 3D‐spaces of variables. Although the graphical skills display real solutions in specified search areas on a graphical monitor, it is not assured that “all” real roots are detected. The total number of identified real and complex roots of simultaneous equations must be generally consistent with that predicted algebraically to ensure that all real and complex roots are captured in MB. In computational algebra, Gröbner bases are employed to convert a set of polynomial equations into single recursively solvable equations and can be plugged into visualization steps. Therefore, Gröbner bases and graphical skills are complementary and can be applied to numerically solve a set of plate/shell structural MBEs.