Manifold-based mass lumping addressing the inertia representation for rotational/torsional degrees of freedom in Kirchhoff plate vibration analysis

Abstract

The vibration analysis of Kirchhoff plates requires robust mass lumping schemes to guarantee numerical stability and accuracy. However, existing methods fail to generate symmetric and positive definite mass matrices when handling rotational degrees of freedom, leading to compromised performance in both time and frequency domains analyses. This study proposes a manifold-based mass lumping scheme that systematically resolves the inertia matrix formulas for rotational/torsional degrees of freedom. By reinterpreting the finite element mesh as a mathematical cover composed of overlapping patches, Hermitian interpolations for plate deflection are derived using partition of unity principles. The manifold-based mass matrix is constructed by integrating the virtual work of inertia forces over these patches, ensuring symmetry and positive definiteness. Numerical benchmarks demonstrate that the manifold-based mass lumping scheme performance can be comparable or better than the consistent mass scheme and other existing mass lumping schemes. This work establishes a unified theory for mass lumping in fourth order plate dynamics, proving that the widely used row-sum method is a special case of the manifold-based framework. The scheme resolves long-standing limitations in rotational/torsional inertia conservation and provides a foundation for extending rigorous mass lumping to 3D shell and nonlinear dynamic analyses.

Graphical abstract

Keywords

Kirchhoff plates / manifold-based mass lumping / symmetric positive definite matrices / rotational/torsional inertia conservation / partition of unity / finite element vibration analysis

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