Quadrature Element Methods in Structural Vibration Analysis

Quadrature Element Methods (QEM) integrate variational principles with high‐order numerical quadrature to solve vibration eigenproblems with enhanced accuracy and efficiency. In these formulations, the weak form of the governing differential equations is discretised using element shape functions and evaluated at optimally chosen quadrature points. This approach permits the direct incorporation of higher‐order derivatives—key for models such as Timoshenko and Euler–Bernoulli beam theories, strain‐gradient elasticity and nonlocal continuum theories—while retaining a compact system size. QEM variants, including the weak form quadrature element method (WQEM) and locally adaptive QEM, offer rapid convergence even on coarse meshes and can accommodate geometric nonlinearity, variable material grading and higher‐order boundary conditions. By reducing numerical dispersion, improving mass‐lumping strategies and enabling superconvergent frequency predictions, QEM has become a versatile tool in structural dynamics, from nanoscale beam resonators to large‐scale aerospace panels.

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Recent studies have extended WQEM to nonlocal and nonlinear graded nanobeams, demonstrating that high-order derivative terms derived from Hamilton’s principle can be efficiently discretised to yield free and forced vibration responses matching differential quadrature benchmarks at reduced computational cost. A locally adaptive weak QEM formulation has been proposed for strain‐gradient Timoshenko and Euler–Bernoulli nanobeams, employing Gauss quadrature to integrate variational matrices and allowing straightforward inclusion of extra derivative degrees of freedom without reformulating the elemental equations. This adaptive method exhibits improved convergence speed and accuracy for both linear and nonlinear frequency predictions under diverse boundary conditions. Further developments in quadrature‐based discretisations have achieved coarse‐mesh superconvergence for shear‐deformable plates and beams, reducing locking phenomena and attaining sixth‐order frequency accuracy when reduced integration is applied to isogeometric splines.

Quadrature Element Methods in Structural Vibration Analysis publication trend

The graph below shows the total number of articles in quadrature element methods in structural vibration analysis across all publications each year (not limited to Nature Index journals).

Technical terms

Quadrature element method (QEM): A numerical scheme that uses weighted integration points to discretise the weak form of governing equations, enabling high-order accuracy with fewer degrees of freedom.

Weak form quadrature element method (WQEM): A QEM variant where the variational statement is directly discretised via quadrature, accommodating higher-order derivatives and nonlinearity.

Timoshenko beam theory: A structural model accounting for both bending and shear deformations, suitable for thick beams at high frequencies.

Strain‐gradient elasticity: A continuum model incorporating higher-order spatial derivatives of strain to capture size-dependent effects at small scales.

Numerical dispersion: Artificial alteration of wave speeds in discrete models, which can be minimised by optimising quadrature weights or element formulations.

References

1. A high-order FEM formulation for free and forced vibration analysis of a nonlocal nonlinear graded Timoshenko nanobeam based on the weak form quadrature element method. Archive of Applied Mechanics (2020).
2. A novel formulation for the weak quadrature element method for solving vibration of strain gradient graded nonlinear nanobeams. Acta Mechanica (2022).
3. Coarse Mesh Superconvergence in Isogeometric Frequency Analysis of Mindlin–Reissner Plates with Reduced Integration and Quadratic Splines. Acta Mechanica Solida Sinica (2022).
4. Optimal reduction of numerical dispersion for wave propagation problems. Part 2: Application to 2-D isogeometric elements. Computer Methods in Applied Mechanics and Engineering (2017).
5. A mathematical theory for mass lumping and its generalization with applications to isogeometric analysis. Computer Methods in Applied Mechanics and Engineering (2023).