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<p>This study aims to develop a novel DQM matrix formulation for 2D non-tensor product basis Hermite polynomials without a mixed second derivative at the corners. This formulation can be used to solve fourth-order systems present in the nanomechanics field and thin plate problems. The proposed formulation is based on a new paradigm of building a 2D Hermite basis using only regular 1D Lagrange polynomials introduced in Part I of this study. This simplifies the implementation of the present formulation as it relies only on the well-documented Lagrange-based DQM. Another critical advantage of the proposed formulation is that, unlike classical implementations, the proposed formulation matches the required physical degrees of freedom while giving access to the analytical expression of the shape functions. To accomplish this, purpose-built transfer matrices between several 2D polynomial bases are developed. To assess the accuracy of the proposed formulation, these new DQM matrices are used to build a strong and a weak Quadrature Element Method formulation for 2D fourth-order systems based on a Gauss–Lobatto–Legendre grid. This ensures a faster convergence for SQEM and provides WQEM formulation with a diagonal mass matrix. A diagonal mass matrix is a significant numerical advantage for WQEM despite the reduced accuracy of Gauss–Lobatto–Legendre integration. An LaDQM formulation is also proposed using a grid without outer corner points. A convergence study is performed for all proposed methods. The accuracy of the proposed methods was evaluated and validated using results from the literature. It is noted here that despite requiring higher mesh density for the highly skew cases the proposed Gauss–Lobatto–Legendre-based WQEM requires less computational power to compute the natural frequency thanks to its diagonal mass matrix. Like most DQ-based methods, the application of the aforementioned formulations to nonlinear systems is of the essence, as DQ-based methods typically demonstrate high accuracy and efficient convergence.</p>

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A new paradigm for deriving the higher-order 2D Hermite polynomial basis: Part II - DQM matrix formulation for the LaDQM and SQEM and WQEM GLL for some fourth-order systems

  • 期刊论文

  • 发表日期：

    2025年10月13日

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  • Mohamed Trabelssi

摘要

This study aims to develop a novel DQM matrix formulation for 2D non-tensor product basis Hermite polynomials without a mixed second derivative at the corners. This formulation can be used to solve fourth-order systems present in the nanomechanics field and thin plate problems. The proposed formulation is based on a new paradigm of building a 2D Hermite basis using only regular 1D Lagrange polynomials introduced in Part I of this study. This simplifies the implementation of the present formulation as it relies only on the well-documented Lagrange-based DQM. Another critical advantage of the proposed formulation is that, unlike classical implementations, the proposed formulation matches the required physical degrees of freedom while giving access to the analytical expression of the shape functions. To accomplish this, purpose-built transfer matrices between several 2D polynomial bases are developed. To assess the accuracy of the proposed formulation, these new DQM matrices are used to build a strong and a weak Quadrature Element Method formulation for 2D fourth-order systems based on a Gauss–Lobatto–Legendre grid. This ensures a faster convergence for SQEM and provides WQEM formulation with a diagonal mass matrix. A diagonal mass matrix is a significant numerical advantage for WQEM despite the reduced accuracy of Gauss–Lobatto–Legendre integration. An LaDQM formulation is also proposed using a grid without outer corner points. A convergence study is performed for all proposed methods. The accuracy of the proposed methods was evaluated and validated using results from the literature. It is noted here that despite requiring higher mesh density for the highly skew cases the proposed Gauss–Lobatto–Legendre-based WQEM requires less computational power to compute the natural frequency thanks to its diagonal mass matrix. Like most DQ-based methods, the application of the aforementioned formulations to nonlinear systems is of the essence, as DQ-based methods typically demonstrate high accuracy and efficient convergence.