[Paper Review] Novel differential quadrature element method for higher order strain gradient elasticity theory

Md. Ishaquddin, S. Gopalakrishnan|arXiv (Cornell University)|Feb 11, 2018

Nonlocal and gradient elasticity in micro/nano structures39 references3 citations

TL;DR

This paper proposes a novel differential quadrature element method based on Lagrange interpolation to solve sixth-order partial differential equations in strain gradient elasticity theory for beams and plates. By treating displacement as the sole degree of freedom in the domain and incorporating slope and curvature at boundaries via modified weighting coefficients, the method efficiently handles non-classical boundary conditions, demonstrating high accuracy in static analysis of gradient elastic beams and plates across various boundary conditions.

ABSTRACT

In this paper, we propose a novel and efficient differential quadrature element based on Lagrange interpolation to solve a sixth order partial differential equations encountered in non-classical beam theories. These non-classical theories render displacement, slope and curvature as degrees of freedom for an Euler-Bernoulli beam. A generalize scheme is presented herein to implementation the multi-degrees degrees of freedom associated with these non-classical theories in a simplified and efficient way. The proposed element has displacement as the only degree of freedom in the domain, whereas, at the boundaries it has displacement, slope and curvature. Further, we extend this methodology and formulate two novel versions of plate element for gradient elasticity theory. In the first version, Lagrange interpolation is assumed in $x$ and $y$ directions and the second version is based on mixed interpolation, with Lagrange interpolation in $x$ direction and Hermite interpolation in $y$ direction. The procedure to compute the modified weighting coefficients by incorporating the classical and non-classical boundary conditions is explained. The efficiency of the proposed elements is demonstrated through numerical examples on static analysis of gradient elastic beams and plates for different boundary conditions.

Motivation & Objective

• To address the challenge of solving sixth-order partial differential equations arising in non-classical beam and plate theories based on strain gradient elasticity.
• To develop a simplified and efficient implementation of multi-degree-of-freedom systems (displacement, slope, curvature) in higher-order continuum theories.
• To extend the differential quadrature element method (DQEM) to gradient elasticity by introducing a new formulation using Lagrange interpolation and mixed interpolation.
• To provide a generalized framework for imposing classical and non-classical boundary conditions through modified weighting coefficients.
• To validate the accuracy and efficiency of the proposed elements through comprehensive numerical benchmarks on beams and plates under various loading and support conditions.

Proposed method

• A novel differential quadrature beam element is formulated using Lagrange interpolation, with displacement as the only degree of freedom in the domain and displacement, slope, and curvature as degrees of freedom at the boundaries.
• Two new plate element formulations are developed: one using Lagrange interpolation in both x and y directions (SgDQE-LL), and another using mixed interpolation—Lagrange in x and Hermite in y (SgDQE-LH)—to ensure C² continuity.
• A generalized procedure is introduced to compute modified weighting coefficients that incorporate both classical and non-classical boundary conditions directly into the element formulation.
• The method leverages the differential quadrature technique to discretize the governing sixth-order partial differential equations, enabling high-accuracy numerical solutions.
• The Dirac-delta function is employed to model point loads accurately in the numerical examples.
• The formulation is validated through static analysis of gradient elastic beams and plates under uniformly distributed and point loads, with results compared against analytical and existing numerical solutions.

Experimental results

Research questions

• RQ1 How can a differential quadrature element be efficiently formulated to solve sixth-order partial differential equations in strain gradient elasticity theory for beams and plates?
• RQ2 What is the most effective interpolation strategy to handle multiple degrees of freedom (displacement, slope, curvature) in higher-order beam and plate models?
• RQ3 How can classical and non-classical boundary conditions be consistently and accurately incorporated into the weighting coefficient computation of the DQEM framework?
• RQ4 To what extent does the proposed method maintain accuracy and convergence for gradient elastic structures under various boundary conditions and scale parameters?
• RQ5 How do the SgDQE-LL and SgDQE-LH formulations compare in performance and accuracy for static analysis of gradient elastic beams and plates?

Key findings

• The proposed SgDQE-LL and SgDQE-LH elements achieve excellent agreement with analytical solutions for gradient plates under uniformly distributed load, with relative errors below 0.1% for deflection and curvature at g/lx = 0.00001.
• For central point loading, the SgDQE-LL and SgDQE-LH results for CFCF, SFSF, and SFCF plates show close agreement with classical solutions at g/lx = 0.00001 and with each other at higher scale parameters.
• In cylindrical bending, the maximum deflection results from both elements match analytical solutions within 1% error across all g/lx values, including g/lx = 0.5.
• The SgDQE-LH element shows slightly better accuracy than SgDQE-LL for higher-order moments and deflections under CFFF and SSSS conditions, particularly at g/lx = 0.5.
• The method successfully captures scale effects, with deflections decreasing significantly as g/lx increases from 0.00001 to 0.5, consistent with theoretical expectations of gradient elasticity.
• The modified weighting coefficient approach enables accurate imposition of non-classical boundary conditions without requiring additional degrees of freedom, enhancing computational efficiency.