[Paper Review] New Nonconforming Elements for Linear Strain Gradient Elastic Model

Hongliang Li, Pingbing Ming+1 more|arXiv (Cornell University)|Sep 4, 2018

Nonlocal and gradient elasticity in micro/nano structures19 references4 citations

TL;DR

This paper proposes new nonconforming finite elements for the linear strain gradient elasticity model, leveraging a novel H²-Korn's inequality that eliminates gradient jump terms in the discrete formulation. The method enables robust, uniformly convergent approximations with optimal convergence rates using simpler elements—such as tensor product NTW and Specht triangles, and a modified Morley element—demonstrating optimal half-order convergence in numerical tests.

ABSTRACT

Based on a new H$^2-$Korn's inequality, we propose new nonconforming elements for the linear strain gradient elastic model. The first group of elements are H$^1-$conforming but H$^2-$nonconforming. The tensor product NTW element [Tai:2001] and the tensor product Specht triangle are two typical representatives. The second element is based on Morley's triangle with a modified elastic strain energy. We proved new interpolation error estimates for all these elements, which are key to prove uniform rates of convergence for the proposed elements. Numerical results are reported and they are consistent with the theoretical prediction.

Motivation & Objective

• To develop simpler, robust nonconforming finite elements for the linear strain gradient elasticity model with uniform convergence across small material parameters.
• To overcome the high degrees of freedom and complexity of C¹ conforming elements by using H¹-conforming but H²-nonconforming elements.
• To establish a new discrete H²-Korn's inequality that excludes gradient jump terms, simplifying element construction and enabling optimal convergence.
• To prove new interpolation error estimates that ensure uniform convergence rates independent of the microscopic material parameter.
• To validate the theoretical findings with numerical experiments on smooth and boundary layer solutions.

Proposed method

• Propose a new H²-Korn's inequality and its discrete analog that exclude gradient jump terms, simplifying the construction of nonconforming elements.
• Construct two families of elements: tensor product of NTW and Specht triangles (9 degrees of freedom, quartic polynomials), and a modified Morley triangle with adjusted elastic strain energy.
• Use the discrete H²-Korn's inequality to prove uniform convergence in the energy norm with respect to the small material parameter ι.
• Derive new interpolation error estimates for all proposed elements, essential for proving optimal convergence rates.
• Implement and test the elements on a model problem with varying mesh sizes and ι = 10⁻⁶ to confirm theoretical convergence rates.
• Utilize the discrete H²-Korn's inequality as a foundation for potential C⁰ penalty methods and future 3D extensions.

Experimental results

Research questions

• RQ1 Can a new discrete H²-Korn's inequality be derived that excludes gradient jump terms, enabling simpler nonconforming element construction for strain gradient elasticity?
• RQ2 Do the proposed nonconforming elements achieve uniform convergence with respect to the small material parameter ι in the energy norm?
• RQ3 Can the modified Morley triangle element converge uniformly when the elastic strain energy is properly adjusted, despite its failure in the standard form?
• RQ4 What are the optimal convergence rates of the proposed elements, and do they match theoretical predictions in numerical experiments?
• RQ5 Can the new H²-Korn's inequality be extended to three-dimensional problems and C⁰ penalty methods for strain gradient models?

Key findings

• The proposed NTW and Specht triangle elements achieve approximately half-order convergence rates (≈0.5) in the energy norm, consistent with theoretical predictions.
• The modified Morley element achieves a convergence rate of approximately 0.6 for the first mesh refinement, approaching the theoretical half-order rate.
• Numerical results confirm uniform convergence with respect to the small material parameter ι = 10⁻⁶ across all tested elements.
• The new discrete H²-Korn's inequality successfully eliminates the need for gradient degrees of freedom along edges, simplifying element construction.
• The interpolation error estimates derived are sharp and directly support the uniform convergence proof for all proposed elements.
• The method enables robust, optimal convergence using simpler elements compared to previous C¹-conforming or high-degree H²-nonconforming approaches.