Locking-free Hybrid High-Order Method for Efficient Linear Elasticity Modeling

Core Concepts

The authors propose a locking-free hybrid high-order (HHO) method for the efficient numerical modeling of linear elasticity problems. The method utilizes a single reconstruction operator for the linear Green strain, avoiding the need for a split in deviatoric and spherical behavior. The a priori error analysis provides quasi-best approximation results with parameter-independent equivalence constants, and the a posteriori error estimates are stabilization-free and robust to the critical Lamé parameter.

Abstract

The content presents a hybrid high-order (HHO) method for the numerical modeling of linear elasticity problems. The key highlights and insights are: Motivation: The HHO method has been successfully applied to linear elasticity, but the classical approach requires a split of the reconstruction terms, which may be motivated by the Stiff Stokes equations. This paper proposes a simpler HHO method that uses a single reconstruction operator for the linear Green strain. Discrete Formulation: The HHO method seeks the discrete solution uh in the ansatz space Vh, which consists of piecewise polynomial functions on the mesh. The method utilizes reconstruction operators to define the discrete bilinear form ah(uh, vh) and the discrete stress σh = Cεhuh. A Priori Error Analysis: The authors establish a quasi-best approximation result (1.3) for the error in the stress σ - σh, the energy norm ∥Iu - uh∥ah, and the stabilization seminorm |uh|s. The analysis relies on a right-inverse operator, a quasi-best approximation result for the stabilization, and a tr-dev-div lemma that provides λ-robust estimates. A Posteriori Error Analysis: Leveraging the quasi-best approximation result, the authors derive a stabilization-free a posteriori error estimator η that is reliable and efficient, with constants that are independent of the Lamé parameter λ. Numerical Benchmarks: The computational results provide empirical evidence for the optimal convergence rates of the a posteriori error estimator in adaptive mesh-refining algorithms, even in the incompressible limit.

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Locking-free hybrid high-order method for linear elasticity

by Carsten Cars... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2404.02768.pdf

Deeper Inquiries

How can the proposed HHO method be extended to nonlinear elasticity problems, and what are the potential challenges in the analysis and implementation

The proposed HHO method can be extended to nonlinear elasticity problems by incorporating nonlinear terms in the governing equations, such as nonlinear stress-strain relationships. This extension would involve modifying the reconstruction operators and stabilization techniques to handle the nonlinearities in the problem. One potential challenge in the analysis and implementation of the nonlinear HHO method is the increased complexity introduced by the nonlinear terms, which may require more sophisticated numerical techniques and error analysis. Additionally, ensuring stability and convergence of the method for nonlinear problems can be more challenging than in the linear case.

What are the limitations of the current analysis, and how can it be further generalized to handle more complex geometries, boundary conditions, or material models

One limitation of the current analysis is the restriction to simplicial meshes, which limits the applicability of the method to more general geometries. To generalize the analysis for more complex geometries, boundary conditions, or material models, the method could be extended to handle polytopal meshes or curved elements. This would require adapting the reconstruction and stabilization techniques to work effectively on these types of meshes. Additionally, incorporating more realistic boundary conditions, such as contact or frictional conditions, would require further development of the method to accurately capture these effects. Furthermore, extending the analysis to handle nonlinear material models would involve considering more complex constitutive laws and their impact on the discretization.

Can the insights from this work on locking-free and λ-robust discretizations be applied to other problems in computational mechanics, such as mixed formulations or multiphysics couplings

The insights from this work on locking-free and λ-robust discretizations can be applied to other problems in computational mechanics, such as mixed formulations or multiphysics couplings. For mixed formulations, the stabilization techniques developed for the HHO method could be adapted to handle mixed problems involving multiple unknowns, such as in fluid-structure interaction or porous media flow. The λ-robustness of the method could also be beneficial in handling problems with highly varying material properties or boundary conditions. Overall, the principles of stabilization-free discretization and robust error estimates can be valuable in a wide range of computational mechanics problems beyond linear elasticity.