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Virtual Element Method Applied to Two Dimensional Axisymmetric Elastic Problems

Paulo Akira F. Enabe, Rodrigo Provasi

TL;DR

This work introduces a Virtual Element Method formulation for two-dimensional axisymmetric linear elasticity, reducing 3D axisymmetric problems to a meridional plane while incorporating the radial weight $r$ in all variational terms. A boundary-based projection operator $oldsymbol{ abla}$ is formulated to map virtual displacements to constant strain fields, augmented by a volumetric correction to capture hoop–radial coupling, and a stabilization term that acts on the non-polynomial space via boundary integrals. An a priori error analysis is developed in weighted Sobolev spaces, establishing interpolation and approximation bounds with norm equivalences and trace inequalities adapted to axisymmetry, and proving optimal convergence rates. Numerical patch tests on polygonal meshes validate the method's accuracy and robustness, highlighting excellent performance for axial and shear states while revealing intrinsic coupling challenges in hoop-related components due to axisymmetric geometry. Overall, the framework provides a rigorous, flexible, and scalable approach for axisymmetric elasticity on general polygonal meshes, with clear directions for enhancements and extensions to more complex material behavior.

Abstract

This work presents a Virtual Element Method (VEM) formulation tailored for two-dimensional axisymmetric problems in linear elasticity. By exploiting the rotational symmetry of the geometry and loading conditions, the problem is reduced to a meridional cross-section, where all fields depend only on the radial and axial coordinates. The method incorporates the radial weight $r$ in both the weak formulation and the interpolation estimates to remain consistent with the physical volume measure of cylindrical coordinates. A projection operator onto constant strain fields is constructed via boundary integrals, and a volumetric correction term is introduced to account for the divergence of the stress field arising from axisymmetry. The stabilization term is designed to act only on the kernel of the projection and is implemented using a boundary-based formulation that guarantees stability without affecting polynomial consistency. Furthermore, an a priori interpolation error estimate is established in a weighted Sobolev space, showing optimal convergence rates. The implementation is validated through patch tests that demonstrate the accuracy, consistency, and robustness of the proposed approach.

Virtual Element Method Applied to Two Dimensional Axisymmetric Elastic Problems

TL;DR

This work introduces a Virtual Element Method formulation for two-dimensional axisymmetric linear elasticity, reducing 3D axisymmetric problems to a meridional plane while incorporating the radial weight in all variational terms. A boundary-based projection operator is formulated to map virtual displacements to constant strain fields, augmented by a volumetric correction to capture hoop–radial coupling, and a stabilization term that acts on the non-polynomial space via boundary integrals. An a priori error analysis is developed in weighted Sobolev spaces, establishing interpolation and approximation bounds with norm equivalences and trace inequalities adapted to axisymmetry, and proving optimal convergence rates. Numerical patch tests on polygonal meshes validate the method's accuracy and robustness, highlighting excellent performance for axial and shear states while revealing intrinsic coupling challenges in hoop-related components due to axisymmetric geometry. Overall, the framework provides a rigorous, flexible, and scalable approach for axisymmetric elasticity on general polygonal meshes, with clear directions for enhancements and extensions to more complex material behavior.

Abstract

This work presents a Virtual Element Method (VEM) formulation tailored for two-dimensional axisymmetric problems in linear elasticity. By exploiting the rotational symmetry of the geometry and loading conditions, the problem is reduced to a meridional cross-section, where all fields depend only on the radial and axial coordinates. The method incorporates the radial weight in both the weak formulation and the interpolation estimates to remain consistent with the physical volume measure of cylindrical coordinates. A projection operator onto constant strain fields is constructed via boundary integrals, and a volumetric correction term is introduced to account for the divergence of the stress field arising from axisymmetry. The stabilization term is designed to act only on the kernel of the projection and is implemented using a boundary-based formulation that guarantees stability without affecting polynomial consistency. Furthermore, an a priori interpolation error estimate is established in a weighted Sobolev space, showing optimal convergence rates. The implementation is validated through patch tests that demonstrate the accuracy, consistency, and robustness of the proposed approach.
Paper Structure (21 sections, 4 theorems, 130 equations, 1 figure, 4 tables, 1 algorithm)

This paper contains 21 sections, 4 theorems, 130 equations, 1 figure, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Linear elastic axisymmetric formulation
  3. Virtual element formulation
  4. Virtual element space
  5. Projection operator
  6. Compute the projection matrix
  7. Stabilization term
  8. Load term
  9. Error analysis
  10. Interpolation error estimate
  11. Approximation error estimate
  12. Numerical Results
  13. Patch Test Description
  14. Patch Test Cases
  15. Test 1: Constant Radial Strain
  16. ...and 6 more sections

Key Result

Theorem 4.1

For each integer $l$, with $0\leq l \leq m$, there exists a constant $C>0$, independent of $h$, such that for $1\leq m \leq k+1$, where the $H^1$-seminorm is weighted by the radial coordinate $r$: and $\Omega$ is assumed to be bounded away from the axis $r=0$, ensuring $r\geq r_0 > 0$.

Figures (1)

  • Figure 1: Annular cylindrical sector defined by the radial bounds $r_{\text{inner}} = 1.0$ and $r_{\text{outer}} = 3.0$, and the axial bounds $z_{\text{min}} = 0.0$ and $z_{\text{max}} = 2.0$ with the corresponding cooridnate system.

Theorems & Definitions (9)

  • Remark 3.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • proof