Source-linked AI summary
A Virtual Element Method for elastic and inelastic problems on polytope meshes
L. Beirão da Veiga, C. Lovadina, D. Mora
TL;DR
The paper addresses nonlinear elastic and inelastic mechanics on general polygonal and polyhedral meshes, mainly in small deformations. It develops a low-order VEM with suitable displacement-gradient treatment, efficient constitutive-law evaluation, and black-box constitutive algorithms; elastic theory and numerical tests support the approach.
Problem
Nonlinear elastic and inelastic problems need a VEM treatment that preserves general polytopal-mesh capabilities while accommodating constitutive laws in structural mechanics.
Method
The method combines a low-order displacement approximation with a suitable treatment of the numerical displacement gradient and independently embedded black-box constitutive algorithms.
Results
The method supports general polygonal and polyhedral meshes, uses few constitutive-law applications, develops elastic theoretical results, and is assessed through numerical tests.
Takeaways & Limitations
The scheme provides a first VEM framework for nonlinear computational mechanics that accommodates general meshes and standard constitutive-law algorithms.
Takeaways & Limitations
Large-deformation elasticity remains a preliminary extension requiring substantially deeper design and analysis.
Abstract
from arXiv · showhide
We present a Virtual Element Method (VEM) for possibly nonlinear elastic and inelastic problems, mainly focusing on a small deformation regime. The numerical scheme is based on a low-order approximation of the displacement field, as well as a suitable treatment of the displacement gradient. The proposed method allows for general polygonal and polyhedral meshes, it is efficient in terms of number of applications of the constitutive law, and it can make use of any standard black-box constitutive law algorithm. Some theoretical results have been developed for the elastic case. Several numerical results within the 2D setting are presented, and a brief discussion on the extension to large deformation problems is included.
1 Introduction
The paper introduces VEM as a promising approach for nonlinear elastic and inelastic structural mechanics on general polygonal and polyhedral meshes. It targets small-deformation problems while seeking efficient constitutive-law evaluations and black-box algorithm integration.
- Motivation: VEM handles general polygonal and polyhedral meshes without explicitly constructing local basis functions or performing complex element integrations.This supports applications involving cracks, inclusions, hanging nodes, moving meshes, and adaptivity.
- Motivation: General polygons can improve robustness to mesh distortion, reduce mesh sensitivity in topology optimization, and assist contact and crack-propagation problems.These advantages are associated with polygonal finite-element approaches in structural mechanics.
- Scope: The paper initiates VEM investigations for nonlinear elastic and inelastic constitutive laws in the small-deformation regime.The considered cases include stable nonlinear elasticity and inelastic laws such as those arising in classical plasticity.
- Scope: The scheme applies the constitutive law only once per mesh element and permits independently embedding a general nonlinear constitutive algorithm as a black box.This is presented as analogous to one-point Gauss quadrature and common engineering finite-element procedures.
- Organization: The paper develops separate continuous and discrete treatments for elastic and inelastic problems, including approximation spaces and projection operators.The outline distinguishes the elastic and inelastic formulations before presenting their VEM discretizations.
2 The continuous problems
The paper formulates small-deformation elastic and rate-independent inelastic boundary-value problems for bodies in two or three dimensions. Elasticity uses a constitutive strain–stress relation, while inelasticity additionally evolves history variables over pseudo-time.
- Elastic problem: The elastic problem seeks displacement in a two- or three-dimensional clamped body under body loading and small deformations.The body occupies Ω and is clamped on part Γ of its boundary.
- Elastic problem: Elastic stresses are related to strains through a material constitutive law involving the displacement gradient.The displacement gradient is denoted by ∇u.
- Elastic problem: The elastic deformation problem is expressed variationally using admissible displacements and variations satisfying homogeneous Dirichlet conditions on Γ.The formulation is written for a body subjected to body load f.
- Inelastic problem: The inelastic problem assumes small deformations and rate-independent behavior, with loading depending on pseudo-time and displacement sought at final time T.The body is likewise clamped on part Γ and occupies a domain in two or three dimensions.
- Inelastic problem: Inelastic constitutive behavior relates strains and stresses while an evolution law updates material history variables over pseudo-time.At each time instant, stresses and displacements satisfy equilibrium and boundary conditions.
- Inelastic problem: The inelastic deformation problem is given in a minimal formulation using admissible displacements, variations, initial history values, and the evolution law.The formulation is intentionally limited to the setting needed for the associated discrete problem.
3 The virtual element approximation
The method constructs low-order virtual displacement spaces on general polygonal and polyhedral meshes, with computable gradient projections and stabilized, P1-consistent local forms. It supports black-box constitutive algorithms while limiting constitutive-law evaluations to one per element.
- Virtual spaces: The discretization uses general polygonal or polyhedral conforming meshes and defines local virtual displacement spaces in two and three dimensions.The three-dimensional construction builds face spaces on polygonal faces, while the two-dimensional spaces use harmonic functions with piecewise-linear boundary traces.
- Virtual spaces: Local virtual functions are harmonic inside each element and continuous with piecewise-linear traces on edges or faces, while their interior values remain implicit.The spaces contain P1(E), and triangular or tetrahedral elements recover the corresponding standard P1 space.
- Degrees of freedom and operators: Vertex values serve as degrees of freedom, determining the boundary values and enabling integration by parts to compute the element-average displacement gradient.The gradient average is obtained from face or edge integrals involving the outward normal.
- Degrees of freedom and operators: The projection Π∇v is piecewise linear, with the elementwise mean gradient and vertex-value average fixing its gradient and constant part.The tensor-valued Π0 projection is also defined elementwise, and the relevant gradient projections are explicitly computable.
- Discrete forms and stabilization: The stabilized local forms are P1-consistent, computable on arbitrary polygonal or polyhedral elements, and avoid non-physical kernels that can produce spurious modes.The stabilization uses positive element-dependent constants αE to account for material constants and nonlinear materials.
- Constitutive treatment: The constitutive law is evaluated only once per element, making the method comparable in cost to finite elements with one-point Gauss integration and especially advantageous for expensive inelastic laws.The scheme is designed to remain compatible with standard black-box constitutive algorithms imported independently of the global discretization.
4 Theoretical results
The theoretical analysis establishes stability and continuity properties for the nonlinear discrete forms, then derives an error bound and linear convergence under mesh regularity and solution smoothness assumptions.
- Assumptions: The analysis assumes constitutive-law hypotheses and polygonal mesh regularity, including star-shaped elements and edge lengths proportional to element diameter.These shape-regularity conditions require each element to be star shaped with respect to a ball of radius ρ ≥ C_sh h_E and each edge to satisfy h_e ≥ C_s h_E.
- Stability and continuity: The discrete bilinear forms satisfy stability and continuity estimates under the stated hypotheses.The bounds control differences of the forms through energy seminorm products and the constitutive-law Lipschitz condition.
- Error analysis: Theorem 4.1 provides an error estimate for the discrete solution relative to interpolation and piecewise-linear comparison functions.The result applies to the solution of Problem (23) for any admissible previous state and comparison functions specified in the theorem.
- Incremental loading: Theorem 4.1 also applies to the incremental nonlinear problem at the final loading step after identifying the theorem variables with the final-step quantities.The required choices are f = f_N and s_N = u^{N−1}, with the discrete solution identified as u^N.
- Convergence: Under u ∈ [H^2(Ω)]^d, Corollary 4.1 yields a linear convergence bound based on standard polygonal approximation estimates.The corollary follows by combining Theorem 4.1 with approximation estimates for V_h,E and P_1(E).
5 Numerical tests
The numerical tests examine convergence and scaling choices for elastic problems, then compare Voronoi-mesh results with reference solutions for plasticity and finite-strain elasticity.
- Elasticity tests: The Hencky-von Mises elasticity test shows first-order convergence in the discrete H1-like norm and quadratic convergence in the discrete L∞ norm.Table 1 reports mesh vertices, convergence rates, and discrete errors for different mesh families.
- Elasticity tests: The benchmark elasticity model likewise exhibits quadratic discrete L∞ convergence and linear convergence in the discrete H1-like norm.This benchmark constitutive choice does not correspond to an elastic material and violates condition (28).
- Scaling tests: Updating the stability scaling is tested for moderate and much larger deformations using two external-force cases on regular Voronoi meshes.Tables 3 and 4 compare relative errors for updated and fixed scaling choices.
- Inelasticity tests: In the von Mises plasticity strip problem, all reported quantities converge toward fine-triangular reference values, while maximum stress needs finer Voronoi meshes for better approximation.The study also finds good agreement between computed and reference plastic consistency parameters.
- Finite-strain elasticity: The preliminary finite-strain elasticity study finds convergence to the reference displacement and comparable deformed shapes across triangular, quadrilateral, and hexagonal Voronoi meshes.The example focuses on neo-Hookean hyperelastic materials.
6 Conclusions
The paper presents a VEM scheme for nonlinear elastic and inelastic problems that supports diverse polygonal meshes and standard constitutive-law algorithms. Numerical tests assess its computational performance, while the authors identify large deformations and additional inelastic models as extensions requiring further investigation.
- The scheme combines a low-order displacement approximation with a suitable treatment of the numerical displacement gradient.
- VEM supports triangular, square, and hexagonal Voronoi meshes in the presented computations.Figure 5 shows deformed bodies for meshes T2, Q2, and V2; Table 6 reports computed displacements for corresponding mesh families.
- The method is efficient in constitutive-law applications and can use standard black-box constitutive-law algorithms.
- The numerical tests assess the computational performance of the proposed methodology.
- Large-deformation problems and additional inelastic cases, including perfect plasticity and damage, remain possible extensions requiring deeper investigation.