Source-linked AI summary
Arbitrary order 2D virtual elements for polygonal meshes: Part II, inelastic problem
Edoardo Artioli, Lourenco Beirão da Veiga, Carlo Lovadina, Elio Sacco
TL;DR
The prior VEM formulation addressed linear 2D elasticity, and this paper examines its extension to arbitrary-order material-nonlinear problems. The authors implement three inelastic constitutive models and report generally greater accuracy and fewer Newton iterations than FEM; they identify small strain and displacement as the setting.
Problem
The prior VEM formulation addressed linear 2D elasticity, and this work examines its extension to material nonlinearity at arbitrary polynomial order.
Source: §1 · p. 2Method
The paper extends the 2D VEM formulation to Maxwell viscoelasticity, Mises plasticity, and a shape memory alloy constitutive law.
Source: §3 · p. 8, §5 · p. 17Results
VEM solutions are generally more accurate than FEM solutions and require fewer iterations when both use the same tangent (Newton) algorithm.
Source: §5 · p. 17Takeaways & Limitations
The authors conclude that the approach’s accuracy and implementation simplicity make it interesting for structural applications, including possible integration into existing commercial FEM codes.
Source: §5 · p. 18The continuous problem is formulated under the assumption of small strain and displacement.
Source: §2.1 · p. 3
Abstract
from arXiv · showhide
The present paper is the second part of a twofold work, whose first part is reported in [3], concerning a newly developed Virtual Element Method (VEM) for 2D continuum problems. The first part of the work proposed a study for linear elastic problem. The aim of this part is to explore the features of the VEM formulation when material nonlinearity is considered, showing that the accuracy and easiness of implementation discovered in the analysis inherent to the first part of the work are still retained. Three different nonlinear constitutive laws are considered in the VEM formulation. In particular, the generalized viscoplastic model, the classical Mises plasticity with isotropic/kinematic hardening and a shape memory alloy (SMA) constitutive law are implemented. The versatility with respect to all the considered nonlinear material constitutive laws is demonstrated through several numerical examples, also remarking that the proposed 2D VEM formulation can be straightforwardly implemented as in a standard nonlinear structural finite element method (FEM) framework.
1 Introduction
This paper extends the VEM formulation from linear 2D elasticity to material-nonlinear inelastic problems at arbitrary polynomial order. It evaluates three constitutive models and retains a constitutive black-box approach compatible with standard nonlinear FEM workflows.
1 Introduction
- Earlier structural-mechanics VEM work covered linear elasticity and, in some cases, low-order inelastic or contact problems; this paper builds on the linear formulation in.The introduction also notes VEM’s use with general polygonal and polyhedral meshes and cites related polytopal-mesh methods.Source: §1 · p. 2
- The formulation extends the Part I VEM for linear 2D elasticity to arbitrary-order problems with nonlinear inelastic constitutive behavior.The numerical assessment covers generalized Maxwell viscoelasticity, von Mises plasticity with linear isotropic/kinematic hardening, and phenomenological shape memory alloy behavior.Source: §1 · p. 2
- A general phenomenological framework accommodates the models, while constitutive algorithms remain independently embeddable as black boxes in a nonlinear FEM-style code.The paper extends higher-order schemes from; an earlier VEM approach in was limited to low order.Source: §1 · pp. 2–3
- The paper assesses the three inelastic models numerically and discusses computational behavior, presenting VEM as an alternative to standard FEM.The authors note that the tests generally differ from those in.Source: §1 · pp. 2–3
2 Statement of the problem
The problem statement sets out a small-strain, small-displacement inelastic continuum model and its polygon-mesh VEM discretization. The formulation supports general constitutive algorithms through pointwise updates, polynomial projections, and stabilization.
2.1 The continuous problem
- The continuous 2D problem assumes small strain and displacement, decomposes total strain into elastic and inelastic parts, and imposes equilibrium with homogeneous displacement boundary conditions.The stress law is coupled to evolution laws for inelastic strain and history variables.Source: §2.1 · pp. 3–4
- The constitutive framework allows stress to depend on strain, strain rates, inelastic strain, and history variables, with evolution laws that may be single- or set-valued.In some models, history variables can be expressed through inelastic strain, allowing their evolution law to be derived from that relation.Source: §2.1 · p. 4
2.2 The virtual element formulation
- The VEM uses polygon-element displacement spaces with polynomial edge traces and polynomial Laplacians, and a strain projection into degree k−1 polynomials.The degree k is the method’s degree of accuracy; the projection approximates strains in the square-integral norm.Source: §2.2 · pp. 5–6
- At each time step, an independent backward-Euler constitutive algorithm takes previous state variables and a tentative strain, then returns updated stress and internal variables.The formulation evaluates this algorithm at selected element points, analogous to Gauss points, and assembles local contributions into the global scheme.Source: §2.2 · pp. 6–7
- Element integration points and weights support the local form, while stabilization uses a parameter based on the mean trace of the tangent matrix at the previous time.The integration rule is chosen to be exact for polynomials up to degree 2(k−1); the stabilization parameter is selected to improve robustness in the inelastic quasi-static setting.Source: §2.2 · pp. 7–8
3 Constitutive models
The section presents three phenomenological constitutive models in an energetic format: generalized Maxwell viscoelasticity, von Mises plasticity with combined hardening, and shape memory alloy behavior.
3.1 Generalized Maxwell isotropic viscoelastic constitutive model
- The generalized Maxwell model represents viscoelasticity through an elastic element in parallel with spring-dashpot elements, whose partial strains evolve under relaxation.For isotropic response, the inelastic strain is purely deviatoric, and the stress-strain relation can also be expressed using a shear-modulus relaxation function.Source: §3.1 · pp. 8–9
3.2 Plasticity model
- The plasticity model uses a von Mises yield condition with linear isotropic and kinematic hardening and an isochoric plastic flow.Its strain is additively split into elastic and plastic parts; the yield stress depends on initial yield stress, isotropic hardening, and accumulated plastic strain.Source: §3.2 · pp. 10–11
3.3 Shape memory alloy constitutive model
- The SMA model treats phase transformation as an inelastic transformation strain constrained by a saturation domain and governed by a stress-based flow rule.The transformation strain is traceless, and the free energy includes elastic, chemical, and transformation-strain contributions.Source: §3.3 · pp. 11–13
4 Numerical results
The numerical section assesses the VEM against displacement-based and mixed finite elements on viscoelastic, plastic, and shape-memory-alloy problems. It also describes polygon integration choices used in the computations.
4 Numerical results
- Classical benchmarks assess VEM accuracy, robustness, and efficient treatment of material nonlinearity against standard displacement-based finite elements, while retaining Newton’s method for equilibrium.The test campaign covers the inelastic constitutive models presented in the preceding section.Source: §4 · p. 14
4.1 On the integration rule for polygons
- For k = 2, midpoint edge quadrature uses 2m integration points per polygon, with shared edge points processed once through an edge-wise mesh loop.The polygon integration discussion also describes centroid-based subdivision into triangles for area integration.Source: §4.1 · p. 14
4.2 Thick-walled viscoelastic cylinder subjected to internal pressure
- For the pressurized viscoelastic cylinder, VEM responses agree excellently with standard displacement finite elements at the same spatial discretization.The benchmark also reports that existing viscoelastic integration algorithms transfer into the VEM framework without modification.Source: §4.2 · p. 15
4.3 Perforated plastic plate
- For the perforated plate, VEM agrees with mixed-FEM reference displacements and quadratic FEM load-displacement curves, while showing a slight iteration-efficiency edge and reported mesh and convergence advantages.The comparison reports average Newton iterations per incremental loading step; the broader benchmark also reports mesh versatility and convergence robustness relative to standard FEM.Source: §4.3 · p. 16
4.4 Shape memory alloy device
- The SMA arch load-displacement curve exhibits the classical shape recovery of the device under the analyzed load-temperature history.The reported results use the quadratic VEM case, k = 2.Source: §4.4 · p. 17
5 Conclusion
The VEM formulation is extended to nonlinear material response and applied to Maxwell viscoplasticity, Mises plasticity, and shape memory alloy behavior. The numerical results support its accuracy and its relatively simple implementation, particularly alongside an existing FEM code.
5 Conclusion
- The nonlinear formulation covers Maxwell viscoplasticity, Mises plasticity, and a shape memory alloy constitutive law.Source: §5 · p. 17
- Material-nonlinearity algorithms can be embedded in VEM as FEM-style black boxes, without substantial changes to their implementation.Polygonal-domain integration is performed by dividing the domain into triangles and applying Gauss quadrature on each triangle.Source: §5 · p. 17
- VEM solutions are generally more accurate than FEM solutions and require fewer iterations when both use the same tangent (Newton) algorithm.Source: §5 · p. 17
- The two-part study presents VEM as accurate and almost simple to implement, especially when a FEM code is available.The authors identify possible inclusion of VEM elements in available commercial FEM codes.Source: §5 · p. 18