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A virtual element method for Kelvin--Voigt viscoelasticity
Utkarsh Rajput, Ankit Kumar, Andres E. Rubiano, Ricardo Ruiz-Baier, Sarvesh Kumar
TL;DR
Predicting viscoelastic response requires models that capture elastic and time-dependent viscous effects, while VEM studies identify open challenges in fully discrete schemes and rigorous error estimates. This paper applies VEM to the Kelvin–Voigt model, establishes well-posedness and optimal a priori error estimates for semi-discrete and fully discrete formulations, and reports numerical validation of convergence rates.
Problem
Prior VEM studies identify fully discrete schemes and rigorous a priori error estimates as open challenges in viscoelastic modeling.
Source: §1 · p. 2Method
Results
The semi-discrete and fully discrete formulations are well-posed and have optimal a priori error estimates; numerical experiments validate theoretical convergence rates on different 2D and 3D meshes.
Source: §1 · p. 3Takeaways & Limitations
The numerical experiments support the proposed VEM formulation's convergence behavior across different mesh types in 2D and 3D.
Source: §1 · p. 3The analysis assumes bounded, simply connected polygonal or polyhedral domains in 2D or 3D and positive physical parameters.
Source: §1.1 · p. 3
Abstract
from arXiv · showhide
Considering the computational advantages of virtual element methods (VEM), this work employs a conforming VEM for the numerical approximation of the Kelvin--Voigt model of viscoelasticity. For clarity and simplicity, we focus on the primal formulation. The spatial discretization is carried out using the virtual element method, while the temporal discretization is handled via the {second-order Crank--Nicolson} scheme. We establish the well-posedness of both the semi-discrete and fully discrete problems and derive {\it a priori} error estimates. Several representative numerical examples are presented to validate the theoretical results and to demonstrate the effectiveness of the proposed formulation.
1 Introduction
Viscoelastic materials exhibit both elastic and viscous behavior, motivating models and numerical methods that capture time-dependent response on complex geometries. This paper applies conforming VEM to the Kelvin–Voigt model and develops its continuous formulation in two and three dimensions.
1 Introduction
- The Kelvin–Voigt model combines immediate elastic deformation with time-dependent viscous response, representing phenomena such as creep and stress relaxation.The material is represented by a spring and dashpot in parallel.Source: §1 · p. 2
- VEM supports polygonal and polyhedral meshes through computable bilinear forms based on degrees of freedom and projection operators, without explicit basis functions.The introduction highlights geometric flexibility as relevant to irregular meshes and heterogeneous coefficients.Source: §1 · p. 2
- The paper combines VEM spatial discretization with Crank–Nicolson time stepping for the Kelvin–Voigt model and derives optimal a priori error estimates for semi-discrete and fully discrete formulations.The authors describe this as an application of VEM to this class of problems.Source: §1 · pp. 2–3
1.1 Continuous problem
- The model problem is posed on bounded, simply connected polygonal or polyhedral domains in d=2,3, with homogeneous Dirichlet boundary conditions.The displacement equation couples inertia, viscous strain-rate response, elastic strain response, and applied load.Source: §1.1 · p. 3
- The paper formulates the continuous problem variationally, seeking displacement in V and testing against every v in V almost everywhere in time.The weak equation includes acceleration, viscous and elastic bilinear forms, and the applied load, with prescribed initial displacement and velocity.Source: §1.1 · p. 4
2 Preliminaries on virtual element discretizations
The preliminaries construct elementwise virtual spaces and degrees of freedom for polygonal meshes, then extend the construction to polyhedra. In three dimensions, the local space is introduced because the L2 projection cannot be computed from the chosen degrees of freedom.
2 Preliminaries on virtual element discretizations
- The elementwise construction assumes each polygon is star-shaped with respect to a sufficiently large ball and has vertices separated by a mesh-scaled minimum distance.The two conditions use constants γ1 and γ2 independent of mesh size.Source: §2 · p. 4
- The scalar local spaces use polynomial boundary traces and degrees of freedom at vertices, internal edge points, and, for k>1, element moments.The vector case is obtained by tensor product of the scalar counterpart.Source: §2 · p. 4
2.1 Three-dimensional spaces
- The three-dimensional degrees of freedom include vertex values, edge values, and, for k>1, face and element moments.The cubic-element illustration distinguishes edge, internal, and face degrees of freedom by color.Source: §2.1 · p. 5
- In three dimensions, the chosen degrees of freedom permit the elliptic projection but not the L2 projection, so the construction introduces a local virtual space.The global discrete space is assembled from these local spaces.Source: §2.1 · p. 6
3 Semi-discretization
The semi-discrete VEM uses computable elementwise bilinear forms and establishes stability, well-posedness, and optimal error estimates under stated regularity assumptions.
3 Semi-discretization
- Elementwise discrete forms are computable using projection operators, and a modified, more robust stabilization is used in the numerical section.The stabilization is symmetric positive definite, with a constant independent of h.Source: §3 · pp. 6–7
- The discrete stiffness forms are not exactly consistent even when one argument is a polynomial, so consistency-error estimates are established.Source: §3 · p. 7
3.1 Semi-discrete formulation
- The semi-discrete solution is unique on [0, T], is stable, and dissipates kinetic-plus-potential energy when the load is zero.The energy-dissipation statement is for f = 0.Source: §3.1 · p. 9
3.2 Error analysis
- The method achieves optimal convergence for displacement in the H1 norm and velocity in the L2 norm under the stated regularity assumptions.The assumptions include u, u′, u′′, and f in L2(0, T; Hk+1(Ω; Rd)).Source: §3.2 · p. 10
- An optimal L2-norm error estimate is also established for displacement under the theorem’s regularity assumptions.The theorem assumes u, u′′, and f in L2(0, T; Hk+1(Ω; Rd)).Source: §3.2 · p. 11
4 Fully discrete case
The fully discrete formulation applies Crank–Nicolson in time to a first-order reformulation of the model. The analysis establishes a fully discrete error bound under regularity assumptions.
4 Fully discrete case
- The model is rewritten as a first-order system using v = u′, and the Crank–Nicolson scheme discretizes it in time.Source: §4 · pp. 12–13
- A fully discrete error bound is established under regularity assumptions on the solution, initial data, and forcing.Source: §4 · p. 13
- The error analysis compares the fully discrete and semi-discrete equations at time-step midpoints and accumulates estimates across steps.Source: §4 · pp. 13–14
5 Numerical experiments
Numerical experiments assess the fully discrete scheme’s convergence across meshes and time steps, then illustrate it in viscoelastic applications.
5 Numerical experiments
- The numerical section defines a computable error and refinement rate, and uses the stabilization term following the D-recipe.The text attributes differences between 2D and 3D error metrics to projection-space definitions in vem++ [14].Source: §5 · p. 15
5.1 Experiment 1: Convergence under h-refinement and ∆t-refinement in two-dimensions
- Across all test cases, spatial convergence is optimal at O(h^(k+1)) for mesh families and degrees k ∈ {1, 2, 3, 4}; temporal tests on meshes in Figure 5.1 confirm optimal second-order convergence at O(∆t^2).Source: §5.1 · pp. 15–16
5.2 Experiment 2: Convergence under h-refinement and ∆t-refinement in three-dimensions
- Figures 5.5 and 5.6 show spatial and time convergence curves for all considered mesh families; across tested spatial meshes, O(h^(k+1)) is achieved, while time discretization displays apparent superconvergence.The cited k = 3 case uses uniform Δt-refinement.Source: §5.2 · pp. 16–17
5.3 Experiment 3: Energy dissipation and damping properties of perforated plates
- For perforated plates, deformation develops under loading and the domain gradually recovers after load removal, illustrating Kelvin–Voigt energy dissipation and damping.Figure 5.8 shows single- and four-perforation domains, with displacement components tracked at p = (0.5, 0.8).Source: §5.3 · p. 19, Figure 5.8 · p. 19
5.4 Experiment 4: Sustained loading of viscoelastic slabs
- Under sustained loading, the slab creeps along the z-direction and relaxes toward its reference configuration after release; average displacement then monotonically decays.The reported response closely matches behavior in.Source: §5.4 · p. 20, Figure 5.9 · p. 20, Figure 5.10 · p. 21
5.5 Experiment 5. Linear Kelvin–Voigt modelling of the viscoelastic response in aneurysmal arterial tissue
- In the aneurysm simulation, peak displacement magnitude reaches 1.5 mm in pathological tissue, within the reported 1–3 mm wall-displacement range [15, Fig. 5].The results describe localized inflation corresponding to the cardiac cycle.Source: §5.5 · pp. 21–22, Figure 5.11 · p. 22, Figure 5.12 · p. 23