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On the Virtual Element Method for Three-Dimensional Elasticity Problems on Arbitrary Polyhedral Meshes
Arun L. Gain, Cameron Talischi, Glaucio H. Paulino
TL;DR
Three-dimensional elasticity on arbitrary polyhedral meshes challenges standard finite elements because their interior interpolation functions are difficult to use. The paper develops first-order VEM using local projections and an energy decomposition, showing exact treatment of linear deformations and patch-test consistency while limiting the exposition to first order.
Problem
Standard finite-element discretizations face challenges on arbitrary polyhedral meshes because their interior interpolation functions need not be explicitly computable or conveniently integrated.
Method
The paper develops first-order VEM with local projection maps that split element deformations into rigid-body, constant-strain, and higher-order components.
Results
The method exactly captures linear-deformation strain energy, passes the first-order patch test, and reduces projection and stiffness computations to geometric quantities.
Takeaways & Limitations
Preserving the uniform-strain energy component provides consistency, while a suitable approximation of higher-order energy supports first-order convergence.
Takeaways & Limitations
The exposition is limited to the first-order formulation, and stability requires a suitable positive-definite higher-order stabilization whose uniform coercivity may not be guaranteed in general.
Abstract
from arXiv · showhide
We explore the recently-proposed Virtual Element Method (VEM) for numerical solution of boundary value problems on arbitrary polyhedral meshes. More specifically, we focus on the elasticity equations in three-dimensions and elaborate upon the key concepts underlying the first-order VEM. While the point of departure is a conforming Galerkin framework, the distinguishing feature of VEM is that it does not require an explicit computation of the trial and test spaces, thereby circumventing a barrier to standard finite element discretizations on arbitrary grids. At the heart of the method is a particular kinematic decomposition of element deformation states which, in turn, leads to a corresponding decomposition of strain energy. By capturing the energy of linear deformations exactly, one can guarantee satisfaction of the engineering patch test and optimal convergence of numerical solutions. The decomposition itself is enabled by local projection maps that appropriately extract the rigid body motion and constant strain components of the deformation. As we show, computing these projection maps and subsequently the local stiffness matrices, in practice, reduces to the computation of purely geometric quantities. In addition to discussing aspects of implementation of the method, we present several numerical studies in order to verify convergence of the VEM and evaluate its performance for various types of meshes.
1. Introduction
The introduction motivates polyhedral discretizations for flexible three-dimensional meshing and presents VEM as a finite-element framework that avoids explicit interior basis functions. It outlines first-order VEM's energy decomposition, geometric implementation, and scope.
- Motivation: Polyhedral meshes offer flexibility for complex-domain embedding, mesh adaptation, and arbitrary element geometries.They can also achieve comparable flow accuracy with fewer cells and unknowns in some applications.
- Motivation: Finite-element extensions to general three-dimensional polyhedral meshes have progressed more slowly than finite-volume and mimetic finite-difference approaches.The difficulty is framed within the broader development of methods for general polyhedral discretizations.
- Virtual Element Method: VEM is a Galerkin method that avoids computing interpolation functions in element interiors while approximating strain energy exactly for linear deformations.Local projection maps split deformation into polynomial and non-polynomial components.
- Scope and implementation: The paper elaborates first-order VEM for elasticity, derives stiffness and projection expressions, and discusses implementation for general polyhedral meshes.The implementation reduces surface-integral quantities to geometric information such as face centroids and areas.
- Scope and implementation: The discussion is limited to the first-order formulation, although VEM provides a general recipe for higher-order and higher-continuity polyhedral elements.The limitation is adopted for clarity, potentially leaving broader technology less visible in this paper.
2. Model problem and discretization
The paper formulates three-dimensional linear elasticity on conforming polyhedral meshes and constructs vertex-based element spaces using barycentric coordinates. These spaces support linear fields, provide first-order approximation, and yield linear Galerkin convergence under the stated assumptions.
- Model problem: The elasticity model seeks the displacement minimizing total potential energy under body forces, surface tractions, and prescribed displacements.The domain boundary is partitioned into displacement and traction segments.
- Galerkin approximation: A conforming discrete space consists of continuous displacement fields whose restrictions to each polyhedron belong to a finite-dimensional smooth element space.Exact essential boundary conditions are assumed feasible in the discrete space; nodal boundary approximation errors are omitted from the presentation.
- Galerkin approximation: Including linear displacement fields gives each element rigid-body motions and constant-strain states, providing first-order approximation with O(h) energy-norm error.The Galerkin error inherits the best-approximation behavior, yielding a linear convergence rate.
- Construction of W(E): The element space uses three vertex-associated degrees of freedom per vertex, with vector basis functions built from scalar barycentric coordinates.Barycentric coordinates satisfy vertex interpolation, vary linearly on edges, and have face behavior determined by incident-face geometry.
- Construction of W(E): For tetrahedra, the linear shape functions are unique and the element space equals P(E).This is stated as a special case of the general barycentric-coordinate construction.
- Construction of W(E): Harmonic lifting constructs polyhedral barycentric coordinates from planar face coordinates while preserving vertex, edge, continuity, and linear-completeness properties.The construction solves a Laplace problem with face boundary data on incident faces and zero data on remaining faces.
3. Virtual Element Method (VEM)
VEM avoids explicit interior basis-function computation on polyhedral elements by projecting deformation states onto rigid-body and constant-strain components, then decomposing strain energy. Exact treatment of linear deformation energy supports the patch test and O(h) consistency, while stability depends on the higher-order stabilization form.
- Method motivation and construction: VEM avoids costly interior basis-function computation and numerical quadrature by constructing discrete forms from projection maps and geometric element information.The implementation reduces projection and stiffness calculations to vertex coordinates and face geometric quantities.
- Kinematic decomposition: The linear displacement space is split into rigid body motions and constant-strain modes, with bases for translations, rotations, axial strains, and shear strains.Projection maps extract these components from arbitrary element deformation states.
- Projection maps: The rigid-body projection is determined by average translations and rotations and can be computed from boundary integrals using nodal vertex values.This converts volumetric quantities into geometric boundary data available on polyhedral faces.
- Energy decomposition: Energy orthogonality decomposes strain energy into constant-strain and higher-order contributions, allowing the constant-strain part to be computed exactly.The projection onto the constant-strain space is uniquely determined by the energy-orthogonality condition.
- Stability: The stabilization form must be positive definite on higher-order deformation modes to prevent zero-energy modes and rank deficiency while preserving polynomial consistency.Its choice does not affect the consistency of the method, provided it controls the higher-order space appropriately.
- Consistency and convergence: Exact capture of linear deformation energy makes the discrete bilinear form pass the first-order patch test and yields O(h) consistency for the energy term.Together with O(h) load-form error from first-order quadrature, this preserves linear convergence.
4. Implementation aspects
The implementation derives projection maps and stiffness matrices from nodal and geometric data, avoiding explicit interior basis functions. The resulting construction uses elasticity-dependent uniform-strain energies, face geometry, and quadrature schemes whose first-order accuracy preserves linear convergence.
- Projection maps and stiffness assembly: The implementation derives discrete projection maps and element stiffness matrices for the VEM formulation before standard global assembly.The projections are represented by matrices acting on nodal degrees of freedom, while the global stiffness matrix follows the usual finite element assembly process.
- Projection maps and stiffness assembly: Nodal deformation representations identify each virtual element state with an array in R3n, enabling matrix representations of rigid-motion and constant-strain projections.The matrices P_R and P_C project onto the ranges of N_R and N_C, and their application to nodal values returns the nodal values of the projected fields.
- Stiffness structure: The matrix D of normalized uniform-deformation strain energies depends only on the elasticity tensor C, not on element geometry.For isotropic materials, D is specified using Young’s modulus and Poisson’s ratio.
- Geometric implementation: Computing the stiffness matrix reduces to four matrices, N_R, N_C, W_R, and W_C, assembled from vertex coordinates, face normals, and surface integrals.The required face data include connectivity, orientation, unit normals, and integrals involving basis functions over element boundaries.
- Geometric implementation: Surface integrals can be computed exactly from face geometry with suitable boundary barycentric coordinates or approximated by first-order nodal quadrature.Nodal quadrature makes the surface value independent of the particular barycentric coordinates used to construct the element space.
- Accuracy and stabilization: Linear polynomial exactness is required for the quadrature maps to preserve the stated projection properties, while the global patch test is still passed exactly.The consistency error remains O(h), and the reported normalized higher-mode energies cluster around 1, supporting γ = 1 for the stabilization choice.
5. Numerical studies
The numerical studies assess VEM accuracy, convergence, patch-test behavior, and stabilization across polyhedral mesh types. VEM passes the displacement patch test and achieves optimal convergence, while shear-beam accuracy depends on the stabilization parameter γ.
- Error measures: The studies measure relative displacement and stress errors using surrogates because the exact VEM basis functions are unavailable.Displacement error approximates the L2 error through nodal quadrature, while stress error uses an element-wise constant stress field.
- Displacement patch test: VEM passes the displacement patch test on CVT, random Voronoi, and meshes containing non-convex elements, with errors near machine precision.Random Voronoi errors are approximately one order of magnitude larger than CVT errors but remain close to machine precision.
- Shear-loaded beam: With γ = 1, VEM exhibits second-order displacement convergence and first-order stress convergence, including on irregular random Voronoi meshes.These optimal rates are reported for all tested mesh types.
- Shear-loaded beam: An optimal γ minimizes displacement error for each mesh type, and the hexahedral VEM solution can be nearly two orders of magnitude more accurate than the corresponding finite element solution.Stress errors are less sensitive to γ, while VEM stress errors on hexahedral meshes are comparable to finite element errors.
- Shear-loaded beam: The optimal γ is fairly stable under refinement, but the authors caution that values observed for the shear-loaded beam should not be generalized to other problems.They recommend γ = 1 for general polyhedral meshes when problem-specific information is unavailable.
6. Concluding remarks
The paper presents VEM for three-dimensional elasticity on polyhedral meshes as an energy-splitting approach that preserves linear deformation energy exactly. This supports patch-test satisfaction, optimal convergence, and a geometric alternative for finite element discretization on arbitrary grids.
- Core method: Local polynomial projections split element strain energy into uniform-strain and higher-order components.The uniform-strain component is preserved exactly, while the higher-order component can use a suitable approximate energy.
- Accuracy and consistency: Exact preservation of uniform-strain energy guarantees the patch test and consistency, while a suitable higher-order approximation can ensure first-order convergence.The same energy splitting can also restore consistency for polyhedral finite elements affected by quadrature error.
- Implication: The method offers an alternative approach for addressing finite element challenges on arbitrary grids.This conclusion is stated in the context of three-dimensional elasticity on polyhedral meshes.
Appendix
The appendix describes polygonal barycentric coordinates used to construct the local VEM element space. Their first-order moments are defined exactly, while only boundary-based average quantities are needed in the method.
- Polygonal coordinates: The barycentric coordinate associated with a polygon vertex equals one there, varies linearly along incident edges, and vanishes at the other vertices.Its interior Laplacian is linear and is constrained to match the zeroth and first moments of a polynomial projection.
- Implementation: VEM does not need explicit interior face coordinates; it uses their average values, computed from the face centroid and boundary geometry.The construction uses unit edge normals and vertex locations to simplify the projection expressions.
- Polygonal coordinates: The coordinate functions are uniquely determined by requiring their first-order moments to equal those of their polynomial projections.This follows from the one-to-one correspondence between the functions' moments and their Laplacians.