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An isogeometric method for the Reissner-Mindlin plate bending problem
L. Beirão da Veiga, A. Buffa, C. Lovadina, M. Martinelli, G. Sangalli
TL;DR
Thin Reissner–Mindlin plates pose a shear-locking challenge for standard low-order finite elements. This paper constructs smooth spline spaces for deflections and rotations whose discrete formulation satisfies the Kirchhoff constraint exactly at zero thickness. The analysis gives a discrete-norm error bound, while high-regularity estimates may be unrealistic in the presence of boundary layers.
Problem
Standard low-order finite element schemes can lose convergence when plate thickness is small relative to other dimensions, as the limiting Kirchhoff constraint is difficult to approximate.
Source: §1 · p. 2Method
The method uses smooth isogeometric deflection and rotation spaces built through a differential-form framework, with the discrete constraint enforced exactly at zero thickness.
Source: §1 · p. 3, §4.3 · p. 12Results
The discrete-norm error is bounded by C times the error of any admissible approximation of the solution and its shear.
Source: §5.2 · p. 17Takeaways & Limitations
The selected approximation spaces contain a subspace that exactly satisfies the Kirchhoff constraint and has optimal approximation features, yielding a formulation built to be locking-free.
Source: §1 · p. 3The regularity required for high-s error estimates may be unrealistic because of boundary layers.
Source: §5.2 · p. 17
Abstract
from arXiv · showhide
We present a new isogeometric method for the discretization of the Reissner-Mindlin plate bending problem. The proposed scheme follows a recent theoretical framework that makes possible to construct a space of smooth discrete deflections $W_h$ and a space of smooth discrete rotations $\Rots_h$ such that the Kirchhoff contstraint is exactly satisfied at the limit. Therefore we obtain a formulation which is natural from the theoretical/mechanical viewpoint and locking free by construction.
1. Introduction
Standard low-order finite elements can lose convergence for thin Reissner–Mindlin plates because the limiting Kirchhoff constraint is difficult to approximate. The paper uses smooth spline spaces to build a locking-free isogeometric formulation with exact CAD geometry capability.
1. Introduction
- As plate thickness becomes small, standard low-order finite elements can lose convergence because the Kirchhoff constraint leaves too small a discrete space to approximate generic plate solutions.This failure is known as shear locking; conventional remedies include mixed formulations or shear-reduction operators.Source: §1 · p. 2
- The proposed framework pairs smooth discrete deflections as 0-forms with smooth discrete rotations as 1-forms, enabling a subspace that exactly satisfies the Kirchhoff constraint and has optimal approximation features.The spaces draw on spline regularity and a framework based on smooth approximations of differential forms.Source: §1 · p. 3
- The resulting formulation is described as mechanically natural, easy to study, and locking-free by construction.Source: §1 · p. 3
- The method retains isogeometric analysis capabilities for exact CAD geometry and flexible choices of polynomial degree and function regularity.The paper notes that exact geometry representation is important in plate analysis.Source: §1 · p. 3
2. The Reissner-Mindlin plate bending problem
The Reissner–Mindlin model describes plate bending under transverse loading using deflection and rotation fields. Its mixed formulation introduces shear stresses, and the continuous problem is stated to be well posed and uniformly stable in thickness.
2. The Reissner-Mindlin plate bending problem
- The plate domain is a bounded Lipschitz open subset of R2, with clamped, simply supported, and free boundary portions and homogeneous conditions assumed for exposition.The combined simply supported and clamped boundary has positive measure to exclude rigid body motions.Source: §2 · pp. 3–4
- The model seeks deflection and rotation fields satisfying a variational equation that balances bending, shear response, and the applied transverse load.The plate thickness is t, the deflection is w, the rotation of normal fibers is θ, and f is the scaled transverse load.Source: §2 · p. 4
- Introducing scaled shear stresses γ = μkt^-2(θ − ∇w) rewrites the plate problem as a mixed variational formulation.Here μ is the shear modulus and k is the shear correction factor.Source: §2 · p. 4
- The continuous problem is well posed and uniformly stable with respect to plate thickness using the stated norms for deflection, rotation, and shear.Source: §2 · p. 5
3. B-splines and piece-wise smooth functions
The paper defines spline spaces through knot vectors and their degree and continuity, then extends them to two-dimensional tensor-product spaces on a mapped plate domain. These constructions provide the smooth approximation spaces used by the method.
3.1. B-spline spaces and piece-wise smooth functions in one dimension
- In one dimension, degree-p B-splines are piecewise polynomials whose continuity at knots is determined by knot multiplicity, with α = p − r continuous derivatives.Open knot vectors repeat endpoint knots p + 1 times; basis functions are nonnegative and form a partition of unity.Source: §3.1 · pp. 5–6
- The one-dimensional spline spaces have piecewise smooth functions with prescribed interelement regularity, and differentiation maps the degree-p+1 spline space surjectively onto the degree-p space with constants as kernel.Source: §3.1 · p. 6
3.2. B-spline spaces and piece-wise smooth functions in two dimensions
- The parametric mesh is assumed shape regular, and the computational domain is represented by a fixed smooth geometric map from the unit-square parametric domain.Tensor-product structure makes shape regularity imply quasi-uniformity.Source: §3.2 · pp. 7–8
- In two dimensions, tensor products of univariate B-spline bases define spline spaces on a rectangular parametric mesh, with regularity specified separately along vertical and horizontal edges.The spaces are characterized by mesh, polynomial degrees, and directional regularities.Source: §3.2 · p. 7
4. Discretization of the scalar and vector fields
The method constructs spline spaces on the parametric and physical domains, then uses them in a discrete Reissner–Mindlin formulation. At zero thickness, the discrete constraint is exact and the formulation reduces to the Kirchhoff plate bending problem.
4.1. Spline spaces on the parametric domain
- Parametric spline spaces impose essential boundary conditions directly; Lemma 4.1 characterizes the zero-boundary shear space.The characterization accounts for the boundary conditions on the deflection and rotation spaces.Source: §4.1 · pp. 8–9
4.2. Spline spaces on the physical domain
- The geometric map carries deflections to the physical domain, while the covariant map carries rotations and shears, preserving tangential-component nullity and mapping gradients to gradients.The gradient mapping property follows from the chain rule.Source: §4.2 · p. 10
- For 1 < s ≤ p + 1, physical-domain projectors approximate deflections and rotations in H1 with errors bounded by Ch^(s−1) times the corresponding Hs seminorms.The estimates hold elementwise for fields with the stated regularity.Source: §4.2 · p. 11
4.3. The discrete problem
- The discrete method uses a Galerkin problem on deflection and rotation spaces, with a mixed shear-stress formulation introduced for stability and convergence analysis.The mixed formulation includes the discrete shear stress as an additional variable.Source: §4.3 · pp. 11–12
- At t = 0, the second discrete equation enforces θh = ∇wh exactly, and the first equation reduces to the Kirchhoff plate bending problem.The reduced problem is stated as a variational equation on the deflection space.Source: §4.3 · p. 12
5. Convergence analysis
The convergence analysis treats a restricted boundary-condition case by coercivity and general boundary conditions in a discrete norm, then establishes improved convergence rates.
5.1. Coercivity-based convergence analysis
- For 1 < s ≤ p, the coercivity-based analysis establishes an error estimate whose constant is independent of h and t under the stated boundary-condition assumption.The shear variables are defined as γ = t^-2(θ − ∇w) and γh = t^-2(θh − ∇wh).Source: §5.1 · p. 13, §5 · p. 13
5.2. Convergence in a discrete norm
- For general boundary conditions, the discrete-norm error is bounded by the best-approximation error with a constant independent of h and t.The corresponding rate corollary requires sufficient solution regularity, which may be unrealistic for high s because of boundary layers.Source: §5.2 · p. 17
- Under sufficient solution regularity, interpolation yields a discrete-norm convergence estimate for general boundary conditions with a constant independent of h and t.The required regularity may be unrealistic for high s because of boundary layers.Source: §5.2 · p. 17
5.3. Improved convergence rates
- For a regular problem, the analysis proves improved convergence results for the L2 rotation error and the deflection error; the higher approximation order of Wh supports an improved rate over Proposition 5.2.The rotation result uses an Aubin-Nitsche argument; the deflection result is stated for any interpolant wI in Wh,0.Source: §5.3 · pp. 18–19
6. Numerical tests
The numerical tests show convergence consistent with the theory on clamped-square and simply supported annular cases, while boundary-layer resolution determines performance in the third case. Layer-adapted meshes restore optimal H1 convergence where uniform refinement is suboptimal until sufficiently fine.
6. Numerical tests
- The experiments use Poisson’s ratio ν = 0.3 and Young’s modulus E = 1.092 · 10^7 N/m^2, with forces and lengths expressed in N and m.The clamped-square test uses a known analytical solution; the annular tests use fine-mesh reference solutions.Source: §6 · p. 19, §6.1 · p. 19, §6.2 · p. 20
- Across tests, approximation spaces are built by elevating degree, repeating knots for continuity, and uniformly inserting knots for h-refinement.For NURBS geometry, degree elevation is performed on the B-spline basis with weights set to one.Source: §6 · p. 19
6.1. Case 1: fully hardly clamped square plate.
- For the clamped square at t = 10^-3, H1 approximation errors for deflection and rotation converge at rates matching the theoretical results.The tests use p = 3, 4 and α = 2, 3 for deflections and rotations, respectively.Source: §6.1 · pp. 19–20
6.2. Case 2: quarter of an annulus with four hardly simply supported sides.
- For the simply supported quarter-annulus, H1 approximation errors at t = 10^-2 and 10^-3 agree excellently with theoretical predictions; a fine-mesh solution supplies the reference.The reference uses h = 1/256, p = 3, and α = 2 for deflections and rotations.Source: §6.2 · p. 20, Figure 3 · p. 22, Figure 4 · p. 23
6.3. Case 3: boundary layer test problem
- Uniform meshes show suboptimal convergence until they resolve the boundary layer, requiring excessive degrees of freedom to reduce error, especially for small plate thicknesses.The curved-boundary layer has characteristic length O(t).Source: §6.3 · pp. 22–23, Figure 6 · p. 24
- Layer-adapted refinement restores the optimal H1-norm convergence rate for the quarter-ring boundary-layer test, unlike uniform refinement that is suboptimal until the layer is resolved.The adapted sequence starts from three elements, with layer elements of width 0.045.Source: §6.3 · pp. 23–24, Figure 6 · p. 24, Figure 7 · p. 25