Source-linked AI summary
Cut Bogner-Fox-Schmit Elements for Plates
Erik Burman, Mats G. Larson, Peter Hansbo
TL;DR
The BFS element’s rectangular-mesh restriction limits its use for smooth plate boundaries, especially in the simply supported setting. The paper develops a cut finite element method with Nitsche enforcement and stabilization; its analysis establishes stability and optimal-order convergence, and numerical results indicate boundary-approximation continuity is not crucial.
Problem
The BFS element is only C1 on rectangular tensor-product elements, limiting its applicability to smooth geometries that do not fit those meshes.
Source: §1 · p. 1Method
Results
The method is stable and optimally order convergent, while numerical results indicate that continuity of the boundary approximation is not crucial.
Source: §6 · p. 11Takeaways & Limitations
The cut formulation greatly increases the practical applicability of the simple rectangular plate element.
Source: §6 · p. 11The presentation focuses on simply supported boundary conditions.
Source: §2 · p. 2
Abstract
from arXiv · showhide
We present and analyze a method for thin plates based on cut Bogner-Fox-Schmit elements, which are $C^1$ elements obtained by taking tensor products of Hermite splines. The formulation is based on Nitsche's method for weak enforcement of essential boundary conditions together with addition of certain stabilization terms that enable us to establish coercivity and stability of the resulting system of linear equations. We also take geometric approximation of the boundary into account and we focus our presentation on the simply supported boundary conditions which is the most sensitive case for geometric approximation of the boundary.
1 Introduction
The paper adapts the Bogner-Fox-Schmit element to smooth boundaries that cut through a tensor-product mesh, using stabilization to obtain a stable method with optimal-order convergence. This addresses the element’s restriction to rectangular meshes and focuses the analysis on simply supported plates.
1 Introduction
- The BFS element is a simple, efficient C1 thin-plate element, but its C1 continuity only on rectangular tensor-product elements limits its applicability.The introduction also cautions against straight-segment boundary discretization because of accuracy issues and Babuška’s paradox for simply supported plates.Source: §1 · p. 1
- A cut finite element version of the BFS element accommodates smooth boundaries cutting the tensor-product mesh arbitrarily, with stabilization and optimal-order convergence.The analysis also accounts for boundary approximation and focuses on simply supported conditions, identified as the most challenging case.Source: §1 · p. 1
- The paper proceeds from the Kirchhoff plate model to method formulation and analysis, then presents numerical illustrations and concluding remarks.The analysis develops consistency and a priori error estimates in energy and L2 norms.Source: §1 · p. 1
2 The Kirchhoff Plate
The section sets out a simply supported Kirchhoff plate model on a smooth domain and its weak formulation. The resulting variational problem is well-posed under the stated assumptions.
2 The Kirchhoff Plate
- The model considers a simply supported thin plate on a domain with smooth boundary.The displacement belongs to the weak-form space H2(Ω) with zero boundary values.Source: §2 · p. 2
- The weak formulation seeks u in V = {v ∈ H2(Ω) : v = 0 on ∂Ω} such that the variational equation holds for every v in V.The load functional is l(v) = (f, v)Ω.Source: §2 · p. 2
- The bilinear form is symmetric, continuous, and coercive on V with the H2(Ω) norm, yielding a unique solution; smooth boundaries and f ∈ L2 also give elliptic regularity.The existence and uniqueness claim follows from the Lax-Milgram theorem.Source: §2 · p. 2
3 The Finite Element Method
The method restricts a tensor-product cubic Hermite BFS space to an active mesh around an approximated domain boundary. Nitsche boundary terms and stabilization control the cut-element formulation.
3 The Finite Element Method
- The underlying BFS space consists of tensor products of cubic Hermite splines on square meshes, restricted to active elements intersecting the approximated domain.The discrete space is the restriction of the BFS space to the active mesh.Source: §3 · pp. 2–3
- The approximated boundary is piecewise smooth; within each element it cuts, the boundary segment is smooth and intersects the element boundary at exactly two points.The set Xh collects points where the approximated boundary is not smooth.Source: §3 · pp. 2–3
- The bilinear form uses Nitsche boundary terms and a penalty term, while the stabilization form supplies necessary control of cut elements.The stabilization parameters β and γ are positive and proportional to κ.Source: §3 · p. 3
4 Error Estimates
The section develops stability and error-estimation tools for the cut finite element method, including interpolation and consistency estimates. Its coercivity result depends on choosing stabilization parameters appropriately.
4.1 Basic Properties of Ah
- For sufficiently large γ, A_h is coercive when δ is small enough and β and γ satisfy the stated lower-bound conditions.The proof uses the stabilization bound to control cut elements.Source: §4.1 · pp. 3–4
4.2 Interpolation
- An extension-based interpolation operator for cut elements is introduced, followed by interpolation error estimates in the stated norms.The operator extends functions from Ω to R² before applying the elementwise interpolant.Source: §4.2 · p. 4
4.3 Consistency Error Estimate
- The consistency analysis gives a bound for the exact solution against finite element test functions and uses that s_h(u,v)=0 for u∈H⁴(Ω).For a C¹ boundary, the boundary term (v,[M_nt])_Xh vanishes.Source: §4.3 · p. 6
4.4 Error Estimates
- The section states error estimates for the finite element solution, including an estimate developed using a discrete dual problem.The supplied theorem statements do not expose the bounds themselves.Source: §4.4 · pp. 8–9
5 Numerics
The numerical section describes constructing cubic boundary and geometry approximations for cut elements, then tests convergence on a uniformly loaded simply supported circular plate.
5.1 Implementation
- The implementation locates cut elements, triangulates the interior for integration, and uses endpoint positions and tangent inclinations to form a C1 spline boundary when needed.The spline step is skipped for a C0 boundary approximation; cubic triangles then interpolate the exact boundary or spline boundary.Source: §5.1 · p. 10
- When the approximated boundary extends beyond an element, that element’s basis functions are also used outside the element.Source: §5.1 · p. 10
5.2 Example
- The example uses a radius-1/2 circular simply supported plate centered at (1/2, 1/2), under uniform load.Source: §5.2 · p. 10
- Normalized L2, H1, and piecewise H2 errors show about O(h1/2) convergence in all norms, slightly higher than optimal.The norms are computed on the discrete geometry, using straight segments for simplicity.Source: §5.2 · p. 10
- Convergence curves for the C0 and C1 boundary approximations are very close, so boundary-approximation continuity seems not crucial.Source: §5.2 · p. 10
6 Concluding Remarks
The cut finite element method for rectangular plate elements accommodates curved boundaries; analysis shows stability and optimal-order convergence.
6 Concluding Remarks
- The method accommodates curved boundaries with a rectangular plate element, and analysis shows it is stable and optimally order convergent.Numerical results indicate that continuity of the boundary approximation is not crucial, increasing the simple rectangular element’s practical applicability.Source: §6 · p. 11