A family of continuous piecewise linear finite elements for thin plate problems is presented. We use standard linear interpolation of the deflection field to reconstruct a discontinuous piecewise quadratic deflection field. This allows us to use discontinuous Galerkin methods for the Kirchhoff-Love plate equation. Three example reconstructions of quadratic functions from linear interpolation triangles are presented: a reconstruction using Morley basis functions, a fully quadratic reconstruction, and a more general least squares approach to a fully quadratic reconstruction. The Morley reconstruction is shown to be equivalent to the Basic Plate Triangle. Given a condition on the reconstruction operator, a priori error estimates are proved in energy norm and $L^2$ norm. Numerical results indicate that the Morley reconstruction/Basic Plate Triangle does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence.
This work focuses on the development of a posteriori error estimates for fourth-order, elliptic, partial differential equations. In particular, we propose a novel algorithm to steer an adaptive simulation in the context of Kirchhoff plates and Kirchhoff-Love shells by exploiting the local refinement capabilities of hierarchical B-splines. The method is based on the solution of an auxiliary residual-like variational problem, formulated by means of a space of localized spline functions. This space is characterized by $C^1$ continuous B-splines with compact support on each active element of the h…
In this work we develop a novel fully discrete version of the plates complex, an exact Hilbert complex relevant for the mixed formulation of fourth-order problems. The derivation of the discrete complex follows the discrete de Rham paradigm, leading to an arbitrary-order construction that applies to meshes composed of general polygonal elements. The discrete plates complex is then used to derive a novel numerical scheme for Kirchhoff--Love plates, for which a full stability and convergence analysis are performed. Extensive numerical tests complete the exposition.
Efficient and accurate numerical algorithms are developed to solve a generalized Kirchhoff-Love plate model subject to three common physical boundary conditions: (i) clamped; (ii) simply supported; and (iii) free. We solve the model equation by discretizing the spatial derivatives using second-order finite-difference schemes, and then advancing the semi-discrete problem in time with either an explicit predictor-corrector or an implicit Newmark-Beta time-stepping algorithm. Stability analysis is conducted for the schemes and the results are used to determine stable time steps in practice. A ser…
Stable and accurate modeling of thin shells requires proper enforcement of all types of boundary conditions. Unfortunately, for Kirchhoff-Love shells, strong enforcement of Dirichlet boundary conditions is difficult because both functional and derivative boundary conditions must be applied. A popular alternative is to employ Nitsche's method to weakly enforce all boundary conditions. However, while many Nitsche-based formulations have been proposed in the literature, they lack comprehensive error analyses and verifications. In fact, existing formulations are variationally inconsistent and yiel…
Analytical properties of polygonal Stokes and BGG Hessian complexes with application to Kirchhoff--Love plates
IMAG, Univ. Montpellier, CNRS（蒙彼利埃大学，法国国家科学研究中心）；School of Mathematics, Monash University（莫纳什大学数学学院）；Mathematical Institute, University of Oxford（牛津大学数学研究所）
The inf-sup condition, also called the Ladyzhenskaya--Babu\v ska--Brezzi (LBB) condition, ensures the existence, uniqueness and well-posedness of a saddle point problem, relative to a partial differential equation. Discretization by the finite element method gives the discrete problem which must satisfy the discrete inf-sup condition. But, depending on the choice of finite elements, the discrete condition may fail. This paper attempts to explain why it fails from an engineer's perspective, and reviews current methods to work around this failure. The last part recalls the mathematical bases.
Penalty methods have proven to be particularly effective for achieving the required $C^1$-continuity in the context of multi-patch isogeometric Kirchhoff-Love shells. Due to their conceptual simplicity, these algorithms are readily applicable to the displacement and rotational coupling of trimmed, non-conforming surfaces. However, the accuracy of the resulting solution depends heavily on the choice of penalty parameters. Furthermore, the selection of these coefficients is generally problem-dependent and is based on a heuristic approach. Moreover, developing a penalty-like procedure that avoids…
We present a comprehensive rotation-free Kirchhoff-Love (KL) shell formulation for peridynamics (PD) that is capable of modeling large elasto-plastic deformations and fracture in thin-walled structures. To remove the need for a predefined global parametric domain, Principal Component Analysis is employed in a meshfree setting to develop a local parameterization of the shell midsurface. The KL shell kinematics is utilized to develop a correspondence-based PD formulation. A bond-stabilization technique is employed to naturally achieve stability of the discrete solution. Only the mid-surface velo…
An isogeometric Galerkin approach for analysing the free vibrations of piezoelectric shells is presented. The shell kinematics is specialised to infinitesimal deformations and follow the Kirchhoff-Love hypothesis. Both the geometry and physical fields are discretised using Catmull-Clark subdivision bases. It provides the required C1 continuous discretisation for the Kirchhoff-Love theory. The crystalline structure of piezoelectric materials is described using an anisotropic constitutive relation. Hamilton's variational principle is applied to the dynamic analysis to derive the weak form of the…
We introduce a Nitsche's method for the numerical approximation of the Kirchhoff-Love plate equation under general Robin-type boundary conditions. We analyze the method by presenting a priori and a posteriori error estimates in mesh-dependent norms. Several numerical examples are given to validate the approach and demonstrate its properties.
In this work, we propose and develop efficient and accurate numerical methods for solving the Kirchhoff-Love plate model in domains with complex geometries. The algorithms proposed here employ curvilinear finite-difference methods for spatial discretization of the governing PDEs on general composite overlapping grids. The coupling of different components of the composite overlapping grid is through numerical interpolations. However, interpolations introduce perturbation to the finite-difference discretization, which causes numerical instability for time-stepping schemes used to advance the res…
We consider an optimal control problem for the steady-state Kirchhoff equation, a prototype for nonlocal partial differential equations, different from fractional powers of closed operators. Existence and uniqueness of solutions of the state equation, existence of global optimal solutions, differentiability of the control-to-state map and first-order necessary optimality conditions are established. The aforementioned results require the controls to be functions in $H^1$ and subject to pointwise upper and lower bounds. In order to obtain the Newton differentiability of the optimality conditions…
A four-field mixed variational principle is proposed for large deformation analysis of Kirchhoff rods with a $C^0$ mixed FE approximations. The core idea behind the approach is to introduce a one-parameter family of points (the centerline) and a separate one-parameter family of orthonormal frames (the Cartan moving frame) that are specified independently. The curvature and torsion of the curve are related to the relative rotation of neighboring frames. The relationship between the frame and the centerline is then enforced at the solution step using a Lagrange multiplier (which plays the role o…
We study the finite element approximation of the Kirchhoff plate equation on domains with curved boundaries using the Hellan-Herrmann-Johnson (HHJ) method. We prove optimal convergence on domains with piecewise $C^{k+1}$ boundary for $k \geq 1$ when using a parametric (curved) HHJ space. Computational results are given that demonstrate optimal convergence and how convergence degrades when curved triangles of insufficient polynomial degree are used. Moreover, we show that the lowest order HHJ method on a polygonal approximation of the disk does not succumb to the classic Babuška paradox, highli…
A method to simulate orthotropic behaviour in thin shell finite elements is proposed. The approach is based on the transformation of shape function derivatives, resulting in a new orthogonal basis aligned to a specified preferred direction for all elements. This transformation is carried out solely in the undeformed state leaving minimal additional impact on the computational effort expended to simulate orthotropic materials compared to isotropic, resulting in a straightforward and highly efficient implementation. This method is implemented for rotation-free triangular shells using the finite …
Unified numerical analysis for thermoelastic diffusion and thermo-poroelasticity of thin plates
Indian Institute of Technology Bombay（印度理工学院孟买分校）；Monash University（蒙纳士大学）；Universidad Adventista de Chile（智利联合基督大学）；Sechenov First Moscow State Medical University（谢切诺夫第一莫斯科国立医科大学）
