Bilayer plates are compound materials that exhibit large bending deformations when exposed to environmental changes that lead to different mechanical responses in the involved materials. In this article a new numerical method which is suitable for simulating the isometric deformation induced by a given material mismatch in a bilayer plate is discussed. A dimensionally reduced formulation of the bending energy is discretized generically in an abstract setting and specified for discrete Kirchhoff triangles; convergence towards the continuous formulation is proved. A practical semi-implicit discrete gradient flow employing a linearization of the isometry constraint is proposed as an iterative method for the minimization of the bending energy; stability and a bound on the violation of the isometry constraint are proved. The incorporation of obstacles is discussed and the practical performance of the method is illustrated with numerical experiments involving the simulation of large bending deformations and investigation of contact phenomena.
We propose a new discontinuous Galerkin (dG) method for a geometrically nonlinear Kirchhoff plate model for large isometric bending deformations. The minimization problem is nonconvex due to the isometry constraint. We present a practical discrete gradient flow that decreases the energy and computes discrete minimizers that satisfy a prescribed discrete isometry defect. We prove $Γ$-convergence of the discrete energies and discrete global minimizers. We document the flexibility and accuracy of the dG method with several numerical experiments.
We present a computational model of thin elastic bilayers that undergo large bending isometric deformations when actuated by non-mechanical stimuli. We propose a discontinuous Galerkin approximation of the variational formulation discussed in Bartels et al. (2018). We showcase its advantages and good computational performance with configurations of interest in both engineering and medicine and either Dirichlet or free boundary conditions.
We develop and analyze an ultraweak variational formulation of the Reissner-Mindlin plate bending model both for the clamped and the soft simply supported cases. We prove well-posedness of the formulation, uniformly with respect to the plate thickness $t$. We also prove weak convergence of the Reissner-Mindlin solution to the solution of the corresponding Kirchhoff-Love model when $t\to 0$. Based on the ultraweak formulation, we introduce a discretization of the discontinuous Petrov-Galerkin type with optimal test functions (DPG) and prove its uniform quasi-optimal convergence. Our theory cove…
We present two new recovery-based a posteriori error estimates for the Hellan--Herrmann--Johnson method in Kirchhoff--Love plate theory. The first error estimator uses a postprocessed deflection and controls the $L^2$ moment error and the discrete $H^2$ deflection error. The second one controls the $L^2\times H^1$ total error and utilizes superconvergent postprocessed moment field and deflection. The effectiveness of the theoretical results is numerically validated in several experiments.
Inspired by recent results on self-avoiding inextensible curves, we propose and experimentally investigate a numerical method for simulating isometric plate bending without self-intersections. We consider a nonlinear two-dimensional Kirchhoff plate model which is augmented via addition of a tangent-point energy. The resulting continuous model energy is finite if and only if the corresponding deformation is injective, i.e. neither includes self-intersections nor self-contact. We propose a finite element method method based on discrete Kirchhoff triangles for the spatial discretization and emplo…
In this work we propose a discretisation method for the Reissner--Mindlin plate bending problem in primitive variables that supports general polygonal meshes and arbitrary order. The method is inspired by a two-dimensional discrete de Rham complex for which key commutation properties hold that enable the cancellation of the contribution to the error linked to the enforcement of the Kirchhoff constraint. Denoting by $k\ge 0$ the polynomial degree for the discrete spaces and by $h$ the meshsize, we derive for the proposed method an error estimate in $h^{k+1}$ for general $k$, as well as a lockin…
In this paper, we propose and analyze the extrapolation method and asymptotically exact a posterior error estimate for eigenvalues of the Morley element. We establish an asymptotic expansion of eigenvalues, and prove an optimal result for this expansion and the corresponding extrapolation method. We also design an asymptotically exact a posterior error estimate and propose new approximate eigenvalues with higher accuracy by utilizing this a posteriori error estimate. Finally, several numerical experiments are considered to confirm the theoretical results and compare the performance of the prop…
Leibniz Universität Hannover（汉诺威莱布尼茨大学）；Bauhaus-Universität Weimar（魏玛包豪斯大学）；Tongji University（同济大学）
In this paper, a deep collocation method (DCM) for thin plate bending problems is proposed. This method takes advantage of computational graphs and backpropagation algorithms involved in deep learning. Besides, the proposed DCM is based on a feedforward deep neural network (DNN) and differs from most previous applications of deep learning for mechanical problems. First, batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries. A loss function is built with the aim that the governing partial differential equations (PDEs) of Kirchhoff …
Static bending and free vibration of cross ply laminated composite plates using NURBS based finite element method and unified formulation
The University of New South Wales（新南威尔士大学）；University of Science, HCMC（胡志明市科学大学）；Faculdade de Engenharia da Universidade do Porto（波尔图大学工程学院）；King Abdulaziz University（阿卜杜勒阿齐兹国王大学）；Politecnico di Torino（都灵理工大学）
Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. In the fifth paper, the usual structural analysis of plates on an elastic foundation is extended to a thorough multiscale analysis for a system of a fourth order linear differential equation (for transverse displacement of the plate) and a second order linear differential equation (for the stress function), where general boundary conditions and a wide spectrum of model parameters are prescribed. For this purpose, the solution function each is…
Analysis of the Discontinuous Petrov-Galerkin Method with Optimal Test Functions for the Reissner-Mindlin Plate Bending Model
King Abdullah University of Science and Technology (KAUST) Center for Numerical Porous Media（阿卜杜拉国王科技大学多孔数值研究中心）；Aalto University School of Engineering Department of Civil and Structural Engineering（阿尔托大学土木与结构工程学院）
We present a discontinuous Petrov-Galerkin (DPG) method with optimal test functions for the Reissner-Mindlin plate bending model. Our method is based on a variational formulation that utilizes a Helmholtz decomposition of the shear force. It produces approximations of the primitive variables and the bending moments. For any canonical selection of boundary conditions the method converges quasi-optimally. In the case of hard-clamped convex plates, we prove that the lowest-order scheme is locking free. Several numerical experiments confirm our results.
Multigrid Methods for Hellan-Herrmann-Johnson Mixed Method of Kirchhoff Plate Bending Problems
University of California at Irvine（加州大学欧文分校）；Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology（北京工业大学北京科学计算与工程研究所）；Peking University（北京大学）；College of Mathematics and Information Science, Wenzhou University（温州大学数学与信息科学学院）
In this paper, we study an elastic bilayer plate composed of a nematic liquid crystal elastomer in the top layer and a nonlinearly elastic material in the bottom layer. While the bottom layer is assumed to be stress-free in the flat reference configuration, the top layer features an eigenstrain that depends on the local liquid crystal orientation. As a consequence, the plate shows non-flat deformations in equilibrium with a geometry that non-trivially depends on the relative thickness and shape of the plate, material parameters, boundary conditions for the deformation, and anchorings of the li…
Leibniz Universität Hannover（莱布尼茨汉诺威大学）；Tongji University（同济大学）；King Saud University（沙特国王大学）；Ton Duc Thang University（同奈大学）
In this paper, we present a deep autoencoder based energy method (DAEM) for the bending, vibration and buckling analysis of Kirchhoff plates. The DAEM exploits the higher order continuity of the DAEM and integrates a deep autoencoder and the minimum total potential principle in one framework yielding an unsupervised feature learning method. The DAEM is a specific type of feedforward deep neural network (DNN) and can also serve as function approximator. With robust feature extraction capacity, the DAEM can more efficiently identify patterns behind the whole energy system, such as the field vari…
In this work we develop new finite element discretisations of the shear-deformable Reissner--Mindlin plate problem based on the Hellinger-Reissner principle of symmetric stresses. Specifically, we use conforming Hu-Zhang elements to discretise the bending moments in the space of symmetric square integrable fields with a square integrable divergence $\boldsymbol{M} \in \mathcal{HZ} \subset H^{\mathrm{sym}}(\mathrm{Div})$. The latter results in highly accurate approximations of the bending moments $\boldsymbol{M}$ and in the rotation field being in the discontinuous Lebesgue space $\boldsymbolϕ …
A reduced model for large deformations of prestrained plates consists of minimizing a second order bending energy subject to a nonconvex metric constraint. The former involves the second fundamental form of the middle plate and the later is a restriction on its first fundamental form. We discuss a formal derivation of this reduced model along with an equivalent formulation that makes it amenable computationally. We propose a local discontinuous Galerkin (LDG) finite element approach that hinges on the notion of reconstructed Hessian. We design discrete gradient flows to minimize the ensuing no…
A local discontinuous Galerkin (LDG) method for approximating large deformations of prestrained plates is introduced and tested on several insightful numerical examples in our previous computational work. This paper presents a numerical analysis of this LDG method, focusing on the free boundary case. The problem consists of minimizing a fourth order bending energy subject to a nonlinear and nonconvex metric constraint. The energy is discretized using LDG and a discrete gradient flow is used for computing discrete minimizers. We first show $Γ$-convergence of the discrete energy to the continuou…
Analysis of functionally graded material plates using triangular elements with cell-based smoothed discrete shear gap method
The University of New South Wales（新南威尔士大学）；Faculdade de Engenharia da Universidade do Porto（波尔图大学工程学院）；King Abdulaziz University（阿卜杜勒阿齐兹国王大学）；Cardiff University（卡迪夫大学）；Politecnico di Torino（都灵理工大学）；Kafrelsheikh University（凯弗雷舒赫大学）
