Tom Gustafsson, Rolf Stenberg, Juha Videman
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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…
Nitsche's method is a well-established approach for weak enforcement of boundary conditions for partial differential equations (PDEs). It has many desirable properties, including the preservation of variational consistency and the fact that it yields symmetric, positive-definite discrete linear systems that are not overly ill-conditioned. In recent years, the method has gained in popularity in a number of areas, including isogeometric analysis, immersed methods, and contact mechanics. However, arriving at a formulation based on Nitsche's method can be a mathematically arduous process, especial…
Nitsche's method method for mixed dimensional analysis: conforming and non-conforming continuum-beam and continuum-plate coupling
本文提出一种 Nitsche 方法，用于解决固体与梁、板等不同力学模型在协调与非协调情况下的耦合问题，支持 Lagrange 单元与高阶 NURBS 等几何单元，适用于多维结构分析。
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…
A posteriori estimates for conforming Kirchhoff plate elements
本文研究了Kirchhoff板弯曲问题的后验估计，考虑了混合边界条件和多种载荷情况，并通过大量数值计算验证了估计器的有效性。
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.
In this paper, we use isogeometric Kirchhoff plates to approximate composite laminates adopting the classical laminate plate theory. Both isogeometric Galerkin and collocation formulations are considered. Within this framework, interlaminar stresses are recovered through an effective post-processing technique based on the direct imposition of equilibrium in strong form, relying on the accuracy and the higher continuity typically granted by isogeometric discretizations. The effectiveness of the proposed approach is proven by extensive numerical tests.
A family of C^0 finite elements for Kirchhoff plates I: Error analysis
本文提出一种基于稳定化方法和自由边界处理技术的C^0有限元位移公式，用于Kirchhoff板模型，并推导了最优先验和后验误差估计。
Additive Schwarz Preconditioners for the Obstacle Problem of Clamped Kirchhoff Plates
本文研究了压紧柯西板障碍问题的离散变分不等式，通过加性Schwarz预条件子提升求解效率，并展示数值结果验证理论分析。
Continuous/Discontinuous Finite Element Modelling of Kirchhoff Plate Structures in $\mathbb{R}^3$ Using Tangential Differential Calculus
本文提出了一种基于切向微分学的连续/不连续有限元方法，用于建模Kirchhoff板结构的平面膜变形，并采用C^0元进行板和膜变形的离散化。
An A Posteriori Analysis of C^0 Interior Penalty Methods for the Obstacle Problem of Clamped Kirchhoff Plates
本文研究了C^0内点惩罚法在固定边Kirchhoff板障碍问题中的后验分析，提出了一种基于残差的误差估计器，并证明其可靠性和有效性，通过数值实验验证了该算法在二次和三次C^0内点惩罚法中的最优性能。
This work focuses on the development of a super-penalty strategy based on the $L^2$-projection of suitable coupling terms to achieve $C^1$-continuity between non-conforming multi-patch isogeometric Kirchhoff plates. In particular, the choice of penalty parameters is driven by the underlying perturbed saddle point problem from which the Lagrange multipliers are eliminated and is performed to guarantee the optimal accuracy of the method. Moreover, by construction, the method does not suffer from locking also on very coarse meshes. We demonstrate the applicability of the proposed coupling algorit…
Fully discrete DPG methods for the Kirchhoff-Love plate bending model
本文扩展了Kirchhoff-Love板弯曲问题的分析和离散化，提出了一种well-posed的公式和准最优DPG离散化方法，包含位移梯度，并构造了Fortin算子以证明最低阶离散方案的well-posed性和准最优收敛性。
A fourth-order multi-scale computational method and its convergence analysis for composite Kirchhoff plates with microscopic periodic configurations
本文提出一种四阶多尺度计算方法，用于高精度模拟具有微观周期性配置的复合柯西板，并通过理论分析和数值实验验证其收敛性和准确性。
$C^0$ Discontinuous Galerkin Methods for a Kirchhoff Plate Contact Problem
本文研究了$C^0$不连续 Galerkin 方法用于解决四阶椭圆变分不等式问题，推导了最优误差估计，并通过数值实验验证了方法的有效性。
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…
Mixed finite elements for Kirchhoff-Love plate bending
提出一种用于Kirchhoff-Love板弯曲的混合有限元方法，通过构建低维局部空间与自由度实现协调性并支持多种边界条件，证明了拟最优收敛性，推导了后验误差估计子，数值结果验证了收敛速率与自适应算法性能。
Quasi-optimal convergence rate for an adaptive hybridizable C0 discontinuous Galerkin method for Kirchhoff plates
本文提出了一种用于Kirchhoff板的自适应混合型C0不连续Galerkin方法，并建立了可靠的后验误差估计器。通过后处理弯矩和离散霍尔姆霍兹分解，建立了准正交性和离散可靠性，研究了自适应方法的收缩性质和复杂度。
Continuous Piecewise Linear Finite Elements for the Kirchhoff-Love Plate Equation
本文提出了一种连续分段线性有限元用于薄板问题，通过线性插值重构二次不连续场，利用不连续Galerkin方法求解Kirchhoff-Love板方程，并展示了三种二次函数重构方法，证明了Morley重构等价于基本板三角形，给出了先验误差估计。
Optimality of adaptive $H(\operatorname{div}\operatorname{div})$ mixed finite element methods for the Kirchhoff-Love plate bending problem
本文针对 Kirchhoff-Love 板弯曲问题，提出可靠的后验误差分析，通过构建保边界条件复形与扩展离散空间确保嵌套性，实现自适应混合有限元方法的最优性，数值实验验证了其最优收敛率。
On a new mixed formulation of Kirchhoff plates on curvilinear polygonal domains
本文提出了一种基于连续求解三个二次问题的离散化方法，用于解决Kirchhoff板弯曲问题，针对曲线多边形域的扩展提出了基于拉格朗日乘数的替代方法。
A hybrid isogeometric approach on multi-patches with applications to Kirchhoff plates and eigenvalue problems
本文提出了一种高阶惩罚技术用于等几何粘结方法，解决了第四阶问题和二阶、四阶方程的本征值问题，并展示了在多补丁设置中处理第四阶问题和自然边界条件的有效性。
Virtual Element for the Buckling Problem of Kirchhoff-Love plates
本文提出一种高阶虚拟元方法（VEM）来求解多边形网格上的四阶板屈曲特征值问题，基于Kirchhoff-Love模型建立变分公式，提出任意阶数k≥2的C^1相容虚拟元离散化方法，并利用Babuska-Osborn抽象谱近似理论证明该方法能正确近似特征值谱并给出最优阶误差估计，最后通过数值实验验证了方法的正确性和理论结果。
Boundary elements for clamped Kirchhoff--Love plates
本文提出了一种用于夹紧Kirchhoff-Love板的Galerkin边界元方法，通过分段光滑边界条件实现最优收敛性。
Scaled boundary isogeometric analysis with C1 coupling for Kirchhoff plate theory
针对多片参数化中强 C1 耦合难题，本文将比例边界等几何分析与 AS-G1 参数化结合以构造 C1 光滑基函数，并在 Kirchhoff 板及复杂修剪形状上验证其有效性。
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 …
A virtual element method for the vibration problem of Kirchhoff plates
本文提出一种虚拟元方法用于Kirchhoff板的振动问题，采用横向位移变分公式，实现H²(Ω)相容离散化，证明了谱的正确逼近和最优误差估计，通过数值实验验证了方法的可靠性。
A robust $C^0$ interior penalty method for a gradient-elastic Kirchhoff plate model
本文针对凸多边形上的梯度弹性Kirchhoff板模型，提出一种结合三角Hermite元与C^0内罚的稳健数值方法，避免高阶形函数或宏元，并证明其最优误差估计、小参数稳健性及无额外正则性假设下的收敛性。
An adaptive Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) method for multi-patch structured Kirchhoff--Love plates
提出一种自适应等几何方法，通过IETI-DP框架和拉格朗日乘子强制C1连续性，求解具有奇异顶点的平面多片域上的双调和方程，数值实验展示了最优收敛性和预条件器性能。
Analytical properties of polygonal Stokes and BGG Hessian complexes with application to Kirchhoff--Love plates
针对多边形网格上的Kirchhoff--Love板问题，提出基于BGG构造的离散Hessian复形任意阶方法，建立一致性与Poincaré不等式，证明其强制性与k+1阶收敛，数值实验验证理论。
Multigrid Methods for Hellan-Herrmann-Johnson Mixed Method of Kirchhoff Plate Bending Problems
本文提出了一种多重网格V-循环方法用于Kirchhoff板弯曲问题的Hellan-Herrmann-Johnson离散化，证明了该方法在网格尺寸变化时的统一收敛性，仅需一次平滑步骤且无需全椭圆正则性，关键在于稳定的核空间分解和加强的柯西-施瓦茨不等式。
A Hybrid High-Order method for Kirchhoff-Love plate bending problems
本文提出了一种新型混合高阶方法，用于求解源于Kirchhoff-爱因斯坦板弯曲行为的四阶椭圆问题，包括双调和方程作为特例。该方法支持在一般多边形网格上任意逼近阶数，并在每个元素内局部重现关键力学平衡关系。
A decomposition result for Kirchhoff plate bending problems and a new discretization approach
本文提出了一种新的混合变分公式推导方法，通过非标准Sobolev空间的正则分解，将四阶问题转化为三个连续求解的二阶问题，从而得到新的离散化方法。
The Immersed Boundary Conformal Method for Kirchhoff-Love and Reissner-Mindlin shells
提出 IBCM 结合高次样条、Nitsche 耦合与高阶积分，在浸入框架中准确高效分析 Kirchhoff-Love 和 Reissner-Mindlin 壳。
A stabilised finite element method for the plate obstacle problem
本文提出了一种稳定的有限元方法用于Kirchhoff板障碍问题，并利用符合$C^1$连续的有限元进行了先验和残差基的后验误差估计，通过数值实验验证了该方法在弹性与刚性障碍情况下的有效性。
A posteriori error estimates for continuous/discontinuous Galerkin approximations of the Kirchhoff-Love buckling problem
本文提出用于Kirchhoff板屈曲问题的后验误差估计，针对最小特征值及对应特征向量，考虑应力张量近似计算的影响，并提供平面应力问题的误差指标。
An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation
本文提出Kirchhoff-Love板弯曲模型的超弱变分形式，并基于该形式引入最优测试函数的断续Petrov-Galerkin型离散化方法，证明了超弱形式的适定性和DPG方案的准最优收敛性。
We consider $C^1$-continuous approximations of the Kirchhoff plate problem in combination with a mesh dependent augmented Lagrangian method on a simply supported Signorini boundary.
A posteriori error estimation for an interior penalty virtual element method of Kirchhoff plates
针对Kirchhoff板弯曲问题，本文构建了内罚虚单元法的残差型后验误差估计，推导了其可靠性与有效性，提出自适应虚单元法，数值实验验证了估计子的鲁棒性与方法的高效性。
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 geo…
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…
A Generalized Finite Element Method for the Obstacle Problem of Plates
针对夹紧Kirchhoff板的位移障碍问题，提出一种广义有限元方法，推导最优误差估计并通过数值结果验证方法性能。
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…
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…
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 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…
Two-Scale Finite Element Approximation of a Homogenized Plate Model
针对微结构薄板的均匀化降维模型，论文结合三维单胞问题的多仿射有限元与二维宏观问题的 DKT 离散，证明收敛性并通过数值及纸张实验验证。
Bilayer Plates: Model Reduction, $Γ$-Convergent Finite Element Approximation and Discrete Gradient Flow
本文研究双层板的弯曲机制，通过小的外部诱导晶格不匹配实现大变形，提出基于Kirchhoff四边形的离散化方法，并证明其Γ-收敛性，通过数值实验验证了能量下降迭代方法的收敛性和降阶模型的性能。
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 discr…
Virtual element methods for Biot-Kirchhoff poroelasticity
本文针对可变形多孔板的 Biot-Kirchhoff 固流耦合问题，提出任意阶协调与非协调虚拟元方法，构造满足正交和最佳逼近性质的增强算子，并证明先验与后验误差估计的鲁棒性，通过数值算例验证了理论结果。
An Interior Penalty coupling strategy for Isogeometric non-conformal Kirchhoff-Love shell patches
本研究提出一种基于对称内罚法的等几何非共形Kirchhoff-Love壳片耦合策略，采用变分一致通量表达式，经多类数值实验验证收敛性接近理论最优，可适配工业复杂结构场景。
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…
Numerical simulation of a fine-tunable Föppl-von Kármán model for foldable and bilayer plates
本研究提出一种基于解耦梯度流的数值方案，结合P1有限元与离散基尔霍夫三角形单元离散化，实现双层板Föppl-von Kármán模型，可分析预应变相关低能构型、曲率反转及可折叠纸板弹性响应等问题。
Unified numerical analysis for thermoelastic diffusion and thermo-poroelasticity of thin plates
研究薄板热弹性扩散和热孔弹性的耦合双曲-抛物系统，用伽辽金方法建立可解性，采用多种数值方法离散，证明全离散格式收敛，通过数值实验验证二维模型有效性及理论收敛率。
Finite element simulation of nonlinear bending models for thin elastic rods and plates
本文讨论了薄弹性杆和板非线性弯曲模型的有限元模拟，通过适配调和映射方法离散等距约束并利用梯度流降低弯曲能量，结合实验验证了数值方法的有效性。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。