Fleurianne Bertrand, Daniele Boffi
We study the approximation of the spectrum of least-squares operators arising from linear elasticity. We consider a two-field (stress/displacement) and a three-field (stress/displacement/vorticity) formulation; other formulations might be analyzed with similar techniques. We prove a priori estimates and we confirm the theoretical results with simple two-dimensional numerical experiments.
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In this paper we study the approximation of eigenvalues arising from the mixed Hellinger--Reissner elasticity problem by using the simple finite element using partial relaxation of $C^0$ vertex continuity of stresses introduced recently by Jun Hu and Rui Ma. We prove that the method converge when a residual type error estimator is considered and that the estimator decays optimally with respect to the number of degrees of freedom.
Immersed finite element method for eigenvalue problems in elasticity
本文提出利用浸入有限元法求解具有界面的弹性本征值问题，通过Crouzeix-Raviart P1非协调元进行高效离散，并通过谱分析方法证明了方法的稳定性和最优收敛性。
Interior penalty discontinuous Galerkin methods for the nearly incompressible elasticity eigenvalue problem with heterogeneous media
针对非均质介质近不可压缩线性弹性特征值问题，提出内部罚间断 Galerkin 方法，证明与 Lamé 常数无关的收敛性并给出可靠高效的后验误差估计子。
A locking-free mixed virtual element discretization for the two dimensional elasticity eigenvalue problem
本文提出并分析了一种混合虚拟元方法，用于二维弹性本征值问题的本征值和本征函数近似，证明了离散解算子在网格尺寸趋于零时收敛到连续对应项，并通过数值实验验证了方法的无锁性和准确逼近谱的能力。
Partition of unity a posteriori error estimators for the linear elasticity eigenvalue problem
本文针对线性弹性特征值问题，提出并检验了基于单位分解和局部残差的后验误差估计器，在不同边界条件下验证了其可靠性与效率，数值实验显示其效率指数优于标准残差估计器。
A method of estimating all eigenvalues of a preconditioned discretized scalar diffusion operator with Dirichlet boundary conditions has been recently introduced in T. Gergelits, K.A. Mardal, B.F. Nielsen, Z. Strakoš: Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator, SIAM Journal on Numerical Analysis 57(3) (2019), 1369-1394. Motivated by this paper, we offer a slightly different approach that extends the previous results in some directions. Namely, we provide bounds on all increasingly ordered eigenvalues of a general diffusion or elastici…
In this paper we study Steklov eigenvalues for the Lamé operator which arise in the theory of linear elasticity. In this eigenproblem the spectral parameter appears in a Robin boundary condition, linking the traction and the displacement. To establish the existence of a countable spectrum for this problem, we present an extension of Korn's inequality. We also show that a proposed conforming Galerkin scheme provides convergent approximations to the true eigenvalues. A standard finite element method is used to conduct numerical experiments on 2D and 3D domains to support our theoretical findings…
Finite element analysis of the nearly incompressible linear elasticity eigenvalue problem with variable coefficients
本文分析弹性–Stokes特征值问题，采用inf-sup稳定有限元并证明适定性，给出与$\lambda$无关的最优特征函数和双倍阶特征值误差估计及后验估计子。
We introduce a pure--stress formulation of the elasticity eigenvalue problem with mixed boundary conditions. We propose an H(div)-based discontinuous Galerkin method that imposes strongly the symmetry of the stress for the discretization of the eigenproblem. Under appropriate assumptions on the mesh and the degree of polynomial approximation, we demonstrate the spectral correctness of the discrete scheme and derive optimal rates of convergence for eigenvalues and eigenfunctions. Finally, we provide numerical examples in two and three dimensions.
In this paper we provide some more details on the numerical analysis and we present some enlightening numerical results related to the spectrum of a finite element least-squares approximation of the linear elasticity formulation introduced recently. We show that, although the formulation is robust in the incompressible limit for the source problem, its spectrum is strongly dependent on the Lamé parameters and on the underlying mesh.
We continue the investigation on the spectrum of operators arising from the discretization of partial differential equations. In this paper we consider a three field formulation recently introduced for the finite element least-squares approximation of linear elasticity. We discuss in particular the distribution of the discrete eigenvalues in the complex plane and how they approximate the positive real eigenvalues of the continuous problem. The dependence of the spectrum on the Lamé parameters is considered as well and its behavior when approaching the incompressible limit.
A Least Squares Method for Linear Elasticity using A Patch Reconstructed Space
本文提出了一种不连续的最小二乘有限元方法来求解线性弹性问题，通过仅每个元素一个未知数的拼接重建得到近似空间，并基于不连续近似在内部面处具有法向连续性，应用L2范数最小二乘原理到应力-位移公式，获得了能量范数下的最优收敛阶，数值结果验证了误差估计，同时展示了该方法在实现上的简便性、鲁棒性和效率提升。
The purpose of this paper is to analyze a mixed method for linear elasticity eigenvalue problem, which approximates numerically the stress, displacement, and rotation, by piecewise $(k+1)$, $k$ and $(k+1)$-th degree polynomial functions ($k\geq 1$), respectively. The numerical eigenfunction of stress is symmetric. By the discrete $H^1$-stability of numerical displacement, we prove an $O(h^{k+2})$ approximation to the $L^{2}$-orthogonal projection of the eigenspace of exact displacement for the eigenvalue problem, with proper regularity assumption. Thus via postprocessing, we obtain a better ap…
In this paper we analyze a posteriori error estimates for a mixed formulation of the linear elasticity eigenvalue problem. A posteriori estimators for the nearly and perfectly compressible elasticity spectral problems are proposed. With a post-process argument, we are able to prove reliability and efficiency for the proposed estimators. The numerical method is based in Raviart-Thomas elements to approximate the pseudostress and piecewise polynomials for the displacement. We illustrate our results with numerical tests.
In this paper we analyze a mixed displacement-pseudostress formulation for the elasticity eigenvalue problem. We propose a finite element method to approximate the pseudostress tensor with Raviart-Thomas elements and the displacement with piecewise polynomials. With the aid of the classic theory for compact operators, we prove that our method is convergent and does not introduce spurious modes. Also, we obtain error estimates for the proposed method. Finally, we report some numerical tests supporting the theoretical results.
A Least Squares Weak Galerkin Framework for Linear Elasticity on Polytopal Meshes
研究线性弹性问题，提出最小二乘弱伽辽金有限元方法，利用弱微分算子处理复杂条件，避免离散inf-sup条件，公式对称正定，有几何灵活性，建立唯一性和误差估计，数值实验验证其性能。
In this paper we analyze a virtual element method for the two dimensional elasticity spectral problem allowing small edges. Under this approach, and with the aid of the theory of compact operators, we prove convergence of the proposed VEM and error estimates, where the influence of the Lamé constants is presented. We present a series of numerical tests to assess the performance of the method where we analyze the effects of the Poisson ratio on the computation of the order of convergence, together with the effects of the stabilization term on the arising of spurious eigenvalues.
The least squares RBF-PU method for linear elasticity in the diaphragm geometry
针对薄膈肌几何结构的线性弹性问题，采用不拟合最小二乘径向基函数单位分解方法进行求解，推导了其高阶收敛的理论证明，可用于模拟膈肌变形以辅助理解机械通气相关并发症。
Mini mixed finite element method for nearly incompressible linear elasticity problems
本文提出基于Mini单元的混合有限元方法，解决近不可压缩线性弹性问题的锁现象，开发非嵌套增量子空间算法，经理论分析和数值实验验证其无锁性与收敛鲁棒性。
The direct-line method for forward and inverse linear elasticity problems of composite materials in general domains with multiple singularities
本文提出结合域分解与直接线方法，解决通用领域复合材料正反向线弹性问题，能处理多重奇点并重建拉梅系数。
In this paper, a discontinuous Galerkin finite element method of Nitsche's version for the Steklov eigenvalue problem in linear elasticity is presented. The a priori error estimates are analyzed under a low regularity condition, and the robustness with respect to nearly incompressible materials (locking-free) is proven. Furthermore, some numerical experiments are reported to show the effectiveness and robustness of the proposed method.
The Least Squares Finite Element Method for Elasticity Interface Problem on Unfitted Mesh
针对非拟合网格上的线性弹性界面问题，提出两类基于不同范数的最小二乘有限元方法，结合幽灵罚技术保证稳定性，推导收敛阶与条件数估计，数值实验验证了方法的精度与鲁棒性。
Mixed methods and lower eigenvalue bounds
本文提出利用混合有限元方法为偏微分算子相关的对称正定特征值问题提供有保证的特征值下界，基于经典相容性条件和紧嵌入局部常数，适用于标量二阶椭圆算子、线弹性和Steklov特征值问题。
Mixed discontinuous Galerkin approximation of the elasticity eigenproblem
本文提出一种用于弹性本征问题混合形式的不连续伽辽金方法，该方法具有降低对称性。分析显示，由于源算子非紧致且方案非一致，离散本征问题不遵循标准谱近似框架。本文证明了该方法能正确近似谱并给出本征值和本征函数的渐近误差估计，通过数值测试验证了理论结果。
In this paper we discuss spectral properties of operators associated with the least-squares finite element approximation of elliptic partial differential equations. The convergence of the discrete eigenvalues and eigenfunctions towards the corresponding continuous eigenmodes is studied and analyzed with the help of appropriate $L^2$ error estimates. A priori and a posteriori estimates are proved.
Least squares spectral element formulation of eigenvalue problems with/without interface : the one dimensional example
提出最小二乘谱元法求解一维特征值问题，涵盖无界面与有界面情形，分析收敛性并展示不同跳跃条件的数值实验。
Robust preconditioning for stochastic Galerkin formulations of parameter-dependent nearly incompressible linear elasticity equations
本文研究了具有不确定空间变化杨氏模量的近不可压缩线性弹性问题，提出了一种新的三场混合变分公式，并利用随机伽辽金混合有限元技术进行近似，同时引入了适用于最小残差法（MINRES）的新型预处理器，证明了预处理后的系统特征值界与离散参数和泊松比无关。
A computational approach to identify the material parameters of the relaxed micromorphic model
本研究提出一种计算方法，通过能量最小二乘拟合确定松弛微形态广义连续介质模型的微观弹性张量与特征长度参数，并对比了该模型与Cosserat-微极模型、经典微形态模型的拟合效果。
We study the numerical approximation of the inverse scattering problem in the two-dimensional homogeneous isotropic linear elasticity with an unknown linear load given by a square matrix. For both backscattering data and fixed-angle scattering data, we show how to obtain numerical approximations of the so-called Born approximations and propose new iterative algorithms that provide sequences of approximations to the unknown load. Numerical evidences of the convergence for not too large loads are also given.
A Preconditioned Riemannian Gauss-Newton Method for Least Squares Inverse Eigenvalue Problems
本文提出了一种Riemannian近似广义牛顿法用于最小二乘反特征值问题，分析了其全局和局部收敛性，并提出预条件共轭梯度法求解Riemannian广义牛顿方程，通过数值实验验证了方法的有效性。
A locking free mixed FEM based on a pure pseudostress based formulation for the elasticity eigenproblem
研究二维和三维弹性特征值问题的新型无锁混合公式，用拟应力张量表示，无需强制对称性。基于标准有限元族张量形式离散空间，证明收敛性等，进行后验误差分析并通过数值测试验证理论结果。
The present paper introduces the analysis of the eigenvalue problem for the elasticity equations when the so called Navier-Lamé system is considered. Such a system introduces the displacement, rotation and pressure of some linear and elastic structure. The analysis of the spectral problem is based in the compact operators theory. A finite element method based in polynomials of degree $k\geq 1$ are considered in order to approximate the eigenfrequencies and eigenfunctions of the system. Convergence and error estimate are presented. An a posteriori error analysis is performed, where the reliabil…
Efficient parameter-robust preconditioners for linear poroelasticity and elasticity in the primal formulation
针对多孔弹性离散化产生的大规模鞍点系统，提出非奇异块三角Schur补预条件子，其特征值簇半径独立于网格尺寸、时间步长和锁定参数，结合无锁定弱Galerkin有限元方法，实现了高效且参数鲁棒的快速迭代求解。
Approximation of the Maxwell eigenvalue problem in a Least-Squares setting
本文在最小二乘有限元框架下，通过将Maxwell旋度旋度方程转化为一阶系统并设计新颖的最小二乘公式，实现了特征解的有效逼近，数值测试验证了方法在奇异解情况下的良好收敛性。
Superlinear Convergence of GMRES for clustered eigenvalues and its application to least squares problems
本文分析GMRES在系数矩阵特征值聚集时的超线性收敛，利用范德蒙德矩阵推导残差上界，并应用于解释最小二乘问题中内迭代预条件GMRES的收敛。
Least-squares spectral methods for ODE eigenvalue problems
本研究提出基于最小二乘公式的ODE及算子特征值问题谱方法，将矩形广义特征值解法扩展到拟矩阵相关对象，具备基函数选择灵活、精度高的特点，还可适配特征值相关边界条件，结合积分方程重写能进一步提升精度。
A Low-Rank QTT-based Finite Element Method for Elasticity Problems
该研究提出结合量化张量列（QTT）格式与区域划分策略的低秩有限元方法，用于求解一般计算域上的二维线弹性问题，可大幅降低内存占用与秩，还通过三个算例验证了性能。
Constraint Energy Minimizing Generalized Multiscale Finite Element Method for high-contrast linear elasticity problem
本文提出一种约束能量最小化的广义多尺度有限元方法，用于解决高对比率线弹性问题，通过构造辅助多尺度空间和松弛最小化问题，提高计算精度和鲁棒性，结合数值实验验证方法有效性。
Randomized Neural Networks with Petrov-Galerkin Methods for Solving Linear Elasticity Problems
本文提出随机神经网络与Petrov-Galerkin方法（RNN-PG方法）求解线弹性问题，通过随机固定隐藏层参数并用最小二乘法确定输出层参数，同时发展混合形式以保证应力张量对称并避免锁定效应，数值实验表明其精度和效率优于有限元法、混合间断Galerkin方法和物理信息神经网络。
Computation of Transmission Eigenvalues for Elastic Waves
本文提出数值方法计算弹性波中最小的传输本征值，通过构造非线性函数和混合有限元方法验证了非线性、非自共轭、四阶问题的求解有效性。
A Riemannian Gradient Descent Method for the Least Squares Inverse Eigenvalue Problem
本文证明 Chen-Chu 的 Lift and Projection 方法等价于黎曼梯度下降，并据此提出更高效实现，在锰-6 非弹性中子散射等逆特征值问题上计算时间改进可超一个数量级。
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。
A two-dimensional fractional-order element-free Galerkin method for nonlocal elasticity and complex domain problems
提出二维分数阶无单元Galerkin方法求解分数阶微分方程，能处理复杂区域并缓解FEM缺点，通过MLS近似有效求解非局部弹性问题并经基准验证。
A Two-level Filtered Subspace Iterative Method for Generalized Hermitian Interior Eigenvalue Problems
提出TLFSI方法，结合两级与滤波子空间迭代，高效计算广义Hermitian内部特征值问题中指定区间内的所有特征对，理论分析与数值实验验证其有效性。
A four-field mixed formulation for incompressible finite elasticity
本文将质量守恒混合应力有限元方法推广至不可压缩有限弹性，通过三场Hu-Washizu原理引入位移梯度和第一Piola-Kirchhoff应力张量避免本构求逆，提出稳定化技术并验证了方法的优异性能。
An Enhanced Shifted QR Algorithm for Efficient Eigenvalue Computation of Square Non-Hermitian Matrices
本文提出一种改进的移位QR算法，显著提升非Hermitian矩阵特征值计算的收敛速度，同时保持准确性，适用于中大型及更大规模矩阵。
Cut Finite Element Methods for Linear Elasticity Problems
本文提出了一种基于固定背景网格的高阶有限元方法，用于解决线弹性问题，通过稳定化项控制边界附近有限元函数导数的跳跃，并推导了误差估计和质量矩阵和刚度矩阵的条件数估计，应用该方法解决位移问题、频率响应问题和特征值问题，并展示了薄壁弯曲结构的数值示例，最后开发了用于纤维增强材料的切割有限元方法，其中纤维模型为桁架和欧拉-伯努利梁的叠加，梁模型导致四阶问题，通过限制体积有限元空间到纤维并结合连续/不连续有限元公式进行离散化，体积材料稳定问题，无需额外稳定化项。
A new low-cost meshfree method for two and three dimensional problems in elasticity
本文提出了一种改进的直接无网格局部 Petrov-Galerkin 方法，用于弹性静力学问题，通过提高计算效率，简化了局部弱形式的积分过程，展示了其在二维和三维弹性问题中的有效性。
Reduced basis isogeometric mortar approximations for eigenvalue problems in vibroacoustics
本文提出利用降阶基底方法模拟小提琴桥的振动，基于正交各向异性线弹性方程的本征值问题，结合等几何接合方法处理复杂几何结构，通过多查询贪心策略选择快照，以求解对称正定矩阵的降阶本征值问题。
We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized nonlinear elliptic partial differential equations (PDEs) based on overlapping subdomains. Our approach reads as a constrained optimization statement that penalizes the jump at the components' interfaces subject to the approximate satisfaction of the PDE in each local subdomain. Furthermore, the approach relies on the decomposition of the local states into a port component -- associated with the solution on interior boundaries -- and a bubble component that vanishes at ports: since the bubble…
Nonconforming tetrahedral mixed finite elements for elasticity
本文提出一种基于四面体网格的非协调有限元逼近可积散度的对称张量空间，用于应力近似，并与分段线性向量场位移空间结合，形成稳定的混合有限元方法，该方法比任何稳定的协调混合有限元方法更简单，且应力和位移均线性收敛。
A characterization for the Fourier multipliers and eigenvalues of linear peridynamic operators is provided. The analysis is presented for state-based peridynamic operators for isotropic homogeneous media in any spatial dimension. We provide explicit formulas for the eigenvalues in terms of the space dimension, the nonlocal parameters, and the material properties. The approach we follow is based on the Fourier multiplier analysis developed for the nonlocal Laplacian. The Fourier multipliers of linear peridynamic operators are second-order tensor fields, which are given through integral represen…
In this paper, a two-scale approach for elastic shape optimization of fine-scale structures in additive manufacturing is investigated. To this end, a free material optimization is performed on the macro-scale using elasticity tensors in a set of microscopically realizable tensors. A database of these realizable tensors and their cost values is obtained with a shape and topology optimization on microscopic cells, working within a fixed set of elasticity tensors samples. This microscopic optimization takes into account manufacturability constraints via predefined material bridges to neighbouring…
In this paper, we analyze and implement the Dirichlet spectral-Galerkin method for approximating simply supported vibrating plate eigenvalues with variable coefficients. This is a Galerkin approximation that uses the approximation space that is the span of finitely many Dirichlet eigenfunctions for the Laplacian. Convergence and error analysis for this method is presented for two and three dimensions. Here we will assume that the domain has either a smooth or Lipschitz boundary with no reentrant corners. An important component of the error analysis is Weyl's law for the Dirichlet eigenvalues. …
Non-Linear Least-Squares Optimization of Rational Filters for the Solution of Interior Eigenvalue Problems
本文提出基于非凸加权最小二乘方案的有理滤波器优化框架，结合FEAST库在多个基准问题中表现优异，通过利用对称性优化过程、梯度下降算法、起始位置选择及约束优化方案提升本征值求解性能。
Numerical methods for rectangular multiparameter eigenvalue problems, with applications to finding optimal ARMA and LTI models
针对新兴的矩形多参数特征值问题（RMEP），提出将其转化为标准多参数特征值问题（MEP）的数值求解方法，给出解的数量与新线性化方案，计算性能优于现有块Macaulay方法，可用于最优ARMA和LTI模型求解。
On Parameter Identification in Three-Dimensional Elasticity and Discretisation with Physics-Informed Neural Networks
本文提出了一种基于物理信息神经网络的三维弹性力学参数识别方法，通过同时估计状态和参数实现稳定逼近，并与传统网格方法进行比较。
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