Computer Methods in Applied Mechanics and Engineering最新文献_第5页

Adaptive finite element phase-field simulation of dynamic brittle fracture 动态脆性断裂的自适应有限元相场模拟

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-16 DOI: 10.1016/j.cma.2025.118642

Hossein Saberi, Alexander Düster

In finite element analysis, mesh refinement is typically employed to improve accuracy by increasing spatial resolution in regions with steep solution gradients. This study presents an adaptive mesh refinement technique for dynamic fracture simulation based on the phase-field method. A multi-level node distance function is introduced using the phase-field variable to control mesh density. As damage evolves, the nodal spacing is adaptively refined according to the prescribed maximum spacing; whenever the computed distance exceeds this threshold, a new field node is introduced at the element center, ensuring the mesh evolves consistently with crack propagation. In addition, material damping effects are incorporated into the phase-field formulation to capture realistic dynamic fracture responses. Time integration is investigated using the Newmark scheme, the generalized-α method, and the backward implicit approach. The results indicate that, while the first two schemes are commonly applied, the backward implicit method provides superior stability in dynamic simulations. Furthermore, a staggered solution strategy is proposed in which both displacement and phase-field variables are iteratively and consistently updated within each solution step. The effectiveness of the proposed methodology is demonstrated through three numerical examples. The responses of elastic energy, kinetic energy, and energy dissipated by crack propagation are evaluated, together with the effects of material damping. The results confirm that the presented approach significantly improves computational efficiency while preserving accuracy and exhibits robust convergence behavior in highly dynamic fracture simulations.

在有限元分析中，网格细化通常是通过提高求解梯度较大区域的空间分辨率来提高精度。提出了一种基于相场法的动态裂缝模拟自适应网格细化技术。引入了多级节点距离函数，利用相场变量控制网格密度。随着损伤的发展，节点间距根据规定的最大间距自适应细化；当计算的距离超过该阈值时，在单元中心引入一个新的场节点，以确保网格的演化与裂纹扩展一致。此外，材料阻尼效应被纳入相场公式，以捕捉真实的动态断裂响应。利用Newmark格式、广义-α方法和后向隐式方法研究了时间积分。结果表明，与前两种方法相比，后向隐式方法在动态仿真中具有更好的稳定性。此外，提出了一种交错求解策略，其中位移和相场变量在每个求解步骤中迭代且一致地更新。通过三个算例验证了所提方法的有效性。计算了弹性能、动能和裂纹扩展耗散能的响应，并考虑了材料阻尼的影响。结果证实，该方法在保持精度的同时显著提高了计算效率，并在高动态裂缝模拟中表现出鲁棒的收敛性。

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引用次数: 0

A large deformation finite element analysis of SE(3)-based beams using dual algebra 用对偶代数对SE(3)基梁进行大变形有限元分析

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-16 DOI: 10.1016/j.cma.2025.118595

Jianqin Yang , Jielong Wang , Peijing Rong

The paper focuses on the formulation of a geometrically exact beam based on the dual representation of the special Euclidean group SE(3). Within the framework of dual algebra, it clarifies the relation between Euler and Wiener-Milenković dual quaternions, and proposes a unified treatment of translations and rotations using these dual quaternions. The paper designs a new interpolation scheme by operating on Euler dual quaternion, and with the application of this scheme, the objectivity of material strain measures is preserved. The effects of objectivity on the accuracy and convergence characteristics of SE(3) beam element are then investigated. Furthermore, the third-order Lie group solver based on the 2-stage Radau IIA algorithm is implemented to solve the dynamic problems of beams. Finally, the geometrically exact beam elements with the number of nodes varying from 2 to 9 are developed, and their capabilities in addressing different dynamic and static problems are verified via typical numerical examples.

本文重点研究了基于特殊欧几里得群SE(3)对偶表示的几何精确梁的表达式。在对偶代数的框架内，澄清了欧拉和维纳-米伦科维奇对偶四元数之间的关系，并提出了使用这些对偶四元数统一处理平移和旋转的方法。利用欧拉对偶四元数设计了一种新的插值格式，该格式的应用保证了材料应变测量的客观性。研究了客观性对SE(3)光束单元精度和收敛特性的影响。在此基础上，采用基于两阶段Radau IIA算法的三阶李群求解器求解梁的动力问题。最后，建立了节点数为2 ~ 9的几何精确梁单元，并通过典型数值算例验证了其解决不同动、静态问题的能力。

引用次数: 0

A peridynamic model for mixed-mode fracture and compression-shear failure in geomaterials 岩土材料混合模式断裂与压剪破坏的周动力模型

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-16 DOI: 10.1016/j.cma.2025.118657

Tao Ni , Jianfu Shao , Bernhard A. Schrefler

In this paper, we propose a tension/compression-aware ordinary state-based peridynamic (OSB-PD) formulation for geomaterials that is thermodynamically consistent and friction-capable. The force-density state is split into volumetric and deviatoric parts: the volumetric response is retained in compression and degrades only in tension, while the deviatoric response degrades in both. A Mohr-Coulomb-type friction state activates post-damage under compression to supply residual sliding resistance. An energy split into positive/negative parts with mode-dependent degradation enforces the Clausius-Duhem inequality; frictional work remains non-conservative. Softening is driven by equivalent normal and shear strains reconstructed via peridynamic differential operator method, with thresholds tied to tensile/shear strengths and Mode I/II fracture energies for transparent calibration. Benchmarks and representative problems confirm accuracy and robustness: single-edge-notched plates in tension and pure shear reproduce theoretical initiation angles and peak loads; a long-shear apparatus recovers Palmer-Rice scaling and a uniform-traction slip surface; flawed-gypsum compression captures observed crack coalescence and post-peak softening; and a pseudo-3D slope develops a continuous shear band consistent with FEM strength-reduction analysis. The framework unifies tensile, mixed-mode, and shear-dominated failure within a single, calibratable model.

在本文中，我们提出了一种张力/压缩感知的基于普通状态的岩土材料围动力学（OSB-PD）公式，该公式具有热力学一致性和摩擦能力。力密度状态分为体积响应和偏差响应两部分：体积响应在压缩状态下保持不变，仅在拉伸状态下退化，而偏差响应在两种状态下都退化。在压缩作用下，损伤后产生莫尔-库仑摩擦状态，提供残余滑动阻力。能量分裂成正/负部分并伴有模式依赖的退化强化了克劳修斯-迪昂不等式；摩擦功是非保守的。软化由等效法向应变和剪切应变驱动，通过动态微分算子方法重建，阈值与拉伸/剪切强度和I/II型断裂能相关，以便透明校准。基准和代表性问题证实了准确性和鲁棒性：单边缺口板在张力和纯剪切下再现了理论起始角和峰值载荷；长剪切装置恢复Palmer-Rice结垢和均匀牵引滑移面；有缺陷的石膏压缩捕获了观察到的裂纹合并和峰后软化；拟三维边坡发育连续剪切带，与有限元强度折减分析结果一致。框架将拉伸、混合模式和剪切主导的破坏统一在一个单一的、可校准的模型中。

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引用次数: 0

A novel finite-strain mixed isogeometric collocation formulation for hyperelastic geometrically exact beams 一种新的超弹性几何精确梁的有限应变混合等几何配置公式

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-15 DOI: 10.1016/j.cma.2025.118641

Diego Ignesti , Giulio Ferri , Alessandro Reali , Ferdinando Auricchio , Josef Kiendl , Enzo Marino

We propose a novel finite-strain, mixed isogeometric collocation formulation for hyperelastic geometrically exact beams. The model supports general three-dimensional hyperelastic materials and reconstructs the cross-sectional deformation without any modification of the fundamental kinematic assumption typically employed for geometrically exact beams. We express the finite strain components in terms of the one-dimensional geometrically exact strain measures, which allows to exploit existing SO(3)-consistent linearization procedures developed for linearly elastic materials. The governing equations in the strong form are discretized using the isogeometric collocation method (IGA-C). Through various numerical examples, we demonstrate the capability of the model to reproduce both large deformations and finite strains, including the cross-sectional ones, without introducing additional kinematic unknowns.

我们提出了一种新的超弹性几何精确梁的有限应变混合等几何配置公式。该模型支持一般的三维超弹性材料，并在不修改通常用于几何精确梁的基本运动学假设的情况下重建截面变形。我们用一维几何上精确的应变测量来表示有限应变分量，这允许利用为线弹性材料开发的现有SO(3)-一致线性化程序。采用等几何配置法（IGA-C）对强形式的控制方程进行离散化。通过各种数值实例，我们证明了该模型在不引入额外的运动学未知数的情况下再现大变形和有限应变（包括截面应变）的能力。

引用次数: 0

Scalable h-adaptive probabilistic solver for time-independent and time-dependent systems 时间无关和依赖系统的可扩展h-自适应概率解算器

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-15 DOI: 10.1016/j.cma.2025.118647

Akshay Thakur , Sawan Kumar , Matthew Zahr , Souvik Chakraborty

Solving partial differential equations (PDEs) within the framework of probabilistic numerics offers a principled approach to quantifying epistemic uncertainty arising from discretization. By leveraging Gaussian process regression and imposing the governing PDE as a constraint at a finite set of collocation points, probabilistic numerics delivers mesh-free solutions at arbitrary locations. However, the high computational cost, which scales cubically with the number of collocation points, remains a critical bottleneck, particularly for large-scale or high-dimensional problems. We propose a scalable enhancement to this paradigm through two key innovations. First, we employ the stochastic dual descent algorithm, which reduces the per-iteration complexity from cubic to linear in the number of collocation points, thereby enabling tractable inference. Second, we exploit a clustering-based active learning strategy that adaptively selects collocation points to maximize information gain while minimizing computational expense. Together, these contributions result in an h-adaptive probabilistic solver that can scale to a large number of collocation points. We demonstrate the efficacy of the proposed solver on benchmark PDEs, including two- and three-dimensional steady-state elliptic problems, as well as a time-dependent parabolic PDE formulated in a space-time setting.

在概率数值的框架内求解偏微分方程（PDEs）为量化离散化引起的认知不确定性提供了一种原则性的方法。通过利用高斯过程回归并在有限的并置点集上施加控制PDE作为约束，概率数值在任意位置提供无网格解决方案。然而，高的计算成本（随搭配点数量的三次增长）仍然是一个关键的瓶颈，特别是对于大规模或高维问题。我们建议通过两个关键创新对该范式进行可扩展的增强。首先，我们采用随机对偶下降算法，将每次迭代的复杂度从三次降低到线性，从而实现易于处理的推理。其次，我们利用基于聚类的主动学习策略，自适应地选择搭配点以最大化信息增益，同时最小化计算开销。总之，这些贡献导致了一个h-自适应概率求解器，它可以扩展到大量的并置点。我们证明了所提出的求解器在基准偏微分方程上的有效性，包括二维和三维稳态椭圆问题，以及在时空设置中制定的时间相关抛物型偏微分方程。

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引用次数: 0

Entropy stable hydrodynamic discretizations of the linearized Boltzmann equation 线性化玻尔兹曼方程的熵稳定水动力离散化

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-13 DOI: 10.1016/j.cma.2025.118650

Frimpong A. Baidoo , Torsten Keßler , Irene M. Gamba , Thomas J.R. Hughes , Michael R.A. Abdelmalik

The implied distribution functions of the Navier-Stokes-Fourier (NSF) equations and the novel Entropy Stable Extension to the Navier-Stokes-Fourier (ESE) equations are computed and compared to a numerically derived distribution function of the Linearized Boltzmann equation for the Stationary Heat Transfer and Couette flow problems in a one-dimensional channel. As the Knudsen number is reduced, both implied distribution functions better match the Linearized Boltzmann distribution especially in the interior of the domain. In most instances, the implied distribution function of the ESE equations yields significantly smaller errors than that of the NSF equations. This is consistent with the assertion that ESE equations are an improvement upon the NSF equations for rarefied gas flows.

计算了一维通道中稳态传热和Couette流动问题的Navier-Stokes-Fourier （NSF）方程的隐含分布函数和对Navier-Stokes-Fourier （ESE）方程的新的熵稳定扩展，并与线性化Boltzmann方程的数值推导分布函数进行了比较。随着Knudsen数的减小，两种隐含分布函数都能更好地匹配线性化的Boltzmann分布，特别是在域的内部。在大多数情况下，ESE方程的隐含分布函数产生的误差明显小于NSF方程的误差。这与ESE方程是对稀薄气体流动的NSF方程的改进的断言是一致的。

引用次数: 0

Real-time forecasting of chaotic dynamics from sparse data and autoencoders 基于稀疏数据和自编码器的混沌动态实时预测

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-13 DOI: 10.1016/j.cma.2025.118600

Elise Özalp , Andrea Nóvoa , Luca Magri

The real-time prediction of chaotic systems requires a nonlinear-reduced order model (ROM) to forecast the dynamics, and a stream of data from sensors to update the ROM. Data-driven ROMs are typically built with a two-step strategy: data compression onto a lower-dimensional latent space, and prediction of the temporal dynamics on it. Although these methods have proven effective, to achieve real-time prediction, there are two key challenges to overcome: (i) ROMs of chaotic systems can become numerically unstable; and (ii) sensors’ data are sparse, i.e., partial, and noisy. To overcome these challenges, we propose an augmented sequential data assimilation (DA) framework based on the Ensemble Kalman filter (EnKF) that updates the latent state of ROM by assimilating noisy and sparse measurements. We demonstrate the proposed DA-ROM framework using a ROM that consists of a convolutional autoencoder (CAE) to compresses the system’s state onto a lower-dimensional latent space, and an echo state network (ESN) formulated as a state-space model to forecast the temporal evolution on the latent space. The DA-CAE-ESN provides a numerically stable and real-time adaptive ROM. The DA-CAE-ESN is tested on spatio-temporally chaotic partial differential equations: the Kuramoto–Sivashinsky equation, and a two-dimensional Navier-Stokes equation (Kolmogorov flow). We show that the method provides accurate and stable forecasts across different levels of noise, sparsity, and sampling rates. As a by-product, the DA-CAE-ESN acts as a localization strategy that mitigates spurious correlations, which arise when applying the EnKF to high-dimensional systems. The DA-CAE-ESN provides a numerically stable method to perform real-time predictions, which opens opportunities for deploying data-driven latent models.

混沌系统的实时预测需要一个非线性降阶模型（ROM）来预测动力学，并需要来自传感器的数据流来更新ROM。数据驱动的ROM通常采用两步策略构建：将数据压缩到低维潜在空间，并预测其上的时间动力学。虽然这些方法已被证明是有效的，但要实现实时预测，有两个关键挑战需要克服：(i)混沌系统的rom可能变得数值不稳定；（ii）传感器的数据是稀疏的，即部分的和有噪声的。为了克服这些挑战，我们提出了一种基于集成卡尔曼滤波器（EnKF）的增强顺序数据同化（DA）框架，该框架通过同化噪声和稀疏测量来更新ROM的潜在状态。我们使用由卷积自编码器（CAE）组成的ROM来将系统状态压缩到低维潜在空间，并使用回声状态网络（ESN）作为状态空间模型来预测潜在空间上的时间演变，从而演示了所提出的DA-ROM框架。DA-CAE-ESN提供了一个数值稳定的实时自适应ROM。DA-CAE-ESN在时空混沌偏微分方程：Kuramoto-Sivashinsky方程和二维Navier-Stokes方程（Kolmogorov流）上进行了测试。我们表明，该方法在不同的噪声水平、稀疏度和采样率下提供了准确和稳定的预测。作为副产品，DA-CAE-ESN作为一种定位策略，可以减轻在将EnKF应用于高维系统时产生的虚假相关性。DA-CAE-ESN提供了一种数值稳定的方法来执行实时预测，这为部署数据驱动的潜在模型提供了机会。

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引用次数: 0

Large-scale topology optimisation of time-dependent thermal conduction using space-time finite elements and a parallel space-time multigrid preconditioner 基于时空有限元和平行时空多网格预调节器的大规模时变热传导拓扑优化

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-13 DOI: 10.1016/j.cma.2025.118605

Joe Alexandersen, Magnus Appel

This paper presents a novel space-time topology optimisation framework for time-dependent thermal conduction problems, aiming to significantly reduce the time-to-solution. By treating time as an additional spatial dimension, we discretise the governing equations using a stabilised continuous Galerkin space-time finite element method. The resulting large all-at-once system is solved using an iterative Krylov solver preconditioned with a parallel space-time multigrid method employing a semi-coarsening strategy. Implemented in a fully parallel computing framework, the method yields a parallel-in-time method that demonstrates excellent scalability on a distributed-memory supercomputer, solving problems up to 4.2 billion degrees of freedom. Comparative studies show up to 52 ×  speed-up over traditional time-stepping approaches, with only moderate increases in total computational cost in terms of core-hours. The framework is validated on benchmark problems with both time-constant and time-varying designs, and its flexibility is demonstrated through variations in material properties. These results establish the proposed space-time method as a promising approach for large-scale time-dependent topology optimisation in thermal applications.

本文提出了一种新的时变热传导问题的时空拓扑优化框架，旨在显著缩短求解时间。通过将时间作为一个额外的空间维度，我们使用稳定连续伽辽金时空有限元方法离散控制方程。采用半粗化策略的平行时空多重网格法作为前置条件，利用迭代Krylov求解器求解得到的大型一次性系统。该方法在完全并行计算框架中实现，产生了一种并行实时方法，在分布式内存超级计算机上展示了出色的可扩展性，可解决高达42亿自由度的问题。比较研究表明，与传统的时间步进方法相比，加速高达52 × ，而以核心小时计算的总计算成本仅略有增加。该框架在时间常数和时变设计的基准问题上得到了验证，并通过材料性能的变化证明了其灵活性。这些结果表明，所提出的时空方法是热应用中大规模时变拓扑优化的一种有前途的方法。

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引用次数: 0

Learning a hyperelastic constitutive model from 3D experimental data 从三维实验数据中学习超弹性本构模型

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-12 DOI: 10.1016/j.cma.2025.118592

M. Bourdyot , M. Compans , R. Langlois , B. Smaniotto , E. Baranger , C. Jailin

A hyperelastic constitutive law is experimentally learned and validated from three-dimensional full-field kinematic data. 3D-printed thermoplastic polyurethane specimens were subjected to monotonic uniaxial tension within a laboratory micro-computed tomography system. Global Digital Volume Correlation yields volumetric displacement fields, reaching up to 35 % axial strain. A Physics-Augmented Neural Network (PANN) architecture, embedding polyconvexity constraints through an input-convex neural network, is trained in an unsupervised manner by enforcing mechanical equilibrium via the EUCLID loss, using only measured displacements and global reaction forces. This three-dimensional formulation eliminates the need for depth-related assumptions inherent in 2D measurements and provides improved representation of boundary conditions, which is critical for equilibrium-based training. For comparison, Neo-Hookean and Saint-Venant-Kirchhoff models are identified in parallel. The PANN achieves superior equilibrium satisfaction, outperforming both linear and Neo-Hookean benchmarks. In addition to the original specimen, a second experiment on a different sample was performed under comparable loading conditions. This configuration provides an independent validation dataset acquired under slightly varying experimental conditions, thereby testing the robustness and transferability of the learned constitutive model. This work presents the first experimental demonstration of unsupervised PANN model training and establishes a practical protocol for data-based hyperelastic characterization from volumetric measurements.

通过实验学习并验证了三维全场运动数据的超弹性本构规律。3d打印的热塑性聚氨酯样品在实验室微计算机断层扫描系统中受到单调单轴张力。全球数字体积相关产生体积位移场，达到35%的轴向应变。物理增强神经网络（PANN）架构通过输入-凸神经网络嵌入多凸约束，通过EUCLID损失强制机械平衡，仅使用测量的位移和全局反作用力，以无监督的方式进行训练。这种三维公式消除了对二维测量中固有的深度相关假设的需要，并提供了改进的边界条件表示，这对于基于平衡的训练至关重要。为了比较，新hookean和Saint-Venant-Kirchhoff模型是并行识别的。PANN实现了卓越的均衡满意度，优于线性和新胡克基准。除原始试样外，在相同的加载条件下对不同的试样进行了第二次实验。这种配置提供了在略有变化的实验条件下获得的独立验证数据集，从而测试了学习的本构模型的鲁棒性和可移植性。这项工作提出了无监督PANN模型训练的第一个实验演示，并建立了一个基于数据的基于体积测量的超弹性表征的实用协议。

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引用次数: 0

Invariant-manifold-based model reduction for geometrically exact beam dynamics 基于不变流形的几何精确梁动力学模型约简

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-12 DOI: 10.1016/j.cma.2025.118643

Yifan Qi , Shilei Han , Minghe Shan , Mingwu Li , Qiang Tian

This paper presents an invariant-manifold (IM)-based reduction approach for geometrically exact beams undergoing finite rigid-body motions and elastic deformations. For a free beam or beam assembly, direct application of standard IM reduction is hindered by infinite system equilibria and secular growth in physical coordinates. To address this challenge, the floating frame of reference (FFR) method is employed to decompose the beam kinematics into a fictitious rigid-body motion and elastic deformation relative to the FFR. Furthermore, externally applied loads are recast as ordinary differential equations for an excitation vector. These steps render the system dynamics independent of the FFR motion tensor and the time variable, yielding an autonomous system with all parameters bounded in the vicinity of the state-space origin. Polynomial parameterization of the target invariant manifold is enabled using the FFR velocity, a set of dominant modal coordinates, and the excitation vector. A hierarchy of cohomological equations is formulated to compute the manifold mapping and the reduced dynamics. An efficient solution strategy is proposed to solve the high-dimensional cohomological equations, where higher-order problems are separated into resonant and non-resonant parts: the resonant part, associated with small-scale singular linear systems, is solved using least-squares minimization, while the non-resonant part is solved sequentially through a hierarchy of linear systems. Numerical results show that the proposed IM-based reduction yields highly accurate predictions of the nonlinear dynamic response of geometrically exact beams using only third–order polynomial expansions of the manifold and reduced dynamics. Moreover, the method achieves substantial computational savings, as dynamic simulation requires integrating only the low-dimensional reduced-order equations rather than the full state dynamics.

本文提出了一种基于不变流形（IM）的几何精确梁在有限刚体运动和弹性变形下的约简方法。对于一个自由梁或梁组合，直接应用标准的IM减少阻碍了无限系统平衡和长期增长的物理坐标。为了解决这一问题，采用浮动参照系（FFR）方法将梁的运动学分解为虚拟的刚体运动和相对于FFR的弹性变形。此外，外部施加的载荷被重铸为激励矢量的常微分方程。这些步骤使系统动力学独立于FFR运动张量和时间变量，从而产生一个所有参数都在状态空间原点附近的自治系统。使用FFR速度、一组主导模态坐标和激励向量实现目标不变流形的多项式参数化。建立了一套上同调方程来计算流形映射和简化动力学。提出了一种求解高维上同调方程的有效策略，其中高阶问题分为谐振部分和非谐振部分：谐振部分与小规模奇异线性系统相关，采用最小二乘最小化方法求解，而非谐振部分则通过线性系统的层次顺序求解。数值结果表明，所提出的基于im的约简方法仅使用流形和约简动力学的三阶多项式展开，就能高精度地预测几何精确梁的非线性动力响应。此外，由于动态模拟只需要积分低维降阶方程，而不需要积分全状态动力学，因此该方法节省了大量的计算量。

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引用次数: 0