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

Eigenstate based homogenization 基于特征态的均匀化

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-24 DOI: 10.1016/j.cma.2025.118718

Jacob Fish, Junhe Cui

The goal of this work is to develop a homogenization approach that achieves the computational efficiency of unresolved (reduced-order) schemes while retaining the accuracy of fully resolved computational homogenization. In the proposed Eigenstate-Based Homogenization (EBH), the fine-scale solution space is represented in terms of eigenstates—such as plastic multipliers or damage parameters—resulting in far fewer unknowns than in finite element discretizations. The representative volume element (RVE) response is expressed as a linear combination of precomputed eigenmodes, together with partition-level state variables updated to satisfy local consistency, enabling accurate recovery of both fine- and coarse-scale fields.

The nonlinear evolution of the state variables is solved using a Controllable Fixed-Point (CFP) iteration. For plasticity, the spectral radius of the iteration matrix scales linearly with the plastic-multiplier increment; for regularized damage, convergence depends on the load increment, maximum damage, and viscous regularization, ranging from sublinear to superlinear. The resulting tangent operator of the nonlinear system of equation of eigenstates has a condition number several orders of magnitude smaller than that of finite-element direct homogenization, yielding substantial computational savings. To further reduce computational cost, we study a truncated variant of the fixed-point method, termed the Truncated Fixed-Point (TFP) scheme, which leverages the rapid initial error decrease of CFP to obtain low-tolerance solutions with far fewer iterations. Its CPU time is comparable to well-known reduced-order homogenization approaches employing one partition per phase with calibrated properties to fit coarse-scale behavior, yet its accuracy approaches that of fully converged CFP solutions.

Numerical studies show that EBH reproduces both fine-scale fields and macroscopic responses in close agreement with fully resolved finite-element simulations, but at a fraction of the computational cost.

本工作的目标是开发一种均质化方法，在保持完全解析计算均质化的准确性的同时，实现未解析（降阶）格式的计算效率。在提出的基于特征态的均质化（EBH）中，精细尺度解空间用特征态（如塑性乘数或损伤参数）表示，导致的未知数比有限元离散化少得多。代表性体积元（RVE）响应表示为预先计算的特征模态的线性组合，以及更新以满足局部一致性的分区级状态变量，从而能够精确恢复细尺度和粗尺度场。采用可控不动点迭代法求解状态变量的非线性演化问题。对于塑性，迭代矩阵的谱半径随塑性乘法器的增量呈线性增长；对于正则化损伤，收敛取决于载荷增量、最大损伤和粘性正则化，范围从亚线性到超线性。所得到的非线性特征态方程系统的正切算子的条件数比有限元直接均匀化的条件数小几个数量级，从而大大节省了计算量。为了进一步减少计算成本，我们研究了一种截断的定点方法，称为截断定点（TFP）方案，该方案利用CFP的快速初始误差减小来获得低公差解，迭代次数少得多。它的CPU时间与众所周知的降阶均匀化方法相当，该方法采用每相位一个分区，具有校准特性以适应粗尺度行为，但其精度接近完全收敛的CFP解决方案。数值研究表明，EBH重现的细尺度场和宏观响应与完全解析的有限元模拟非常接近，但计算成本只是其中的一小部分。

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引用次数: 0

On the prospects of interpolatory spline bases for accurate mass lumping strategies in isogeometric analysis 等几何分析中精确质量集总策略的插值样条基展望

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-23 DOI: 10.1016/j.cma.2026.118762

Yannis Voet , Espen Sande

While interpolatory bases such as the Lagrange basis form the cornerstone of classical finite element methods, they have been replaced in the more general finite element setting of isogeometric analysis in favor of other desirable properties. Yet, interpolation is a key property for devising accurate mass lumping strategies that are ubiquitous in explicit dynamic analyses of structures. In this article, we explore the possibility of restoring interpolation for spline bases within isogeometric analysis for the purpose of mass lumping. Although reminiscent of the spectral element method, this technique comes with its lot of surprises and challenges, which are critically assessed.

虽然诸如拉格朗日基之类的插值基构成了经典有限元方法的基石，但在更一般的等几何分析的有限元设置中，它们已被其他理想性质所取代。然而，插值是设计精确的质量集总策略的关键属性，在结构的显式动力分析中无处不在。在这篇文章中，我们探讨了在等几何分析中为质量集总目的恢复样条基插值的可能性。虽然让人想起光谱元素方法，但这项技术带来了许多惊喜和挑战，这些都是经过严格评估的。

引用次数: 0

Topology-aware stress analysis in shell structures 壳结构的拓扑感知应力分析

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-23 DOI: 10.1016/j.cma.2026.118770

Junpeng Wang , Yingjian Liu , Jun Wu , Rüdiger Westermann

We present a stable and accurate algorithm for tracing principal stress lines (PSLs) in shell structures, applicable to both first- and second-order triangular and quadrilateral elements. The algorithm operates directly in the isoparametric space of the elements, leveraging their inherent shape functions to account for curved geometry without resorting to artificial subdivision. This approach enables, for the first time, a consistent stress topology analysis for shell elements, including a rigorous treatment of stress degeneracies. Our PSL seeding strategy integrates stress topology with the curved shell surface, ensuring a uniform and consistent PSL distribution. We evaluate the algorithm’s performance through a series of numerical experiments, demonstrating its utility for advanced stress analysis. Furthermore, we demonstrate the generation of a globally consistent, space-filling PSL structure, which is directly applicable to downstream tasks such as lightweight structural design. To support practical use, we provide a publicly available MATLAB implementation. The implementation features a unified file interface that supports diverse mesh types and is compatible with standard finite element method (FEM) output, offering a versatile tool for stress investigation and design evaluation in computational mechanics. The code is available at https://github.com/PSLer/PSLshell.

本文提出了一种稳定、准确的壳结构主应力线追踪算法，适用于一阶和二阶三角形和四边形单元。该算法直接在元素的等参空间中操作，利用其固有的形状函数来解释弯曲的几何形状，而无需诉诸人工细分。该方法首次实现了对壳单元的一致应力拓扑分析，包括对应力退化的严格处理。我们的PSL播种策略将应力拓扑与弯曲的壳表面相结合，确保PSL分布均匀一致。我们通过一系列的数值实验来评估算法的性能，证明了它在高级应力分析中的实用性。此外，我们还展示了一种全球一致的、填充空间的PSL结构的生成，它直接适用于轻量化结构设计等下游任务。为了支持实际使用，我们提供了一个公开可用的MATLAB实现。该实现具有统一的文件接口，支持多种网格类型，并与标准有限元方法（FEM）输出兼容，为计算力学中的应力调查和设计评估提供了一个通用的工具。代码可在https://github.com/PSLer/PSLshell上获得。

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引用次数: 0

Isogeometric block BDDC/FETI-DP preconditioners for the three-field Biot’s consolidation model 三场Biot固结模型的等几何块BDDC/FETI-DP预调节器

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-23 DOI: 10.1016/j.cma.2026.118757

Hanyu Chu , Luca Franco Pavarino , Stefano Zampini

This paper presents a block dual-primal preconditioner for the three-field mixed isogeometric discretization of the stationary Biot’s consolidation model, formulated in terms of displacement, pressure, and total pressure. After decomposing the computational domain into subdomains and eliminating the displacement variables and the interior components of pressure and total pressure within each subdomain, we reduce the problem to a symmetric positive definite system for the subdomain interface unknowns and the Lagrange multiplier. The reduced system is solved by a preconditioned conjugate gradient method with a block preconditioner, based on a Balancing Domain Decomposition by Constraints (BDDC) method with deluxe scaling for the interface block and a FETI-DP preconditioner for the Lagrange multiplier block. We prove that the algorithm is scalable with respect to the number of subdomains and achieves a quasi-optimal convergence rate bound that is polylogarithmic in the ratio of subdomain to element sizes and robust with respect to the model parameters. Numerical experiments confirm the efficiency of the proposed preconditioner, even in the presence of discontinuous Lamé parameters, and illustrate its robustness with respect to the spline polynomial degree, regularity, and domain deformation.

本文提出了固定Biot固结模型的三场混合等几何离散化的块双原预条件，用位移、压力和总压力表示。将计算域分解为子域，消去每个子域内的位移变量、压力和总压力的内部分量，将问题简化为子域界面未知数和拉格朗日乘子的对称正定系统。基于基于约束的平衡域分解（BDDC）方法和基于拉格朗日乘法器块的FETI-DP预调节器，采用带块预调节器的预条件共轭梯度法求解了简化后的系统。我们证明了该算法在子域数量上是可扩展的，并实现了一个拟最优收敛速度界，该界在子域与元素大小的比值上是多对数的，并且对模型参数具有鲁棒性。数值实验证实了所提出的预调节器的有效性，即使在存在不连续lam参数的情况下，也证明了它在样条多项式度、规则性和域变形方面的鲁棒性。

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引用次数: 0

The inference of Fokker-Planck equations via transport maps 通过输运图的福克-普朗克方程的推断

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-22 DOI: 10.1016/j.cma.2026.118760

Saem Han , Krishna Garikipati

We present a framework, which, from the trajectories detailing the spatiotemporal dynamics of a population, simultaneously reconstructs a transport map as well as the Fokker-Planck equation governing the coarse-grained probability distribution. Leveraging the Knothe-Rosenblatt rearrangement, we model the transport map from a fixed reference distribution to the target distribution, and derive the velocity fields of the flows from the trajectory of transport maps. Exploiting the velocity fields, we circumvent spatial gradients to infer the Fokker-Planck equation’s potential and diffusivity. The sparsity of trajectories injects uncertainty, which we treat in a Bayesian setting using variational inference. The approach is applied to inferring the Fokker-Planck dynamics in spaces of up to five dimensions, demonstrating both accurate identification of the system and efficiency with respect to data size.

我们提出了一个框架，该框架从详细描述人口时空动态的轨迹中，同时重建了运输图以及控制粗粒度概率分布的福克-普朗克方程。利用Knothe-Rosenblatt重排法，我们建立了从固定参考分布到目标分布的输运图模型，并从输运图的轨迹推导出了流的速度场。利用速度场，我们绕过空间梯度来推断Fokker-Planck方程的势和扩散率。轨迹的稀疏性注入了不确定性，我们使用变分推理在贝叶斯设置中处理。该方法被应用于在多达五个维度的空间中推断福克-普朗克动力学，证明了系统的准确识别和数据大小方面的效率。

引用次数: 0

Reinforcement learning closures for underresolved partial differential equations using synthetic data 使用合成数据的欠解偏微分方程的强化学习闭包

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-22 DOI: 10.1016/j.cma.2026.118767

Lothar Heimbach , Sebastian Kaltenbach , Petr Karnakov , Francis J. Alexander , Petros Koumoutsakos

Partial Differential Equations (PDEs) describe phenomena ranging from turbulence and epidemics to quantum mechanics and financial markets. Despite recent advances in computational science, solving such PDEs for real-world applications remains prohibitively expensive because of the necessity of resolving a broad range of spatiotemporal scales. In turn, practitioners often rely on coarse-grained approximations of the original PDEs, trading off accuracy for reduced computational resources. To mitigate the loss of detail inherent in such approximations, closure models are employed to represent unresolved spatiotemporal interactions. We present a framework for developing closure models for PDEs using synthetic data acquired through the method of manufactured solutions. These data are used in conjunction with reinforcement learning to provide closures for coarse-grained PDEs. We illustrate the efficacy of our method using the one-dimensional and two-dimensional Burgers’ equations and the two-dimensional advection equation. Moreover, we demonstrate that closure models trained for inhomogeneous PDEs can be effectively generalized to homogeneous PDEs. The results demonstrate the potential for developing accurate and computationally efficient closure models for systems with scarce data.

偏微分方程（PDEs）描述从湍流和流行病到量子力学和金融市场的各种现象。尽管计算科学在最近取得了进展，但由于需要解决大范围的时空尺度，因此在实际应用中解决这种偏微分方程仍然非常昂贵。反过来，从业者通常依赖于原始偏微分方程的粗粒度近似值，以降低计算资源的准确性为代价。为了减轻这种近似中固有的细节损失，闭合模型被用来表示未解决的时空相互作用。我们提出了一个框架，利用通过制造解决方案的方法获得的合成数据来开发pde的闭合模型。这些数据与强化学习一起使用，为粗粒度的pde提供闭包。我们用一维和二维Burgers方程以及二维平流方程来说明我们的方法的有效性。此外，我们证明了针对非齐次偏微分方程训练的闭包模型可以有效地推广到齐次偏微分方程。研究结果表明，为具有稀缺数据的系统开发精确且计算效率高的闭包模型具有潜力。

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引用次数: 0

Simultaneous multi-objective optimization of tri-directional graded material and porosity distributions for multi-patch structures using adaptive isogeometric analysis and Nitsche’s method 基于自适应等几何分析和Nitsche方法的三方向梯度材料和多孔性分布同步多目标优化

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-22 DOI: 10.1016/j.cma.2026.118758

Guangshuai Gu , Pin Wen , Jie Gao , Fei Chen

This study presents a unified framework for the simultaneous multi-objective optimization of material, porosity, and geometry with tri-directional functionally graded distributions in multi-patch complex structures. The elastodynamic governing equations are derived based on three-dimensional elasticity theory, with Nitsche’s method employed to accurately enforce coupling constraints across non-conforming interfaces in multi-patch geometries. PHT-based adaptive isogeometric analysis is introduced to substantially reduce computational cost through appropriate mesh refinement. This numerical methodology is deeply integrated with the non-dominated sorting genetic algorithm III, enabling an efficient exploration of the Pareto front within an extensive design space, aiming to enhance the fundamental frequency of free vibration and reduce structural mass. An independent hexahedral design domain is introduced to decouple the design mesh from the analysis mesh, enabling more flexible and generalizable design representation. Several numerical examples are presented to demonstrate the effectiveness of the proposed approach. The results indicate that increased design mesh resolution with more design variables yields superior optimization outcomes by generating more refined material and porosity distributions, albeit at the cost of reduced population diversity. Furthermore, compared to non-porous and uniform porosity configurations, the simultaneous optimization of material and porosity distributions achieves remarkable structural overall performance enhancement by substantially increasing fundamental frequency while effectively reducing structural mass.

该研究提出了一个统一的框架，用于同时对多斑块复杂结构中具有三方向功能梯度分布的材料、孔隙度和几何形状进行多目标优化。基于三维弹性理论推导了弹性动力学控制方程，并采用Nitsche方法在多块几何形状的非一致性界面上精确执行耦合约束。引入基于pht的自适应等几何分析，通过适当的网格细化，大大降低了计算成本。该数值方法与非主导排序遗传算法III深度集成，能够在广泛的设计空间内有效地探索帕累托前沿，旨在提高自由振动的基本频率并减少结构质量。引入独立的六面体设计域，实现了设计网格与分析网格的解耦，实现了更灵活、更通用的设计表示。算例验证了该方法的有效性。结果表明，增加的设计网格分辨率和更多的设计变量可以产生更精细的材料和孔隙度分布，从而产生更好的优化结果，尽管以减少种群多样性为代价。此外，与无孔和均匀孔隙度配置相比，材料和孔隙度分布的同时优化通过大幅提高基频而有效降低结构质量，实现了显著的结构整体性能增强。

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引用次数: 0

A locking-free SPH solid-shell formulation for modeling shell structures with general constitutive models 用一般本构模型模拟壳结构的无锁紧SPH固壳公式

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-22 DOI: 10.1016/j.cma.2026.118766

Yao Lu , Wenhao Shen , Moubin Liu

Shell structures composed of diverse materials play a fundamental role in modern engineering, and their accurate numerical modeling is essential for predicting structural performance and failure. However, conventional shell formulations often involve assumptions such as straight normals, constant transverse shear strain, and plane-stress condition, and these assumptions limit their ability to capture realistic 3D deformation and general constitutive responses. To address these limitations, this study proposes a novel smoothed particle hydrodynamics (SPH) solid-shell formulation. In contrast to existing SPH shell models derived from classical shell theories, the proposed formulation explicitly resolves the through-thickness field by introducing displacement degrees of freedom on the top and bottom surfaces. This concept preserves full 3D kinematics and provides a unified interface for various constitutive models. Furthermore, an enhanced assumed strain (EAS) approach is incorporated into SPH for the first time by introducing an independent transverse normal strain at each particle. The enhancement effectively mitigates thickness locking while maintaining full compatibility with general constitutive models. The proposed formulation is validated through a series of benchmark tests covering elasticity, isotropic and anisotropic hyperelasticity, viscoelasticity, and elastoplasticity, demonstrating its rapid convergence, high accuracy, and strong robustness. It establishes a versatile and reliable framework for simulating shell structures exhibiting strong nonlinearity and complex material behavior.

由多种材料组成的壳结构在现代工程中起着至关重要的作用，其精确的数值模拟对于预测结构性能和破坏至关重要。然而，传统的壳体公式通常涉及直法向、恒定横向剪切应变和平面应力条件等假设，这些假设限制了它们捕捉真实的3D变形和一般本构响应的能力。为了解决这些限制，本研究提出了一种新的光滑颗粒流体力学（SPH）固体壳公式。与现有的基于经典壳理论的SPH壳模型相比，本文提出的公式通过在顶、底表面引入位移自由度来明确求解全厚场。这个概念保留了完整的3D运动学，并为各种本构模型提供了统一的接口。此外，通过在每个粒子处引入独立的横向法向应变，首次将增强假设应变（EAS）方法纳入SPH。增强有效地减轻了厚度锁定，同时保持与一般本构模型的完全兼容性。通过弹性、各向同性和各向异性超弹性、粘弹性和弹塑性等一系列基准测试验证了该公式的收敛速度快、精度高、鲁棒性强。它为模拟具有强非线性和复杂材料特性的壳结构建立了一个通用和可靠的框架。

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引用次数: 0

Computing leaky waves in semi-analytical waveguide models by exponential residual relaxation 用指数残差松弛法计算半解析波导模型中的漏波

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-22 DOI: 10.1016/j.cma.2026.118763

Hauke Gravenkamp , Bor Plestenjak , Daniel A. Kiefer

Semi-analytical methods for modeling guided waves in structures of constant cross-section yield frequency-dependent polynomial eigenvalue problems for the wavenumbers and mode shapes. Solving these eigenvalue problems over a range of frequencies results in continuous eigencurves. Recent research has shown that eigencurves of differentiable parameter-dependent eigenvalue problems can alternatively be computed as solutions to a system of ordinary differential equations (ODEs) obtained by postulating an exponentially decaying residual of a modal solution. Starting from an approximate initial guess of the eigenvalue and eigenvector at a given frequency, the complete eigencurve is obtained using standard numerical ODE solvers. We exploit this idea to develop an efficient method for computing the dispersion curves of plate structures coupled to unbounded solid or fluid media. In these scenarios, the approach is particularly useful because the boundary conditions give rise to nonlinear terms that severely hinder the application of traditional solvers. We discuss suitable approximations of the nonlinearity for obtaining initial values, analyze computational costs and robustness of the proposed algorithm, and verify results by comparison against existing methods.

等截面结构中导波的半解析方法产生了波数和模态振型的频率相关多项式特征值问题。在一定频率范围内求解这些特征值问题得到连续的特征曲线。最近的研究表明，可微分参数相关特征值问题的特征曲线可以通过假设模态解的指数衰减残差得到常微分方程系统的解来计算。从给定频率下的特征值和特征向量的近似初始猜测开始，使用标准的数值ODE求解器获得完整的特征曲线。我们利用这一思想开发了一种计算板结构与无界固体或流体介质耦合的色散曲线的有效方法。在这些情况下，该方法特别有用，因为边界条件会产生非线性项，严重阻碍了传统求解器的应用。我们讨论了获得初始值的合适的非线性近似，分析了所提出算法的计算成本和鲁棒性，并通过与现有方法的比较验证了结果。

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引用次数: 0

Training deep physics-informed Kolmogorov–Arnold networks 训练深度物理信息Kolmogorov-Arnold网络

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-21 DOI: 10.1016/j.cma.2026.118761

Spyros Rigas , Fotios Anagnostopoulos , Michalis Papachristou , Georgios Alexandridis

Since their introduction, Kolmogorov–Arnold Networks (KANs) have been successfully applied across several domains, with physics-informed machine learning (PIML) emerging as one of the areas where they have thrived. In the PIML setting, Chebyshev-based physics-informed KANs (cPIKANs) have become the standard due to their computational efficiency. However, like their multilayer perceptron-based counterparts, cPIKANs face significant challenges when scaled to depth, leading to training instabilities that limit their applicability to several PDE problems. To address this, we propose a basis-agnostic, Glorot-like initialization scheme that preserves activation variance and yields substantial improvements in stability and accuracy over the default initialization of cPIKANs. Inspired by the PirateNet architecture, we further introduce Residual-Gated Adaptive KANs (RGA KANs), designed to mitigate divergence in deep cPIKANs where initialization alone is not sufficient. Through empirical tests and information bottleneck analysis, we show that RGA KANs successfully traverse all training phases, unlike baseline cPIKANs, which stagnate in the diffusion phase in specific PDE settings. Evaluations on nine standard forward PDE benchmarks under a fixed training pipeline with adaptive components demonstrate that RGA KANs consistently outperform parameter-matched cPIKANs and PirateNets—often by several orders of magnitude—while remaining stable in settings where the others diverge.

自推出以来，Kolmogorov-Arnold网络（KANs）已经成功地应用于多个领域，其中物理信息机器学习（PIML）成为它们蓬勃发展的领域之一。在PIML环境中，基于chebyhev的物理信息的KANs （cPIKANs）由于其计算效率已成为标准。然而，与基于多层感知器的同类方法一样，cPIKANs在扩展到深度时面临重大挑战，导致训练不稳定性，限制了它们对一些PDE问题的适用性。为了解决这个问题，我们提出了一种与基无关的、类似于gloot的初始化方案，该方案保留了激活方差，并比cPIKANs的默认初始化在稳定性和准确性方面有了实质性的改进。受PirateNet架构的启发，我们进一步引入了残余门控自适应KANs (RGA KANs)，旨在缓解深度cPIKANs的分歧，其中初始化本身是不够的。通过实证测试和信息瓶颈分析，我们发现RGA KANs成功地遍历了所有训练阶段，而不像基线cPIKANs，在特定的PDE设置中停滞在扩散阶段。在具有自适应组件的固定训练管道下对9个标准前向PDE基准进行的评估表明，RGA KANs始终优于参数匹配的cPIKANs和piratenet -通常是几个数量级-同时在其他设置中保持稳定。

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引用次数: 0