Computer Methods in Applied Mechanics and Engineering最新文献_第4页

Data generation with optimal experimental design for operator learning 数据生成与优化实验设计算子学习

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-18 DOI: 10.1016/j.cma.2025.118675

Xingzi Xu , Johann Guilleminot , Vahid Tarokh

Partial differential equations (PDEs) are fundamental to modeling complex physical phenomena across scientific disciplines. While operator learning offers a promising alternative to conventional PDE solvers, it generally requires substantial high-fidelity training data, resulting in significant computational costs. Existing approaches typically sample PDE parameters uniformly or heuristically, which can be inefficient in computational resources and may lead to suboptimal neural operator performance. Leveraging functional encoding, we propose a systematic framework that adapts (finite-dimensional) optimal experimental design (OED) principles for generating informative training datasets while minimizing computational costs. The OED framework employs physics-infused informativeness metrics—solution variance, energy dissipation, and high-frequency spectral content—to guide strategic parameter sampling using Bayesian inference with adaptive acquisition strategies. The framework achieves substantial improvements in data efficiency through computationally lightweight OED operations that incur negligible overhead compared to expensive PDE simulations. We empirically demonstrate that strategic parameter sampling guided by informativeness metrics significantly outperforms uniform random sampling strategies across multiple PDE benchmarks, including Burgers, Darcy, and Navier-Stokes equations. Comprehensive evaluations using both Fourier Neural Operators (FNO) and Deep Operator Networks (DeepONet), two leading neural operator architectures, confirm the generality of the approach. Using equivalent computational budgets, the method achieves substantially lower validation errors by concentrating simulations on parameter regions that provide maximum learning value. We provide a principled yet practical approach to training data generation that reduces the computational barrier to deploying neural operators for complex parametric PDEs. We release an open-source implementation to make this data-efficient operator learning framework accessible.

偏微分方程（PDEs）是跨科学学科建模复杂物理现象的基础。虽然操作员学习是传统PDE求解器的一个很有前途的替代方案，但它通常需要大量高保真度的训练数据，从而导致大量的计算成本。现有的方法通常是统一或启发式地对PDE参数进行采样，这可能会导致计算资源的效率低下，并可能导致神经算子性能的次优。利用功能编码，我们提出了一个系统框架，该框架适应（有限维）最佳实验设计（OED）原则，以生成信息丰富的训练数据集，同时最小化计算成本。OED框架采用物理注入的信息度量（解决方案方差、能量耗散和高频频谱内容），使用贝叶斯推理和自适应采集策略指导策略参数采样。该框架通过计算轻量级的OED操作实现了数据效率的实质性改进，与昂贵的PDE模拟相比，这些操作带来的开销可以忽略不计。我们通过经验证明，在多个PDE基准（包括Burgers、Darcy和Navier-Stokes方程）中，由信息度量指导的策略参数抽样显著优于均匀随机抽样策略。使用傅里叶神经算子（FNO）和深度算子网络（DeepONet）这两种领先的神经算子架构进行综合评估，证实了该方法的通用性。使用等效的计算预算，该方法通过将模拟集中在提供最大学习价值的参数区域上，大大降低了验证误差。我们提供了一种原则而实用的方法来训练数据生成，减少了为复杂参数偏微分方程部署神经算子的计算障碍。我们发布了一个开源实现，使这个数据高效的操作员学习框架易于访问。

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引用次数: 0

BubbleOKAN: A physics-informed interpretable neural operator for high-frequency bubble dynamics BubbleOKAN：用于高频气泡动力学的物理信息可解释神经算子

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-18 DOI: 10.1016/j.cma.2025.118667

Yunhao Zhang , Sidharth S. Menon , Lin Cheng , Aswin Gnanaskandan , Ameya D. Jagtap

In this work, we employ physics-informed neural operators to map pressure profiles from an input function space to the corresponding bubble radius responses. Our approach employs a two-step DeepONet architecture. To address the intrinsic spectral bias of deep learning models, our model incorporates the Rowdy adaptive activation function, enhancing the representation of high-frequency features. Moreover, we introduce the Kolmogorov-Arnold network (KAN) based two-step DeepOKAN model, which enhances interpretability (often lacking in conventional multilayer perceptron architectures) while efficiently capturing high-frequency bubble dynamics without explicit utilization of activation functions in any form. We particularly investigate the use of spline basis functions in combination with radial basis functions (RBF) within our architecture, as they demonstrate superior performance in constructing a universal basis for approximating high-frequency bubble dynamics compared to alternative formulations. Furthermore, we emphasize on the performance bottleneck of RBF while learning the high frequency bubble dynamics and showcase the advantage of using spline basis function for the trunk network in overcoming this inherent spectral bias. The model is systematically evaluated across three representative scenarios: (1) bubble dynamics governed by the Rayleigh-Plesset equation with a single initial radius, (2) bubble dynamics governed by the Keller-Miksis equation with a single initial radius, and (3) Keller-Miksis dynamics with multiple initial radii. We also compare our results with state-of-the-art neural operators, including Fourier Neural Operators, Wavelet Neural Operators, OFormer, and Convolutional Neural Operators. Our findings demonstrate that the two-step DeepOKAN accurately captures both low- and high-frequency behaviors, and offers a promising alternative to conventional numerical solvers. The two-step DeepOKAN code is available at https://github.com/ParamIntelligence/Two-Step-DeepOKAN.

在这项工作中，我们使用物理信息神经算子将压力剖面从输入函数空间映射到相应的气泡半径响应。我们的方法采用两步DeepONet架构。为了解决深度学习模型固有的频谱偏差，我们的模型结合了Rowdy自适应激活函数，增强了高频特征的表示。此外，我们引入了基于Kolmogorov-Arnold网络（KAN）的两步DeepOKAN模型，该模型增强了可解释性（通常缺乏传统的多层感知器架构），同时有效地捕获高频气泡动态，而无需显式地利用任何形式的激活函数。我们特别研究了样条基函数与径向基函数（RBF）在我们架构中的结合使用，因为与其他公式相比，它们在构建近似高频气泡动力学的通用基础方面表现出优越的性能。此外，我们强调了RBF在学习高频气泡动力学时的性能瓶颈，并展示了在主干网络中使用样条基函数克服这种固有频谱偏差的优势。该模型在三种典型场景下进行了系统评估：(1)由单一初始半径的瑞利-普莱塞特方程控制的气泡动力学，(2)由单一初始半径的凯勒-米克西斯方程控制的气泡动力学，以及(3)具有多个初始半径的凯勒-米克西斯动力学。我们还将我们的结果与最先进的神经算子进行了比较，包括傅里叶神经算子、小波神经算子、OFormer和卷积神经算子。我们的研究结果表明，两步DeepOKAN准确地捕获了低频和高频行为，并为传统的数值求解器提供了一个有希望的替代方案。DeepOKAN的两步代码可在https://github.com/ParamIntelligence/Two-Step-DeepOKAN上获得。

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引用次数: 0

An adaptive multiresolution vortex particle-mesh method for the simulation of unbounded incompressible flows 一种模拟无界不可压缩流动的自适应多分辨率涡旋粒子网格方法

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-18 DOI: 10.1016/j.cma.2025.118638

Pierre Balty, Matthieu Duponcheel, Philippe Chatelain

Simulations of incompressible convection-dominated flows, such as vortical flows or wake flows, require accurate and efficient numerical methods. Among the various approaches, the vortex particle-mesh (VPM) method stands out for its ability to combine the strengths of mesh-based methods, such as efficient elliptic solvers and finite difference stencils, with the advantages of particle methods, which minimize numerical dispersion and dissipation errors. However, most VPM implementations consider a uniform grid. This can lead to a prohibitive computational cost, especially in three dimensions. This challenge can be addressed effectively through adaptive mesh refinement (AMR), which significantly reduces simulation costs by dynamically adjusting the local grid resolution to meet the requirements of the underlying physics. In this work, we present a novel VPM method that leverages the benefits of AMR. Building upon murphy, a wavelet-based block-structured AMR framework, we propose a new approach to represent the particles that allows for their refinement or coarsening. We show that our method achieves high-order accuracy, grid adaptation, and CFL relaxation at the same time. We then demonstrate the scalability of our method from 128 to 16,384 cores, and compare its performance with a uniform resolution framework for the simulation of aircraft wake vortices. We show that despite the increased complexity due to the overhead of the AMR operations, the resulting gains in computational efficiency and reductions in memory footprint are substantial.

不可压缩对流主导流的模拟，如涡旋流或尾流，需要精确和有效的数值方法。在众多方法中，涡旋粒子网格（vortex particle-mesh， VPM）方法以其将基于网格的方法（如高效椭圆求解法和有限差分模板法）的优点与粒子方法的优点相结合而脱颖而出，使数值色散和耗散误差最小化。然而，大多数VPM实现都考虑统一网格。这可能导致令人望而却步的计算成本，特别是在三维空间中。这一挑战可以通过自适应网格细化（AMR）来有效解决，AMR通过动态调整局部网格分辨率来满足底层物理的要求，从而显著降低仿真成本。在这项工作中，我们提出了一种新的VPM方法，利用AMR的好处。在基于小波的块结构AMR框架murphy的基础上，我们提出了一种新的方法来表示允许其细化或粗化的粒子。结果表明，该方法同时实现了高阶精度、网格自适应和CFL松弛。然后，我们展示了我们的方法的可扩展性，从128到16,384核，并比较其性能与统一分辨率框架的模拟飞机尾流涡。我们表明，尽管由于AMR操作的开销而增加了复杂性，但由此带来的计算效率的提高和内存占用的减少是实质性的。

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引用次数: 0

A new data-driven energy-stable evolve-filter-relax model for turbulent flow simulation 一种新的数据驱动的能量稳定演化滤波松弛模型

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-17 DOI: 10.1016/j.cma.2025.118654

Anna Ivagnes , Toby van Gastelen , Syver Døving Agdestein , Benjamin Sanderse , Giovanni Stabile , Gianluigi Rozza

We present a novel approach to define the filter and relax steps in the evolve-filter-relax (EFR) framework for simulating turbulent flows. The EFR main advantages are its ease of implementation and computational efficiency. However, as it only contains two parameters (one for the filter step and one for the relax step) its flexibility is rather limited. In this work, we propose a data-driven approach in which the optimal filter is found based on DNS data in the frequency domain. The optimization step is computationally efficient and only involves one-dimensional least-squares problems for each wavenumber. Across both decaying turbulence and Kolmogorov flow, our learned filter decisively outperforms the standard differential filter and the Smagorinsky model, yielding significantly improved accuracy in energy spectra and in the temporal evolution of both energy and enstrophy. In addition, the relax parameter is determined by requiring energy and/or enstrophy conservation, which enforces stability of the method and reduces the appearance of numerical wiggles, especially when the filter is built in scarce data regimes. Applying the learned filter is also more computationally efficient compared to traditional differential filters, as it circumvents solving a linear system.

我们提出了一种新的方法来定义演化-滤波-松弛（EFR）框架中的滤波和松弛步骤，用于模拟湍流。EFR的主要优点是易于实现和计算效率高。然而，由于它只包含两个参数（一个用于过滤步骤，一个用于松弛步骤），其灵活性相当有限。在这项工作中，我们提出了一种数据驱动的方法，其中基于频域的DNS数据找到最优滤波器。优化步骤计算效率高，并且只涉及每个波数的一维最小二乘问题。在衰减湍流和Kolmogorov流中，我们的学习滤波器明显优于标准微分滤波器和Smagorinsky模型，在能谱和能量和熵的时间演化中显著提高了精度。此外，松弛参数是通过要求能量和/或熵守恒来确定的，这加强了方法的稳定性，减少了数值波动的出现，特别是当滤波器建立在稀缺数据区域时。与传统的微分滤波器相比，应用学习滤波器的计算效率也更高，因为它避免了求解线性系统。

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引用次数: 0

Topology optimization of multi-cracked structures with an adaptive penalty strategy for stress intensity factors 基于应力强度因子自适应惩罚策略的多裂纹结构拓扑优化

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-17 DOI: 10.1016/j.cma.2025.118653

Changhao Yang , Bin Xu , Zeyu Wu , Zunyi Duan , Si Zeng , Xiaodong Huang

This paper addresses key challenges in the topology optimization of multi-cracked structures, including crack removal, coupling effects, and the singular behavior of stress intensity factors (SIFs). A new optimization framework is developed by integrating the Solid Isotropic Material with Penalization (SIMP) method and the extended finite element method (XFEM). Within this framework, an adaptive penalty strategy for stress intensity factors (SIF-APS) is proposed. By employing the density field in the crack-tip region as a carrier, SIF-APS combines crack geometry and material distribution so that it adaptively penalizes SIFs at crack tips. This strategy effectively mitigates the numerical issues associated with element deactivation in conventional methods, which often lead to distorted fracture responses as well as singularities in SIFs. Furthermore, the proposed framework explicitly incorporates multi-crack coupling effects into the optimization model. This establishes a general and versatile approach for multi-cracked structures. Numerical validation through both benchmark problems and an engineering-inspired case confirmed the effectiveness of the method in complex crack scenarios. The results demonstrate that the proposed approach significantly enhances the resistance of structures against cracking while achieving lightweight design, and adaptively captures the coupling effects induced by variations in crack spacings, lengths, orientations, and positions.

本文讨论了多裂纹结构拓扑优化中的关键挑战，包括裂纹去除、耦合效应和应力强度因子（SIFs）的奇异行为。将固体各向同性材料惩罚法（SIMP）与扩展有限元法（XFEM）相结合，建立了一种新的优化框架。在此框架下，提出了应力强度因子自适应惩罚策略（SIF-APS）。SIF-APS利用裂纹尖端区域的密度场作为载体，结合裂纹几何形状和材料分布，自适应地惩罚裂纹尖端的sif。该策略有效地缓解了传统方法中与元件失活相关的数值问题，这些问题通常会导致裂缝响应扭曲以及SIFs中的奇异性。此外，该框架明确地将多裂纹耦合效应纳入优化模型。这为多裂纹结构建立了一种通用的通用方法。通过基准问题和工程实例的数值验证，验证了该方法在复杂裂纹情况下的有效性。结果表明，该方法在实现轻量化设计的同时显著提高了结构的抗裂性，并自适应地捕捉了裂纹间距、长度、方向和位置变化引起的耦合效应。

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引用次数: 0

Resolved CFD-DEM for high-fidelity multiphase flow modeling in porous media of arbitrary geometry 任意几何多孔介质中高保真多相流建模的离散CFD-DEM

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-17 DOI: 10.1016/j.cma.2025.118676

Tao Yu, Jidong Zhao

The design of resilient coastal infrastructure requires high-fidelity modelling of complex interactions between waves, porous structures, and mobile seabed. To address this need, we develop a novel computational framework that couples Computational Fluid Dynamics (CFD) and the Discrete Element Method (DEM), explicitly integrating a resolved porous media module. This approach enables direct numerical simulation of multiphase flows and their particle-scale interactions with both stationary and mobile porous structures, such as breakwaters or armor units. The model is rigorously validated against six benchmark cases, demonstrating robust capabilities in capturing permeability, capillary effects, and fluid–solid momentum exchange. We further apply the framework to large-scale coastal scenario featuring realistic wave generation, curved and trapezoidal seawalls, and over one hundred mobile cubic armor units. The simulations provide deep insights into critical processes like wave reflection, entrapped air dynamics, and drag-induced energy dissipation. The simulation results quantitatively show that porous structures significantly enhance wave energy dissipation, leading to superior wave attenuation. This integrated framework represents a significant advancement for high-fidelity, efficient simulations of fluid–structure interactions in dynamic and porous coastal environments, with great potential for coastal engineering design and environmental fluid mechanics.

弹性沿海基础设施的设计需要对波浪、多孔结构和移动海床之间的复杂相互作用进行高保真建模。为了满足这一需求，我们开发了一种新的计算框架，将计算流体动力学（CFD）和离散元法（DEM）结合在一起，明确地集成了一个可分解的多孔介质模块。这种方法可以直接数值模拟多相流及其与固定和移动多孔结构（如防波堤或装甲单元）的颗粒级相互作用。该模型针对六个基准案例进行了严格验证，证明了在捕获渗透率、毛细效应和流固动量交换方面的强大能力。我们进一步将该框架应用于大规模的海岸场景，包括逼真的波浪产生，弯曲和梯形海堤，以及100多个移动立方装甲单位。模拟提供了深入了解关键过程，如波反射，困住空气动力学和阻力诱导的能量耗散。模拟结果定量地表明，多孔结构显著地增强了波的能量耗散，导致了较好的波衰减。这个综合框架代表了在动态和多孔海岸环境中高保真、高效模拟流固相互作用的重大进步，在海岸工程设计和环境流体力学方面具有巨大潜力。

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引用次数: 0

Primal-dual splitting methods for phase-field surfactant model with moving contact lines 具有运动接触线的相场表面活性剂模型的原始-对偶分裂方法

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-17 DOI: 10.1016/j.cma.2025.118670

Wei Wu , Zhen Zhang , Chaozhen Wei

Surfactants have important effects on the dynamics of droplets on solid surfaces, which have inspired many industrial applications. The phase-field surfactant model with moving contact lines (PFS-MCL) has been employed to investigate the complex droplet dynamics with surfactants, while its numerical simulation remains challenging due to the coupling of gradient flows with respect to transport distances involving nonlinear and degenerate mobilities. We propose a novel structure-preserving variational scheme for the PFS-MCL model with the dynamic boundary condition based on the minimizing movement scheme and optimal transport theory for Wasserstein gradient flows. The proposed scheme consists of a series of convex minimization problems and can be efficiently solved by our proposed primal-dual splitting method and its accelerated version. By respecting the underlying PDE’s variational structure with respect to the transport distance, the proposed scheme is proved to inherit the desirable properties including original energy dissipation, bound-preserving, and mass conservation. Through a suite of numerical simulations, we validate the performance of the proposed scheme and investigate the effects of surfactants on the droplet dynamics.

表面活性剂对固体表面液滴的动力学有重要影响，这激发了许多工业应用。带移动接触线的相场表面活性剂模型（PFS-MCL）已被用于研究具有表面活性剂的复杂液滴动力学，但由于涉及非线性和退化流动性的输运距离的梯度流动耦合，其数值模拟仍然具有挑战性。基于Wasserstein梯度流的最小运动格式和最优输运理论，提出了具有动态边界条件的PFS-MCL模型的一种新的保结构变分格式。该方案由一系列凸极小化问题组成，可通过本文提出的原对偶分裂方法及其加速版本进行有效求解。通过尊重底层PDE随输运距离的变分结构，证明了该方案继承了原始能量耗散、保界和质量守恒等特性。通过一系列的数值模拟，我们验证了所提出方案的性能，并研究了表面活性剂对液滴动力学的影响。

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引用次数: 0

Sparse narrow-band topology optimization for large-scale thermal-fluid applications 大规模热流体应用的稀疏窄带拓扑优化

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-12-17 DOI: 10.1016/j.cma.2025.118655

Vladislav Pimanov , Alexandre T R Guibert , John-Paul Sabino , Michael Stoia , H. Alicia Kim

We propose a fluid-based topology optimization methodology for convective heat-transfer problems that can manage an extensive number of design variables, enabling the fine geometric features required for the next generation of heat-exchangers design. Building on the classical Borrvall-Petersson formulation for the Stokes flow, we introduce an optimization algorithm that focuses computational effort on the fluid-solid interface, where it is most needed. To address the high cost of repeated forward and adjoint analyses and to avoid leakage through nominally solid regions, we exclude fictitious solid voxels from the analysis by imposing the no-slip boundary conditions in the vicinity of the fluid-solid interface. In contrast to the prior approaches, the fictitious solids are also excluded from the global optimization problem via reducing it to a sequence of local narrow-band subproblems with a variable design space. The contribution of our method is that large-scale optimization can be solved efficiently by continuous simplex method while reliably obtaining binary designs without additional filtering or projection. We demonstrate efficiency of the method on multiple examples, including the optimization of a two-fluid heat exchanger at

P e = 10^{4}

on a 370³ grid comprising 5 × 10⁷ design variables using only a single desktop workstation.

我们提出了一种基于流体的对流传热问题拓扑优化方法，该方法可以管理大量的设计变量，从而实现下一代换热器设计所需的精细几何特征。在Stokes流的经典borrvallpetersson公式的基础上，我们引入了一种优化算法，该算法将计算精力集中在最需要的流固界面上。为了解决重复正演和伴随分析的高成本问题，并避免通过名义上的固体区域泄漏，我们通过在流固界面附近施加无滑移边界条件，将虚拟的固体体素从分析中排除。与之前的方法相反，虚拟实体也通过将全局优化问题简化为具有可变设计空间的局部窄带子问题序列而被排除在全局优化问题之外。该方法的贡献在于，在不需要额外滤波和投影的情况下，可以通过连续单纯形法有效地求解大规模优化问题，同时可靠地得到二值设计。我们通过多个实例证明了该方法的有效性，包括仅使用单个桌面工作站在3703网格上优化Pe=104的双流体热交换器，该网格包含5 × 107个设计变量。

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引用次数: 0

Data-augmented predictive deep neural network: Enhancing the extrapolation capabilities of non-intrusive surrogate models 数据增强预测深度神经网络：增强非侵入性代理模型的外推能力

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.2025.118604

Shuwen Sun , Lihong Feng , Peter Benner

Numerically solving a large parametric nonlinear dynamical system is challenging due to its high complexity and the high computational costs associated with this task. In recent years, machine-learning-aided surrogates are being actively researched. However, many methods fail in accurately generalizing in the entire time interval [0, T], when the training data is available only in a training time interval [0, T₀], with T₀ < T.

To improve the extrapolation capabilities of the surrogate models in the entire time domain, we propose a new deep learning framework, where kernel dynamic mode decomposition (KDMD) is employed to evolve the dynamics of the latent space generated by the encoder part of a convolutional autoencoder (CAE). After adding the KDMD-decoder-extrapolated data into the original data set, we train the CAE along with a feed-forward deep neural network using the augmented data. The trained network can predict future states outside the training time interval at any out-of-training parameter samples. The proposed method is tested for two numerical examples: a FitzHugh-Nagumo model and a model of incompressible flow past a circular obstacle. Numerical results show accurate and fast prediction performance in both the time and the parameter domain.

数值求解大型参数非线性动力系统是一项具有挑战性的任务，因为它具有高复杂性和高计算成本。近年来，机器学习辅助替代物正在被积极研究。然而，当训练数据仅在一个训练时间区间[0，T0]内可用，T0 <； T时，许多方法无法在整个时间区间[0，T]内准确泛化。为了提高代理模型在整个时域内的外推能力，我们提出了一种新的深度学习框架，该框架采用核动态模式分解（KDMD）来演化卷积自编码器（CAE）编码器部分生成的潜在空间的动态。在将kdmd解码器外推数据添加到原始数据集中后，我们使用增强数据与前馈深度神经网络一起训练CAE。训练后的网络可以预测任何训练外参数样本在训练时间间隔之外的未来状态。通过FitzHugh-Nagumo模型和不可压缩过圆障碍物模型两个数值算例对该方法进行了验证。数值结果表明，该方法在时间域和参数域均具有准确、快速的预测效果。

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引用次数: 0

Scalable augmented Lagrangian preconditioners for fictitious domain problems 虚拟域问题的可伸缩增广拉格朗日预条件

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.2025.118522

Michele Benzi , Marco Feder , Luca Heltai , Federica Mugnaioni

We present preconditioning techniques to solve linear systems of equations with a block two-by-two and three-by-three structure arising from finite element discretizations of the fictitious domain method with Lagrange multipliers. In particular, we propose two augmented Lagrangian-based preconditioners to accelerate the convergence of iterative solvers for such classes of linear systems. We consider two relevant examples to illustrate the performance of these preconditioners when used in conjunction with flexible GMRES: the Poisson and the Stokes fictitious domain problems. A spectral analysis is established for both exact and inexact versions of the preconditioners. We show the effectiveness of the proposed approach and the robustness of our preconditioning strategy through extensive numerical tests in both two and three dimensions.

利用拉格朗日乘子虚域法的有限元离散化，提出了一种预处理技术来求解具有块二乘二和三乘三结构的线性方程组。特别地，我们提出了两个基于增广拉格朗日的预条件来加速这类线性系统的迭代解的收敛。我们考虑了两个相关的例子来说明这些前置条件在与柔性GMRES结合使用时的性能：泊松和斯托克斯虚拟域问题。建立了精确和不精确版本的预调节器的光谱分析。我们通过广泛的二维和三维数值测试证明了所提出方法的有效性和我们的预处理策略的鲁棒性。

引用次数: 0