Computer Methods in Applied Mechanics and Engineering最新文献_第9页

An end-to-end deep learning method for solving nonlocal Allen–Cahn and Cahn–Hilliard phase-field models 求解非局部Allen-Cahn和Cahn-Hilliard相场模型的端到端深度学习方法

IF 6.9 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-01-17 DOI: 10.1016/j.cma.2024.117721

Yuwei Geng , Olena Burkovska , Lili Ju , Guannan Zhang , Max Gunzburger

We propose an efficient end-to-end deep learning method for solving nonlocal Allen–Cahn (AC) and Cahn–Hilliard (CH) phase-field models. One motivation for this effort emanates from the fact that discretized partial differential equation-based AC or CH phase-field models result in diffuse interfaces between phases, with the only recourse for remediation is to severely refine the spatial grids in the vicinity of the true moving sharp interface whose width is determined by a grid-independent parameter that is substantially larger than the local grid size. In this work, we introduce non-mass conserving nonlocal AC or CH phase-field models with regular, logarithmic, or obstacle double-well potentials. Because of non-locality, some of these models feature totally sharp interfaces separating phases. The discretization of such models can lead to a transition between phases whose width is only a single grid cell wide. Another motivation is to use deep learning approaches to ameliorate the otherwise high cost of solving discretized nonlocal phase-field models. To this end, loss functions of the customized neural networks are defined using the residual of the fully discrete approximations of the AC or CH models, which results from applying a Fourier collocation method and a temporal semi-implicit approximation. To address the long-range interactions in the models, we tailor the architecture of the neural network by incorporating a nonlocal kernel as an input channel to the neural network model. We then provide the results of extensive computational experiments to illustrate the accuracy, predictive capabilities, and cost reductions of the proposed method.

我们提出了一种有效的端到端深度学习方法来求解非局部Allen-Cahn （AC）和Cahn-Hilliard （CH）相场模型。这项工作的一个动机源于这样一个事实，即基于偏微分方程的离散化AC或CH相场模型会导致相位之间的扩散界面，而唯一的补救办法是严格细化真正移动尖锐界面附近的空间网格，其宽度由一个与网格无关的参数决定，该参数远远大于局部网格尺寸。在这项工作中，我们引入了具有规则、对数或障碍双阱势的非质量守恒非局部AC或CH相场模型。由于非局部性，这些模型中的一些具有完全尖锐的界面分离阶段。这种模型的离散化可以导致相位之间的过渡，其宽度仅为单个网格单元的宽度。另一个动机是使用深度学习方法来改善求解离散非局部相场模型的高成本。为此，使用AC或CH模型的完全离散近似的残差来定义自定义神经网络的损失函数，这是应用傅里叶搭配方法和时间半隐式近似的结果。为了解决模型中的远程交互，我们通过将非局部核作为神经网络模型的输入通道来定制神经网络的体系结构。然后，我们提供了广泛的计算实验结果，以说明所提出方法的准确性、预测能力和成本降低。

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引用次数: 0

An improved explicit MPM formulation and its coupling scheme with FEM 改进的显式质点模型及其与有限元的耦合形式

IF 6.9 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-01-17 DOI: 10.1016/j.cma.2025.117734

Xi-Wen Zhou , Yin-Fu Jin , Kai-Yuan He , Zhen-Yu Yin

Accurately imposing boundary conditions and contact constraints in the Material Point Method (MPM) and its coupling with the Finite Element Method (FEM-MPM) is challenging, especially when dealing with complex geometrical shapes and misalignment between material boundaries and the computational grid. To address these issues, an improved explicit penalty formulation based on particle positions is developed to effectively impose Dirichlet boundary conditions and tie-contact constraints in both MPM and FEM-MPM coupling. Specifically, the concepts of boundary reference points and tied reference points are introduced to discretize the penalty terms associated with Dirichlet boundary conditions and tied contact constraints, respectively. These methods are straightforward to implement and highly suitable for explicit computational frameworks. A dimensionless penalty factor selection scheme is designed to avoid excessive tunning and minimize the decrease in stable time step. Additionally, contact forces are formulated as a conservative force field, ensuring energy conservation during MPM-Rigid and FEM-MPM collisions, which enhance numerical performances. Moreover, Dirichlet boundary conditions and contact constraints are discretized on material points, improving compatibility with complex geometrical shapes. The proposed explicit computational framework is straightforward to implement in both Updated Lagrangian and Total Lagrangian formulations, broadening its applicability to various engineering problems. Finally, the robustness, accuracy, and efficiency of the proposed approach are demonstrated through a series of numerical experiments, showcasing precise implementation of irregular boundary conditions, accurate calculation of contact forces, and good energy conservation.

在材料点法（MPM）及其与有限元法（FEM-MPM）的耦合中准确施加边界条件和接触约束是具有挑战性的，特别是在处理复杂的几何形状和材料边界与计算网格之间的不对齐时。为了解决这些问题，开发了一种改进的基于粒子位置的显式惩罚公式，以有效地在MPM和FEM-MPM耦合中施加Dirichlet边界条件和捆绑接触约束。具体地说，引入了边界参考点和捆绑参考点的概念，分别离散了与Dirichlet边界条件和捆绑接触约束相关的罚项。这些方法易于实现，非常适合显式计算框架。设计了一种无量纲惩罚因子选择方案，避免了过度调谐，使稳定时间步长减少到最小。此外，接触力被表述为一个保守力场，确保了在mpm -刚性和FEM-MPM碰撞过程中的能量守恒，从而提高了数值性能。此外，Dirichlet边界条件和接触约束在材料点上离散化，提高了与复杂几何形状的相容性。所提出的显式计算框架在更新拉格朗日和总拉格朗日公式中都易于实现，从而扩大了其对各种工程问题的适用性。最后，通过一系列数值实验证明了该方法的鲁棒性、准确性和高效性，表明该方法能够精确实现不规则边界条件，准确计算接触力，并具有良好的能量守恒性。

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引用次数: 0

Fully-discrete decoupled Subdivision-based IGA-IEQ-ZEC numerical scheme for the binary surfactant phase-field model coupled with Darcy flow equations on Surfaces 基于细分的完全离散解耦 IGA-IEQ-ZEC 数值方案，用于表面达西流方程耦合的二元表面活性剂相场模型

IF 6.9 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-01-17 DOI: 10.1016/j.cma.2025.117733

Qing Pan , Yunqing Huang , Timon Rabczuk , Yin Yang , Xiaofeng Yang

In this paper, we present a comprehensive numerical investigation of the binary phase-field surfactant model coupled with the Darcy flow equation to explore the impact of surfactant addition on the evolution of Saffman–Taylor fingering patterns within a Hele-Shaw cell on surfaces. We develop an efficient and robust spatiotemporal discretization framework that effectively addresses the highly nonlinear terms arising from the strong coupling structure inherent to the model on surface geometries. For the spatial discretization, we employ the recently developed subdivision-based isogeometric analysis (IGA), which provides the advantages of hierarchical refinability and adaptability to arbitrary topologies. This approach eliminates geometric errors associated with surface approximation and reduces additional approximation errors introduced by the numerical schemes. For the temporal discretization, we integrate the Invariant Energy Quadratization (IEQ) method – used to linearize the nonlinear potential – with the Zero-Energy-Contribution (ZEC) decoupling approach, which facilitates fully decoupled computations. The resulting fully discrete numerical framework possesses several desirable properties, including geometric exactness, compatibility with arbitrary topologies, linearity, second-order temporal accuracy, full decoupling, and unconditional energy stability. Additionally, we rigorously establish the unconditional energy stability of the scheme within this work. Furthermore, we perform several numerical experiments to demonstrate the accuracy and robustness of our method, including simulations of benchmark Saffman–Taylor fingering instability to evaluate the weakening effects of surfactants on surface tension.

在本文中，我们对二元相场表面活性剂模型与达西流动方程进行了全面的数值研究，以探索表面活性剂的添加对表面上Hele-Shaw细胞内Saffman-Taylor指动模式演变的影响。我们开发了一个高效和稳健的时空离散化框架，有效地解决了由表面几何形状模型固有的强耦合结构引起的高度非线性项。对于空间离散化，我们采用了最近发展起来的基于细分的等几何分析（IGA），它具有层次可细化性和对任意拓扑的适应性。该方法消除了曲面近似带来的几何误差，减少了数值格式带来的附加近似误差。对于时间离散化，我们将用于线性化非线性势的不变能量二次化（IEQ）方法与零能量贡献（ZEC）解耦方法相结合，从而实现了完全解耦计算。所得到的完全离散数值框架具有几个理想的特性，包括几何精度、与任意拓扑的兼容性、线性、二阶时间精度、完全解耦和无条件能量稳定性。此外，我们严格地建立了该方案的无条件能量稳定性。此外，我们进行了几个数值实验来证明我们的方法的准确性和鲁棒性，包括模拟基准的Saffman-Taylor指指不稳定性来评估表面活性剂对表面张力的减弱作用。

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引用次数: 0

Modeling of cardiac fibers as oriented liquid crystals 心脏纤维定向液晶的建模

IF 6.9 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-01-16 DOI: 10.1016/j.cma.2024.117710

Nicolás A. Barnafi , Axel Osses

In this work we propose a mathematical model that describes the orientation of ventricular cardiac fibers. These fibers are commonly computed as the normalized gradient of certain harmonic potentials, so our work consisted in finding the equations that such a vector field satisfies, considering the unitary norm constraint. The resulting equations belong to the Frank–Oseen theory of nematic liquid crystals, which yield a bulk of mathematical properties to the cardiac fibers, such as the characterization of singularities. The numerical methods available in literature are computationally expensive and not sufficiently robust for the complex geometries obtained from the human heart, so we also propose a preconditioned projected gradient descent scheme that circumvents these difficulties in the tested scenarios. The resulting model further confirms recent experimental observations of liquid crystal behavior of soft tissue, and provides an accurate mathematical description of such behavior.

在这项工作中，我们提出了一个数学模型来描述心室心脏纤维的方向。这些纤维通常被计算为某些谐波势的归一化梯度，因此我们的工作包括在考虑酉范数约束的情况下找到这样一个矢量场满足的方程。由此产生的方程属于弗兰克-奥西恩的向列液晶理论，该理论为心脏纤维提供了大量的数学性质，例如奇点的表征。文献中可用的数值方法计算成本高，并且对于从人类心脏获得的复杂几何图形不够鲁棒，因此我们还提出了一种预置投影梯度下降方案，该方案在测试场景中绕过了这些困难。所得到的模型进一步证实了最近对软组织液晶行为的实验观察，并提供了这种行为的精确数学描述。

引用次数: 0

Metamaterial design with vibroacoustic bandgaps through topology optimization 基于拓扑优化的振动声带隙超材料设计

IF 6.9 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-01-16 DOI: 10.1016/j.cma.2025.117744

Vanessa Cool , Ole Sigmund , Niels Aage

Metamaterials have shown potential to achieve strong noise or vibration reduction in predefined frequency ranges. Targeting both wave types simultaneously remains, however, a cumbersome design task requiring complex geometries which often only enable a wide bandgap for one type while limited attenuation for the other. To overcome this hurdle, this work presents a 2D topology optimization framework to obtain broadband vibroacoustic bandgaps, simultaneously targeting acoustic and structural waves. Although bandgap topology optimization is a matured area of research, this work differentiates itself by including both physics simultaneously during the optimization resulting in novel vibroacoustic unit cell geometries. The intricate multi-physical metamaterial designs achieve broad frequency zones of simultaneous acoustic and structural attenuation. During the optimization, both volume and connectivity constraints are used to ensure lightweight, functional designs without material islands. Moreover, a zipper methodology is presented to enlarge the chances of achieving broad bandgaps. With both weakly and strongly coupled vibroacoustic case studies, the versatility of the framework is shown.

超材料已经显示出在预定频率范围内实现强噪声或减振的潜力。然而，同时瞄准两种波类型仍然是一项繁琐的设计任务，需要复杂的几何形状，通常只能实现一种波类型的宽带隙，而另一种波类型的衰减有限。为了克服这一障碍，这项工作提出了一个二维拓扑优化框架，以获得宽带振动声带隙，同时针对声波和结构波。虽然带隙拓扑优化是一个成熟的研究领域，但这项工作的区别在于，在优化过程中同时包含了两种物理特性，从而产生了新的振声单元胞几何形状。复杂的多物理超材料设计实现了同时声学和结构衰减的宽频率区域。在优化过程中，体积和连接性都受到了限制，以确保轻量化，功能设计没有材料孤岛。此外，提出了一种拉链方法，以扩大实现宽带隙的机会。通过弱耦合和强耦合振动声学案例研究，表明了该框架的多功能性。

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引用次数: 0

Analysis of stress intensity factor oscillations in 3D cracks using domain integrals and the extended finite element method 应用区域积分和扩展有限元法分析三维裂纹中的应力强度因子振荡

IF 6.9 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-01-16 DOI: 10.1016/j.cma.2025.117739

Vicente F. González-Albuixech , Eugenio Giner , Anthony Gravouil

Fracture-related failure of structural integrity can be evaluated using stress intensity factors (SIFs), and complex fractured geometries can be modeled using the extended finite element method (XFEM). Typically, domain integrals — especially

J

-integrals and interaction integrals — are used to compute SIFs. Although these integrals produce accurate estimates with the finite element method, they exhibit oscillations in the finite element support mesh when using XFEM due to their sensitivity to various topological relationships between the enrichment zone, domain definition and mesh. These oscillations can jeopardize accuracy, stability, and robustness of XFEM, influencing the convergence rate of SIFs. The domain integration solution has already undergone several changes to reduce these impacts, but no comparison or consideration of topological influence has been made. Here, we study some of the elements that lead to the unwanted behavior observed in various domain integration definitions, which incorporate corrections for curved and nonplanar cracks in examples with relatively coarse meshes. Consequently, various methodological limitations are discussed along with recommendations and suggestions.

裂缝相关的结构完整性破坏可以使用应力强度因子（SIFs）进行评估，复杂的裂缝几何形状可以使用扩展有限元法（XFEM）进行建模。通常，域积分——尤其是j积分和相互作用积分——用于计算sif。虽然这些积分用有限元法得到了准确的估计，但由于它们对富集带、域定义和网格之间的各种拓扑关系的敏感性，在使用XFEM时，它们在有限元支撑网格中表现出振荡。这些振荡会危及XFEM的精度、稳定性和鲁棒性，影响SIFs的收敛速度。为了减少这些影响，已经对域集成方案进行了多次修改，但没有对拓扑影响进行比较或考虑。在这里，我们研究了一些导致在各种领域积分定义中观察到的不良行为的元素，这些定义包括对具有相对粗糙网格的示例中的弯曲和非平面裂纹的修正。因此，讨论了各种方法的局限性，并提出了建议和建议。

{"title":"Analysis of stress intensity factor oscillations in 3D cracks using domain integrals and the extended finite element method","authors":"Vicente F. González-Albuixech , Eugenio Giner , Anthony Gravouil","doi":"10.1016/j.cma.2025.117739","DOIUrl":"10.1016/j.cma.2025.117739","url":null,"abstract":"<div><div>Fracture-related failure of structural integrity can be evaluated using stress intensity factors (SIFs), and complex fractured geometries can be modeled using the extended finite element method (XFEM). Typically, domain integrals — especially <span><math><mi>J</mi></math></span>-integrals and interaction integrals — are used to compute SIFs. Although these integrals produce accurate estimates with the finite element method, they exhibit oscillations in the finite element support mesh when using XFEM due to their sensitivity to various topological relationships between the enrichment zone, domain definition and mesh. These oscillations can jeopardize accuracy, stability, and robustness of XFEM, influencing the convergence rate of SIFs. The domain integration solution has already undergone several changes to reduce these impacts, but no comparison or consideration of topological influence has been made. Here, we study some of the elements that lead to the unwanted behavior observed in various domain integration definitions, which incorporate corrections for curved and nonplanar cracks in examples with relatively coarse meshes. Consequently, various methodological limitations are discussed along with recommendations and suggestions.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"436 ","pages":"Article 117739"},"PeriodicalIF":6.9,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142987903","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Learning implicit yield surface models with uncertainty quantification for noisy datasets 学习带有不确定性量化的噪声数据集的隐式屈服面模型

IF 6.9 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-01-15 DOI: 10.1016/j.cma.2025.117738

Donovan Birky , John Emery , Craig Hamel , Jacob Hochhalter

Materials often exhibit stochastic mechanical behaviors due to their inherent intrinsic variability. Data acquisition also introduces extrinsic noise into data. To learn yield surface models under uncertainty, we present a method that uses genetic programming based symbolic regression (GPSR) and a multi-objective fitness function (MOSR). Previous works have demonstrated using an implicit fitness metric in GPSR that compares the partial derivatives of proposed models with those of the data, allowing the generation of mechanics-guided, implicit yield surface models. MOSR adds to that a Bayesian fitness metric to simultaneously quantify parameter uncertainty. We test this method on benchmark implicit and physical test problems to demonstrate MOSR’s efficacy in finding implicit model forms on noisy data compared to the conventional implicit fitness metric. The results show that the MOSR algorithm prevents overfitting to noisy data, improves parameter estimates on data even with no noise present, and reduces model complexity, improving overall model interpretability. The MOSR method affords the ability to learn new and improved yield surface models while simultaneously quantifying the uncertainty in model parameters, leading to enhanced model interpretability.

材料由于其固有的可变性，往往表现出随机的力学行为。数据采集也会给数据带来外部噪声。为了学习不确定条件下的产量面模型，提出了一种基于遗传规划的符号回归（GPSR）和多目标适应度函数（MOSR）的方法。先前的研究已经证明了在GPSR中使用隐式适应度度量，将所提出模型的偏导数与数据的偏导数进行比较，从而可以生成力学指导的隐式屈服面模型。MOSR在此基础上增加了贝叶斯适应度度量，同时量化了参数的不确定性。我们在基准隐式和物理测试问题上测试了该方法，以证明与传统隐式适应度度量相比，MOSR在噪声数据上寻找隐式模型形式的有效性。结果表明，MOSR算法防止了对有噪声数据的过拟合，提高了对无噪声数据的参数估计，降低了模型复杂度，提高了整体模型的可解释性。MOSR方法提供了学习新的和改进的产量面模型的能力，同时量化了模型参数的不确定性，从而提高了模型的可解释性。

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引用次数: 0

Constructing boundary-identical microstructures via guided diffusion for fast multiscale topology optimization 基于引导扩散构建边界相同微结构的快速多尺度拓扑优化

IF 6.9 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-01-14 DOI: 10.1016/j.cma.2025.117735

Jingxuan Feng , Lili Wang , Xiaoya Zhai , Kai Chen , Wenming Wu , Ligang Liu , Xiao-Ming Fu

Hierarchical structures exhibit critical features across multiple scales. However, designing multiscale structures demands significant computational resources, and ensuring connectivity between microstructures remains a key challenge. To address these issues, large-range, boundary-identical microstructure datasets are successfully constructed, where the microstructures share the same boundaries and exhibit a wide range of elastic moduli. This approach enables highly efficient multiscale topology optimization. Central to our technique adopts a deep generative model, guided diffusion, to generate microstructures under the two conditions, including the specified boundary and homogenized elastic tensor. We generate the desired datasets using active learning approaches, where microstructures with diverse elastic moduli are iteratively added to the dataset, which is then retrained. After that, sixteen boundary-identical microstructure datasets with wide ranges of elastic modulus are constructed. We demonstrate the effectiveness and practicability of the obtained datasets over various multiscale design examples. Specifically, in the design of a mechanical cloak, we utilize macrostructures with 30 × 30 elements and microstructures filled with 256 × 256 elements. The entire reverse design process is completed within one minute, significantly enhancing the efficiency of the multiscale topology optimization.

等级结构在多个尺度上表现出关键特征。然而，设计多尺度结构需要大量的计算资源，并且确保微观结构之间的连接仍然是一个关键挑战。为了解决这些问题，成功构建了大范围，边界相同的微观结构数据集，其中微观结构共享相同的边界并表现出大范围的弹性模量。该方法实现了高效的多尺度拓扑优化。我们技术的核心是采用一种深度生成模型，即引导扩散，在两种条件下生成微观结构，包括指定边界和均质弹性张量。我们使用主动学习方法生成所需的数据集，其中迭代地将具有不同弹性模量的微结构添加到数据集中，然后重新训练数据集。在此基础上，构建了16个弹性模量范围大、边界相同的微观结构数据集。我们通过各种多尺度设计实例证明了所获得的数据集的有效性和实用性。具体来说，在机械斗篷的设计中，我们使用了30 × 30单元的宏观结构和256 × 256单元的微观结构。整个反设计过程在1分钟内完成，大大提高了多尺度拓扑优化的效率。

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引用次数: 0

SUPG-stabilized time-DG finite and virtual elements for the time-dependent advection–diffusion equation 用于时变平流扩散方程的 SUPG 稳定时间-DG 有限元和虚拟元

IF 6.9 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-01-14 DOI: 10.1016/j.cma.2024.117722

L. Beirão da Veiga , F. Dassi , S. Gómez

We carry out a stability and convergence analysis for the fully discrete scheme obtained by combining a finite or virtual element spatial discretization with the upwind-discontinuous Galerkin time-stepping applied to the time-dependent advection–diffusion equation. A space–time streamline-upwind Petrov–Galerkin term is used to stabilize the method. More precisely, we show that the method is inf–sup stable with constant independent of the diffusion coefficient, which ensures the robustness of the method in the convection- and diffusion-dominated regimes. Moreover, we prove optimal convergence rates in both regimes for the error in the energy norm. An important feature of the presented analysis is the control in the full

L^{2} (0, T; L^{2} (Ω))

norm without the need of introducing an artificial reaction term in the model. We finally present some numerical experiments in

(3 + 1)

-dimensions that validate our theoretical results.

将有限元或虚元空间离散化与逆风不连续Galerkin时间步进相结合，对时变平流扩散方程的完全离散格式进行了稳定性和收敛性分析。利用时空流线逆风彼得罗夫-伽辽金项稳定该方法。更准确地说，我们证明了该方法是稳定的，常数与扩散系数无关，这保证了该方法在对流和扩散占主导地位的情况下的鲁棒性。此外，对于能量范数误差，我们证明了两种情况下的最优收敛速率。所提出的分析的一个重要特征是控制在完整的L2（0，T;L2（Ω））范数中，而不需要在模型中引入人工反应项。最后给出了（3+1）维的数值实验，验证了我们的理论结果。

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引用次数: 0

DeepOKAN: Deep operator network based on Kolmogorov Arnold networks for mechanics problems DeepOKAN：基于力学问题Kolmogorov Arnold网络的深度算子网络

IF 6.9 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2025-01-14 DOI: 10.1016/j.cma.2024.117699

Diab W. Abueidda , Panos Pantidis , Mostafa E. Mobasher

The modern digital engineering design often requires costly repeated simulations for different scenarios. The prediction capability of neural networks (NNs) makes them suitable surrogates for providing design insights. However, only a few NNs can efficiently handle complex engineering scenario predictions. We introduce a new version of the neural operators called DeepOKAN, which utilizes Kolmogorov Arnold networks (KANs) rather than the conventional neural network architectures. Our DeepOKAN uses Gaussian radial basis functions (RBFs) rather than the B-splines. RBFs offer good approximation properties and are typically computationally fast. The KAN architecture, combined with RBFs, allows DeepOKANs to represent better intricate relationships between input parameters and output fields, resulting in more accurate predictions across various mechanics problems. Specifically, we evaluate DeepOKAN’s performance on several mechanics problems, including 1D sinusoidal waves, 2D orthotropic elasticity, and transient Poisson’s problem, consistently achieving lower training losses and more accurate predictions compared to traditional DeepONets. This approach should pave the way for further improving the performance of neural operators.

现代数字工程设计往往需要对不同场景进行昂贵的重复模拟。神经网络（NNs）的预测能力使其成为提供设计见解的合适替代品。然而，只有少数神经网络可以有效地处理复杂的工程场景预测。我们引入了一种名为DeepOKAN的新版本的神经算子，它利用Kolmogorov Arnold网络（KANs）而不是传统的神经网络架构。我们的DeepOKAN使用高斯径向基函数（rbf）而不是b样条。rbf提供了良好的近似特性，并且通常计算速度很快。KAN架构与rbf相结合，使DeepOKANs能够表示输入参数和输出字段之间更复杂的关系，从而对各种力学问题进行更准确的预测。具体来说，我们评估了DeepOKAN在几个力学问题上的性能，包括1D正弦波、2D正交异性弹性和瞬态泊松问题，与传统deeponet相比，始终实现更低的训练损失和更准确的预测。这种方法将为进一步提高神经算子的性能铺平道路。

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引用次数: 0