Engineering Analysis with Boundary Elements最新文献_第6页

A spectral approach for fast evaluation of 3D thermomagnetoelectroelastic Green’s functions and their derivatives in the boundary element method 边界元法中快速求解三维热磁电弹性格林函数及其导数的谱法

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-03-01 Epub Date: 2026-01-13 DOI: 10.1016/j.enganabound.2026.106647

Viktoriya Pasternak , Heorhiy Sulym , Andrii Korniichuk , Iaroslav Pasternak

This paper presents a spectral method for the efficient evaluation of Green’s functions in three-dimensional anisotropic thermoelastic and thermomagnetoelectroelastic problems. The method expands the Green’s function kernel in spherical harmonics, reducing its integral representation to a finite sum containing only odd- or even-degree harmonic coefficients. This eliminates the need for mesh-based evaluation and interpolation, significantly improving computational speed and numerical robustness. The approach requires precomputation of only a small set of spectral coefficients, after which Green’s functions and their spatial derivatives can be evaluated rapidly at arbitrary target points. Moreover, the precomputed coefficients depend only on the material properties, can be calculated to the desired accuracy, and can be reused to analyze different solid geometries composed of the same material. The resulting formulation is well suited for boundary element methods (BEM) and other integral equation schemes. Numerical experiments demonstrate fast spectral convergence of the spherical harmonic series, while performance benchmarks show that the proposed approach reduces the overall BEM computation time for the considered problems by approximately a factor of two. The method provides an efficient and scalable tool for simulating complex multiphysics phenomena in anisotropic solids.

本文提出了三维各向异性热弹性和热磁电弹性问题中格林函数的有效求值的谱方法。该方法将球面谐波中的格林函数核展开，将其积分表示化为只包含奇次或偶次谐波系数的有限和。这消除了基于网格的评估和插值的需要，显著提高了计算速度和数值鲁棒性。该方法只需要预先计算一小部分谱系数，然后可以在任意目标点快速计算格林函数及其空间导数。此外，预先计算的系数仅取决于材料的性质，可以计算到所需的精度，并且可以重复使用来分析由相同材料组成的不同固体几何形状。所得公式非常适合于边界元法（BEM）和其他积分方程格式。数值实验表明，该方法具有快速的谱收敛性，而性能基准测试表明，该方法将所考虑问题的总边界元计算时间减少了大约两倍。该方法为模拟各向异性固体中复杂的多物理场现象提供了一种有效且可扩展的工具。

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引用次数: 0

A novel quasi-interpolation radial integration BEM for non-homogeneous problems 非齐次问题的一种新的准插值径向积分边界元

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-03-01 Epub Date: 2026-01-16 DOI: 10.1016/j.enganabound.2026.106645

Bin Hu , Cong Li

A novel quasi-interpolation radial integration boundary element method (QIRIBEM) is proposed for solving non-homogeneous problems in this study. For domain integrals involving unknowns, the conventional methods use compactly supported radial basis functions (CSRBFs) to interpolate these unknowns directly. The direct interpolation scheme requires the introduction of an interpolation matrix, followed by matrix inversion and matrix multiplication operations. To overcome this limitation, a quasi-interpolator based on CSRBFs is first introduced to approximate the unknowns, where the unknown serves as the interpolation coefficient, eliminating the requirement for the interpolation matrix. Subsequently, a new interaction list is proposed to improve the computational efficiency of constructing the quasi-interpolator. Finally, the quasi-interpolator is incorporated into the radial integration method to transform domain integrals into boundary integrals. In contrast to the direct interpolation radial integration boundary element method (DIRIBEM), the proposed method achieves good accuracy by use a smaller supported domain and avoid a series of calculation and storage operations related to the interpolation matrix, which can save considerable computational time and memory spaces for large-scale models.

提出了一种求解非齐次问题的准插值径向积分边界元法。对于涉及未知数的域积分，传统方法使用紧支持径向基函数（csrbf）直接插值这些未知数。直接插补方案需要引入插补矩阵，然后进行矩阵反演和矩阵乘法运算。为了克服这一限制，首先引入基于csrbf的准插值器来逼近未知量，其中未知量作为插值系数，消除了对插值矩阵的要求。为了提高拟插值器的计算效率，提出了一种新的相互作用表。最后，将拟插值器引入到径向积分法中，将域积分转化为边界积分。与直接插值径向积分边界元法（DIRIBEM）相比，该方法利用更小的支持域，避免了与插值矩阵相关的一系列计算和存储操作，获得了较好的精度，可为大规模模型节省大量的计算时间和存储空间。

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引用次数: 0

Graph-based compactly supported radial basis function neural network 基于图的紧支持径向基函数神经网络

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-03-01 Epub Date: 2026-01-15 DOI: 10.1016/j.enganabound.2026.106644

Hongjin Ren , Dengao Li , Hongen Jia , Ruiping Niu , Hongbin Wang

In this paper, a novel graph-based compactly supported radial basis function physics-informed neural network (G-CS-RBN) is proposed for partial differential equations. Compactly supported radial basis functions are employed to replace the linear interpolation to construct an efficient one-hidden-layer neural network. An adaptive support radius is proposed that allows each point to automatically learn its local support according to the loss function. This overcomes the drawback of the traditional numerical compactly supported radial basis functions using fixed empirical support radius, which restricts the accuracy of models. A graph structure is used to store the collocation points and their respective center points to improve the interpretability of the network, storage efficiency, and network learning. Besides, the adaptive center point is also suggested to aid the adaptive support radius, which can further boost the performance of G-CS-RBN. Finally, extensive numerical experiments on 2D and 3D PDEs demonstrate that G-CS-RBN achieves consistently better accuracy and efficiency compared with classical numerical CS-RBF methods and standard PINNs, while showing improved robustness across different PDEs.

本文提出了一种新的基于图的紧支持径向基函数物理信息神经网络（G-CS-RBN）。采用紧支持径向基函数代替线性插值，构造了高效的单隐层神经网络。提出了一种自适应支持半径，使每个点能够根据损失函数自动学习其局部支持。这克服了传统数值紧支撑径向基函数采用固定经验支持半径的缺点，限制了模型的精度。采用图结构存储并置点及其各自的中心点，提高了网络的可解释性、存储效率和网络学习能力。此外，还提出了自适应中心点来辅助自适应支撑半径，进一步提高了G-CS-RBN的性能。最后，在二维和三维偏微分方程上进行的大量数值实验表明，与经典的数值CS-RBF方法和标准pin相比，G-CS-RBN方法的精度和效率始终更高，同时在不同偏微分方程上表现出更好的鲁棒性。

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引用次数: 0

Multi-physics smoothed particle hydrodynamics (SPH) model for icing channel in cold regions 寒区结冰通道的多物理场光滑粒子流体力学模型

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-03-01 Epub Date: 2026-01-19 DOI: 10.1016/j.enganabound.2025.106623

Yuxiao Li , Haitao Wu , Shenglong Gu , Chi Zhang , Songdong Shao

This study proposed a multi-physics mesh-free Smoothed Particle Hydrodynamics (SPH)-based model for ice cover growth, which is designed to simulate ice dynamic evolutions in a cold-regional water conveyance channel. The model addressed the limitations of grid-based methods in multiphase interface treatment, free-surface tracking, and phase-change coupling. A three-dimensional (3D) computational model integrating the heat transfer, flow, and phase changes has been developed via modification of the open-source Smoothed Particle Hydrodynamics for industrial complex systems (SPHinXsys) framework. Natural convection is incorporated to dynamically simulate the ice cover growth in conveyance channels. Benchmark validations include two-dimensional (2D) Stefan problems and natural convection in square cavity. Simulations of three-dimensional channel flows with natural convection showed good agreement with the grid-based method. In the proposed multi-physics ice growth simulations, the maximum root mean square error of ice thickness is 1.93 mm with the growth rate deviations below 2.19 %. These results confirm the model’s capability for complex flows and its potential applications in cold-region anti-icing designs.

本文提出了一种基于多物理场无网格光滑粒子水动力学（SPH）的冰盖生长模型，用于模拟寒冷地区输水通道的冰动力演变。该模型解决了基于网格的方法在多相界面处理、自由表面跟踪和相变耦合方面的局限性。通过修改开放源代码的工业复杂系统平滑粒子流体动力学（SPHinXsys）框架，开发了一个集成传热、流动和相变的三维（3D）计算模型。采用自然对流的方法，动态模拟了输送通道中冰盖的增长。基准验证包括二维（2D） Stefan问题和方形腔内的自然对流。对自然对流的三维通道流动的模拟结果与基于网格的方法吻合较好。在多物理场冰生长模拟中，冰厚度的最大均方根误差为1.93 mm，生长速率偏差小于2.19%。这些结果证实了该模型对复杂流动的能力及其在寒冷地区防冰设计中的潜在应用。

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引用次数: 0

A novel multi-loading deep energy method for 3D hyperelastic problems using a separable structure 基于可分离结构的三维超弹性问题多载荷深能新方法

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-02-01 Epub Date: 2025-12-05 DOI: 10.1016/j.enganabound.2025.106567

Linh Nguyen Hoai Le, Nha Thanh Nguyen

In recent years, physics-informed neural networks (PINNs) have become an active research field in computational mechanics. Among the PINN variants, the energy-based PINN, known as the deep energy method (DEM), is regarded as particularly effective. Unlike traditional PINNs, DEM merges multiple objective functions into one energy formulation, making convergence easier. Requiring only lower-order derivatives, DEM is also simpler to implement. This paper introduces a multi-loading model for nonlinear analysis that incorporates load factors as inputs during training. By training with a small set of loads, the resulting network can predict approximated fields for any load factor. To enhance efficiency, a separable structure is employed as a low-rank approximator, and its performance is compared with conventional approaches. Due to the well-conditioned and convex nature of hyperelastic energy functions, the model focuses on hyperelastic materials. A regression-based criterion is developed to estimate convergence. To assess robustness, three-dimensional problems are analyzed. The network is trained over a wide range of load factors, and the Adaptive Moment Estimation (Adam) optimizer is used. Results are compared with reference data for validation. Findings demonstrate that the proposed multi-loading model provides accurate, highly correlated predictions in both interpolation and extrapolation regions.

近年来，物理信息神经网络（pinn）已成为计算力学领域的一个活跃研究领域。在PINN变体中，基于能量的PINN，称为深能量法（DEM），被认为是特别有效的。与传统的pinn不同，DEM将多个目标函数合并到一个能量公式中，使收敛更容易。只需要低阶导数，DEM也更容易实现。本文介绍了一种多载荷非线性分析模型，该模型在训练过程中将载荷因子作为输入。通过使用一小组负载进行训练，得到的网络可以预测任何负载因子的近似场。为了提高效率，采用可分离结构作为低秩逼近器，并与传统逼近方法进行了性能比较。由于超弹性能量函数的良好条件和凸性，该模型主要研究超弹性材料。提出了一种基于回归的准则来估计收敛性。为了评估鲁棒性，分析了三维问题。采用自适应矩估计（Adam）优化器对网络进行了大范围负载因子的训练。结果与参考数据进行了对比验证。研究结果表明，所提出的多载荷模型在内插和外推区域都提供了准确的、高度相关的预测。

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引用次数: 0

Evaluating singular and near-singular integrals on C2 smooth surfaces with quadratic geometric approximation and closed form expressions 用二次几何近似和封闭形式表达式求C2光滑曲面上的奇异积分和近奇异积分

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-02-01 Epub Date: 2025-12-04 DOI: 10.1016/j.enganabound.2025.106588

Andrew Zheng, Spyros Alexakis, Adam R. Stinchcombe

Fredholm integral equations that appear in Boundary Element Methods often involve integrals with weakly singular kernels. Once the domain of integration is discretized into flat triangular elements, these weakly singular kernels become strongly singular or near-singular. Common methods to compute these integrals when the kernel is a Green’s function include coordinate transformations, polar coordinates with closed analytic formulas, and singularity extraction. However, these methods do not generalize well to the normal derivatives of Green’s functions due to the strongly singular behavior of these functions on triangular elements. We provide methods to integrate both the Green’s function and its normal derivative on smooth surfaces discretized by triangular elements in three dimensions for many commonly encountered differential operators. For strongly singular integrals involving normal derivatives of Green’s functions, we introduce a more refined approximation called Quadratic Surface Approximation. By using geometric information of the true surface of integration in combination with push-forward maps, it is significantly more accurate than the naive method of setting the singular integrals to zero, while being faster than adaptive refinement methods. We provide an algorithm for explicit computations on triangles, and present necessary analytic formulas that the algorithm requires in the appendix.

边界元法中出现的Fredholm积分方程通常涉及弱奇异核积分。一旦将积分域离散成平面三角元，这些弱奇异核就变成强奇异或近奇异核。当核函数为格林函数时，计算这些积分的常用方法包括坐标变换、带封闭解析公式的极坐标和奇点提取。然而，由于格林函数在三角元上的强奇异性，这些方法不能很好地推广到格林函数的正规导数。对于许多常见的微分算子，我们提供了在三维三角形离散光滑表面上对格林函数及其法向导数进行积分的方法。对于涉及格林函数法向导数的强奇异积分，我们引入了一种更精细的近似，称为二次曲面近似。利用积分真曲面的几何信息与推前映射相结合，其精度明显高于将奇异积分设为零的朴素方法，同时速度也快于自适应细化方法。我们在附录中提供了一种对三角形进行显式计算的算法，并给出了该算法所需的必要解析公式。

{"title":"Evaluating singular and near-singular integrals on C2 smooth surfaces with quadratic geometric approximation and closed form expressions","authors":"Andrew Zheng, Spyros Alexakis, Adam R. Stinchcombe","doi":"10.1016/j.enganabound.2025.106588","DOIUrl":"10.1016/j.enganabound.2025.106588","url":null,"abstract":"<div><div>Fredholm integral equations that appear in Boundary Element Methods often involve integrals with weakly singular kernels. Once the domain of integration is discretized into flat triangular elements, these weakly singular kernels become strongly singular or near-singular. Common methods to compute these integrals when the kernel is a Green’s function include coordinate transformations, polar coordinates with closed analytic formulas, and singularity extraction. However, these methods do not generalize well to the normal derivatives of Green’s functions due to the strongly singular behavior of these functions on triangular elements. We provide methods to integrate both the Green’s function and its normal derivative on smooth surfaces discretized by triangular elements in three dimensions for many commonly encountered differential operators. For strongly singular integrals involving normal derivatives of Green’s functions, we introduce a more refined approximation called Quadratic Surface Approximation. By using geometric information of the true surface of integration in combination with push-forward maps, it is significantly more accurate than the naive method of setting the singular integrals to zero, while being faster than adaptive refinement methods. We provide an algorithm for explicit computations on triangles, and present necessary analytic formulas that the algorithm requires in the appendix.</div></div>","PeriodicalId":51039,"journal":{"name":"Engineering Analysis with Boundary Elements","volume":"183 ","pages":"Article 106588"},"PeriodicalIF":4.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

A Trefftz collocation method for multiple interacting cylindrical nanofibers with the Steigmann-Ogden interface model 基于Steigmann-Ogden界面模型的多圆柱形纳米纤维的Trefftz配点法

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-02-01 Epub Date: 2025-12-05 DOI: 10.1016/j.enganabound.2025.106563

Cheng Chen , Yezeng Huang , Junbo Wang , Leiting Dong

In this study, a Trefftz collocation method (TCM) is proposed for modeling multiple interacting cylindrical nanofibers with the Steigmann-Ogden interface model. The Trefftz trial displacement fields are assumed in terms of the Papkovich-Neuber (P-N) solutions with the cylindrical harmonics as P-N potentials. The non-singular and singular harmonic functions from multiple source points are included for investigating multiple interacting nanofibers. The displacement continuity and the stress jump across the matrix/nanofiber interfaces and the boundary conditions are satisfied by the collocation method. To avoid the ill-conditioning of the resulting system of equations, a two-step scaling method is implemented: first, three characteristic lengths are introduced to scale the Trefftz trial functions; and second, a multi-scale characteristic length is adopted to scale each column of the coefficient matrix to equalize the norm. The good agreements between the numerical results and analytical solutions demonstrate the accuracy of the proposed method in simulating the interface effects of nanofiber composites. The numerical results reveal that the stress distributions in nanofibers are significantly size-dependent, which are influenced by interface elasticity parameters. Besides, the interactions of multiple nanofibers are also studied. The TCM developed in this work can be contributed to the design of advanced nanofiber composites.

本文提出了一种基于Steigmann-Ogden界面模型的多圆柱形纳米纤维的Trefftz配置方法（TCM）。用Papkovich-Neuber （P-N）解来假设Trefftz试验位移场，其柱面谐波为P-N势。采用多源点的非奇异和奇异谐波函数来研究多相互作用的纳米纤维。配置方法满足了基体/纳米纤维界面间的位移连续性和应力跳变，并满足了边界条件。为了避免结果方程组的病态化，实现了两步缩放方法：首先，引入三个特征长度对Trefftz试函数进行缩放；其次，采用多尺度特征长度对系数矩阵的每一列进行尺度化，使范数相等。数值结果与解析解吻合较好，证明了该方法在模拟纳米纤维复合材料界面效应方面的准确性。数值结果表明，纳米纤维中的应力分布具有明显的尺寸依赖性，受界面弹性参数的影响。此外，还研究了多种纳米纤维的相互作用。本研究所建立的纳米复合材料理论可以为高级纳米纤维复合材料的设计做出贡献。

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引用次数: 0

Axisymmetric crack in transversely isotropic FGM Full-space under tractions on crack surfaces 横向各向同性FGM全空间裂纹表面牵拉作用下的轴对称裂纹

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-02-01 Epub Date: 2025-12-08 DOI: 10.1016/j.enganabound.2025.106587

Sha Xiao , Zhongqi Quentin Yue

This paper develops a discontinuous displacement method to analyze an axisymmetric crack in an infinite functionally graded material (FGM) with transverse isotropy. The fundamental solution for a transversely isotropic layered solid of infinite extent subjected to point concentrated loads is employed to establish boundary integral equations only over one side of crack surfaces. Both continuous and discontinuous elements are employed to discretize the crack surface. A discretization technique is utilized to model arbitrary variations of the FGM along a graded direction. Verifications are conducted to illustrate that the numerical solutions presented by the proposed approach are in very good agreement with the classical analytical solutions. Examples are given for penny-shaped cracks in isotropic and transversely isotropic FGMs under different types of tractions, and the stress intensity factors (SIF) for two types of elastic media are analyzed and compared in detail. Results illustrate the effect of the heterogeneity and anisotropy of the materials on the fracture behavior of crack problems.

本文提出了一种分析具有横向各向同性的无限功能梯度材料轴对称裂纹的不连续位移法。利用横向各向同性无限宽层状固体受点集中荷载作用的基本解，建立了仅在裂纹表面一侧的边界积分方程。采用连续单元和不连续单元对裂纹表面进行离散化。利用离散化技术对梯度方向上FGM的任意变化进行了建模。验证表明，所提出的数值解与经典解析解非常吻合。给出了各向同性和横向各向同性fgm中便士形裂纹在不同牵引力作用下的算例，并对两种弹性介质的应力强度因子进行了详细的分析和比较。结果说明了材料的非均质性和各向异性对裂纹问题断裂行为的影响。

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引用次数: 0

An improved scaled boundary finite element method for thermoelastic analysis of plate elements under various boundary conditions 改进的尺度边界有限元法用于各种边界条件下的板单元热弹性分析

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-02-01 Epub Date: 2025-12-23 DOI: 10.1016/j.enganabound.2025.106611

Chenxi Ji , Jun Liu , Wenbin Ye , Lei Gan , Wei Yu , Haibo Wang , Tugen Feng , Jie Ren , Zhi Liu

In this paper, a modified scaled boundary finite element method (SBFEM) is developed to investigate the thermoelastic behavior of plate structures under thermal loading. The computational framework is based on three-dimensional elasticity theory and implemented via an enhanced SBFEM formulation. In this enhanced approach, the geometry is constructed by translating a two-dimensional mesh along the thickness direction, a concept equivalent to placing the scaling center of the SBFEM at infinity. This technique eliminates geometric errors introduced by discretization, thereby improving modeling accuracy, while retaining the essential advantages of the conventional SBFEM—only the boundary of the computational domain requires discretization, and analytical solution capability is maintained through the thickness. The thermoelastic response of the system is subsequently solved using the precise integration method. The accuracy and reliability of the proposed method are verified through comparisons with existing benchmark solutions. A parametric study is further conducted to systematically examine the effects of geometric parameters, boundary conditions, types of thermal loading, and thermo-mechanical coupling on the resulting thermal deformation and stress fields. Numerical results demonstrate that these parameters have a significant influence on the thermoelastic response of the plates.

本文提出了一种改进的尺度边界有限元法（SBFEM）来研究板结构在热载荷作用下的热弹性行为。计算框架基于三维弹性理论，并通过改进的SBFEM公式实现。在这种增强的方法中，几何结构是通过沿厚度方向平移二维网格来构建的，这一概念相当于将SBFEM的缩放中心放置在无穷远处。该技术消除了离散化带来的几何误差，从而提高了建模精度，同时保留了传统sbfem的本质优点-仅计算域的边界需要离散化，并且通过厚度保持解析解能力。然后用精确积分法求解了系统的热弹性响应。通过与已有基准解的比较，验证了所提方法的准确性和可靠性。进一步进行了参数化研究，系统地考察了几何参数、边界条件、热载荷类型和热-机械耦合对产生的热变形和应力场的影响。数值结果表明，这些参数对板的热弹性响应有显著影响。

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引用次数: 0

A deep energy method for solid mechanics based on a generalized multiplier approach 基于广义乘子法的固体力学深能法

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-02-01 Epub Date: 2025-12-19 DOI: 10.1016/j.enganabound.2025.106606

Ye Ouyang , Wei Jiang , Wei Du , Jintao Zhang , Yong Chen , Hong Zheng

The deep energy method provides a variational framework for solving mechanical problems governed by higher-order partial differential equations. In this framework, the imposition of boundary conditions is a crucial step. Boundary conditions are usually enforced using the classical penalty function method and its variants. However, the penalty factor directly affects the computational accuracy and efficiency. If the boundary conditions are imposed using the Lagrangian multiplier method, the resulting solution may not coincide with the optimal solution of the original constrained problem, especially when the Lagrangian is not strictly convex in the primal variables. To overcome this limitation, the generalized multiplier method is used to impose the boundary conditions. The neural-network loss function is constructed within the generalized multiplier framework. Finally, gradient-based optimization is employed to update the network parameters until the loss satisfies a prescribed tolerance. Numerical results show that the proposed method achieves higher accuracy and better solution efficiency than neural networks based on the classical penalty method, the L₁ exact penalty method, and the Lagrange multiplier method. The training process is also more stable.

深能量法为求解由高阶偏微分方程控制的力学问题提供了一个变分框架。在这个框架中，强加边界条件是关键的一步。边界条件通常使用经典罚函数方法及其变体来实现。然而，惩罚因子直接影响计算精度和效率。如果使用拉格朗日乘子法施加边界条件，得到的解可能与原约束问题的最优解不一致，特别是当拉格朗日在原始变量中不是严格凸时。为了克服这一局限性，采用广义乘子法施加边界条件。在广义乘子框架内构造神经网络损失函数。最后，采用基于梯度的优化方法更新网络参数，直到损失满足规定的容差。数值结果表明，该方法比基于经典惩罚法、L1精确惩罚法和拉格朗日乘子法的神经网络具有更高的精度和更好的求解效率。训练过程也更加稳定。

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引用次数: 0