Engineering Analysis with Boundary Elements最新文献

High-order meshless approach for solving the 3D nonlinear Bratu problem 求解三维非线性Bratu问题的高阶无网格方法

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-01-17 DOI: 10.1016/j.enganabound.2026.106651

El-Houssaine El-Asri , Abdeljalil Tri , Bouazza Braikat , Hamid Zahrouni

In this work, we study the nonlinear Bratu problem in three dimensions (3D), known for its complexity due to the presence of multiple solutions and bifurcations. We employ the Method of Fundamental Solutions (MFS) and Radial Basis Functions (RBF), combined with a High Order Continuation Method (HOCM), to compute the entire solution branch. The nonlinearity of the problem is handled by reformulating it into a sequence of linear problems using a Taylor series expansion. The resulting linear problems are approximated using MFS and RBF. The core idea is to represent the solution as a linear combination of fundamental solutions associated with source points located outside the domain, along with a particular solution constructed at collocation points within the domain. This approach enables the solution to be computed branch by branch through the continuation method. Bifurcation points are detected using a scalar indicator along the solution branches, based on a common tangent operator. This methodology allows for the efficient resolution of nonlinear problems in complex three-dimensional geometries. Our results demonstrate the accuracy and effectiveness of the proposed approach in solving the nonlinear Bratu equation.

在这项工作中，我们研究了三维（3D）的非线性Bratu问题，由于存在多个解和分支而以其复杂性而闻名。我们采用基本解法（MFS）和径向基函数法（RBF），结合高阶延拓法（HOCM）来计算整个解分支。该问题的非线性是通过使用泰勒级数展开将其重新表述为一系列线性问题来处理的。用MFS和RBF对得到的线性问题进行近似。其核心思想是将解表示为与位于域外的源点相关的基本解的线性组合，以及在域内的并置点构造的特定解。这种方法可以通过延拓法逐个分支地计算解。根据公切算子，沿着解分支使用标量指示器来检测分岔点。这种方法允许在复杂的三维几何非线性问题的有效解决。我们的结果证明了该方法在求解非线性Bratu方程中的准确性和有效性。

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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 : 2026-01-16 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

Scattering of capillary-gravity waves by surface-piercing porous barriers in the presence of uniform current over a porous sea bed 在多孔海床上存在均匀水流时，穿透表面的多孔屏障对毛细管重力波的散射

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-01-15 DOI: 10.1016/j.enganabound.2026.106646

Gagan Sahoo, Harekrushna Behera, Tai-Wen Hsu

Capillary–gravity waves, influenced by both surface tension and gravity, interact strongly with marine structures, especially in the presence of uniform currents. Despite extensive studies on wave scattering by porous structures, the combined effects of surface tension, current, and porous barriers over a porous bottom remain insufficiently explored. This study examines the scattering of such waves by two thin surface-piercing porous barriers in the presence of a uniform current over a porous sea bed. A linear wave–structure interaction model is solved numerically through a hybrid Boundary Element-Finite Difference Method (BEM–FDM) and analytically through an eigenfunction expansion combined with a least-squares approach. The hybrid BEM–FDM efficiently handles higher-order boundary conditions that cannot be directly addressed by conventional BEM, while the analytical method eliminates the need for eigenfunction orthogonality and explicit mode coupling. The effects of surface tension, current velocity and direction, porous effect parameters of barriers as well as bottom, barrier length and spacing between them on reflection, transmission, and energy dissipation are analyzed. Results show that surface tension enhances reflection and dissipation while reducing transmission. Current direction strongly affects scattering: following currents enhance transmission, whereas opposing currents increase reflection and dissipation. Longer barriers and larger porous-effect parameters of both porous barriers and porous bottom enhance energy dissipation, while spacing between porous barriers induce interference driven oscillations.

毛细重力波受表面张力和重力的影响，与海洋结构有强烈的相互作用，特别是在有均匀洋流的情况下。尽管对多孔结构的波散射进行了广泛的研究，但表面张力、电流和多孔底部上的多孔屏障的综合影响仍然没有得到充分的探索。本研究考察了在多孔海床上均匀水流存在的情况下，两个薄的穿透表面的多孔屏障对这种波的散射。采用混合边界元-有限差分法（BEM-FDM）对线性波-结构相互作用模型进行数值求解，采用特征函数展开结合最小二乘方法对线性波-结构相互作用模型进行解析求解。该方法有效地处理了传统边界元法无法直接解决的高阶边界条件，而解析法消除了特征函数正交性和显式模态耦合的需要。分析了表面张力、电流速度和方向、阻挡层及底部的多孔效应参数、阻挡层长度和阻挡层间距对反射、透射和能量耗散的影响。结果表明，表面张力增强了反射和耗散，降低了透射。电流方向强烈影响散射：顺电流增强透射，相反电流增强反射和耗散。更长的势垒和更大的孔效应参数增强了能量耗散，而多孔势垒之间的间距引起干涉驱动振荡。

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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 : 2026-01-15 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

Mc-PINN for solving Conservative Allen–Cahn equations using Metropolis–Hasting based sampling Mc-PINN用于求解基于Metropolis-Hasting采样的保守Allen-Cahn方程

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-01-14 DOI: 10.1016/j.enganabound.2026.106643

Anjali Singh, Rajen Kumar Sinha

The conservative Allen–Cahn (CAC) equation is a second-order nonlinear partial differential equation that models phase separation in binary mixtures while preserving the total volume. Physics-informed neural networks (PINNs) have demonstrated considerable success in approximating solutions to various classes of partial differential equations; however, their application to CAC models remains challenging. These difficulties stem from the presence of a small interfacial parameter

ϵ

in the nonlinear term

F^{'} (u)

and highly nonlinear mass-correction terms

G_{i}

,

i = 1, 2, 3

, which significantly degrade the approximation accuracy and mass conservation properties of standard PINNs. In this work, we propose a novel hybrid mass-constrained physics-informed neural network (Mc-PINN) framework for efficiently and accurately solving three types of CAC models in convex polygonal domains. The proposed method integrates deep learning with operator-splitting strategies to decompose the original CAC equations into simpler subproblems. One subproblem admits an analytical solution, while the other is solved using the Mc-PINN scheme. To further enhance efficiency, a Metropolis–Hastings based adaptive sampling strategy is introduced. In addition, we derive error estimates for the proposed method applied to all three CAC models. Numerical experiments demonstrate the robustness, accuracy, and mass-conservation capability of the proposed framework.

保守的Allen-Cahn （CAC）方程是一个二阶非线性偏微分方程，它在保持总体积的情况下模拟二元混合物的相分离。物理信息神经网络（pinn）在逼近各种偏微分方程的解方面取得了相当大的成功；然而，它们在CAC模型中的应用仍然具有挑战性。这些困难源于非线性项F ' (u)中存在一个小的界面参数，以及高度非线性的质量校正项Gi， i=1,2,3，这显著降低了标准pin的近似精度和质量守恒特性。在这项工作中，我们提出了一种新的混合质量约束物理信息神经网络（Mc-PINN）框架，用于有效和准确地求解凸多边形域中的三种类型的CAC模型。该方法将深度学习与算子分裂策略相结合，将原始CAC方程分解为更简单的子问题。其中一个子问题允许解析解，而另一个子问题则使用Mc-PINN方案求解。为了进一步提高效率，引入了一种基于Metropolis-Hastings的自适应采样策略。此外，我们推导了适用于所有三种CAC模型的方法的误差估计。数值实验证明了该框架的鲁棒性、准确性和质量守恒性。

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引用次数: 0

Crack analysis of the foundation gallery within an asphalt concrete core dam based on 3D SBFEM-PFM 基于三维SBFEM-PFM的沥青混凝土心坝基础廊裂缝分析

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-01-13 DOI: 10.1016/j.enganabound.2026.106641

Yue Zhuo , Kai Chen , Degao Zou , Shanlin Tian , Shiyong Wu , Shi Zhang

Fracture analysis of anti-seepage systems in earth-rock dams involves pronounced material nonlinearity, frictional contact behavior, and convergence difficulties, which remain long-standing challenges in geotechnical engineering. In this study, A nonlinear 3D SBFEM–PFM formulation was created by embedding the phase-field model (PFM) into the scaled boundary finite element method (SBFEM) and implementing an intra-element phase-field interpolation scheme. Base on independently developed GEODYNA, a unified and parallelized framework was established through various coupling schemes, including FEM-SBFEM, phase-field, and nonlinear contact algorithm. The method is validated against a classical benchmark and subsequently applied to the world’s highest asphalt concrete core rockfill dam (ACCRD) on the overburden to simulate full-process cracking of gallery. Cracks were identified along the inner surface of the gallery and on the outer surfaces of both banks, with reservoir impoundment exhibiting opposing effects on crack width. Additionally, the severity of structural damage was significantly influenced by the interface characteristics between the gallery and surrounding geomaterial. This study signifies the inaugural implementation of PFM to achieve a full-process simulation of stress accumulation, crack initiation, and progressive propagation within dam anti-seepage systems. High risk zones were precisely identified, and the practical optimization measure was suggested, providing an innovative and effective approach for structural analysis in related geotechnical engineering applications.

土石坝防渗系统的断裂分析涉及明显的材料非线性、摩擦接触行为和收敛困难，是岩土工程中长期存在的难题。本研究通过将相场模型（PFM）嵌入到比例边界有限元法（SBFEM）中，并实现单元内相场插值方案，建立了三维SBFEM - PFM非线性公式。在自主开发的GEODYNA基础上，通过FEM-SBFEM、相场法、非线性接触算法等多种耦合方案，建立了统一的并行化框架。通过经典基准验证了该方法的有效性，并将其应用于世界上最高的沥青混凝土堆芯坝（ACCRD）覆盖层上，模拟了廊道的全过程开裂。沿廊道的内表面和两岸的外表面发现了裂缝，水库蓄水对裂缝宽度的影响相反。此外，廊道与周围岩土材料之间的界面特征对结构损伤的严重程度有显著影响。该研究标志着PFM首次实现了大坝防渗系统内应力积累、裂缝萌生和渐进扩展的全过程模拟。准确识别出高风险区，并提出切实可行的优化措施，为相关岩土工程应用中的结构分析提供了一种创新有效的方法。

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引用次数: 0

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 : 2026-01-13 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 coupled finite element-virtual element method for thermomechanical analysis of electronic packaging structures 电子封装结构热力分析的有限元-虚元耦合方法

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-01-12 DOI: 10.1016/j.enganabound.2026.106640

Yanpeng Gong , Sishuai Li , Yue Mei , Bingbing Xu , Fei Qin , Xiaoying Zhuang , Timon Rabczuk

This study presents a finite element and virtual element (FE-VE) coupled method for thermomechanical analysis in electronic packaging structures. The approach partitions computational domains strategically, employing FEM for regular geometries to maximize computational efficiency and VEM for complex shapes to enhance geometric flexibility. Interface compatibility is maintained through coincident nodal correspondence, ensuring solution continuity across domain boundaries while reducing meshing complexity and computational overhead. Validation through electronic packaging applications demonstrates reasonable agreement with reference solutions and acceptable convergence characteristics across varying mesh densities. The method effectively captures thermal distributions and stress concentrations in multi-material systems, establishing a practical computational framework for electronic packaging analysis involving complex geometries. Source codes are available at https://github.com/yanpeng-gong/FeVeCoupled-ElectronicPackaging.

提出了一种用于电子封装结构热力学分析的有限元与虚元耦合方法。该方法对计算域进行了战略性划分，对规则几何图形采用有限元法以提高计算效率，对复杂形状采用VEM法以提高几何灵活性。接口兼容性通过一致的节点通信保持，确保解决方案跨域边界的连续性，同时降低网格复杂性和计算开销。通过电子封装应用的验证证明了与参考解决方案的合理一致，以及在不同网格密度下可接受的收敛特性。该方法有效地捕获了多材料系统中的热分布和应力集中，为涉及复杂几何形状的电子封装分析建立了实用的计算框架。源代码可从https://github.com/yanpeng-gong/FeVeCoupled-ElectronicPackaging获得。

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引用次数: 0

An explicit numerical manifold method with timestep scaling for quasi-static unsaturated hydro-mechanical coupling problems 准静态非饱和水-力耦合问题的时间步标化显式数值流形方法

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-01-10 DOI: 10.1016/j.enganabound.2026.106639

Junfeng Li , Yongtao Yang , Xiaodong Fu , Wenan Wu , Hangtian Song

Hydro-mechanical coupling in unsaturated media is of significant engineering relevance. To efficiently address quasi-static, long-term hydro-mechanical problems that are typically computationally expensive for conventional explicit schemes, a timestep scaling technique is integrated into the explicit numerical manifold method (NMM). A key advantage of this technique is that it effectively overcomes the strict stability limit of the mechanical solver. This approach aligns the critical timestep size of the mechanical field with that of the hydro field, bridging the significant timescale disparity between the two fields. This enables a unified and large timestep size within a staggered solution strategy, thereby drastically reducing the computational cost. Additionally, a simplified pressure-based explicit algorithm for unsaturated flow is embedded into the hydro-mechanical coupling framework. Verification and application examples confirm the substantial acceleration and high accuracy of the proposed explicit NMM with timestep scaling for hydro-mechanical coupling problems. Furthermore, its potential for extension to other multiphysics problems warrants continued investigation.

非饱和介质中的水-力耦合具有重要的工程意义。为了有效地解决准静态、长期的流体力学问题，在显式数值流形方法（NMM）中集成了时间步长缩放技术。该方法的一个关键优点是有效地克服了机械解算器的严格稳定性限制。这种方法将机械场的临界时间步长与水力场的关键时间步长保持一致，弥合了两个领域之间显著的时间尺度差异。这使得在交错解决方案策略中实现统一的大时间步长，从而大大降低了计算成本。此外，将一种简化的基于压力的非饱和流显式算法嵌入到水-力耦合框架中。验证和应用实例表明，所提出的带时间步长标度的显式NMM具有较高的加速度和精度。此外，它扩展到其他多物理场问题的潜力值得继续研究。

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引用次数: 0

Modeling of magneto-electro-elastic solids with complex cutouts by the numerical manifold method 采用数值流形方法模拟具有复杂切口的磁电弹性固体

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-01-09 DOI: 10.1016/j.enganabound.2025.106625

Chenliang Li, Donglin Guo, Huihua Zhang

Magneto-electro-elastic (MEE) materials are pivotal in a multitude of fields. In this work, the numerical manifold method (NMM) is innovatively extended to establish 2-D numerical models for perforated MEE solids. The unique dual-cover system, namely, the mathematical cover and the physical cover, enables the NMM to discretize the physical domain with non-conforming mathematical covers straightforwardly. The governing equations and the boundary conditions for hole problems of MEE materials are firstly introduced. Then, by taking into account the governing equations, boundary conditions, and the NMM field approximations, the NMM global discrete equations are derived using the weighted residual method. Through three benchmark examples, the precision of the proposed method is verified, and it is then applied to study two more complex cases, where the effects of hole configurations, loading conditions, and polarization directions on the field quantities of perforated MEE materials are further examined.

磁电弹性（MEE）材料在许多领域都是至关重要的。本文创新性地扩展了数值流形方法（NMM），建立了多孔MEE固体的二维数值模型。独特的双覆盖系统，即数学覆盖和物理覆盖，使NMM可以直接离散不符合数学覆盖的物理域。首先介绍了MEE材料空穴问题的控制方程和边界条件。然后，通过考虑控制方程、边界条件和NMM场近似，利用加权残差法推导了NMM全局离散方程。通过三个基准算例验证了该方法的精度，并将其应用于两种更复杂的情况，进一步考察了孔构型、载荷条件和极化方向对多孔MEE材料场量的影响。

引用次数: 0