Computer Methods in Applied Mechanics and Engineering最新文献_第10页

A continuous topology optimization framework using an explicit binarization constraint 使用显式二值化约束的连续拓扑优化框架

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-07 DOI: 10.1016/j.cma.2025.118724

Tao Xu , Yi Min Xie , Jie Yang

Achieving crisp, manufacturable black-and-white (0/1) designs from element-based topology optimization methods is a critical and long-standing challenge. This paper introduces a novel continuous framework, termed Explicit Binarization Topology Optimization (EBTO), that addresses this challenge by treating binarization not as an implicit byproduct but as a direct mathematical constraint. The proposed method achieves this by introducing an explicit constraint formulated using a tunable function that directly measures and controls the global “greyness” of the design. This approach fundamentally decouples the binarization mechanism from the material model, allowing for the use of a linear material interpolation scheme that simplifies sensitivity analysis and provides a clearer physical interpretation for optimization problems. The versatility and robustness of the EBTO framework are demonstrated through a comprehensive set of 2D and 3D numerical examples, including compliance minimization, compliant mechanism design, and challenging stress-based optimization problems. The results consistently show that the proposed method generates clear 0/1 solutions with excellent structural performance, demonstrating superior results in benchmark cases compared to established methods. Furthermore, a set of guiding principles for formulating such explicit constraints is established, providing a foundation for future advancements in this class of topology optimization methods.

通过基于元素的拓扑优化方法实现清晰、可制造的黑白（0/1）设计是一个关键且长期存在的挑战。本文介绍了一种新的连续框架，称为显式二值化拓扑优化（EBTO），它通过将二值化视为直接的数学约束而不是隐含的副产品来解决这一挑战。提出的方法通过引入显式约束来实现这一目标，该约束使用可调函数来直接测量和控制设计的全局“灰色度”。这种方法从根本上将二值化机制与材料模型解耦，允许使用线性材料插值方案，简化灵敏度分析，并为优化问题提供更清晰的物理解释。EBTO框架的多功能性和鲁棒性通过一组全面的2D和3D数值示例得到了证明，包括合规最小化、合规机构设计和具有挑战性的基于应力的优化问题。结果一致表明，本文方法生成的0/1解清晰，结构性能优异，在基准案例中效果优于已有方法。此外，还建立了一套用于制定此类显式约束的指导原则，为该类拓扑优化方法的未来发展奠定了基础。

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引用次数: 0

Concurrent optimization of layups and stiffeners for stiffened laminated composite structures considering manufacturing constraints 考虑制造约束的加筋层合复合材料结构层合层和加筋并行优化

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-07 DOI: 10.1016/j.cma.2025.118711

Shanwei Li , Zongliang Du , Zunyi Duan , Yibo Jia , Chang Liu , Zhifu Ge , Xuefeng Mu , Xu Guo

Aiming to optimize thin-walled composite structures commonly used in the aerospace industry, the concurrent optimization of stiffeners and layup angles in stiffened laminated composite structures is achieved by combining the Moving Morphable Component (MMC) method and the Shape Function with Penalization (SFP) method. Based on the proposed explicit topology optimization framework for stiffened laminate structures, optimization formulations for maximizing stiffness and maximizing fundamental natural frequency, considering additive manufacturing constraints, are proposed. An efficient numerical algorithm is established, with analytical sensitivity analysis results derived. The constraints considered include stiffener thickness limits and laminate layup requirements. In addition, by fully leveraging the larger design space of synergies between stiffeners and laminates, it is demonstrated that the design result obtained through concurrent optimization exhibits better structural performance than the stiffened laminate structures achieved through the sequential optimization of stiffener and laminate layup angles.

针对航空航天中常用的薄壁复合材料结构优化问题，采用移动变形分量法（MMC）和形状函数惩罚法（SFP）相结合的方法，实现了加筋层合复合材料结构加筋和层合角的并行优化。基于所提出的加筋层压结构显式拓扑优化框架，提出了考虑增材制造约束条件下刚度最大化和基频最大化的优化公式。建立了一种高效的数值算法，并给出了解析灵敏度分析结果。考虑的约束条件包括加劲板厚度限制和层压板铺设要求。同时，充分利用加强筋与层合板协同作用的更大设计空间，证明通过并行优化获得的设计结果比通过加强筋与层合板叠加角顺序优化获得的加筋层合板结构具有更好的结构性能。

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引用次数: 0

An efficient three-step subgrid stabilized method for the steady natural convection equations 稳定自然对流方程的一种有效的三步子网格稳定方法

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-07 DOI: 10.1016/j.cma.2025.118710

Bo Zheng , Chuanqin Zheng , Yueqiang Shang , Yinnian He

This article is concerned with an efficient three-step subgrid stabilized method for the steady natural convection equations with moderate to high Rayleigh numbers in which two levels of finite element meshes are required. Within this method, we first solve one small, nonlinear coarse mesh natural convection problem with subgrid stabilizations, and then solve two large, subgrid stabilized and Newton-linearized fine mesh problems which have identical coefficient matrices with only different right-hand sides. This indicates that we avoid reassembly of the linearized problem in the third step, making the presented method easy-to-implement. Among them, both the coarse mesh and fine mesh problems are stabilized by the subgrid-scale models defined by two elliptic projections into lower-order finite element spaces of the velocity and temperature to improve the stability and convergence and thus, our present method has broad potential applications in simulating moderate to high Rayleigh number flows. Under the weak uniqueness condition, error bounds of the approximate solutions from the proposed method are strictly established. Scalings of the algorithmic parameters concerning the mesh sizes and stabilization parameters are also derived. In the end, manufactured solution examples are carried out to conform the theoretical analysis, showing a higher precision of the stabilized solutions calculated by our present three-step subgrid stabilized method than that of its counterpart method only staying at the second step. Besides, some numerical simulations and results are presented for the buoyancy-driven square cavity flow, sinusoidal hot cylinder flow and the isolated island problem with the Rayleigh numbers up to

R a = 10^{7}

representative of the convection-dominated regime to illustrate the high efficiency and effectiveness of our present method.

本文研究了需要两层有限元网格的中高瑞利数定常自然对流方程的高效三步亚网格稳定方法。在该方法中，我们首先求解一个具有子网格稳定化的小型非线性粗网格自然对流问题，然后求解两个具有相同系数矩阵，只是右侧不同的大型子网格稳定化牛顿线性化细网格问题。这表明我们在第三步中避免了线性化问题的重组，使所提出的方法易于实现。其中，粗网格和细网格问题均采用由两个椭圆投影到速度和温度的低阶有限元空间中定义的亚网格尺度模型来稳定，从而提高了稳定性和收敛性，因此本方法在模拟中高瑞利数流动方面具有广泛的潜在应用前景。在弱唯一性条件下，严格建立了该方法近似解的误差界。本文还推导了网格尺寸和稳定参数的算法参数比例。最后，通过算例验证了理论分析的正确性，结果表明，采用三步亚网格稳定法计算的稳定解比只停留在第二步的方法具有更高的精度。此外，通过浮力驱动的方腔流、正弦热柱体流和瑞利数高达Ra=107的对流占主导地位的孤岛问题的数值模拟和结果，说明了本文方法的高效性和有效性。

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引用次数: 0

Unsupervised Constitutive Model Discovery from Sparse and Noisy Data 基于稀疏和噪声数据的无监督本构模型发现

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-07 DOI: 10.1016/j.cma.2025.118722

Vahab Knauf Narouie , Jorge-Humberto Urrea-Quintero , Fehmi Cirak , Henning Wessels

Recently, unsupervised constitutive model discovery has gained attention through frameworks based on the Virtual Fields Method (VFM), most prominently the EUCLID approach. However, the performance of VFM-based approaches, including EUCLID, is affected by measurement noise and data sparsity, which are unavoidable in practice. The statistical finite element method (statFEM) offers a complementary perspective by providing a Bayesian framework for assimilating noisy and sparse measurements to reconstruct the full-field displacement response, together with quantified uncertainty. While statFEM recovers displacement fields under uncertainty, it does not strictly enforce consistency with constitutive relations. In this work, we integrate statFEM with unsupervised constitutive model discovery in the EUCLID framework, yielding statFEM–EUCLID. The framework is demonstrated for isotropic hyperelastic materials. The results show that this integration reduces sensitivity to noise and data sparsity, while ensuring that the reconstructed fields remain consistent with both equilibrium and constitutive laws.

近年来，基于虚拟场方法（VFM）的框架（最突出的是EUCLID方法）引起了无监督本构模型发现的关注。然而，包括EUCLID在内的基于vfm的方法的性能受到测量噪声和数据稀疏性的影响，这在实践中是不可避免的。统计有限元法（statFEM）提供了一个互补的视角，它提供了一个贝叶斯框架，用于吸收噪声和稀疏测量来重建全场位移响应，以及量化的不确定性。statFEM恢复不确定条件下的位移场，但并不严格遵守本构关系。在这项工作中，我们将statFEM与EUCLID框架中的无监督本构模型发现相结合，得到statFEM - EUCLID。对各向同性超弹性材料的框架进行了论证。结果表明，该方法降低了对噪声的敏感性和数据稀疏性，同时保证了重构场符合平衡和本构律。

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引用次数: 0

HERMES: A fast transient heat transfer solver for metal additive manufacturing HERMES：用于金属增材制造的快速瞬态传热求解器

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-06 DOI: 10.1016/j.cma.2025.118673

Hikmet Alperen Aydin , George Biros

Achieving precise thermal simulations of laser powder bed fusion (LPBF) in additive manufacturing is essential to predict microstructure evolution and finished part properties. Due to LPBF’s multiscale nature, where components are built at a scale of centimeters with melt pools that are only micrometers in size, these simulations present substantial complexity. Using a uniform grid is unfeasible because of the immense number of grid points, which can reach hundreds of billions. Adaptive mesh refinement (AMR) techniques are often employed, but general-purpose solvers do not fully exploit the structure of the LPBF problem.

We present HERMES, a GPU-accelerated nonlinear transient heat transfer AMR solver specifically designed for LPBF conditions. It employs a three-level structured mesh that moves concurrently with the laser in the global frame, effectively resolving the temperature field of the melt pool. HERMES accommodates arbitrary laser paths, multilayer processing, varying laser speeds, and temperature-dependent melt pool nonlinearities, allowing the rapid extraction of thermal parameters that influence microstructure, such as the thermal gradient G and solidification velocity R. Compared to a recent advanced LPBF solver, HERMES is more than 10 ×  faster. We demonstrate the solver’s capabilities by successfully simulating the entire print of a centimeter-scale part with a complex multilayer laser path in under an hour using a single GPU, achieving 1% accuracy in thermal gradients and cooling rates. This underscores HERMES’s applicability in realistic LPBF challenges. In total, HERMES balances computational efficiency with accuracy, providing a state-of-the-art tool for predictive thermal and microstructure modeling in metal additive manufacturing. HERMES is open-source and can be accessed on GitHub.

实现增材制造激光粉末床熔合（LPBF）过程的精确热模拟对于预测增材制造过程的微观结构演变和成品性能至关重要。由于LPBF的多尺度性质，其中的组件是在厘米尺度上建造的，而熔池的大小只有微米，因此这些模拟呈现出相当大的复杂性。使用统一的网格是不可行的，因为网格点的数量巨大，可以达到数千亿。自适应网格细化（AMR）技术经常被采用，但通用求解器不能充分利用LPBF问题的结构。我们提出了专为LPBF条件设计的gpu加速非线性瞬态传热AMR求解器HERMES。它采用三层结构网格，在全局框架内与激光同步运动，有效地求解了熔池的温度场。HERMES适用于任意激光路径、多层加工、不同的激光速度和温度相关的熔池非线性，允许快速提取影响微观结构的热参数，如热梯度G和凝固速度r。与最近先进的LPBF求解器相比，HERMES的速度要快10 × 以上。我们通过使用单个GPU在一小时内成功模拟具有复杂多层激光路径的厘米级部件的整个打印，从而证明了求解器的能力，在热梯度和冷却速率方面实现了1%的精度。这强调了HERMES在现实LPBF挑战中的适用性。总的来说，HERMES平衡了计算效率和准确性，为金属增材制造中的预测热和微观结构建模提供了最先进的工具。HERMES是开源的，可以在GitHub上访问。

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引用次数: 0

A physics-informed neural network framework for simulating creep buckling in growing viscoelastic biological tissues 模拟生长粘弹性生物组织蠕变屈曲的物理信息神经网络框架

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-06 DOI: 10.1016/j.cma.2025.118715

Zhongya Lin , Jinshuai Bai , Shuang Li , Xindong Chen , Bo Li , Xi-Qiao Feng

Modeling viscoelastic behavior is crucial in engineering and biomechanics, where materials undergo time-dependent deformations, including stress relaxation, creep buckling and biological tissue development. Traditional numerical methods, like the finite element method, often require explicit meshing, artificial perturbations or embedding customised programs to capture these phenomena, adding computational complexity. In this study, we develop an energy-based physics-informed neural network (PINN) framework using an incremental approach to model viscoelastic creep, stress relaxation, buckling, and growth-induced morphogenesis. Physics consistency is ensured by training neural networks to minimize the system’s potential energy functional, implicitly satisfying equilibrium and constitutive laws. We demonstrate that this framework can naturally capture creep buckling without pre-imposed imperfections, leveraging inherent training dynamics to trigger instabilities. Furthermore, we extend our framework to biological tissue growth and morphogenesis, predicting both uniform expansion and differential growth-induced buckling in cylindrical structures. Results show that the energy-based PINN effectively predicts viscoelastic instabilities, post-buckling evolution and tissue morphological evolution, offering a promising alternative to traditional methods. This study demonstrates that PINN can be a flexible robust tool for modeling complex, time-dependent material behavior, opening possible applications in structural engineering, soft materials, and tissue development.

粘弹性行为建模在工程和生物力学中是至关重要的，在这些领域中，材料会经历随时间的变形，包括应力松弛、蠕变屈曲和生物组织发育。传统的数值方法，如有限元法，通常需要明确的网格划分，人工扰动或嵌入定制程序来捕获这些现象，增加了计算复杂性。在这项研究中，我们开发了一个基于能量的物理信息神经网络（PINN）框架，使用增量方法来模拟粘弹性蠕变、应力松弛、屈曲和生长诱导的形态发生。物理一致性是通过训练神经网络最小化系统的势能泛函，隐式满足平衡和本构定律来保证的。我们证明，该框架可以自然地捕获蠕变屈曲，没有预先施加的缺陷，利用固有的训练动态来触发不稳定。此外，我们将我们的框架扩展到生物组织的生长和形态发生，预测圆柱形结构中的均匀膨胀和差异生长诱导的屈曲。结果表明，基于能量的PINN能有效预测粘弹性不稳定性、屈曲后演化和组织形态演化，为传统方法提供了一种有希望的替代方法。这项研究表明，PINN可以成为一种灵活可靠的工具，用于模拟复杂的、随时间变化的材料行为，在结构工程、软材料和组织发育方面开辟了可能的应用。

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引用次数: 0

An inter-scale stiffness propagation method with nonintrusive modeling of stochastic porosity in unidirectional composites 单向复合材料随机孔隙度的尺度间刚度传播非侵入建模方法

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-06 DOI: 10.1016/j.cma.2025.118720

Yu-Cheng Yang , Zi-Qian Wang , Jian-Jun Gou , Xiao-Bing Ma , Chun-Lin Gong

Porosity is a primary source of stiffness uncertainty in fiber-reinforced composites. However, explicitly modeling pores with prescribed geometry at the composite-scale leads to prohibitive computational cost for uncertainty quantification. This study proposes an inter-scale stiffness propagation method linking matrix-scale stochastic porosity to stiffness uncertainty of unidirectional fiber-reinforced (UD) composites. In such nonintrusive modeling of porosity, the local volume effect strongly influences the quantification accuracy. Pores in the matrix are modeled as spheres distributed by a Poisson point process. Their radius follows a truncated Gaussian law, leading to a porosity field whose covariance follows a Matérn-type form independent of local volume. The decay of porosity variance with increasing volume size, attributed to local volume averaging, is confirmed, indicating a similar behavior in finite element (FE) homogenization at the matrix-scale. The variance of matrix stiffness is found to decrease with growing local volume size, and its consistent negative correlation with porosity is thereby established. The stiffness-porosity joint distribution is then constructed by the conditional Gaussian mapping method. Finally, the stiffness calculation model at the composite-scale is developed, and the uncertainty induced by pores at the matrix-scale is quantified by Monte Carlo simulation. The results show that the nonintrusive modeling of stochastic porosity enables reliable stiffness propagation and efficient pore-induced uncertainty quantification.

孔隙率是纤维增强复合材料刚度不确定性的主要来源。然而，在复合尺度上明确地用规定的几何形状建模孔隙会导致不确定性量化的计算成本过高。本文提出了一种将单向纤维增强（UD）复合材料的基体尺度随机孔隙度与刚度不确定性联系起来的尺度间刚度传播方法。在这种非侵入式孔隙度建模中，局部体积效应严重影响定量精度。孔隙在基质中被建模为球体，通过泊松点过程进行分布。它们的半径遵循截断高斯定律，导致孔隙度场的协方差遵循独立于局部体积的matsamrn型形式。孔隙度随体积大小的增加而衰减，归因于局部体积平均，表明在矩阵尺度上的有限元（FE）均质化具有类似的行为。基体刚度方差随局部体积尺寸的增大而减小，并与孔隙率呈一致的负相关关系。然后采用条件高斯映射法构造刚度-孔隙度节理分布。最后，建立了复合尺度下的刚度计算模型，并通过蒙特卡罗模拟量化了基体尺度下孔隙引起的不确定性。结果表明，随机孔隙度的非侵入式建模能够实现可靠的刚度传播和有效的孔隙诱导不确定性量化。

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引用次数: 0

Error estimates and graded mesh refinement for isogeometric analysis in the vicinity of polar corners 极角附近等距分析的误差估计和梯度网格细化

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-06 DOI: 10.1016/j.cma.2025.118695

Thomas Apel, Philipp Zilk

Isogeometric analysis (IGA) combines exact geometric representations with higher-order accuracy for the numerical solution of partial differential equations. However, in geometrically complex settings – such as domains with corner singularities or non-standard parameterizations – these advantages may not be fully realized by standard IGA techniques. In particular, commonly used NURBS parameterizations can result in polar mappings, where one edge of the parametric domain is collapsed onto a single point, known as the polar point. Although widely used in computer-aided design, such configurations lack a full convergence theory. Additionally, reduced solution regularity near corners can significantly limit the performance of standard IGA, as higher-order convergence is no longer attainable.

In this work, both challenges are addressed by analyzing parameterizations in which the polar point coincides with a corner of the physical domain. To tackle the resulting singularity, a simple and effective local refinement strategy is proposed based on mesh grading toward the collapsed edge. This produces a locally refined mesh in the vicinity of the polar corner that accurately captures the singular behavior of the PDE solution.

To support this strategy, a numerical analysis tailored to polar domains with corners is developed. The framework includes the definition of polar function spaces on the parametric domain, a quasi-interpolant for polar splines, and the derivation of error estimates in weighted Sobolev norms. Optimal convergence is proven for smooth solutions under uniform refinement and for singular solutions using appropriately graded meshes. Numerical experiments on benchmark domains confirm the theoretical predictions and demonstrate the practical efficiency of the proposed method.

等几何分析（IGA）结合了精确的几何表示和偏微分方程数值解的高阶精度。然而，在几何上复杂的环境中-例如具有角点奇点或非标准参数化的域-这些优势可能无法通过标准IGA技术完全实现。特别是，常用的NURBS参数化可以导致极坐标映射，其中参数化域的一条边被折叠成一个点，称为极坐标点。虽然在计算机辅助设计中得到了广泛的应用，但这种构型缺乏完整的收敛理论。此外，在拐角附近降低的解正则性会极大地限制标准IGA的性能，因为不再能够实现高阶收敛。在这项工作中，这两个挑战都是通过分析参数化来解决的，其中极点与物理域的一个角落重合。为了解决由此产生的奇异性，提出了一种简单有效的基于网格向崩塌边缘分级的局部细化策略。这在极角附近产生了一个局部细化的网格，可以准确地捕获PDE解决方案的奇异行为。为了支持这一策略，开发了适合具有角的极域的数值分析。该框架包括参数域上极坐标函数空间的定义，极坐标样条的拟插值，以及加权Sobolev范数误差估计的推导。证明了均匀细化下的光滑解和适当分级网格下的奇异解的最优收敛性。在基准域上的数值实验验证了理论预测，并验证了该方法的实际有效性。

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引用次数: 0

Smoothed particle hydrodynamics of anisotropic diffusions 各向异性扩散的光滑粒子流体力学

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-05 DOI: 10.1016/j.cma.2025.118697

Xiaojing Tang, Xiangyu Hu, Oskar Haidn

Diffusion problems with anisotropic features arise in various scientific and engineering disciplines. However, accurately describing these processes in numerical simulations poses challenges due to the erroneous approximation and the appearance of spurious oscillations and negative concentrations. As a Lagrangian mesh-less method, Smoothed Particle Hydrodynamics (SPH) offers unified framework for addressing multi-physics problems involving diffusion process. Despite these benefits, its application to anisotropic diffusion remains limited, due to the lack of accurate and stable particle approximation. In this paper, we developed a complete Hessian matrix formulation for anisotropic diffusion to achieve 2nd-order accuracy and numerical stability. Specifically, the 2nd-order reproducing approximation of the complete Hessian matrix is used for the coordinate transformation to obtain the anisotropic diffusion operator. The formulation is further applied for simulating diffusion processes in thin structures, where anisotropic resolution is handled with the adaptive SPH method.

To validate the proposed formulation, firstly, anisotropic contaminant diffusion in a fluid is simulated, showing strong consistency with analytical solutions and producing smooth, oscillation-free results even in the presence of discontinuities. Secondly, the diffusion of a physical field with a predefined initial distribution is modeled using anisotropic resolution and the adaptive SPH method. Across various anisotropic ratios, the accuracy is examined in relation to the truncation error. Finally, this formulation is applied to modeling practical applications, including fluid-structure interaction in a thin porous membrane and anisotropic transmembrane potential transport in the left ventricle. These results demonstrate that the proposed formulation can accurately and stably solve complex anisotropic diffusion problems and diffusion problem using anisotropic kernel and spatial resolutions in diverse physical settings.

具有各向异性特征的扩散问题出现在各种科学和工程学科中。然而，由于错误的近似和虚假振荡和负浓度的出现，在数值模拟中准确描述这些过程带来了挑战。光滑粒子流体动力学（SPH）作为一种拉格朗日无网格方法，为解决涉及扩散过程的多物理场问题提供了统一的框架。尽管有这些优点，但由于缺乏精确和稳定的粒子近似，它在各向异性扩散中的应用仍然有限。在本文中，我们建立了一个完整的各向异性扩散的Hessian矩阵公式，以达到二阶精度和数值稳定性。具体来说，利用完全Hessian矩阵的二阶再现近似进行坐标变换，得到各向异性扩散算子。该公式进一步应用于薄结构中扩散过程的模拟，其中采用自适应SPH方法处理各向异性分辨率。为了验证所提出的公式，首先，模拟了流体中各向异性污染物的扩散，显示出与解析解的强一致性，即使在存在不连续的情况下也能产生光滑、无振荡的结果。其次，利用各向异性分辨率和自适应SPH方法对具有预定义初始分布的物理场的扩散进行建模；在不同的各向异性比率下，准确度与截断误差的关系进行了检验。最后，将该公式应用于模拟实际应用，包括薄多孔膜中的流固相互作用和左心室各向异性跨膜电位传递。结果表明，该公式可以准确稳定地求解复杂的各向异性扩散问题，以及在不同物理环境下使用各向异性核和空间分辨率的扩散问题。

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引用次数: 0

Rational-adaptive tip element with implicit enrichment for efficient crack modeling 具有隐式富集的合理自适应尖端单元，用于有效的裂纹建模

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-05 DOI: 10.1016/j.cma.2025.118716

Quan Wang , Hao Yu , YiLun Zhong , XiuYuan Chen , JiYun Sun , Sundararajan Natarajan , HengAn Wu

Cracks in materials induce highly localized displacement gradients in the vicinity of the crack tip, posing significant challenges for numerical simulation. Traditional approaches typically rely on local mesh refinement or enrichment techniques. The former substantially increases the number of degrees of freedom, resulting in computational inefficiency. The latter alleviates the computational burden but generally depends on explicit asymptotic solutions, which are unavailable in most situations and motivates further improvement and development. In this study, we propose a novel Rational-Adaptive Tip Element (RATE) for finite element method (FEM) and boundary element method (BEM) as a natural extension of enrichment techniques, introducing a rational enrichment scheme implicitly defined through weight parameters. The enrichment functions are constructed using rational polynomials formed by the weighted normalization of the partition of unity polynomial bases. A scaled mapping is employed to accurately capture the singular behavior near the crack tip, enabling precise approximation of tip fields without requiring explicit asymptotic expressions. This formulation facilitates indirect fitting of a wide range of crack tip field behaviors, thereby extending its applicability to problems with complex or unknown asymptotics. The effectiveness and robustness of the RATE are demonstrated through benchmark problems. It is observed that the RATE yields highly accurate results even on coarse meshes. Furthermore, it is shown that the method significantly reduces the degrees of freedom for accurate modeling of crack tip asymptotics, offering an efficient numerical tool for analyzing cracked structures.

材料裂纹在裂纹尖端附近会产生高度局部化的位移梯度，这对数值模拟提出了重大挑战。传统的方法通常依赖于局部网格细化或富集技术。前者大大增加了自由度的数量，导致计算效率低下。后者减轻了计算负担，但通常依赖于显式渐近解，这在大多数情况下是不可用的，并激励进一步的改进和发展。在这项研究中，我们提出了一种新的合理自适应尖端单元（RATE）作为富集技术的自然扩展，用于有限元法（FEM）和边界元法（BEM），引入了一种通过权参数隐式定义的合理富集方案。利用单位多项式基划分的加权归一化所形成的有理多项式构造富集函数。采用比例映射来精确捕获裂纹尖端附近的奇异行为，使尖端场的精确逼近不需要显式的渐近表达式。该公式有助于间接拟合大范围的裂纹尖端场行为，从而将其适用于具有复杂或未知渐近性的问题。通过基准问题验证了该方法的有效性和鲁棒性。观察到，即使在粗糙的网格上，RATE也能产生高精度的结果。此外，该方法显著降低了精确模拟裂纹尖端渐近的自由度，为分析裂纹结构提供了一种有效的数值工具。

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引用次数: 0