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

A smoothly varying quadrature approach for 3D IgA-BEM discretizations: Application to Stokes flow simulations 三维IgA-BEM离散化的光滑变正交方法：在Stokes流模拟中的应用

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-30 DOI: 10.1016/j.cma.2026.118773

Cesare Bracco , Francesco Patrizi , Alessandra Sestini

We introduce a novel quadrature strategy for Isogeometric Analysis (IgA) boundary element discretizations, specifically tailored to collocation methods. Thanks to the dimensionality reduction and the natural handling of unbounded domains, boundary integral formulations are particularly appealing in the IgA framework. However, they require the evaluation of boundary integrals whose kernels exhibit singular or nearly singular behavior. Even when the kernel is not singular, its numerical evaluation becomes challenging whenever the integration region lies close to a collocation point. These integrals of polar and nearly singular functions represent the main computational difficulty of IgA-BEM and motivate the development of efficient and accurate quadrature rules. Unlike traditional methods that classify integrals as singular, nearly singular, or regular, our approach employs a desingularizing change of variables that smoothly adapts to the physical distance from singularities in the boundary integral kernels. The transformation intensifies near the polar point and progressively weakens when integrating over portions of the domain that are farther from it, ultimately leaving the integrand unchanged in the limit of a diametrically opposed region. This automatic calibration enhances accuracy and robustness by eliminating the traditional classification step, to which the approximation quality is often highly sensitive. Moreover, integration is performed directly over B-spline supports rather than over individual elements, reducing computational cost, particularly for higher-degree splines. The proposed method is validated through boundary element benchmarks for the three dimensional Stokes problem, where we achieve excellent convergence rates.

我们引入了一种新的正交策略，用于等几何分析（IgA）边界元离散化，专门针对搭配方法。由于降维和无界域的自然处理，边界积分公式在IgA框架中特别有吸引力。然而，它们要求计算核表现出奇异或近似奇异行为的边界积分。即使核函数不是奇异的，当积分区域靠近一个配点时，核函数的数值计算也会变得困难。这些极函数和近奇异函数的积分代表了IgA-BEM的主要计算困难，并推动了高效准确的求积分规则的发展。与将积分分类为奇异、近奇异或正则的传统方法不同，我们的方法采用了一种去奇异化的变量变化，可以平滑地适应边界积分核中与奇点的物理距离。变换在极点附近增强，在离极点较远的区域上积分时逐渐减弱，最终使被积函数在完全相反区域的极限内保持不变。这种自动校准消除了传统的分类步骤，从而提高了精度和鲁棒性，而传统的分类步骤往往对逼近质量非常敏感。此外，积分直接在b样条支撑上执行，而不是在单个元素上执行，减少了计算成本，特别是对于高次样条。通过对三维Stokes问题的边界元基准测试验证了该方法的有效性，并取得了较好的收敛速度。

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引用次数: 0

Shape-topology-layout optimization of stiffened shell structures via the Feature-Driven Optimization (FDO) method 基于特征驱动优化（FDO）方法的加筋壳结构形状拓扑布局优化

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-29 DOI: 10.1016/j.cma.2026.118771

Hualin ZHANG, Weihong ZHANG

In this work, an integrated shape-topology-layout optimization method is developed for stiffened shell structures to match the designs of the skin, openings and stiffeners. The proposed method takes full advantage of IsoGeometric Analysis (IGA), Finite Cell Method (FCM) and Feature-Driven Optimization (FDO). Specifically, the IGA enables accurate structural analysis and facilitates shape optimization of the skin by directly adjusting the control points of Non-Uniform Rational B-Splines (NURBS) skin surface. The FCM enables the use of the fixed structured mesh during the optimization process. The FDO adaptively inserts openings and stiffener features over the NURBS parametric domain for the optimization of skin topology and stiffener layout as easily as for 2D structures. Both implicit and parametric descriptions are jointly used to model openings and stiffener features. The implicit form concerns the level set function (LSF) in favor of the description of topological changes of the skin and the rapid identification of intersections of stiffeners. The parametric form concerns the use of a small number of design variables to produce optimized results with clear and smooth boundaries. Meanwhile, rigidity contributions of stiffeners are merged into the skin surface without the need of mesh conforming between the stiffeners and the skin surface. Representative examples are presented to demonstrate the effectiveness and advantages of the proposed design approach.

本文提出了一种针对加筋壳结构的形状-拓扑-布局集成优化方法，以匹配蒙皮、开口和加筋的设计。该方法充分利用了等几何分析（IGA）、有限单元法（FCM）和特征驱动优化（FDO）的优点。具体来说，IGA通过直接调整非均匀有理b样条（NURBS）蒙皮表面的控制点，实现了精确的结构分析，有利于蒙皮的形状优化。FCM允许在优化过程中使用固定结构网格。FDO自适应地在NURBS参数域上插入开口和加强筋特征，以优化蒙皮拓扑和加强筋布局，就像对二维结构一样容易。隐式描述和参数描述共同用于模拟开口和加筋特征。隐式形式涉及水平集函数（LSF），有利于描述皮肤的拓扑变化和快速识别加强筋的交叉点。参数化形式涉及到使用少量的设计变量来产生具有清晰和平滑边界的优化结果。同时，加强筋的刚度贡献被合并到蒙皮表面，而不需要加强筋与蒙皮表面之间的网格一致性。通过实例验证了所提设计方法的有效性和优越性。

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引用次数: 0

An interface-aware, conservative δR-Plus-SPH for granular slide–water interaction across subaerial–subaqueous regimes 一个界面感知的，保守的δ r - + sph，用于颗粒滑块-水在水下-水下的相互作用

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-29 DOI: 10.1016/j.cma.2026.118756

Yangfan MA , Mitsuteru ASAI , Zheng HAN , Bin SU , Guangqi CHEN

Modeling landslide-generated tsunamis using Smoothed Particle Hydrodynamics (SPH) is hindered by multiphase interactions, large density ratios, and regime transitions that promote excessive numerical dissipation, interface smearing, and divergence errors. This study develops a conservative, interface-aware δR+-SPH framework that couples incremental density diffusion to suppress oscillations with a low-dissipation Riemann flux for energy preservation. Particle disorder is mitigated through a two-stage Optimized Particle Shifting (OPS) for intra-phase regularity and Volume-Conservation Shifting (VCS) to control long-time volume drift. Incompressibility is enforced by combining Velocity-divergence Error Mitigation (VEM) with Hyperbolic/Parabolic Divergence Cleaning (HPDC), which together control both transient and cumulative divergence. The granular phase employs a regime-consistent visco-inertial rheology that unifies rate-dependent friction with effective-pressure regulation across dry, transitional, and submerged regimes. Verification and validation against two-fluid hydrostatics, a rotating patch, immersed granular collapse, and granular slide–tsunami benchmarks confirm that δR+-SPH achieves sharper interfaces, stronger energy and volume conservation, and reduced divergence compared to existing SPH variants. The numerical campaign also yields practical, phase-aware guidelines for shifting and stabilization that balance numerical accuracy and physical fidelity. Collectively, the framework improves the predictive reliability of landslide–tsunami simulations, supporting robust hazard assessment and engineering design.

利用光滑粒子流体力学（SPH）模拟滑坡产生的海啸受到多相相互作用、大密度比和状态转变的阻碍，这些转变会导致过度的数值耗散、界面涂抹和发散误差。本研究开发了一个保守的，界面感知的δR+-SPH框架，该框架将增量密度扩散与低耗散黎曼通量耦合以抑制振荡以保持能量。通过两阶段优化粒子移动（OPS）和体积守恒移动（VCS）来控制长时间体积漂移，减轻了粒子的无序性。不可压缩性是通过结合速度发散误差缓解（VEM）和双曲/抛物线发散清理（HPDC）来实现的，它们一起控制瞬时发散和累积发散。颗粒相采用一致的粘惯性流变学，将速率相关的摩擦与干燥、过渡和淹没状态下的有效压力调节结合起来。对双流体静力学、旋转斑块、浸没颗粒坍塌和颗粒滑动海啸基准的验证和验证证实，与现有的SPH变体相比，δR+-SPH具有更清晰的界面，更强的能量和体积守恒，以及更小的散度。数值运动也产生实用的，相位敏感的指导方针的转移和稳定，平衡数值精度和物理保真度。总的来说，该框架提高了滑坡-海啸模拟预测的可靠性，为可靠的灾害评估和工程设计提供了支持。

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引用次数: 0

Surrogate-enhanced higher order eigenstrain-based reduced order homogenization for polycrystal plasticity 多晶塑性的替代增强高阶本征应变降阶均匀化

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-29 DOI: 10.1016/j.cma.2026.118772

Aslan Nasirov, Caglar Oskay

Reduced-order models (ROMs) are frequently employed within concurrent multiscale frameworks to enhance the efficiency of nonlinear simulations and to mitigate the substantial computational expense associated with direct nonlinear homogenization schemes. Among these, transformation field analysis (TFA)-based ROMs have gained particular prominence due to their reliance on linear elastic simulations for model construction and their characteristically low dimensionality. However, extensive studies have demonstrated that these models often exhibit an artificially stiff response when applied across a broad spectrum of materials. In the present study, a higher order eigenstrain homogenization scheme is presented to mitigate the overly stiff response predicted by these models. Another challenge with the incorporation of nonuniform or higher order basis within models based on TFA is the expensive integration of the constitutive response due to nonuniform fields. A novel approach to compute these integrals is established that reduces the number of integration parameters over complex three-dimensional domains without additional spatial constraints. It is demonstrated that the linear modes can soften the elasto-plastic response and that the proposed integration method achieves approximately an order-of-magnitude improvement in computational efficiency compared to brute-force integration over a background mesh.

在并行多尺度框架中经常使用降阶模型（ROMs）来提高非线性模拟的效率，并减少与直接非线性均匀化方案相关的大量计算费用。其中，基于变换场分析（TFA）的rom由于依赖于线性弹性模拟来构建模型以及其低维的特点而获得了特别突出的地位。然而，广泛的研究表明，当应用于广泛的材料时，这些模型往往表现出人为的僵硬响应。在本研究中，提出了一种高阶特征应变均匀化方案来减轻这些模型预测的过刚性响应。在基于TFA的模型中加入非均匀或高阶基的另一个挑战是由于非均匀场引起的本构响应的昂贵积分。建立了一种计算这些积分的新方法，减少了复杂三维域上积分参数的数量，而没有额外的空间约束。结果表明，线性模式可以软化弹塑性响应，并且与背景网格上的蛮力积分相比，所提出的积分方法的计算效率提高了大约一个数量级。

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引用次数: 0

Stability analyses and instability mitigation for the material point method 物质点法的稳定性分析与不稳定性缓解

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-29 DOI: 10.1016/j.cma.2026.118784

Wen-Chia Yang , Deborah L. Sulsky

This study investigates the numerical stability of the Material Point Method (MPM) by analyzing the energy behavior and spectral properties of single-step updates. An energy-based analysis is first performed to quantify the energy variation introduced during each time step, followed by a spectral analysis that identifies critical time step constraints through the amplification matrix. Closed-form expressions for the critical integration parameter and the non-dimensional critical time step are derived, highlighting their dependence on mass parameters and particle distribution. The stability analyses identify key factors affecting the stability limits in MPM, including mass matrix selection, velocity projection, partially occupied grid cells, and integration errors in the particle-based formulation. Numerical experiments validate the analytical predictions and reveal the influence of particle-based integration errors on stability. A simple stabilization coefficient is proposed, which modifies shape function gradients in partially filled edge cells, significantly extending the stable time step range without increasing computational cost. The proposed framework offers practical guidelines for selecting stable time steps and enhancing the robustness of MPM simulations.

通过分析单步更新的能量行为和谱特性，研究了物质点法的数值稳定性。首先进行基于能量的分析，以量化每个时间步长期间引入的能量变化，然后进行光谱分析，通过放大矩阵确定关键时间步长约束。导出了临界积分参数和无因次临界时间步长的封闭表达式，强调了它们对质量参数和粒子分布的依赖。稳定性分析确定了影响MPM稳定性极限的关键因素，包括质量矩阵选择、速度投影、部分占用的网格单元以及基于颗粒的公式中的积分误差。数值实验验证了分析预测，揭示了基于粒子的积分误差对稳定性的影响。提出了一种简单的稳定系数，该系数可以修改部分填充边缘单元的形状函数梯度，在不增加计算成本的情况下显著延长稳定时间步长范围。提出的框架为选择稳定的时间步长和增强MPM仿真的鲁棒性提供了实用的指导。

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引用次数: 0

A multiscale lattice Boltzmann model for simulating Stokes to pre-Darcy flow 模拟Stokes - pre-Darcy流动的多尺度晶格Boltzmann模型

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-29 DOI: 10.1016/j.cma.2026.118775

Catherine Choquet, Théo Coiffard

This paper presents a unified numerical modeling framework for simulating fluid flow across heterogeneous media and multiple flow regimes, from very low-velocity porous flows to free-fluid Navier-Stokes regimes. The proposed approach builds upon the Lattice-Boltzmann (LB) method, exploiting its kinetic formulation and inherent multiscale character. Unlike conventional continuum models that rely on distinct partial differential equations (Darcy, Brinkman, Forchheimer, or Navier-Stokes) and require complex coupling strategies at interfaces, the present scheme introduces a scaling parameter

θ = ϵ^{α}

(with ϵ the Knudsen number and

α \in R_{+}

) to incorporate the effects of both microscopic structure and observation scale within a single LB formulation. We show that adjusting α, even abruptly, enables simulations in highly heterogeneous media without invoking separate PDE models and interface conditions, or introducing ad hoc force terms. Theoretical analysis based on Chapman-Enskog expansions demonstrates that the proposed LB scheme recovers well-known continuum (PDE) limits under appropriate scaling. Numerical benchmarks validate its accuracy and stability across Darcy, Brinkman, Forchheimer, and Stokes regimes, as well as intermediate transitions, confirming the potential of the method as a fully kinetic and genuinely multiscale alternative to traditional PDE-based approaches.

本文提出了一个统一的数值模拟框架，用于模拟非均质介质和多种流动形式的流体流动，从极低速多孔流动到自由流体Navier-Stokes流动。提出的方法建立在晶格-玻尔兹曼（LB）方法的基础上，利用其动力学公式和固有的多尺度特征。与依赖不同偏微分方程（Darcy, Brinkman， Forchheimer或Navier-Stokes）的传统连续体模型不同，并且需要在界面处采用复杂的耦合策略，本方案引入了一个缩放参数θ=ϵα（其中Knudsen数为λ， α∈R+），以将微观结构和观察尺度的影响纳入单个LB公式中。我们表明，调整α，即使是突然调整，也可以在高度异构的介质中进行模拟，而无需调用单独的PDE模型和界面条件，或引入特别的力项。基于Chapman-Enskog展开的理论分析表明，在适当的尺度下，所提出的LB方案可以恢复众所周知的连续体（PDE）极限。数值基准验证了其在Darcy， Brinkman， Forchheimer和Stokes体系以及中间过渡中的准确性和稳定性，证实了该方法作为传统基于pde的方法的完全动力学和真正的多尺度替代方案的潜力。

{"title":"A multiscale lattice Boltzmann model for simulating Stokes to pre-Darcy flow","authors":"Catherine Choquet, Théo Coiffard","doi":"10.1016/j.cma.2026.118775","DOIUrl":"10.1016/j.cma.2026.118775","url":null,"abstract":"<div><div>This paper presents a unified numerical modeling framework for simulating fluid flow across heterogeneous media and multiple flow regimes, from very low-velocity porous flows to free-fluid Navier-Stokes regimes. The proposed approach builds upon the Lattice-Boltzmann (LB) method, exploiting its kinetic formulation and inherent multiscale character. Unlike conventional continuum models that rely on distinct partial differential equations (Darcy, Brinkman, Forchheimer, or Navier-Stokes) and require complex coupling strategies at interfaces, the present scheme introduces a scaling parameter <span><math><mrow><mi>θ</mi><mo>=</mo><msup><mi>ϵ</mi><mi>α</mi></msup></mrow></math></span> (with ϵ the Knudsen number and <span><math><mrow><mi>α</mi><mo>∈</mo><msub><mi>R</mi><mo>+</mo></msub></mrow></math></span>) to incorporate the effects of both microscopic structure and observation scale within a single LB formulation. We show that adjusting <em>α</em>, even abruptly, enables simulations in highly heterogeneous media without invoking separate PDE models and interface conditions, or introducing <em>ad hoc</em> force terms. Theoretical analysis based on Chapman-Enskog expansions demonstrates that the proposed LB scheme recovers well-known continuum (PDE) limits under appropriate scaling. Numerical benchmarks validate its accuracy and stability across Darcy, Brinkman, Forchheimer, and Stokes regimes, as well as intermediate transitions, confirming the potential of the method as a fully kinetic and genuinely multiscale alternative to traditional PDE-based approaches.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"452 ","pages":"Article 118775"},"PeriodicalIF":7.3,"publicationDate":"2026-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146071613","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

A bridging-domain approach for multiscale modeling of anisotropic fracture in large-scale heterogeneous structures 大型非均质结构中各向异性裂缝多尺度建模的桥域方法

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-28 DOI: 10.1016/j.cma.2026.118774

Zakaria Chafia , Julien Yvonnet , Jérémy Bleyer

The prediction of the mechanical response of strongly heterogeneous structures containing defects critically depends on accurately capturing crack nucleation at micro scale. Fully resolved (high-fidelity) models are costly, whereas homogenized approaches may fail to represent initiation near heterogeneities. An efficient multiscale method is proposed in this work to simulate crack nucleation and propagation by bridging a high-fidelity micro-subdomain, dedicated to initiation, with a homogenized macro-subdomain used for propagation. The two subdomains overlap, may be discretized with nonconforming meshes, and are coupled through an energy-based formulation. The main contribution lies in the use, at the macro scale, of a surrogate anisotropic damage model constructed offline within the DDHAD (Data-Driven Harmonic Analysis of Damage) framework. This model reproduces direction-dependent crack propagation, while nucleation is resolved at the micro scale by the high-fidelity model. Significant computational speed-ups are achieved as compared to high-resolution simulations of the entire structure, and by accurately capturing initiation of the cracks in the microstructure. Examples on heterogeneous media exhibiting strong preferred crack orientations are presented to illustrate the potential of the approach.

含缺陷强非均相结构的力学响应预测关键取决于在微观尺度上准确捕捉裂纹形核。完全分辨（高保真度）的模型是昂贵的，而均质化的方法可能无法表示接近异质的起始。本文提出了一种有效的多尺度方法，通过桥接用于裂纹萌生的高保真微观子域和用于裂纹扩展的均匀化宏观子域来模拟裂纹的成核和扩展。两个子域重叠，可以用不一致的网格离散，并通过基于能量的公式耦合。主要贡献在于，在宏观尺度上，在DDHAD（数据驱动的损伤谐波分析）框架下离线构建了替代各向异性损伤模型。该模型再现了依赖于方向的裂纹扩展，而高保真模型在微观尺度上解决了成核问题。与整个结构的高分辨率模拟相比，通过准确捕获微观结构中裂纹的起始，实现了显著的计算速度提高。在非均质介质中表现出强优先裂纹取向的例子说明了该方法的潜力。

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引用次数: 0

Symmetrisation and hyperbolicity of first-order conservation laws in large strain compressible viscoelasticity using the smoothed particle hydrodynamics method 大应变可压缩粘弹性一阶守恒律的对称性和双曲性

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-28 DOI: 10.1016/j.cma.2026.118742

Chun Hean Lee , Antonio J. Gil , Tadas Jaugielavičius , Thomas Richardson , Sébastien Boyaval , Damien Violeau , Javier Bonet

This paper presents a new first-order hyperbolic framework with relaxation (or dissipation) terms for large strain viscoelastic solids. The framework is based on a compressible Maxwell-type viscoelastic model and integrates linear momentum conservation, geometric conservation laws, and evolution equations for internal variables. First, we propose a polyconvex strain energy function that is jointly convex with respect to the deformation measures and internal variables. Second, we introduce a generalised convex entropy function to symmetrise the hyperbolic system in terms of dual conjugate (entropy) variables. Third, we demonstrate that the system is hyperbolic (i.e., real wave speeds) under all deformation states, and that the relaxation terms correctly capture viscoelastic dissipation. Fourth, we present an upwinding Smoothed Particle Hydrodynamics (SPH) [1–3] scheme that enforces the second law of thermodynamics semi-discretely and uses the time rate of the generalised convex entropy to monitor internal dissipation and stabilise the simulation. Finally, the proposed framework is validated through numerical examples and benchmarked against the in-house Updated Reference Lagragian SPH [2,3] and vertex-centred finite volume [4–7] algorithms, demonstrating stability, accuracy, and consistent energy dissipation.

针对大应变粘弹性固体，提出了一种新的带松弛项的一阶双曲框架。该框架基于可压缩麦克斯韦型粘弹性模型，并集成了线性动量守恒、几何守恒定律和内部变量的演化方程。首先，我们提出了一个多凸应变能函数，该函数相对于变形量和内部变量是联合凸的。其次，我们引入了一个广义凸熵函数，以对偶共轭（熵）变量来对称双曲系统。第三，我们证明了系统在所有变形状态下都是双曲的（即实际波速），并且松弛项正确地捕获了粘弹性耗散。第四，我们提出了一种上旋光滑粒子流体力学（SPH）[1-3]方案，该方案半离散地执行热力学第二定律，并使用广义凸熵的时间率来监测内部耗散并稳定模拟。最后，通过数值算例验证了所提出的框架，并与内部更新的参考Lagragian SPH[2,3]和以顶点为中心的有限体积[4-7]算法进行了基准测试，证明了该框架的稳定性、准确性和一致的能量消耗。

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引用次数: 0

An Isogeometric Tearing and Interconnecting (IETI) method for solving high order partial differential equations over planar multi-patch geometries 求解平面多块几何高阶偏微分方程的等几何撕裂互连方法

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-27 DOI: 10.1016/j.cma.2026.118769

Mario Kapl , Aljaž Kosmač , Vito Vitrih

We present a novel method for solving high-order partial differential equations (PDEs) over planar multi-patch geometries with possibly extraordinary vertices demonstrated on the basis of the polyharmonic equation of order m, m ≥ 1, which is a particular linear elliptic PDE of order 2m. Our approach is based on the concept of Isogeometric Tearing and Interconnecting (IETI) and allows to couple the numerical solution of the PDE with C^s-smoothness,

s \geq m - 1

, across the edges of the multi-patch geometry. The proposed technique relies on the use of a particular class of multi-patch geometries, called bilinear-like G^s multi-patch parameterizations, to represent the multi-patch domain. The coupling between the neighboring patches is done via the use of Lagrange multipliers and leads to a saddle point problem, which can be solved first by a small dual problem for a subset of the Lagrange multipliers followed by local, parallelizable problems on the single patches for the coefficients of the numerical solution. Several numerical examples for the polyharmonic equation of order

m = 1

,

m = 2

and

m = 3

, i.e. for the Poisson’s, the biharmonic and the triharmonic equation, respectively, are shown to demonstrate the potential of our IETI method for solving high-order problems over planar multi-patch geometries with possibly extraordinary vertices.

基于m阶的多谐方程，m ≥ 1，提出了一种求解平面多斑几何上可能存在异常顶点的高阶偏微分方程的新方法，该方法是一个特殊的2m阶线性椭圆偏微分方程。我们的方法基于等几何撕裂和互连（IETI）的概念，并允许将PDE的数值解与cs -平滑度（s≥m−1）耦合在多块几何形状的边缘上。所提出的技术依赖于使用一种特殊的多斑块几何形状，称为双线性Gs多斑块参数化，来表示多斑块域。邻近斑块之间的耦合是通过使用拉格朗日乘子来完成的，并导致一个鞍点问题，该问题可以首先通过拉格朗日乘子子集的小对偶问题来解决，然后在数值解的系数的单个斑块上局部并行化问题来解决。本文给出了m=1阶、m=2阶和m=3阶多谐方程（即泊松方程、双谐方程和三谐方程）的几个数值例子，以证明我们的IETI方法在解决平面多斑几何上可能存在异常顶点的高阶问题方面的潜力。

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引用次数: 0

Numerically robust local continuum damage models with softening response via convex relaxation 具有凸松弛软化响应的数值鲁棒局部连续损伤模型

IF 7.3 1区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Computer Methods in Applied Mechanics and Engineering

Pub Date : 2026-01-25 DOI: 10.1016/j.cma.2026.118759

Celine Lauff , Matti Schneider , Thomas Böhlke

Continuum damage mechanics is characterized by mesh-dependent results unless specific countermeasures are taken. The most popular remedies involve introducing either nonlocality via filtering or a gradient extension for the damage variable(s). Such approaches have their limitations, e.g., they are hard to integrate into conventional finite-element codes, involve parameters that are non-trivial to determine experimentally and are incompatible with a scale transition that is both physically and mathematically sensible. The work at hand considers an alternative route to obtain mesh-independent damage models, namely via convex relaxation. Such convex damage models were considered before, but they are usually not capable of representing softening behavior. Schwarz et al. (Continuum Mech. Thermodyn., 33, pp. 69–95, 2021) proposed such a strategy by considering the convex envelope of a rate-limited simple damage model, i.e., an isotropic damage model without tension-compression anisotropy at small strains. However, they were not able to compute the envelope explicitly and provided an approximation only. In the work at hand, we introduce a number of conditions on the damage-degradation function which permit us to compute the convex envelope analytically for a large class of damage-degradation functions used in small-strain isotropic damage models. Interestingly, the obtained models involve a one-dimensional damaged microstructure, i.e., damage distributions emerge naturally. The resulting model is structurally simple and purely local, i.e., gradient-free, thermodynamically consistent and readily integrated into standard finite-element codes via traditional user subroutines. We discuss the computational and solid mechanical aspects of the ensuing model and demonstrate its numerical robustness via dedicated computational experiments. We also show that the model permits to be homogenized by considering a representative volume element study for an industrial-scale fiber-reinforced composite.

连续损伤力学的特点是网格依赖的结果，除非采取具体的对策。最流行的补救措施包括通过滤波或对损伤变量进行梯度扩展来引入非定域性。这种方法有其局限性，例如，它们很难集成到传统的有限元代码中，涉及的参数在实验中是不平凡的，并且与物理和数学上都合理的尺度转换不兼容。手头的工作考虑了另一种途径来获得网格无关的损伤模型，即通过凸松弛。以前考虑过这种凸损伤模型，但它们通常不能代表软化行为。Schwarz等人（连续介质力学）。Thermodyn。， 33, pp. 69-95, 2021)通过考虑速率受限简单损伤模型（即小应变下无拉压各向异性的各向同性损伤模型）的凸包线提出了这种策略。然而，他们不能明确地计算包络线，只能提供一个近似值。在手头的工作中，我们引入了一些关于损伤退化函数的条件，这些条件允许我们解析地计算用于小应变各向同性损伤模型的一类损伤退化函数的凸包络。有趣的是，所获得的模型涉及一维损伤微观结构，即损伤分布自然出现。所得模型结构简单，纯局部，即无梯度，热力学一致，并易于通过传统用户子程序集成到标准有限元代码中。我们讨论了随后模型的计算和实体力学方面，并通过专门的计算实验证明了其数值鲁棒性。我们还表明，通过考虑工业规模的纤维增强复合材料的代表性体积单元研究，该模型允许均质化。

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引用次数: 0