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Shape-performance coupled digital twin based on heterogeneous data from multiple sources: a scissor lift platform example 基于多源异构数据的形状-性能耦合数字孪生:以剪式升降平台为例
IF 8.7 2区 工程技术 Q1 Mathematics Pub Date : 2024-07-29 DOI: 10.1007/s00366-024-02035-6
Hongjiang Lu, Zenggui Gao, Yanning Sun, Chaojia Gao, Zifeng Xu, Yunjie Pan, Lilan Liu

Digital twin, a concept of establishing mapping linkages between physical and digital areas using digital technology to achieve instantaneous information transfer for monitoring, optimization or decision-making. Digital twins has emerged as a crucial instrument for ensuring structural safety. However, achieving real-time prediction in time series for structural safety monitoring is challenging, as is the dynamic synthesis of heterogeneous data from numerous sources. This study presents a shape-performance coupled digital twin (SPC-DT) model that integrates heterogeneous data from various sources. The model combines structural analysis, reduced-order processing, and artificial intelligence techniques to incorporate geometric, performance, and sensor data. The aim is to enable dynamic monitoring of structural performance. Furthermore, the deployment of physical space and digital space was accomplished by constructing the SPC-DT model of the scissor lift platform as an illustrative example. The model's effectiveness was validated by a comparison of the measured results, the finite element calculation results, and the SPC-DT model prediction findings. Correlation and error analyses were conducted as part of this verification process. The time required for doing a performance study of complex heavy machinery is greatly decreased by the SPC-DT model. For instance, the SPC-DT prediction saves over 255 times the time cost in the structural prediction of a scissor lift when compared to finite element calculation. This creates a new opportunity for mechanical structure and system safety monitoring.

数字孪生,是指利用数字技术在物理区域和数字区域之间建立映射联系,以实现即时信息传输,从而进行监控、优化或决策。数字孪生已成为确保结构安全的重要工具。然而,在结构安全监控中实现时间序列的实时预测具有挑战性,对来自众多来源的异构数据进行动态合成也是如此。本研究提出了一种形状-性能耦合数字孪生(SPC-DT)模型,该模型整合了来自不同来源的异构数据。该模型结合了结构分析、降阶处理和人工智能技术,整合了几何、性能和传感器数据。其目的是实现对结构性能的动态监测。此外,通过构建剪叉式升降平台的 SPC-DT 模型作为示例,完成了物理空间和数字空间的部署。通过比较测量结果、有限元计算结果和 SPC-DT 模型预测结果,验证了模型的有效性。作为验证过程的一部分,还进行了相关性和误差分析。SPC-DT 模型大大减少了对复杂重型机械进行性能研究的时间。例如,与有限元计算相比,SPC-DT 预测为剪叉式升降机的结构预测节省了超过 255 倍的时间成本。这为机械结构和系统安全监测创造了新的机遇。
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引用次数: 0
Updated Lagrangian particle hydrodynamics (ULPH) simulations of underwater bubble motions in three-dimensional space 三维空间水下气泡运动的最新拉格朗日粒子流体力学(ULPH)模拟
IF 8.7 2区 工程技术 Q1 Mathematics Pub Date : 2024-07-27 DOI: 10.1007/s00366-024-02032-9
Xingyu Kan, Jiale Yan, Shaofan Li, Jingzhu Wang, Yiwei Wang, Yonggang Chen

Rising bubbles are often encountered in many engineering fields and have diverse applications. A thorough understanding of bubble rising phenomenon is crucial in these engineering applications. In this study, we employ the developed updated Lagrangian particle hydrodynamics (ULPH) multiphase flow model to investigate the dynamic behavior of bubble flow in quiescent liquids, including bubble rise, deformation, fragmentation, and coalescence. First, a comprehensive numerical study of the influences of computational domain dimensions and fluid/bubble density ratios at the multiphase interface on bubble dynamics is conducted. Subsequently, a variety of scenarios featuring single bubble rising in viscous fluid media are examined. The ULPH simulation results are validated against experimental data, the Level-set (LS) method and Lattice Boltzmann Method (LBM) results. Furthermore, results of three calculations are presented, including dynamic characterization of two horizontal coaxial bubbles, three vertical coaxial bubbles and a single bubble in the presence of an obstacle. The results indicate that the established ULPH multiphase flow model is effective in accurately simulating dynamic characteristics of rising bubbles under various conditions, affirming its applicability in engineering analyses.

许多工程领域经常会遇到气泡上升现象,其应用也多种多样。全面了解气泡上升现象对这些工程应用至关重要。在本研究中,我们采用开发的更新拉格朗日粒子流体力学(ULPH)多相流模型来研究静态液体中气泡流动的动态行为,包括气泡上升、变形、破碎和凝聚。首先,对多相界面的计算域尺寸和流体/气泡密度比对气泡动力学的影响进行了全面的数值研究。随后,研究了粘性流体介质中单个气泡上升的各种情况。ULPH 模拟结果与实验数据、水平集(LS)方法和晶格玻尔兹曼方法(LBM)结果进行了验证。此外,还展示了三个计算结果,包括两个水平同轴气泡、三个垂直同轴气泡和一个有障碍物存在的单个气泡的动态特性。结果表明,已建立的 ULPH 多相流模型能有效准确地模拟上升气泡在各种条件下的动态特性,肯定了其在工程分析中的适用性。
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引用次数: 0
A fixed-stress splitting method for nonlinear poroelasticity 非线性孔弹性的定应力分裂方法
IF 8.7 2区 工程技术 Q1 Mathematics Pub Date : 2024-07-26 DOI: 10.1007/s00366-024-02030-x
Johannes Kraus, Kundan Kumar, Maria Lymbery, Florin A. Radu

In this paper we consider a nonlinear poroelasticity model that describes the quasi-static mechanical behaviour of a fluid-saturated porous medium whose permeability depends on the divergence of the displacement. Such nonlinear models are typically used to study biological structures like tissues, organs, cartilage and bones, which are known for a nonlinear dependence of their permeability/hydraulic conductivity on solid dilatation. We formulate (extend to the present situation) one of the most popular splitting schemes, namely the fixed-stress split method for the iterative solution of the coupled problem. The method is proven to converge linearly for sufficiently small time steps under standard assumptions. The error contraction factor then is strictly less than one, independent of the Lamé parameters, Biot and storage coefficients if the hydraulic conductivity is a strictly positive and Lipschitz-continuous function.

在本文中,我们考虑了一种非线性孔弹性模型,该模型描述了渗透性取决于位移发散的流体饱和多孔介质的准静态力学行为。这种非线性模型通常用于研究组织、器官、软骨和骨骼等生物结构,这些结构的渗透率/液压传导性与固体扩张呈非线性关系。我们制定了(扩展到当前情况下的)一种最流行的分割方案,即用于迭代求解耦合问题的固定应力分割法。在标准假设条件下,该方法可在足够小的时间步长内线性收敛。如果水力传导性是严格的正函数和 Lipschitz 连续函数,则误差收缩因子严格小于 1,与 Lamé 参数、Biot 和存储系数无关。
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引用次数: 0
Fast descriptor-based 2D and 3D microstructure reconstruction using the Portilla–Simoncelli algorithm 使用波蒂利亚-西蒙切利算法快速重建基于描述符的二维和三维微观结构
IF 8.7 2区 工程技术 Q1 Mathematics Pub Date : 2024-07-22 DOI: 10.1007/s00366-024-02026-7
Paul Seibert, Alexander Raßloff, Karl Kalina, Markus Kästner

Reconstructing microstructures from statistical descriptors is a key enabler of computer-based inverse materials design. In the Yeong–Torquato algorithm and other common methods, the problem is approached by formulating it as an optimization problem in the space of possible microstructures. In this case, the error between the desired microstructure and the current reconstruction is measured in terms of a descriptor. As an alternative, descriptors can be regarded as constraints defining subspaces or regions in the microstructure space. Given a set of descriptors, a valid microstructure can be obtained by sequentially projecting onto these subspaces. This is done in the Portilla–Simoncelli algorithm, which is well known in the field of texture synthesis. Noting the algorithm’s potential, the present work aims at introducing it to microstructure reconstruction. After exploring its capabilities and limitations in 2D, a dimensionality expansion is developed for reconstructing 3D volumes from 2D reference data. The resulting method is extremely efficient, as it allows for high-resolution reconstructions on conventional laptops. Various numerical experiments are conducted to demonstrate its versatility and scalability. Finally, the method is validated by comparing homogenized mechanical properties of original and reconstructed 3D microstructures.

根据统计描述符重构微观结构是基于计算机的反向材料设计的关键因素。在 Yeong-Torquato 算法和其他常用方法中,问题是通过在可能的微观结构空间中将其表述为优化问题来解决的。在这种情况下,所需的微观结构与当前重建之间的误差用描述符来衡量。另一种方法是将描述符视为微观结构空间中定义子空间或区域的约束条件。给定一组描述符,按顺序投影到这些子空间,就能得到有效的微观结构。在纹理合成领域广为人知的 Portilla-Simoncelli 算法就是这样实现的。注意到该算法的潜力,本研究旨在将其引入微观结构重建。在探索了该算法在二维领域的能力和局限性后,我们开发了一种维度扩展方法,用于从二维参考数据重建三维体积。由此产生的方法非常高效,因为它可以在传统笔记本电脑上进行高分辨率重建。为了证明该方法的多功能性和可扩展性,我们进行了各种数值实验。最后,通过比较原始三维微结构和重建三维微结构的均质机械性能,验证了该方法的有效性。
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引用次数: 0
An adaptive time integration procedure for automated extended-explicit/implicit hybrid analyses 用于自动扩展显式/隐式混合分析的自适应时间积分程序
IF 8.7 2区 工程技术 Q1 Mathematics Pub Date : 2024-07-13 DOI: 10.1007/s00366-024-02025-8
Delfim Soares

This paper introduces a new explicit-implicit time-marching formulation, presenting a novel hybrid approach for wave propagation analysis. The proposed solution algorithm employs a set of simple, single-step, single-solver, truly self-starting recurrence relationships, which incorporate three time-integration parameters. These parameters are adaptively evaluated for each element of the adopted spatial discretization, taking into account the local characteristics of the model and a user defined parameter. They enable automated extended-explicit/implicit and non-dissipative/dissipative elements to be established, allowing enhanced hybrid analyses to be straightforwardly performed. The proposed formulation is highly accurate, efficient, and very simple to implement and to apply, avoiding complex coupling procedures and interface treatments that are typically considered for mixed explicit/implicit analyses. The new technique is also very versatile, allowing the user to locally control the numerical properties of the adopted time-integration procedure and, consequently, to elaborate very sophisticated solution strategies. Numerical results are presented at the end of the paper, illustrating the good performance and the effectiveness of the proposed novel approach, which combines the best features (such as stability, reduced solver efforts etc.) of both implicit and explicit formulations.

本文介绍了一种新的显式-隐式时间行进公式,为波传播分析提供了一种新颖的混合方法。所提出的求解算法采用了一套简单、单步、单求解器、真正自启动的递推关系,其中包含三个时间积分参数。这些参数针对所采用的空间离散化的每个元素进行自适应评估,同时考虑到模型的局部特征和用户定义的参数。通过这些参数,可以自动建立扩展显式/隐式和非耗散/耗散元素,从而可以直接进行增强型混合分析。所提出的公式非常精确、高效,而且实施和应用非常简单,避免了通常在显式/隐式混合分析中需要考虑的复杂耦合程序和界面处理。新技术还具有很强的通用性,允许用户对所采用的时间积分程序的数值特性进行局部控制,从而制定出非常复杂的求解策略。本文最后介绍了数值结果,说明了所提出的新方法的良好性能和有效性,该方法结合了隐式和显式计算的最佳特点(如稳定性、降低求解难度等)。
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引用次数: 0
Dimensional reduction technique for the prediction of global and local responses of unidirectional composite with matrix nonlinearity and varying fiber packing geometry 用于预测具有基体非线性和不同纤维填料几何形状的单向复合材料的整体和局部响应的降维技术
IF 8.7 2区 工程技术 Q1 Mathematics Pub Date : 2024-07-13 DOI: 10.1007/s00366-024-02024-9
A. Jamnongpipatkul, F. Naets, F. A. Gilabert

The problem associated with the computational homogenization of composite materials often results in expensive computational cost that prevents engineers from comprehensive study for better understanding of composite behaviors, especially when nonlinear effects are considered. While variation in local fiber arrangements has pronounced effect on damage initiation and failure mechanisms in composite, an attempt to reduce the computational cost for the parametric study of such a problem seems to be absent. This paper demonstrates the capability of a model order reduction (MOR) framework to accelerate the parametric study of the unidirectional composite with a plastic constitutive material model for matrix with the varying fiber distribution in the microstructure as the parameter of interest. The MOR framework used in this work is based on the construction of the reduced order basis (ROB) by proper orthogonal decomposition and then the reduced order model (ROM) by Galerkin projection. The concept of local ROB is incorporated which helps decreasing further the dimension of the ROM and, thus, the computational cost. The results from the RVE-based high-fidelity finite element analysis and from the ROM are compared to assess the efficiency and accuracy of the approach. Notable computational gain is achieved with the potential to improve further in the future work. The error in the global response is less than 10% while the local stress fields in the critical regions can be captured well which paves way for the extension to consider the process of damage initiation and evolution as the source of nonlinearity in the future.

与复合材料计算均质化相关的问题往往会导致昂贵的计算成本,妨碍工程师为更好地理解复合材料行为而进行全面研究,尤其是在考虑非线性效应时。虽然局部纤维排列的变化对复合材料的损伤起始和失效机理有明显影响,但在对此类问题进行参数化研究时,似乎还没有降低计算成本的尝试。本文展示了模型阶次缩减(MOR)框架的能力,该框架可加速单向复合材料的参数研究,基体采用塑性组成材料模型,微观结构中的纤维分布变化为相关参数。本研究采用的 MOR 框架基于通过适当的正交分解构建降阶基础 (ROB),然后通过 Galerkin 投影构建降阶模型 (ROM)。局部 ROB 概念的加入有助于进一步降低 ROM 的维度,从而降低计算成本。对基于 RVE 的高保真有限元分析和 ROM 的结果进行了比较,以评估该方法的效率和精度。计算结果显著提高,并有可能在今后的工作中进一步改进。全局响应的误差小于 10%,而临界区域的局部应力场可以很好地捕捉,这为将来扩展到将损伤的发生和演变过程视为非线性源铺平了道路。
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引用次数: 0
A deep neural network for operator learning enhanced by attention and gating mechanisms for long-time forecasting of tumor growth 利用注意力和门控机制增强运算器学习的深度神经网络,用于肿瘤生长的长期预测
IF 8.7 2区 工程技术 Q1 Mathematics Pub Date : 2024-07-10 DOI: 10.1007/s00366-024-02003-0
Qijing Chen, He Li, Xiaoning Zheng

Forecasting tumor progression and assessing the uncertainty of predictions play a crucial role in clinical settings, especially for determining disease outlook and making informed decisions about treatment approaches. In this work, we propose TGM-ONets, a deep neural operator learning (PI-DeepONet) based computational framework, which combines bioimaging and tumor growth modeling (TGM) for enhanced prediction of tumor growth. Deep neural operators have recently emerged as a powerful tool for learning the solution maps between the function spaces, and they have demonstrated their generalization capability in making predictions based on unseen input instances once trained. Incorporating the physics laws into the loss function of the deep neural operator can significantly reduce the amount of the training data. The novelties of the design of TGM-ONets include the employment of a convolutional block attention module (CBAM) and a gating mechanism (i.e., mixture of experts (MoE)) to extract the features of the input images. Our results show that the TGM-ONets not only can capture the detailed morphological characteristics of the mild and aggressive tumors within and outside the training domain but also can be used to predict the long-term dynamics of both mild and aggressive tumor growth for up to 6 months with a maximum error of less than 6.7 (times 10^{-2}) for unseen input instances with two or three snapshots added. We also systematically study the effects of the number of training snapshots and noisy data on the performance of TGM-ONets as well as quantify the uncertainty of the model predictions. We demonstrate the efficiency and accuracy by comparing the performance of TGM-ONets with three state-of-the-art (SOTA) baseline models. In summary, we propose a new deep learning model capable of integrating the TGM and sequential observations of tumor morphology to improve the current approaches for predicting tumor growth and thus provide an advanced computational tool for patient-specific tumor prognosis.

预测肿瘤进展和评估预测的不确定性在临床环境中发挥着至关重要的作用,尤其是在确定疾病前景和就治疗方法做出明智决策方面。在这项工作中,我们提出了一种基于深度神经算子学习(PI-DeepONet)的计算框架--TGM-ONets,它结合了生物成像和肿瘤生长建模(TGM),用于增强肿瘤生长预测。最近,深度神经算子已成为学习函数空间之间解映射的强大工具,它们在训练后根据未见输入实例进行预测方面已证明了自己的泛化能力。将物理定律纳入深度神经算子的损失函数可以大大减少训练数据量。TGM-ONets 的设计新颖之处在于采用了卷积块注意模块(CBAM)和门控机制(即专家混合物(MoE))来提取输入图像的特征。我们的研究结果表明,TGM-ONets不仅能捕捉到训练域内外轻度和侵袭性肿瘤的详细形态特征,还能用于预测轻度和侵袭性肿瘤长达6个月的长期生长动态,对于添加了两到三个快照的未见输入实例,最大误差小于6.7(times 10^{-2})。我们还系统地研究了训练快照数量和噪声数据对 TGM-ONets 性能的影响,并量化了模型预测的不确定性。我们将 TGM-ONets 的性能与三个最先进的(SOTA)基线模型进行了比较,从而证明了其效率和准确性。总之,我们提出了一种新的深度学习模型,该模型能够整合 TGM 和肿瘤形态学的连续观察结果,从而改进当前预测肿瘤生长的方法,为患者特异性肿瘤预后提供先进的计算工具。
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引用次数: 0
Analyzing elastic half-spaces with cavities under wave loads using an RK dynamic infinite meshfree method 利用 RK 动态无限无网格法分析波载荷下带空腔的弹性半空间
IF 8.7 2区 工程技术 Q1 Mathematics Pub Date : 2024-07-06 DOI: 10.1007/s00366-024-02021-y
Kuan-Chung Lin, Ting-Wei Chen

This study introduces a novel dynamic infinite meshfree method, termed RK-DIMM (reproducing kernel dynamic infinite meshfree method), which is specifically developed for analyzing elastic half-spaces with cavities under the influence of both P-waves and SV-waves. RK-DIMM integrates the principles of reproducing kernel particle methods with dynamic infinite element techniques to enhance computational efficiency and accuracy in wave propagation simulations. The method partitions the infinite domain into near and far domains using artificial boundaries, utilizing RK in the near domain and DIMM in the far domain. Through the application of stabilized conforming nodal integration and naturally stabilized nodal integration, RK-DIMM achieves accurate and stable solutions. Our rigorous benchmark comparisons have confirmed the method’s exceptional ability to simulate wave dissipation and reflections with high accuracy and computational efficiency. RK-DIMM has proven to be highly effective in mimicking soil responses to synthetic earthquake forces, closely aligning with analytical predictions, and has demonstrated robust performance in scenarios involving underground cavities. Furthermore, its application to real earthquake data, particularly the 1999 Chi-Chi earthquake, underscores its practical utility and relevance. The results from this study highlight RK-DIMM’s potential as a transformative tool in computational geomechanics, significantly enhancing the precision and reliability of seismic impact assessments on civil infrastructures.

本研究介绍了一种新颖的动态无限无网格方法,称为 RK-DIMM(再现内核动态无限无网格方法),专门用于分析 P 波和 SV 波影响下带空腔的弹性半空间。RK-DIMM 将再现内核粒子法原理与动态无限元技术相结合,提高了波传播模拟的计算效率和精度。该方法利用人工边界将无限域划分为近域和远域,在近域利用 RK,在远域利用 DIMM。通过应用稳定顺应节点积分和自然稳定节点积分,RK-DIMM 实现了精确稳定的求解。通过严格的基准比较,我们证实了该方法在高精度和高计算效率地模拟波浪消散和反射方面的卓越能力。事实证明,RK-DIMM 在模拟土壤对合成地震力的响应方面非常有效,与分析预测结果非常吻合,并在涉及地下空洞的情况下表现出强大的性能。此外,它在实际地震数据中的应用,尤其是在 1999 年芝芝地震中的应用,凸显了它的实用性和相关性。这项研究的结果凸显了 RK-DIMM 作为计算地质力学变革性工具的潜力,可显著提高民用基础设施地震影响评估的精度和可靠性。
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引用次数: 0
The generation of tetrahedral meshes for NURBS-enhanced FEM 生成用于 NURBS 增强有限元的四面体网格
IF 8.7 2区 工程技术 Q1 Mathematics Pub Date : 2024-07-05 DOI: 10.1007/s00366-024-02004-z
Xi Zou, Sui Bun Lo, Ruben Sevilla, Oubay Hassan, Kenneth Morgan

This work presents the first method for generating tetrahedral-based volume meshes dedicated to the NURBS-enhanced finite element method (NEFEM). Built upon the developed method of generating feature-independent surface meshes tailored for NEFEM, the proposed mesh generation scheme is able to grow volume elements that inherit the feature-independence by using the surface mesh as the initial boundary discretisation. Therefore, the generated tetrahedral elements may contain triangular faces that span across multiple NURBS surfaces whilst maintaining the exact boundary description. The proposed strategy completely eliminates the need for de-featuring complex watertight CAD models. At the same time, it eliminates the uncertainty originated from the simplification of CAD models adopted in industrial practice and the error introduced by traditional isoparametric mesh generators that produce polynomial approximations of the true boundary representation. Thanks to the capability of having element faces traversing multiple geometric surfaces, small geometric features in the CAD model no longer restrict the minimum element size, and the user-required mesh spacing in the generated mesh is better satisfied than in traditional meshes that require local refinement. To demonstrate the ability of the proposed approach, a variety of CAD geometries are meshed with the proposed strategy, including examples relevant to the fluid dynamics, wave propagation and solid mechanics communities.

本研究提出了第一种生成基于四面体的体积网格的方法,专用于 NURBS 增强有限元法(NEFEM)。基于为 NEFEM 量身定制的、与特征无关的曲面网格生成方法,所提出的网格生成方案能够通过使用曲面网格作为初始边界离散化,生成继承了特征无关性的体积元素。因此,生成的四面体元素可以包含跨越多个 NURBS 表面的三角形面,同时保持精确的边界描述。所提出的策略完全消除了对复杂的不透水 CAD 模型进行去拟合的需要。同时,它还消除了工业实践中采用的 CAD 模型简化所带来的不确定性,以及传统的等参数网格生成器所带来的误差,这些生成器会生成真实边界表示的多项式近似值。由于元素面能够穿越多个几何表面,CAD 模型中的小几何特征不再限制最小元素尺寸,与需要局部细化的传统网格相比,生成的网格能更好地满足用户对网格间距的要求。为了证明所提方法的能力,使用所提策略对各种 CAD 几何图形进行了网格划分,其中包括与流体动力学、波传播和固体力学领域相关的示例。
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引用次数: 0
Adaptive phase-field total Lagrangian material point method for evaluating dynamic fracture of soft material 用于评估软材料动态断裂的自适应相场总拉格朗日材料点法
IF 8.7 2区 工程技术 Q1 Mathematics Pub Date : 2024-07-03 DOI: 10.1007/s00366-024-02019-6
Yonggang Zheng, Shun Zhang, Weilong Yang, Zijian Zhang, Hongfei Ye, Hongwu Zhang

An adaptive phase-field total Lagrangian material point method (APTLMPM) is proposed in this paper for effectively simulating the dynamic fracture of two-dimensional soft materials with finite deformation. In this method, the governing equations for the fracture of soft materials are derived by integrating the phase-field fracture model with the total Lagrangian material point method (TLMPM), and corresponding discrete equations are then formulated with explicit time integration. To address the significant computational issue in terms of memory and processing time, an adaptive technique for dynamically splitting particles and background grids in the phase-field TLMPM is proposed, based on the phase-field values of the particles. To further maintain continuity of the physical field throughout the computational process and consider the characteristics of the field update, an information remapping strategy is developed. Several representative numerical examples are presented to demonstrate the accuracy and efficiency of the proposed APTLMPM by comparing the simulation results with experimental data and those as obtained with other numerical methods.

本文提出了一种自适应相场全拉格朗日材料点法(APTLMPM),用于有效模拟具有有限变形的二维软材料的动态断裂。在该方法中,通过将相场断裂模型与总拉格朗日材料点法(TLMPM)进行积分,推导出软材料断裂的控制方程,然后通过显式时间积分来制定相应的离散方程。为了解决内存和处理时间方面的重大计算问题,提出了一种基于粒子相场值的自适应技术,用于动态分割相场 TLMPM 中的粒子和背景网格。为了在整个计算过程中进一步保持物理场的连续性,并考虑到场更新的特点,还开发了一种信息重映射策略。通过将模拟结果与实验数据和其他数值方法得出的结果进行比较,介绍了几个具有代表性的数值示例,以证明所提出的 APTLMPM 的准确性和效率。
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引用次数: 0
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