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A novel data compression method for GPU accelerated large-scale isogeometric topology optimization with order-ascending strategy 一种新的GPU数据压缩方法加速了大规模等高几何拓扑优化
IF 3.5 3区 工程技术 Q1 MATHEMATICS, APPLIED Pub Date : 2025-12-18 DOI: 10.1016/j.finel.2025.104503
Tao Nie, Jianli Liu, Wanpeng Zhao, Tao Zhang, Ruichen Zhang, Jinpeng Han, Zhaohui Xia
This paper aims to address the common challenges of storage overhead and computational inefficiencies that arise in isogeometric topology optimization (ITO) when dealing with large-scale problems. To tackle these issues, the paper proposes a novel framework that combines a highly efficient data storage strategy with Graphics Processing Unit (GPU) accelerated optimization. By utilizing control point pairs and removing redundant matrix storage, the Isogeometric Compressed Sparse Row (IGA-CSR) technique effectively reduces storage requirements. Furthermore, the paper presents an order-ascending optimization strategy to avoid intensive calculations caused by large degrees of freedom in the early stage. What's more, the introduction of Graphics Processing Unit further improves the optimization process. Combining these methods, an efficient optimization framework is proposed, which allows efficient optimization even for problems that involve tens of millions of degrees of freedom via single NVIDIA GeForce RTX 3090 GPU with 24 GB. Validation through two 3D benchmark examples reveals that the IGA-CSR method shows the best performance comparing with existing methods in memory consumption. At the same time, it enhances computational efficiency about 65.4 % comparing with conventional second-order isogeometric topology optimization via GPU acceleration.
本文旨在解决在处理大规模问题时等几何拓扑优化(ITO)中出现的存储开销和计算效率低下的常见挑战。为了解决这些问题,本文提出了一个将高效数据存储策略与图形处理单元(GPU)加速优化相结合的新框架。等几何压缩稀疏行(IGA-CSR)技术通过利用控制点对和去除冗余矩阵存储,有效地降低了存储需求。在此基础上,提出了一种递进优化策略,避免了前期由于自由度过大而导致的计算量过大。此外,图形处理单元的引入进一步改善了优化过程。结合这些方法,提出了一个高效的优化框架,即使在涉及数千万个自由度的问题上,也可以通过单个NVIDIA GeForce RTX 3090 24 GB GPU进行高效优化。通过两个三维基准算例的验证表明,IGA-CSR方法在内存消耗方面比现有方法表现出最好的性能。同时,与传统的二阶等几何拓扑优化算法相比,该算法的计算效率提高了65.4%。
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引用次数: 0
Partition of Unity-based four-node tetrahedral element for nonlinear structural analysis of nearly incompressible hyperelastic materials 近乎不可压缩超弹性材料非线性结构分析中基于单位的四节点四面体单元划分
IF 3.5 3区 工程技术 Q1 MATHEMATICS, APPLIED Pub Date : 2025-12-17 DOI: 10.1016/j.finel.2025.104464
Taejung Lim , Minh-Chien Trinh , Hyungmin Jun
This study introduces a refined four-node tetrahedral finite element employing the Partition of Unity method for nonlinear static and modal analysis of nearly incompressible hyperelastic materials. The proposed Partition of Unity-based element effectively reduces volumetric locking and improves solution accuracy without increasing the number of nodes. The Partition of Unity method enriches the displacement field by incorporating additional polynomial basis functions, enabling higher-order displacement approximation, and effectively alleviating volumetric locking. Mooney–Rivlin and Neo-Hookean material models are integrated with the penalty method, ensuring robust handling of nearly incompressible behavior. Large deformations are addressed using a total Lagrangian formulation. In addition, a displacement-based direct iterative nonlinear modal analysis procedure is employed to derive nonlinear natural frequencies and corresponding mode shapes. In nonlinear static analysis, the proposed element is validated through various numerical cases including blocks under compression, cylinders under large deformation, mesh distortion sensitivity analysis, and tires under compression. The present element effectively alleviates the volumetric locking phenomenon and provides excellent performance even when using a coarse mesh. Nonlinear modal analysis has been performed on cases such as free vibration of distorted plates, truncated cylindrical shells, and hyperelastic soft robots. The proposed elements effectively capture nonlinear natural frequencies and mode shapes even with distorted and coarse meshes.
本文介绍了一种改进的四节点四面体有限元,该有限元采用统一分割法对几乎不可压缩的超弹性材料进行了非线性静力和模态分析。该方法在不增加节点数量的情况下,有效地减少了体积锁定,提高了求解精度。统一分割法通过加入额外的多项式基函数丰富了位移场,实现了高阶位移逼近,有效缓解了体积锁定。Mooney-Rivlin和Neo-Hookean材料模型与惩罚方法相结合,确保了几乎不可压缩行为的稳健处理。使用全拉格朗日公式来处理大变形。此外,采用基于位移的直接迭代非线性模态分析方法推导了非线性固有频率和相应的模态振型。在非线性静力分析中,通过压缩砌块、大变形圆柱体、网格畸变敏感性分析和轮胎压缩等多种数值案例对所提出的单元进行了验证。该元件有效地缓解了体积锁紧现象,即使在使用粗网格时也能提供出色的性能。本文对变形板、截短圆柱壳和超弹性软体机器人的自由振动进行了非线性模态分析。提出的单元即使在扭曲和粗糙的网格中也能有效地捕获非线性固有频率和模态振型。
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引用次数: 0
An axisymmetric finite-volume method for thermal stress problems in heterogeneous materials 非均质材料热应力问题的轴对称有限体积法
IF 3.5 3区 工程技术 Q1 MATHEMATICS, APPLIED Pub Date : 2025-12-16 DOI: 10.1016/j.finel.2025.104498
Chenqi Li , Lingkuan Xuan , Jingfeng Gong , Hongyu Guo , Le Gu
To reduce the computational cost of axisymmetric problems and to extend the applicability of the cell-vertex finite volume method (CV-FVM), this paper develops an axisymmetric cell-vertex finite volume method (ACV-FVM) for transient thermal stress analysis in heterogeneous materials with axisymmetric structures. The reduced two-dimensional domain is discretized using 3-node triangular ring elements and 4-node quadrilateral ring elements. A numerical solver based on the ACV-FVM is implemented in C++ and applied to solve thermo-mechanical coupling problems involving homogeneous materials, multilayered materials, functionally graded materials, and materials with temperature-dependent properties. The numerical results show good agreement with analytical solutions and other numerical results. The findings indicate that, compared to nodal-based output schemes, element-center-based output significantly suppresses spurious stress oscillations in multilayered materials. The proposed method has been successfully applied to the thermal stress analysis of a cylinder liner with thermal barrier coatings. Results reveal that temperature-dependent material properties lead to an approximate 1.5 % increase in temperature and a 3.4 % increase in thermal stress at the same location, highlighting the necessity of considering temperature-dependent thermo-mechanical behavior in such analyses.
为了降低轴对称问题的计算成本,扩大胞-顶点有限体积法(CV-FVM)的适用性,本文提出了一种用于轴对称非均质材料瞬态热应力分析的胞-顶点有限体积法(ACV-FVM)。采用3节点三角形环单元和4节点四边形环单元对二维域进行离散化。采用c++语言实现了基于ACV-FVM的数值求解器,并将其应用于均质材料、多层材料、功能梯度材料和温度相关材料的热-力耦合问题的求解。数值结果与解析解和其他数值结果吻合较好。研究结果表明,与基于节点的输出方案相比,基于单元中心的输出方案显著抑制了多层材料中的虚假应力振荡。该方法已成功地应用于热障涂层气缸套的热应力分析。结果表明,在同一位置,温度相关的材料性能导致温度升高约1.5%,热应力增加3.4%,突出了在此类分析中考虑温度相关的热力学行为的必要性。
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引用次数: 0
The Gradient Complete Stabilization Method (GCSM) for scalar diffusive–convective–reactive problems 标量扩散-对流-反应问题的梯度完全稳定方法
IF 3.5 3区 工程技术 Q1 MATHEMATICS, APPLIED Pub Date : 2025-12-14 DOI: 10.1016/j.finel.2025.104501
E.G. Dutra do Carmo , E.F. Fontes Jr. , M.F.F. Santos , W.J. Mansur
Purely convective and convective–diffusive problems with dominant convection, presenting high gradients in directions misaligned with the convective field, are typically stabilized using nonlinear methods, even when the underlying problem is linear. This not only leads to an increase in computational cost but also degrades the accuracy of the gradient of the approximate solution. Therefore, it is desirable to obtain a method that completely stabilizes the approximated solution gradient while ensuring optimal approximation rates for it. In this sense, the Gradient Complete Stabilization Method (GCSM) is proposed in this paper. A rigorous mathematical analysis of the method is performed by elaborating the variational formulation of the diffusive–convective–reactive problem. A robust set of theorems is defined and proved, including the Fundamental Identity Theorem, which plays a central role in enabling gradient stabilization with optimal convergence rates. Several numerical experiments are conducted, comparing accuracy from GCSM against a classic discontinuity capture method, the Consistent Approximate Upwind (CAU). The results demonstrate a marked improvement in performance achieved by the proposed method, especially in the final example, which involves both internal and external boundary layers. In this case, the GCSM delivers solutions that are nearly oscillation-free.
具有优势对流的纯对流和对流扩散问题,在与对流场不对齐的方向上呈现高梯度,通常使用非线性方法来稳定,即使潜在问题是线性的。这不仅会导致计算成本的增加,而且会降低近似解的梯度精度。因此,我们希望找到一种既能使近似解梯度完全稳定,又能保证最优近似速率的方法。在这个意义上,本文提出了梯度完全稳定方法(GCSM)。通过阐述扩散-对流-反应问题的变分公式,对该方法进行了严格的数学分析。定义并证明了一组鲁棒定理,其中包括基本恒等定理,它在实现具有最优收敛速率的梯度镇定中起着核心作用。通过几个数值实验,比较了GCSM与经典的不连续捕获方法一致近似迎风(CAU)的精度。结果表明,该方法的性能有了显著的提高,特别是在最后的例子中,同时涉及到内部和外部边界层。在这种情况下,GCSM提供了几乎无振荡的解决方案。
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引用次数: 0
Regularizing the linearly extrapolated BDF2 scheme for incompressible flows with time relaxation 带时间松弛的不可压缩流线性外推BDF2格式的正则化
IF 3.5 3区 工程技术 Q1 MATHEMATICS, APPLIED Pub Date : 2025-12-10 DOI: 10.1016/j.finel.2025.104491
Sean Breckling , Jorge Reyes , Sidney Shields , Clifford Watkins
This paper presents a highly-efficient finite element scheme for the time relaxation model (TRM). The efficiency is achieved through the second-order BDF2 time-stepping scheme with linear extrapolation (BDF2LE). The accuracy of the scheme is also greatly enhanced through the use of the divergence-free Scott-Vogeulis finite elements, and van Cittert approximate deconvolution. A complete finite element analysis is provided, which includes rigorous proofs for the stability, well-possessedness, and convergence of both velocity and pressure solutions. We also demonstrate that the inclusion of the linear time relaxation term preserves the long-time stability of the unregularized BDF2LE scheme. Finally, numerical experiments are presented that demonstrate the added stability and accuracy that time relaxation can provide.
本文提出了一种求解时间松弛模型(TRM)的高效有限元方案。通过线性外推的二阶BDF2时间步进方案(BDF2LE)实现了效率。通过使用无散度的Scott-Vogeulis有限元和van Cittert近似反卷积,该方案的精度也大大提高。给出了完整的有限元分析,包括速度解和压力解的稳定性、完备性和收敛性的严格证明。我们还证明了线性时间松弛项的加入保留了非正则BDF2LE格式的长期稳定性。最后,给出了数值实验,证明了时间松弛可以提供额外的稳定性和准确性。
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引用次数: 0
Integral constitutive equations based temporal finite element modeling for the static viscoelastic problem 基于积分本构方程的静态粘弹性问题时间有限元建模
IF 3.5 3区 工程技术 Q1 MATHEMATICS, APPLIED Pub Date : 2025-12-05 DOI: 10.1016/j.finel.2025.104490
Fengling Chen, Yiqian He, Haitian Yang
A stepwise spatial–temporal finite element algorithm is developed to provide a general numerical tool for solving static viscoelastic problems with integral constitutive equations. The displacement, strain and stress are formulated by the hybrid basis functions based Temporal Finite Element Method (TFEM), and are incorporated into the constitutive relations. The framework is established based on the virtual work principle and the weighted residual technique, and is convenient to cooperate with kinds of numerical schemes for boundary value problems such as FEM and SBFEM. Two criteria are proposed to numerically evaluate error propagation during the step-marching process, which can be used to determine appropriate time-step sizes for prescribed temporal shape functions and spatial FE meshes. Compared with the TFEM algorithm based on differential viscoelastic constitutive equations, the present approach overcomes the order-restriction limitation by employing integral constitutive equations with Prony-series based relaxation moduli. Numerical examples demonstrate the capability and accuracy of the proposed method in handling viscoelastic problems involving material heterogeneity, stress singularity, various relaxation moduli, and different loading forms. The obtained results with various configurations of temporal shape functions and step sizes, exhibit good agreement with analytical solutions and ABAQUS simulations.
提出了一种分步时空有限元算法,为求解具有积分本构方程的静态粘弹性问题提供了一种通用的数值工具。位移、应变和应力由基于混合基函数的时间有限元法(TFEM)表示,并纳入本构关系。该框架是基于虚功原理和加权残差技术建立的,可方便地与有限元法、单轴有限元法等边值问题的多种数值格式配合使用。提出了两种步进过程误差传播数值评价准则,可用于确定指定时间形状函数和空间有限元网格的适当时间步长。与基于粘弹性微分本构方程的TFEM算法相比,该方法采用基于prony级数的松弛模量的积分本构方程,克服了阶数限制。数值算例验证了该方法处理材料非均质性、应力奇异性、不同松弛模量和不同加载形式等粘弹性问题的能力和准确性。所得结果与解析解和ABAQUS仿真结果吻合较好。
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引用次数: 0
Unsymmetric Serendipity finite elements: Performance analysis 非对称偶然性有限元:性能分析
IF 3.5 3区 工程技术 Q1 MATHEMATICS, APPLIED Pub Date : 2025-12-03 DOI: 10.1016/j.finel.2025.104487
S. Eisenträger , E. Woschke , E.T. Ooi
This paper presents a comparative analysis of the conventional finite element method (FEM) and the unsymmetric finite element method (UFEM) for Serendipity elements (p3), focusing on two factors: (i) achievable accuracy and (ii) computational costs. The UFEM, based on a Petrov–Galerkin formulation, uses metric shape functions as trial functions and parametric shape functions as test functions. This unique approach enhances the resistance against mesh distortion, as it ensures polynomial completeness of the Ansatz space of unsymmetric finite elements. Hence, higher accuracy can be achieved in complex geometries. However, the unsymmetric nature of UFEM leads to increased computational costs as a result of the added complexity of solving the resulting system of equations. This study provides a quantitative evaluation of the computational burden associated with achieving specific error thresholds for both methods. By analyzing a range of benchmark problems, we identify scenarios in which each method performs optimally, offering practical insights for selecting the appropriate approach based on accuracy demands and computational constraints. Our findings suggest that, while UFEM can produce superior accuracy, its computational efficiency depends on application-specific requirements and available resources.
本文对Serendipity单元(p≤3)的传统有限元法(FEM)和非对称有限元法(UFEM)进行了比较分析,重点关注两个因素:(i)可实现的精度和(ii)计算成本。UFEM基于Petrov-Galerkin公式,使用度量形状函数作为试验函数,参数形状函数作为测试函数。这种独特的方法增强了对网格变形的抵抗,因为它保证了非对称有限元的Ansatz空间的多项式完备性。因此,在复杂的几何形状中可以达到更高的精度。然而,UFEM的非对称性质导致计算成本的增加,这是由于求解所得到的方程组的复杂性增加的结果。本研究提供了与实现两种方法的特定误差阈值相关的计算负担的定量评估。通过分析一系列基准问题,我们确定了每种方法执行最佳的场景,为基于精度要求和计算约束选择合适的方法提供了实用的见解。我们的研究结果表明,虽然UFEM可以产生更高的精度,但其计算效率取决于特定应用的要求和可用资源。
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引用次数: 0
Reduced-Order Modelling for Thermal–Mechanical Analysis of Power Electronic Modules 电力电子模块热力学分析的降阶建模
IF 3.5 3区 工程技术 Q1 MATHEMATICS, APPLIED Pub Date : 2025-11-26 DOI: 10.1016/j.finel.2025.104488
Sheikh Hassan , Stoyan Stoyanov , Pushparajah Rajaguru , Christopher Bailey
This paper introduces a compact and time-efficient reduced-order modelling method for conducting thermal–mechanical analyses and studying material nonlinearities in power electronic modules (PEMs). Thermal–mechanical analyses in reduced-order modelling research typically follow a sequential coupling approach, where the thermal model is solved first, allowing the resulting temperature distributions to serve as loads in the mechanical system. In this study, a direct coupling method is employed for the thermomechanical analysis, enabling the simultaneous evaluation of the thermal and structural governing equations to determine thermal and directional deformation distributions, with temperature and deformations as the degrees of freedom (DOFs) of the coupled system. A novel approach, utilising the Krylov subspace-based model order reduction (MOR) process, the Newmark and Newton–Raphson algorithms within the reduced-order modelling framework, have been developed for analysing material nonlinearity in PEMs. The time domain responses, i.e., the transient ROM solutions, align remarkably well with the corresponding FOM solutions. The inelastic strains and plastic work results demonstrate strong consistency for materials having time-independent (plasticity) and time-dependent (creep and viscoplasticity) nonlinearities. Responses of the reduced-order model (ROM) in the frequency (Laplace) domain are analysed in contrast to its full-order model (FOM) to evaluate its characteristics and show suitability within the required expansion points. The MOR process provides a significantly compact ROM order of just 20×20 for reduced-dimensional computation, achieving up to an 83% reduction in computational time compared to its FOM order of approximately 400,000×400,000. The reduced-order modelling approach is implemented using the MATLAB coding environment.
本文介绍了一种紧凑、省时的降阶建模方法,用于电力电子模块的热力学分析和材料非线性研究。在降阶建模研究中,热-力学分析通常遵循顺序耦合方法,首先求解热模型,允许得到的温度分布作为机械系统中的载荷。在本研究中,采用直接耦合方法进行热力学分析,可以同时评估热控制方程和结构控制方程,以温度和变形作为耦合系统的自由度(DOFs),确定热和定向变形分布。利用基于Krylov子空间的模型降阶(MOR)过程,在降阶建模框架内的Newmark和Newton-Raphson算法,开发了一种新的方法来分析PEMs中的材料非线性。时域响应,即瞬态ROM解,与相应的FOM解非常一致。非弹性应变和塑性功结果表明具有时间无关(塑性)和时间相关(蠕变和粘塑性)非线性的材料具有很强的一致性。将降阶模型(ROM)在频率(拉普拉斯)域中的响应与全阶模型(FOM)进行对比分析,以评估其特性并显示其在所需扩展点内的适用性。MOR过程为降维计算提供了一个非常紧凑的ROM顺序,仅为20×20,与大约400,000×400,000的FOM顺序相比,计算时间减少了83%。在MATLAB编码环境下实现了降阶建模方法。
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引用次数: 0
3D simulation of residual stresses induced by ElectroMagnetic pulse Peening process 电磁脉冲强化过程中残余应力的三维模拟
IF 3.5 3区 工程技术 Q1 MATHEMATICS, APPLIED Pub Date : 2025-11-20 DOI: 10.1016/j.finel.2025.104486
Komlavi Mawuli Senyo , Loup Plantevin , Thibaut Chaise , Eric Feulvarch , Jean-Michel Bergheau , Daniel Nélias
Compression techniques such as shot peening, laser shock peening, and water jet peening are commonly employed to induce residual compressive stresses in mechanical components. These residual stresses play a crucial role in preventing the initiation and propagation of cracks. An innovative method known as the ElectroMagnetic pulse Peening (EMP) process utilizes magnetic forces to introduce residual compressive stresses in mechanical components. The EMP process shares similarities with the ElectroMagnetic Forming (EMF) process, which has been extensively studied through numerical and experimental investigations. Existing numerical studies predominantly feature axisymmetric 2D simulations, with limited availability of 3D simulations due to numerical constraints regarding computing time and resources. Since the EMP process shares similarities with EMF, similar challenges arise with respect to computational resources and time. This paper presents an innovative approach for the 3D simulation of residual stresses induced by the EMP process, based on efficient 2D axisymmetric calculations of the electromagnetic fields. The main objective of this approach is to simulate the mechanical impact of electromagnetic pulses applied by sweeping a surface, in order to analyze the stress distribution in the overlapping regions. First, the 2D model used to simulate electromagnetic phenomena is presented, and the 2D-to-3D transfer technique developed is detailed for computing residual stresses in 3D. Subsequently, the validity of this approach is established through a comparative study between 2D and 3D mechanical results for a single electromagnetic pulse. Finally, a multiple-pulse simulation is conducted to investigate the effect of overlapping treatment regions on an AA6061 aluminum alloy. The outcomes of this study are discussed in terms of the residual stresses at the subsurface.
压缩技术,如喷丸强化、激光冲击强化和水射流强化,通常用于在机械部件中产生残余压应力。这些残余应力在防止裂纹的萌生和扩展方面起着至关重要的作用。一种被称为电磁脉冲强化(EMP)工艺的创新方法利用磁力在机械部件中引入残余压应力。EMP过程与电磁成形(EMF)过程有相似之处,后者已经通过数值和实验研究得到了广泛的研究。现有的数值研究主要以轴对称二维模拟为特征,由于计算时间和资源的数值限制,三维模拟的可用性有限。由于EMP过程与EMF有相似之处,因此在计算资源和时间方面也出现了类似的挑战。本文提出了一种基于电磁场二维轴对称计算的电磁脉冲过程残余应力三维模拟方法。该方法的主要目的是模拟电磁脉冲扫面产生的机械冲击,以分析重叠区域的应力分布。首先,提出了用于电磁现象模拟的二维模型,并详细介绍了用于计算三维残余应力的二维到三维传递技术。随后,通过对单个电磁脉冲的二维和三维力学结果的对比研究,验证了该方法的有效性。最后,通过多脉冲模拟研究了重叠处理区域对AA6061铝合金的影响。本文从地下残余应力的角度对研究结果进行了讨论。
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引用次数: 0
Efficient and accurate multiscale modelling of TRIP steels: Advanced numerical strategies and experimental validation 高效和准确的TRIP钢多尺度建模:先进的数值策略和实验验证
IF 3.5 3区 工程技术 Q1 MATHEMATICS, APPLIED Pub Date : 2025-11-20 DOI: 10.1016/j.finel.2025.104484
R.P. Cardoso Coelho, F.M. Andrade Pires
This work presents methodological and computational advances to a multiscale micromechanical framework for modelling transformation-induced plasticity (TRIP) steels, explicitly coupling martensitic phase transformation and multi-phase crystallographic slip within an RVE-based homogenisation setting. Building upon the framework of Cardoso Coelho et al. (2023), we introduce a series of enhancements aimed at improving computational efficiency, expanding modelling capabilities, and increasing predictive fidelity. The numerical implementation is restructured to exploit cache-optimised data access, branchless return-mapping algorithms, selective Jacobian assembly, and explicitly vectorised linear solvers, resulting in significant reductions in computational cost, particularly for martensite-dominated loading scenarios. A mixed stress–strain-driven homogenisation scheme is formulated, enabling independent control of specific stress and strain components, thus improving the representation of experimentally observed strain-controlled uniaxial tests. Model calibration and validation are performed against experimental data using a composite Bayesian optimisation strategy, showing excellent agreement with measured stress–strain responses and a consistent prediction of martensite volume fraction evolution. Additional energetic contributions are investigated to refine the description of transformation kinetics, further enhancing model accuracy. Overall, this work delivers a robust, high-performance multiscale computational framework for TRIP steels, advancing predictive modelling capabilities for phase-transforming materials and supporting more reliable virtual material design.
这项工作为模拟相变诱发塑性(TRIP)钢的多尺度微力学框架提供了方法和计算上的进步,在基于rve的均质化设置中明确地耦合了马氏体相变和多相晶体滑移。在Cardoso Coelho等人(2023)的框架基础上,我们引入了一系列旨在提高计算效率、扩展建模能力和提高预测保真度的增强功能。数值实现进行了重组,以利用缓存优化的数据访问、无分支返回映射算法、选择性雅可比装配和明确的矢量线性求解器,从而显著降低了计算成本,特别是对于马氏体为主的加载场景。制定了混合应力-应变驱动的均质化方案,能够独立控制特定的应力和应变成分,从而改善了实验观察到的应变控制单轴试验的表现。使用复合贝叶斯优化策略对实验数据进行模型校准和验证,显示与测量的应力-应变响应和马氏体体积分数演变的一致预测非常一致。研究了额外的能量贡献,以完善转化动力学的描述,进一步提高模型的准确性。总的来说,这项工作为TRIP钢提供了一个强大的、高性能的多尺度计算框架,提高了相变材料的预测建模能力,并支持更可靠的虚拟材料设计。
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引用次数: 0
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Finite Elements in Analysis and Design
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