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A moving least square immersed boundary method for SPH with thin-walled rigid structures 薄壁刚性结构 SPH 的移动最小平方沉浸边界法
IF 2.8 3区 工程技术 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2024-06-24 DOI: 10.1007/s40571-024-00721-2
Zhuolin Wang, Zichao Jiang, Yi Zhang, Gengchao Yang, Trevor Hocksun Kwan, Yuhui Chen, Qinghe Yao

This paper presents a novel method for smoothed particle hydrodynamics (SPH) with thin-walled rigid structures. Inspired by the direct forcing immersed boundary method, this method employs a moving least square method for the velocity interpolation instead of the linear interpolation. It reduces oscillations due to changing relative positions between fluid grids and structures. It also simplifies thin-walled rigid structure simulations by eliminating the need for multiple layers of boundary particles, and improves computational accuracy and stability in three-dimensional scenarios. Results of the impulsively started plate test demonstrate that the proposed method obtains smooth velocity and pressure, as well as a good match to the references results of the vortex wake development. Results of the flow past cylinder test show that the proposed method avoids mutual interference on both side of the boundary, while accurately calculating the forces acting on structure. By comparing to linear least square direct forcing scheme and the diffusive direction scheme, advantages of lower oscillation and higher accuracy are proven. Results of flow past a sphere further indicate the stability of the proposed method for three-dimensional simulations.

本文提出了一种新的薄壁刚性结构平滑粒子流体力学(SPH)方法。受直接强迫沉浸边界法的启发,该方法采用移动最小平方法进行速度插值,而不是线性插值。它减少了由于流体网格和结构之间相对位置变化引起的振荡。它还通过消除对多层边界颗粒的需求简化了薄壁刚性结构模拟,并提高了三维场景下的计算精度和稳定性。脉冲启动板试验结果表明,所提出的方法可以获得平稳的速度和压力,并与涡流尾流发展的参考结果很好地匹配。流过圆柱体试验的结果表明,提出的方法避免了边界两侧的相互干扰,同时准确计算了作用在结构上的力。通过与线性最小二乘法直接强迫方案和扩散方向方案的比较,证明了该方法具有振荡更小、精度更高的优点。流过球体的结果进一步证明了所提方法在三维模拟中的稳定性。
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引用次数: 0
Influence of intermediate principal stress and rolling resistance on the shearing response of sand: a micromechanical investigation 中间主应力和滚动阻力对沙子剪切响应的影响:微机械研究
IF 2.8 3区 工程技术 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2024-06-24 DOI: 10.1007/s40571-024-00782-3
S. V. Dharani Raj, Mousumi Mukherjee, Andres Alfonso Peña-Olarte, Roberto Cudmani

Existing literature on true triaxial and torsional shear tests indicate that the mechanical response of a granular assembly is significantly influenced by the magnitude of the intermediate principal stress ratio. The present study aims to explore the mechanism behind such effects in reference to the particle-level interaction using 3D DEM simulations. In this regard, true triaxial numerical simulations have been carried out with constant minor principal stress and varying (b) values employing rolling resistance-type contact model to mimic particle shape. The numerical simulations have been validated against the true triaxial experiments reported in the literature for dense Santa Monica beach sand. The macro-level shearing response of the granular assembly has been examined in terms of the evolution of stress ratio and volumetric strain for different rolling resistance coefficients. Further, such macro-level response has been assessed in reference to the micro-scale attributes, e.g. average contact force, number of interparticle contacts, mechanical coordination number, contact normal orientation, and fabric tensor as well as meso-scale attribute like strong contact force network. Lade’s failure surface has been adopted to represent the stress and fabric at peak state in the octahedral plane, and mathematical expressions have been proposed relating the failure surface parameters to the rolling resistance coefficient.

Graphical abstract

有关真实三轴和扭剪试验的现有文献表明,颗粒组件的机械响应受到中间主应力比大小的显著影响。本研究旨在利用三维 DEM 模拟,参照颗粒级相互作用,探索这种影响背后的机制。为此,采用滚动阻力型接触模型来模拟颗粒形状,在次主应力恒定、(b())值变化的情况下进行了真正的三轴数值模拟。数值模拟结果与文献中报道的针对致密圣莫尼卡海滩砂的真实三轴实验结果进行了验证。通过不同滚动阻力系数下应力比和体积应变的演变,研究了颗粒组件的宏观剪切响应。此外,还参考了微观尺度属性(如平均接触力、颗粒间接触数量、机械配合数、接触法线方向和织物张量)以及中观尺度属性(如强接触力网络),对这种宏观响应进行了评估。采用拉德失效面来表示八面体峰值状态下的应力和织物,并提出了失效面参数与滚动阻力系数之间的数学表达式。
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引用次数: 0
Efficient explicit time integration algorithms for non-spherical granular dynamics on group S(3) S(3) 组上非球形颗粒动力学的高效显式时间积分算法
IF 2.8 3区 工程技术 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2024-06-23 DOI: 10.1007/s40571-024-00780-5
Zonglin Li, Ju Chen, Qiang Tian, Haiyan Hu

Discrete element method (DEM) is a powerful tool for the dynamic simulation of irregular non-spherical particle systems. The efficient integration of the rotational motions of numerous particles in DEM poses a big challenge. This paper presents six explicit time integration algorithms, comprising three first-order algorithms and three second-order algorithms, for the rotational motions of non-spherical particles based on the theory of unit quaternion group S(3). The proposed algorithms based on Cayley map do not contain any transcendental function and have high efficiency. The numerical examples underscore the superiority of the first-order symplectic Euler Cayley algorithm (SECay) and the second-order central difference Cayley algorithm (CDCay) in terms of both efficiency and accuracy. In the testing cases of granular systems, SECay and CDCay demonstrate approximately 80% reduction in computational time for the time integration part, compared to the improved predictor–corrector direct multiplication method (IPCDM). Therefore, SECay and CDCay emerge as promising tools for non-spherical DEM simulations.

离散元法(DEM)是对不规则非球形粒子系统进行动态模拟的有力工具。如何在 DEM 中有效地积分众多粒子的旋转运动是一个巨大的挑战。本文基于单位四元数组 S(3) 理论,针对非球形粒子的旋转运动提出了六种显式时间积分算法,包括三种一阶算法和三种二阶算法。所提出的基于 Cayley 映射的算法不包含任何超越函数,具有很高的效率。数值实例凸显了一阶交映欧拉 Cayley 算法(SECay)和二阶中心差分 Cayley 算法(CDCay)在效率和精度方面的优越性。在颗粒系统的测试案例中,与改进的预测器-校正器直接乘法(IPCDM)相比,SECay 和 CDCay 在时间积分部分的计算时间减少了约 80%。因此,SECay 和 CDCay 成为非球形 DEM 仿真的理想工具。
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引用次数: 0
Assessing the effect of layered spatial variability on soil behavior via DEM simulation 通过 DEM 模拟评估分层空间变化对土壤行为的影响
IF 2.8 3区 工程技术 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2024-06-18 DOI: 10.1007/s40571-024-00779-y
De-Yun Liu, Han-qiao Che, Ci Wang, Yuan Chen

Motivated by the spatial variability observed in geological profiles, this study explored the feasibility of using discrete element method (DEM) to capture the effect of layered spatial variability into overall soil performance. The spatial variability of packing densities, particle Young’s modulus (E), and frictional properties (μ) within specimens was studied. It was observed that samples with similar overall void ratios exhibited comparable small-strain stiffness and shearing behaviors. In contrast, the coordination number and particle stress transmission demonstrated significant sensitivity to the layer-wise spatial variability in packing densities. Regarding the spatial variability effect of particle-scale E values, this study illustrates that spatial variability strongly affects the stiffness contributions of individual layers. Specifically, layers with higher E values are capable of transferring much stress and stiffness. For the spatial variability effect of frictional property, a degree of consistency in shearing behaviors was observed among specimens with similar average frictional characteristics, while layers with lower frictional property were identified as potential initial failure junctures. Overall, this study validates the utility of employing a DEM code for analyzing both the macroscopic behavior and localized vulnerabilities within complex granular systems, presenting profound implications for engineering practices.

受地质剖面中观察到的空间变异性的启发,本研究探索了使用离散元素法(DEM)捕捉分层空间变异性对整体土壤性能影响的可行性。研究了试样内堆积密度、颗粒杨氏模量(E)和摩擦特性(μ)的空间变化。研究发现,总体空隙率相似的试样表现出相似的小应变刚度和剪切行为。相比之下,配位数和颗粒应力传递对堆积密度的层间空间变化非常敏感。关于颗粒尺度 E 值的空间变异效应,本研究表明,空间变异会强烈影响单个层的刚度贡献。具体来说,E 值较高的层能够传递较大的应力和刚度。在摩擦特性的空间变异效应方面,在平均摩擦特性相似的试样中观察到了一定程度的剪切行为一致性,而摩擦特性较低的层被确定为潜在的初始失效连接点。总之,这项研究验证了使用 DEM 代码分析复杂颗粒系统的宏观行为和局部脆弱性的实用性,并对工程实践产生了深远影响。
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引用次数: 0
A Cosserat bond-based correspondence model and the investigation of microstructure effect on crack propagation 基于 Cosserat 键的对应模型及微观结构对裂纹扩展影响的研究
IF 2.8 3区 工程技术 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2024-06-13 DOI: 10.1007/s40571-024-00785-0
Zhuang Chen, Xihua Chu, Diansen Yang

Microstructure plays a significant role in the fracture behavior of the material. To analyze the microstructure effect on crack propagation, a peridynamic model named Cosserat bond-based correspondence model (CBBCM) is proposed based on the Cosserat continuum theory and bond-based correspondence model. In CBBCM, the peridynamic (PD) force and moment are obtained by Cosserat constitutive equations through the relation between PD forces/moment and the stress/couple stress. Such relation is derived according to the bond relation of the Cosserat peridynamic model. To validate the proposed CBBCM, three numerical examples are presented, and the comparison shows good agreement between the CBBCM and the experimental observation and numerical results. The numerical convergence studies of m-convergence and δ-convergence are made to demonstrate the proposed CBBCM. The microstructure effect of crack propagation is analyzed by applying different Cosserat shear modulus and internal length scales in the simulation. The results indicate that the Cosserat shear modulus has an impact on the crack propagation and the crack propagates slower with a greater Cosserat shear modulus. The internal length scale has little impact on the crack path and only influences the local damage distribution.

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引用次数: 0
Evaluation of rock pillar failure mechanisms under uniaxial compression: impact of joint number and joint angle 评估单轴压缩下的岩柱破坏机制:节理数量和节理角度的影响
IF 2.8 3区 工程技术 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2024-06-12 DOI: 10.1007/s40571-024-00787-y
Hadi Haeri, Vahab Sarfarazi, Lei Zhou, Hosein Karimi Javid, Kaveh Asgari, Ali Elahi

Experimental and discrete element methods were used to investigate the effects of joint number and joint angle on the failure behavior of rock pillars under the uniaxial compressive test. Gypsum samples with dimensions of 200 mm × 200 mm × 50 mm were prepared. The model material had a tensile strength of 1 MPa. Embedded notches with a length of 6 cm were utilized within the samples to determine its compressive strength. In constant notch length, the number of notches was one, two, and three. In the experimental test, the angle of the diagonal plane related to the horizontal axis was 0°, 30°, 60°, and 90°. In the numerical test, the angles of the diagonal plane related to the horizontal axis were 0°, 15°, 30°, 45°, 60°, 75°, and 90°. The axial load was applied to the model at a rate of 0.05 mm/min. The results show that the failure process was mostly governed by both the non-persistent notch angle and notch number. The compressive strengths of the specimens were related to the fracture pattern and failure mechanism of the discontinuities. It was shown that the shear behavior of discontinuities is related to the number of the induced tensile cracks which are increased by increasing the notch angle. The strength of samples increases by increasing both the notch angle and notch number. The failure pattern and failure strength are similar in both methods, i.e., the experimental testing and the numerical simulation methods.

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引用次数: 0
Modeling failure of hyperelastic solids interacting with fluids 超弹性固体与流体相互作用的失效建模
IF 2.8 3区 工程技术 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2024-06-11 DOI: 10.1007/s40571-024-00784-1
Phanindra Paravastu, Srikanth Vedantam

In this paper, we present a coupled smoothed particle hydrodynamics and a constitutively informed particle dynamics model of a hyperelastic material. The proposed coupling strategy accurately transmits the forces between the fluid and solid, conserving momentum strictly. Using this model, we simulate the role of fluid impact on failure of a nonlinear elastic solid in the case of a canonical dam break problem. The location of the notch on the failure of the obstacle is found to be critical. Several modes of failure of the obstacle are observed, depending on the location of the notch and the height of the obstacle. Under some conditions, multiple failures of the obstacle occur simultaneously.

{"title":"Modeling failure of hyperelastic solids interacting with fluids","authors":"Phanindra Paravastu,&nbsp;Srikanth Vedantam","doi":"10.1007/s40571-024-00784-1","DOIUrl":"10.1007/s40571-024-00784-1","url":null,"abstract":"<div><p>In this paper, we present a coupled smoothed particle hydrodynamics and a constitutively informed particle dynamics model of a hyperelastic material. The proposed coupling strategy accurately transmits the forces between the fluid and solid, conserving momentum strictly. Using this model, we simulate the role of fluid impact on failure of a nonlinear elastic solid in the case of a canonical dam break problem. The location of the notch on the failure of the obstacle is found to be critical. Several modes of failure of the obstacle are observed, depending on the location of the notch and the height of the obstacle. Under some conditions, multiple failures of the obstacle occur simultaneously.</p></div>","PeriodicalId":524,"journal":{"name":"Computational Particle Mechanics","volume":"12 1","pages":"153 - 164"},"PeriodicalIF":2.8,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141357440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Rock joint filling breakage under three different instant loading conditions; physical test and PFC simulation 三种不同瞬时加载条件下的岩石节理充填破损;物理试验和 PFC 模拟
IF 2.8 3区 工程技术 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2024-06-10 DOI: 10.1007/s40571-024-00786-z
Lei Zhou, Vahab Sarfarazi, Hadi Haeri, Armin Shahbazian, Arsham Moayedi Far, Mohammad Fatehi Marji

In this investigation, the influences of filling shapes on the failure mechanisms of rock masses under uniaxial compressive tests were analyzed using experimental and numerical simulation methods. For this purpose, a gypsum filling slab with three different shapes (isosceles trapezoid shape, triangle shape, and square shape) was inserted between the two granite specimens. In this regard, three different gypsum conditions were prepared; (1) gypsum containing a hole, (2) intact gypsum, and (3) gypsum containing the grout. Nine models were subjected to compression load with an axial load rate of 0.05 mm/min. Moreover, PFC2D was also employed to conduct numerical simulations of the models containing gypsum filling. In this matter, eight distinct types of soft gypsum filling were created; (1) concave shape, (2) semi-concave shape, (3) isosceles trapezoid shape, (4) triangle shape, (5) square shape, (6) bit shape, (7) lozenge shape, and (8) trapezoid. In addition, three different conditions were considered, i.e., gypsum containing a hole, intact gypsum, and gypsum containing the grout. The Brazilian tensile strength of gypsum and grout was 0.4 and 1 MPa, respectively. According to the results obtained, the failure process was predominantly controlled by the filling shape and filling conditions. With regard to the compressive strength, the fracture pattern and the failure mechanisms associated with the filling were found to play the main role. It was concluded that the compressive behavior of the filling is highly affected by the number of generated tensile cracks. As for hits of the acoustic emission (AE), a few AE hits were captured in the preliminary phase of loading, followed by a rapid growth in AE hits when the applied load reached its highest value. Furthermore, a large number of AE hits were observed during the stress drop.

{"title":"Rock joint filling breakage under three different instant loading conditions; physical test and PFC simulation","authors":"Lei Zhou,&nbsp;Vahab Sarfarazi,&nbsp;Hadi Haeri,&nbsp;Armin Shahbazian,&nbsp;Arsham Moayedi Far,&nbsp;Mohammad Fatehi Marji","doi":"10.1007/s40571-024-00786-z","DOIUrl":"10.1007/s40571-024-00786-z","url":null,"abstract":"<div><p>In this investigation, the influences of filling shapes on the failure mechanisms of rock masses under uniaxial compressive tests were analyzed using experimental and numerical simulation methods. For this purpose, a gypsum filling slab with three different shapes (isosceles trapezoid shape, triangle shape, and square shape) was inserted between the two granite specimens. In this regard, three different gypsum conditions were prepared; (1) gypsum containing a hole, (2) intact gypsum, and (3) gypsum containing the grout. Nine models were subjected to compression load with an axial load rate of 0.05 mm/min. Moreover, PFC2D was also employed to conduct numerical simulations of the models containing gypsum filling. In this matter, eight distinct types of soft gypsum filling were created; (1) concave shape, (2) semi-concave shape, (3) isosceles trapezoid shape, (4) triangle shape, (5) square shape, (6) bit shape, (7) lozenge shape, and (8) trapezoid. In addition, three different conditions were considered, i.e., gypsum containing a hole, intact gypsum, and gypsum containing the grout. The Brazilian tensile strength of gypsum and grout was 0.4 and 1 MPa, respectively. According to the results obtained, the failure process was predominantly controlled by the filling shape and filling conditions. With regard to the compressive strength, the fracture pattern and the failure mechanisms associated with the filling were found to play the main role. It was concluded that the compressive behavior of the filling is highly affected by the number of generated tensile cracks. As for hits of the acoustic emission (AE), a few AE hits were captured in the preliminary phase of loading, followed by a rapid growth in AE hits when the applied load reached its highest value. Furthermore, a large number of AE hits were observed during the stress drop.</p></div>","PeriodicalId":524,"journal":{"name":"Computational Particle Mechanics","volume":"12 1","pages":"183 - 204"},"PeriodicalIF":2.8,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141364134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
The effect of particle shape on the dynamics of spherical projectile impacting into granular media 颗粒形状对球形弹丸撞击颗粒介质动力学的影响
IF 2.8 3区 工程技术 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2024-06-07 DOI: 10.1007/s40571-024-00745-8
Xingli Zhang, Honghua Zhao, Yifan Wang, Dashuai Zhang, Yuntian Bai

To investigate the effect of particle shape on the dynamics of projectile impact into granular media, this study conducted discrete element method (DEM) numerical simulation using Particle Flow Code (PFC) software. Three particle shapes of granular materials were selected, where spherical particles represented by ball elements, ellipsoidal particles and irregular particles generated by clumps according to a certain template profile. The microscopic contact parameters were calibrated by laboratory tests and numerical simulations of standard direct shear tests. On this basis, the DEM model of a spherical projectile impact into the granular bed was established and laboratory tests were conducted. The test data matched well with the numerical results, verifying the reasonableness and accuracy of the numerical model. Analysing the results by varying the parameters shows that the impact process can be divided into three stages: impact, penetration and collapse. The particle shape does not affect the final penetration depth of the projectile as a power-law function of the initial velocity, and all follow the generalised Poncelet law. The difference in the peak impact force indicates that non-spherical particles have better cushioning capacity, and the analysis of the energy evolution during impact shows that there is significant variability in the effect of particle shape on the energy dissipation of the system. Finally, the internal response of the granular media during the impact process is elucidated by the results of porosity and coordination number. The force chains of granular materials undergo fracture and recombination during the impact process, and the particle shape significantly affects the structural distribution and evolution of the force chains.

为了研究颗粒形状对弹丸撞击颗粒介质动力学的影响,利用粒子流代码(particle Flow Code, PFC)软件进行离散元法(DEM)数值模拟。选取颗粒材料的三种颗粒形状,其中以球元为代表的球形颗粒、椭球形颗粒和按一定模板轮廓由团块生成的不规则颗粒。通过室内试验和标准直剪试验的数值模拟,标定了细观接触参数。在此基础上,建立了球形弹丸撞击颗粒床的DEM模型,并进行了室内试验。试验数据与数值结果吻合较好,验证了数值模型的合理性和准确性。通过对参数变化结果的分析表明,冲击过程可分为冲击、侵彻和崩塌三个阶段。颗粒形状对弹丸最终侵彻深度的影响与初速度成幂函数关系,均遵循广义庞塞莱定律。峰值冲击力的差异表明非球形颗粒具有更好的缓冲能力,冲击过程中的能量演化分析表明,颗粒形状对系统能量耗散的影响存在显著的可变性。最后,通过孔隙率和配位数的计算结果,阐明了颗粒介质在冲击过程中的内部响应。颗粒状材料的力链在冲击过程中发生断裂和重组,颗粒形状对力链的结构分布和演化有显著影响。
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引用次数: 0
Easily porting material point methods codes to GPU 轻松将材料点方法代码移植到 GPU
IF 2.8 3区 工程技术 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2024-06-05 DOI: 10.1007/s40571-024-00768-1
Edward Buckland, Vinh Phu Nguyen, Alban de Vaucorbeil

The material point method (MPM) is computationally costly and highly parallelisable. With the plateauing of Moore’s law and recent advances in parallel computing, scientists without formal programming training might face challenges in developing fast scientific codes for their research. Parallel programming is intrinsically different to serial programming and may seem daunting to certain scientists, in particular for GPUs. However, recent developments in GPU application programming interfaces (APIs) have made it easier than ever to port codes to GPU. This paper explains how we ported our modular C++ MPM code Karamelo to GPU without using low-level hardware APIs like CUDA or OpenCL. We aimed to develop a code that has abstracted parallelism and is therefore hardware agnostic. We first present an investigation of a variety of GPU APIs, comparing ease of use, hardware support and performance in an MPM context. Then, the porting process of Karamelo to the Kokkos ecosystem is detailed, discussing key design patterns and challenges. Finally, our parallel C++ code running on GPU is shown to be up to 85 times faster than on CPU. Since Kokkos also supports Python and Fortran, the principles presented therein can also be applied to codes written in these languages.

材料点法(MPM)计算成本高,并行性强。随着摩尔定律趋于稳定以及并行计算的最新进展,没有接受过正规编程培训的科学家在为其研究开发快速科学代码时可能会面临挑战。并行编程在本质上不同于串行编程,对于某些科学家来说可能会望而生畏,尤其是对于 GPU 而言。然而,GPU 应用编程接口(API)的最新发展使得将代码移植到 GPU 比以往任何时候都更加容易。本文介绍了我们如何在不使用 CUDA 或 OpenCL 等低级硬件 API 的情况下将模块化 C++ MPM 代码 Karamelo 移植到 GPU。我们的目标是开发一种具有抽象并行性的代码,因此与硬件无关。我们首先对各种 GPU API 进行了调查,比较了在 MPM 环境下的易用性、硬件支持和性能。然后,详细介绍将 Karamelo 移植到 Kokkos 生态系统的过程,讨论关键设计模式和挑战。最后,我们在 GPU 上运行的并行 C++ 代码比在 CPU 上运行的速度快 85 倍。由于 Kokkos 还支持 Python 和 Fortran,因此其中介绍的原理也可应用于用这些语言编写的代码。
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引用次数: 0
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Computational Particle Mechanics
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