International Journal for Numerical Methods in Fluids最新文献

{"title":"A GPU Accelerated Mixed-Precision Finite Difference Informed Random Walker (FDiRW) Solver for Strongly Inhomogeneous Diffusion Problems","authors":"Zirui Mao,&nbsp;Shenyang Hu,&nbsp;Ang Li","doi":"10.1002/fld.5394","DOIUrl":"https://doi.org/10.1002/fld.5394","url":null,"abstract":"<div>\u0000 \u0000 <p>In nature, many complex multi-physics coupling problems exhibit significant diffusivity inhomogeneity, where one process occurs several orders of magnitude faster than others temporally. Simulating rapid diffusion alongside slower processes demands intensive computational resources due to the necessity for small time steps. To address these computational challenges, we have developed an efficient numerical solver named Finite Difference informed Random Walker (FDiRW). In this study, we propose a GPU-accelerated, mixed-precision configuration for the FDiRW solver to maximize efficiency through GPU multi-threaded parallel computation and lower precision computation. Numerical evaluation results reveal that the proposed GPU-accelerated mixed-precision FDiRW solver can achieve a 117× speedup over the CPU baseline, while an additional 1.75× speedup is achieved by employing lower precision GPU computation. Notably, for large model sizes, the GPU-accelerated mixed-precision FDiRW solver demonstrates strong scaling with the number of nodes used in simulation. When simulating radionuclide absorption processes by porous wasteform particles with a medium-sized model of 192 × 192 × 192, this approach reduces the total computational time to 10 min, enabling the simulation of larger systems with strongly inhomogeneous diffusivity.</p>\u0000 </div>","PeriodicalId":50348,"journal":{"name":"International Journal for Numerical Methods in Fluids","volume":"97 8","pages":"1104-1119"},"PeriodicalIF":1.7,"publicationDate":"2025-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144524736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-Hydrostatic Model for Simulating Moving Bottom-Generated Waves: A Shallow Water Extension With Quadratic Vertical Pressure Profile","authors":"Kemal Firdaus,&nbsp;Jörn Behrens","doi":"10.1002/fld.5393","DOIUrl":"https://doi.org/10.1002/fld.5393","url":null,"abstract":"<p>We formulate a depth-averaged non-hydrostatic model to solve wave equations with generation by a moving bottom. This model is built on the shallow water equations, which are widely used in tsunami wave modelling. An extension leads to two additional unknowns to be solved: vertical momentum and non-hydrostatic pressure. We show that a linear vertical velocity assumption turns out to give us a quadratic pressure relation, which is equivalent to Boussinesq-type equations, the Green-Naghdi equations specifically, making it suitable for weakly dispersive cases. However, this extension involves a time derivative of an unknown parameter, rendering the solution by a projection method ambiguous. In this study, we derive an alternative form of the elliptic system of equations to avoid such ambiguity. The new set of equations satisfies the desired solubility property, while also consistently representing the non-flat moving topography wave generation. Validations are performed using several test cases based on the previous experiments and a high-fidelity simulation. First, we show the efficiency of our model in solving a vertical movement, which represents an undersea earthquake-generated tsunami. Following that, we demonstrate the accuracy of the model for landslide-generated waves. Finally, we compare the performance of our novel set of equations with the linear and simplified quadratic pressure profiles.</p>","PeriodicalId":50348,"journal":{"name":"International Journal for Numerical Methods in Fluids","volume":"97 8","pages":"1093-1103"},"PeriodicalIF":1.7,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/fld.5393","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144524994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An Extended Height-Function Method for 3D VOF Simulations of Wetting Phenomena on Super-Hydrophilic and Hydrophobic Surfaces","authors":"Kenya Kitada,&nbsp;Ryoichi Kurose","doi":"10.1002/fld.5391","DOIUrl":"https://doi.org/10.1002/fld.5391","url":null,"abstract":"<div>\u0000 \u0000 <p>An extended height-function (HF) method that can be consistently utilized for 3D volume of fluid (VOF) simulations of wetting phenomena on super-hydrophilic and super-hydrophobic surfaces, is proposed. First, the standard HF method is briefly explained. Then, 2D and 3D HF methods that reflect the contact angles reported so far are described, with their limitations discussed. Finally, specific treatments of contact line identification and HF construction reflecting the contact angle boundary condition, required to overcome such limitations, are presented in detail. Numerical tests for a sessile droplet reveal that the contact line identification and HF construction are conducted appropriately with respect to the imposed contact angles ranging from <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <msup>\u0000 <mrow>\u0000 <mn>5</mn>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>∘</mo>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$$ 1{5}^{circ } $$</annotation>\u0000 </semantics></math> to <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>16</mn>\u0000 <msup>\u0000 <mrow>\u0000 <mn>5</mn>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>∘</mo>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$$ 16{5}^{circ } $$</annotation>\u0000 </semantics></math> in the proposed numerical scheme. Additionally, the present method shows approximately first- or second-order convergence of the curvature at the contact line for a wide range of contact angles. Moreover, simulations of droplet spreading driven by surface tension reveal that the proposed method can reasonably reproduce the behavior of a droplet reaching an equilibrium state defined by an imposed contact angle.</p>\u0000 </div>","PeriodicalId":50348,"journal":{"name":"International Journal for Numerical Methods in Fluids","volume":"97 7","pages":"1073-1091"},"PeriodicalIF":1.7,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144197161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cavitation Implementation Algorithms Based on Pressure Projection Method for Incompressible Flows With Three-Phase Interactions","authors":"Zhaoyuan Wang,&nbsp;Sungtek Park,&nbsp;Frederick Stern","doi":"10.1002/fld.5389","DOIUrl":"https://doi.org/10.1002/fld.5389","url":null,"abstract":"<div>\u0000 \u0000 <p>In the present study, a cavitation implementation algorithm is developed using a pressure-based method for incompressible flows with three-phase interactions. Central to this implementation algorithm is the treatment of the velocity jump due to the phase change, which is included in both the cavitation transport and pressure equations. The velocity jump, as a function of the phase change rate, is added as a source term to the pressure Poisson equation. A non-conservative form of the vapor transport equation is derived, and the velocity divergence is replaced by a term related to the mass phase change rate. An algorithm for the three-phase (air, water, and vapor) interactions is also developed. The VOF method is modified and used to identify the ‘dry’ (air) phase and the ‘wet’ (water/vapor mixture) phase, since the cavitation can only occur inside the water phase. The liquid volume fraction is used to distinguish water and vapor phases. The numerical results of the 2D NACA66MOD and 3D Delft Twist 11 hydrofoils show good agreement with the experimental measurement. The forced unsteady cavitation flows are investigated using a pitching foil with the results compared with the experimental observations. Air–water interface effect on the cavitation is investigated using the NACA66MOD hydrofoil. The code is applied to simulate a surface piercing super cavitating hydrofoil with both ventilation and cavitation involved.</p>\u0000 </div>","PeriodicalId":50348,"journal":{"name":"International Journal for Numerical Methods in Fluids","volume":"97 7","pages":"1057-1072"},"PeriodicalIF":1.7,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144197405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"SDF-Guided Point Cloud Generation Framework for Mesh-Free CFD","authors":"Tao Zhang,&nbsp;George N. Barakos","doi":"10.1002/fld.5390","DOIUrl":"https://doi.org/10.1002/fld.5390","url":null,"abstract":"<p>Meshing is a bottleneck of CFD workflows, especially when complex geometries are considered. Mesh-free methods could be a promising solution, but the lack of high-quality point cloud generation methods for boundary layers has hindered their popularity and applications. This work presents a novel point cloud generation framework for near- and off-body regions. The novelty of the method is the introduction of the Signed Distance Function (SDF) to guide advancing point layers in the near-body region. Insertion/removal mechanisms of points, collocation search approach, and point cloud quality metrics were also proposed. These ensure high-quality boundary layer resolution in the near-body region, regardless of the complexity and topology of the geometry. For the off-body region, Cartesian points are employed for smooth and adaptive point distributions. Compared to conventional advancing front point generation, the proposed method ensures surface-norm point distributions with consistent layer structures, which are critical for boundary layer resolution. Compared to the strand mesh generation, the current method presents much greater flexibility with few restrictions on inter-layer connections. The proposed approach is tested for various 2D and 3D benchmark geometries, along with mesh-free modeling results using the generated point clouds. The results demonstrate an important step towards a fully automated, adaptive, and mesh-free CFD workflow for complex engineering applications.</p>","PeriodicalId":50348,"journal":{"name":"International Journal for Numerical Methods in Fluids","volume":"97 7","pages":"1035-1056"},"PeriodicalIF":1.7,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/fld.5390","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144197636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Local and Parallel Mixed-Precision Finite Element Methods for the Time-Dependent Incompressible Flows","authors":"Qingfang Liu,&nbsp;Jian Su,&nbsp;Baotong Li","doi":"10.1002/fld.5388","DOIUrl":"https://doi.org/10.1002/fld.5388","url":null,"abstract":"<div>\u0000 \u0000 <p>In this article, a local and parallel mixed-precision finite element method is applied for solving the time-dependent incompressible flows. We decompose the solution into the large eddy components and small eddy components based on two-grid method. The analysis shows that the small eddy components carry little part of the total energy compared with the large eddy components. In view of this character, we first obtain the large eddy components by solving the standard nonlinear equation using the high-precision solvers globally in the coarse mesh space, then get the small eddy components by solving a series of local linearized residual equation using the low-precision solvers locally and parallel based on the partition of unity. The performance advantages of the mixed-precision methods are tested with respect to speedups over a high-precision implementation in time and less storage requirements in space.</p>\u0000 </div>","PeriodicalId":50348,"journal":{"name":"International Journal for Numerical Methods in Fluids","volume":"97 7","pages":"1023-1034"},"PeriodicalIF":1.7,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144197041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A High-Order Hybrid-Spectral Incompressible Navier–Stokes Model for Non-Linear Water Waves","authors":"Anders Melander,&nbsp;Max Ebstrup Bitsch,&nbsp;Dong Chen,&nbsp;Allan Peter Engsig-Karup","doi":"10.1002/fld.5387","DOIUrl":"https://doi.org/10.1002/fld.5387","url":null,"abstract":"<p>We present a new high-order accurate computational fluid dynamics model based on the incompressible Navier–Stokes equations with a free surface for the accurate simulation of non-linear and dispersive water waves in the time domain. The spatial discretization is based on Chebyshev polynomials in the vertical direction and a Fourier basis in the horizontal direction, allowing for the use of the fast Chebyshev and Fourier transforms for the efficient computation of spatial derivatives. The temporal discretization is done through a generalized low-storage explicit fourth-order Runge–Kutta, and for the scheme to conserve mass and achieve high-order accuracy, a velocity-pressure coupling needs to be satisfied at all Runge–Kutta stages. This results in the emergence of a Poisson pressure problem that constitutes a geometric conservation law for mass conservation. The occurring Poisson problem is proposed to be solved efficiently via an accelerated iterative solver based on a geometric <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 <annotation>$$ p $$</annotation>\u0000 </semantics></math>-multigrid scheme, which takes advantage of the high-order polynomial basis in the spatial discretization and hence distinguishes itself from conventional low-order numerical schemes. We present numerical experiments for validation of the scheme in the context of numerical wave tanks demonstrating that the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 <annotation>$$ p $$</annotation>\u0000 </semantics></math>-multigrid accelerated numerical scheme can effectively solve the Poisson problem that constitute the computational bottleneck, that the model can achieve the desired spectral convergence, and is capable of simulating wave-propagation over non-flat bottoms with excellent agreement in comparison to experimental results.</p>","PeriodicalId":50348,"journal":{"name":"International Journal for Numerical Methods in Fluids","volume":"97 6","pages":"1009-1021"},"PeriodicalIF":1.7,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/fld.5387","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143897170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analyzing 3D Steady Variable Coefficients Convection–Diffusion-Reaction Equations via a Hybrid Element-Free Galerkin Method","authors":"Jiao Zhang,&nbsp;Yi-Chen Yang,&nbsp;Feng-Bin Liu,&nbsp;Heng Cheng","doi":"10.1002/fld.5386","DOIUrl":"https://doi.org/10.1002/fld.5386","url":null,"abstract":"<div>\u0000 \u0000 <p>This study introduces a hybrid element-free Galerkin (HEFG) method to analyze the 3D steady convection–diffusion-reaction equation. By introducing the dimension-splitting method, the governing equation can be split into 2D form in each layer. The 2D form can be solved using the improved element-free Galerkin (IEFG) method with improved moving least-squares (IMLS) approximation as shape function, and discretized equations of 2D form are derived. The finite difference method (FDM) is selected to handle first- and second-order derivatives in the splitting direction. Thus, new 2D discretized equations in each plane are derived, and the final solved equation of the original 3D problem is obtained by coupling these 2D discretized equations. In numerical examples, we study the astringency of the HEFG method by examining the impact of layer and node on relative errors, and the computing time and accuracy of numerical solutions are compared with the dimension-coupling method (DCM), IEFG method, and exact one. The HEFG method can significantly reduce the calculation times of the IEFG method. Compared with the DCM, the advantage of the proposed method is its shorter computing time when dealing with essential boundaries in a splitting direction.</p>\u0000 </div>","PeriodicalId":50348,"journal":{"name":"International Journal for Numerical Methods in Fluids","volume":"97 6","pages":"996-1008"},"PeriodicalIF":1.7,"publicationDate":"2025-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143897265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Continuous Adjoint to Proudman's Formula for Aeroacoustic Shape Optimization","authors":"M. Erfan Farhikhteh,&nbsp;E. M. Papoutsis Kiachagias,&nbsp;K. C. Giannakoglou","doi":"10.1002/fld.5378","DOIUrl":"https://doi.org/10.1002/fld.5378","url":null,"abstract":"<p>This paper presents an approach for aeroacoustic optimization through the reduction of acoustic sources, based on the integration of Proudman's formula into a continuous adjoint framework coupled with the Reynolds-averaged Navier–Stokes equations, for the first-time. The development includes the adjoint to the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>−</mo>\u0000 <mi>ω</mi>\u0000 <mspace></mspace>\u0000 <mi>S</mi>\u0000 <mi>S</mi>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation>$$ k-omega kern0.3em SST $$</annotation>\u0000 </semantics></math> turbulence model. Here, Proudman's formula is used to compute acoustic emissions of turbulent flows around aerodynamic bodies using the turbulent kinetic energy and specific rate of dissipation. Broadband noise generation through Proudman's formula is initially validated for a case including the flow around an isolated airfoil. Subsequently, the sensitivity derivatives of an objective function quantifying acoustic sources are verified against finite differences, with optimizations of two isolated airfoils and the MEXICO wind turbine following. Optimizations are conducted by extending the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mi>d</mi>\u0000 <mi>j</mi>\u0000 <mi>o</mi>\u0000 <mi>i</mi>\u0000 <mi>n</mi>\u0000 <mi>t</mi>\u0000 <mi>O</mi>\u0000 <mi>p</mi>\u0000 <mi>t</mi>\u0000 <mi>i</mi>\u0000 <mi>m</mi>\u0000 <mi>i</mi>\u0000 <mi>s</mi>\u0000 <mi>a</mi>\u0000 <mi>t</mi>\u0000 <mi>i</mi>\u0000 <mi>o</mi>\u0000 <mi>n</mi>\u0000 <mi>F</mi>\u0000 <mi>o</mi>\u0000 <mi>a</mi>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation>$$ adjointOptimisationFoam $$</annotation>\u0000 </semantics></math> tool in OpenFOAM, developed and made publicly available by the group. During the optimization, constraints on the lift force, the drag force, the pitching moment coefficient, the torque, the trailing edge thickness, and airfoil volume are imposed, depending on the case. The geometries and grids are parameterized using PARSEC and morphing boxes based on volumetric B-Splines. The optimizations result in shapes with reduced acoustic sources while preserving aerodynamic efficiency, highlighting the effectiveness of the proposed method and programmed software.</p>","PeriodicalId":50348,"journal":{"name":"International Journal for Numerical Methods in Fluids","volume":"97 6","pages":"966-984"},"PeriodicalIF":1.7,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/fld.5378","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143896807","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A WPOD-Kriging Reduced-Order Method for Parametric CFD Simulations","authors":"Zhehao Xia,&nbsp;Yizhong Wu","doi":"10.1002/fld.5383","DOIUrl":"https://doi.org/10.1002/fld.5383","url":null,"abstract":"<div>\u0000 \u0000 <p>High-fidelity computational fluid dynamics (CFD) simulation usually carries a heavy computational burden, especially for parametric CFD simulations requiring multiple calculations. To address this challenge, researchers have developed reduced-order modeling (ROM) to significantly decrease the computational burden by building a simplified model. This article proposes a hybrid method of weighted proper orthogonal decomposition and Kriging, a novel reduced-order method. This method improves the accuracy of the reduced-order model by assigning appropriate weights to the samples while estimating the specific design parameters. The main innovation of this work is the development of the optimized method for generating the weights. Firstly, the leave-one-out method is employed to divide the samples into the training set and test set, and the multivariate Gaussian distribution is used to convert the Euclidean distance between the training set and test set into weight. Then, we adopt the WPOD-Kriging method to construct a reduced-order model using the training set. This model is compared with the test set to obtain the error. By repeatedly resetting the training set and the test set, we receive multiple errors and average them to calculate the global error. This process involves an important parameter, which is the covariance matrix of multivariate Gaussian distribution. We can generate the optimal covariance matrix by minimizing the global error to achieve the optimized method for generating the weights. The efficacy of the WPOD-Kriging method is validated through three parametric CFD simulations. Compared to other similar approaches, the proposed method offers a more accurate reduced-order model.</p>\u0000 </div>","PeriodicalId":50348,"journal":{"name":"International Journal for Numerical Methods in Fluids","volume":"97 6","pages":"985-995"},"PeriodicalIF":1.7,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143896806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}