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multigrid algorithm for Immersed Finite Element Discretizations of Elliptic Interface Problems

JOURNAL OF SCIENTIFIC COMPUTING 2024年第1期98卷 1-35页

作者： Chu, Hanyu Song, Yongzhong Ji, Haifeng Cai, Ying Nanjing Normal Univ Sch Math Sci Key Lab NSLSCS Jiangsu Int Joint Lab Big Data Modeling Comp & App Nanjing 210023 Peoples R China Nanjing Univ Posts & Telecommun Sch Sci Nanjing 210023 Peoples R China Beijing Computat Sci Res Ctr Beijing 100193 Peoples R China

This paper is devoted to analyzing multigrid algorithm for solving elliptic interface problems discretized using the partially penalized immersed finite element (PPIFE). By taking the average values of nodal variables and integral variables, we construct intergrid transfer operators for the P-1 partially penalized immersed finite element (P-1-PPIFE) and the Crouzeix-Raviart partially penalized immersed finite element (CR-PPIFE), which satisfy certain stable approximation properties. An extra interface correction procedure is added in the smoothing steps to ensure the robustness of multigrid algorithm. We prove that the convergence of W-cycle multigrid algorithm and the condition number of the variable V-cycle as preconditioner are optimal by verifying the regularity-approximation assumption, which means that the convergence rate of algorithm is independent of mesh level, mesh size, and the position of the interface relative to the mesh. Numerical experiments illustrate the convergence of our algorithms using the W-cycle, V-cycle, and preconditioned conjugate gradient algorithm (PCG) with the V-cycle.

关键词： Elliptic interface problems Immersed finite element multigrid algorithm Interface transfer operators Interface correction procedure

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p-multigrid high-order discontinuous Galerkin solver for three-dimensional compressible turbulent flows

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JOURNAL OF COMPUTATIONAL PHYSICS 2025年 525卷

作者： Bulgarini, D. Ghidoni, A. Mantecca, E. Noventa, G. Univ Brescia Dept Mech & Ind Engn Via Branze 36 I-25123 Brescia Italy

The study of turbulent flows through steady-state simulations based on the Reynolds-averaged Navier-Stokes equations and turbulence models can be considered the workhorse in different scientific and industrial applications. Among the different numerical approaches, discontinuous Galerkin methods demonstrated to be perfectly suited for high-order accurate numerical solutions on structured or arbitrary unstructured and non-conforming meshes, and high-performance computing with massively parallel processing. However, their computational cost increases rapidly when the solution is discretized with higher-order polynomial approximations. For this reason, many research efforts have been devoted to overcome this drawback. Literature shows many applications of p-multigrid algorithms for the solution of Euler and Navier-Stokes equations, while few works report the solution of the Reynolds-averaged Navier-Stokes equations with pmultigrid algorithms. In fact, different authors highlighted a lack of performance for the stiffness associated with the discretized RANS equations, and for highly stretched meshes, typically used for an accurate resolution of turbulent boundary layers. This work presents the implementation of an improved p-multigrid algorithm based on the nonlinear full approximation scheme in a discontinuous Galerkin solver for the solution of the three-dimensional and compressible Reynolds-Average Navier-Stokes equations. The performance of the algorithm with different smoothers is compared with the implicit (single-order) time integration on many test cases with different flow conditions, domains, and meshes, showing an average reduction of the computing time around 75% with respect to single-order implicit solvers.

关键词： Discontinuous Galerkin multigrid algorithm RANS equations Compressible flows

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Assembling a Learnable Mumford--Shah Type Model with multigrid Technique for Image Segmentation

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SIAM JOURNAL ON IMAGING SCIENCES 2024年第2期17卷 1007-1039页

作者： Meng, Junying Guo, Weihong Liu, Jun Yang, Mingrui Beijing Normal Univ Sch Math Sci Lab Math & Complex Syst Minist Educ China Beijing 100875 Peoples R China Case Western Reserve Univ Dept Math Appl Math & Stat Cleveland OH 44106 USA Beijing Normal Univ Sch Math Sci Lab Math & Complex Syst Minist Educ China Beijing 100875 Peoples R China Cleveland Clin Lerner Res Inst Cleveland OH 44195 USA

The classical Mumford--Shah (MS) model has been successful in some medical image segmentation tasks, providing segmentation results with smooth boundaries of objects. However, the MS model, which operates at the pixel level of the images, faces challenges when dealing with medical images with low contrast or unclear edges. In this paper, we begin by using a feature extractor to capture high -dimensional deep features that contain more comprehensive semantic information than pixel -level data alone. Inspired by the MS model, we develop a variational model that incorporates threshold dynamics (TD) regularization for segmenting each feature. We obtain the final segmentation result for the original image by assembling segmentation results of all the features. This process results in MS-MGNet, a lightweight trainable segmentation network with a similar architecture to many encoder -decoder networks. The intermediate layers of MS-MGNet are designed by unrolling the numerical scheme based on the multigrid method for solving the variational model. We provide interpretability for the encoder -decoder architecture by elucidating the roles of each layer and offering explanations of the underlying mathematical models. By incorporating the TD regularizer, we integrate spatial priors from the variational models into the network architecture, resulting in better segmentation results with smoother edges and a certain robustness to noise. Compared to some relevant methods, experimental results on the selected data sets with low contrast or unclear edges show that the proposed method can achieve better segmentation performance with fewer parameters, even when trained on smaller data sets.

关键词： image segmentation deep neural network variational prior multigrid algorithm

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Application of multigrid for the Solution of Compressible Reynolds Equation of Gas Foil Bearings

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TRIBOLOGY TRANSACTIONS 2023年第3期66卷 431-442页

作者： Xu, Kefan Zhang, Guanghui Han, Jiazhen Gong, Wenjie Huang, Yanzhong Guo, Yang Harbin Inst Technol Sch Energy Sci & Engn Harbin Peoples R China

The multigrid algorithm can reduce the iterative residuals quickly, but it is rarely used in research about gas foil bearings. This article adopts the Reynolds equation as the pressure governing equation of bump-type foil journal bearings to study the effect of different numerical methods on the execution time of the algorithm program, including the direct method, the iterative method, and the multigrid algorithm. The results indicate that the algorithm performs best, based on the nonlinear discretization scheme and multigrid, and the execution time is generally reduced by 50% compared to the traditional direct method under the same operating parameters. For the case of a large eccentricity ratio, the execution time can even be reduced by up to 68%. The multigrid algorithm is greatly affected by the cycle type parameter and the level numbers of the grid. V-cycle and larger grid-level numbers are recommended on the premise of program convergence. In summary, the multigrid algorithm is an excellent numerical solution that can balance computational accuracy and efficiency when solving the gas lubrication problem of foil bearings.

关键词： Gas foil journal bearing multigrid algorithm iterative method Reynolds equation

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An extrapolation accelerated multiscale Newton-MG method for fourth-order compact discretizations of semilinear Poisson equations

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2022年 113卷 189-197页

作者： Hu, Hongling Li, Ming Pan, Kejia Wu, Pinxia Hunan Normal Univ Sch Math & Stat MOE LCSM Changsha 410081 Hunan Peoples R China Honghe Univ Dept Math & Stat Mengzi 661199 Yunnan Peoples R China Cent South Univ Sch Math & Stat HNP LAMA Changsha 410083 Hunan Peoples R China

An extrapolation accelerated multiscale Newton-multigrid (EMNMG) method is proposed to solve twodimensional semilinear Poisson equations. The nine-point fourth-order compact schemes are used to approximate the nonlinear Poisson equations. In order to accelerate Newton-MG method for calculating the finite difference (FD) solution on the finest grid, a quite good initial guess is constructed from the fourth-order FD solutions at two coarse levels by using Richardson extrapolation and bi-quartic polynomial interpolation, which greatly reduces the number of Newton iterations required. A completed extrapolation technique is adopted to generate a sixth-order extrapolated solution on entire finest grid cheaply. Numerical results are given to show that our proposed EMNMG algorithm can achieve high accuracy and keep less cost simultaneously.

关键词： Extrapolation multigrid algorithm Fourth-order compact scheme Poisson equation Nonlinear term

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Progress in Computational Fluid Dynamics Discretizations algorithms and Solvers for Aerodynamic Flows

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AIAA JOURNAL 2021年第12期59卷 5374-5397页

作者： Mavriplis, Dimitri J. Univ Wyoming Dept Mech Engn Laramie WY 82071 USA

An overview of developments in computational fluid dynamic discretizations and solution techniques for aerodynamic applications over the past five-to-six-year time frame is given in this paper. This paper begins with a survey of community efforts that have contributed to advancing the state-of-the-art of computational aerodynamics. Subsequently, an overview of developments in Reynolds-averaged Navier-Stokes methods is given, including the use of higher-order discretizations. This is followed by a look at recent developments in scale-resolving methods from an algorithmic point of view. The paper concludes with prospects for further advances toward long-term milestones and Grand Challenge projects as outlined in the CFD2030 report.

关键词： Direct Numerical Simulation Incompressible Flow Reynolds Averaged Navier Stokes Aerodynamic Configurations Adaptive Mesh Refinement Finite Volume Method High Performance Computing multigrid algorithm FaSTAR Aerospace Engineering

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An efficient time-dependent auxiliary variable approach for the three-phase conservative Allen-Cahn fluids

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APPLIED MATHEMATICS AND COMPUTATION 2023年 438卷

作者： Tan, Zhijun Yang, Junxiang Chen, Jianjun Kim, Junseok Sun Yat sen Univ Sch Comp Sci & Engn Guangzhou 510275 Peoples R China Sun Yat sen Univ Guangdong Prov Key Lab Computat Sci Guangzhou 510275 Peoples R China Korea Univ Dept Math Seoul 02841 South Korea

Based on a time-dependent auxiliary variable approach, we propose linear, totally decou-pled, and energy dissipative methods for the three-phase conservative Allen-Cahn (CAC) fluid system. The three-phase CAC equation has been extensively applied in the simulation of multi-component fluid flows because of the following advantages: (i) Total mass is con-served, (ii) Topological change of the interface can be implicitly captured. Compared with the ternary Cahn-Hilliard (CH) model, the CAC-type model is simple to solve. When we solve the CAC model by using the classical scalar auxiliary variable (SAV) approach, extra computational time is needed because we must decouple the local and non-local variables. The variant of SAV approach considered in the present study not only leads to linear and energy stable schemes, but also achieves highly efficient computation. Linear and decou-pled equations need to be updated at each time step. We adopt the linear multigrid al-gorithm to speed up the convergence. Extensive numerical experiments with and without fluid flows are conducted to validate the temporal accuracy, mass conservation, and energy law.(c) 2022 Elsevier Inc. All rights reserved.

关键词： Three-phase conservative allen-Cahn fluids Finite difference method multigrid algorithm Dissipation law

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Simple and Efficient Acceleration of Existing multigrid algorithms

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AIAA JOURNAL 2019年第6期57卷 2244-2247页

作者： Citro, Vincenzo Univ Salerno Dept Ind Engn Via Giovanni Paolo II I-84084 Fisciano SA Italy

A simple, robust, and efficient procedure to accelerate multigrid algorithms is discussed in detail. The approach used in this study is based on the BoostConv routine recently proposed by Citro et al. ("Efficient Stabilization and Acceleration of Numerical Simulation of Fluid Flows by Residual Recombination," Journal of Computational Physics, Vol. 344, 2017, pp. 234-246). The present study starts from the idea that an existing multigrid code can be improved without any coding effort: a simple call to a black-box subroutine is enough to accelerate the iterative procedure. As a consequence, the proposed approach is well suited to be applied in real-life applications where the complexity of numerical codes is quite far from textbook standards. The application of BoostConv to a model problem is presented here. The algorithm is able to reduce the number of iterations required to get the target accuracy without increasing the computational time of the original code. As a consequence, BoostConv can be used on simple models, characterized by few degrees of freedom, or large numerical problems arising from the discretization of three-dimensional problems.

关键词： multigrid algorithm Numerical Simulation Black Box multigrid Methods MATLAB Navier Stokes Equations Finite Difference Approximation Dirichlet Boundary Condition Transonic Flow

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Efficient preconditioning of a high-order solver for multiple physics

Efficient preconditioning of a high-order solver for multipl...

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AIAA Aviation Forum and Exposition

作者： Franciolini, Matteo Garai, Anirban Murman, Scott NASA Ames Res Ctr NASA Adv Supercomp Div Moffett Field CA 94035 USA NASA Ames Res Ctr NASA Adv Supercomp Div Sci & Technol Corp Moffett Field CA 94035 USA

ISBN: (数字)9781624106101

ISBN: (纸本)9781624106101

This work addresses preconditioning approaches for an implicit high-order solver framework applied to multiple physics. The solver is based on a space-time spectral element method and matrix-free Newton-Krylov solver developed at NASA over the recent years. Within this context, most preconditioning methods are impractical, as the computational time and memory requirements scale poorly with increasing polynomial orders. To improve computational efficiency, we first describe a novel entity-based Block Jacobi preconditioner for the continuous-Galerkin solution of the linear-elasticity and linear-shell equations. Second, we introduce a multigrid algorithm to further reduce time-to-solution on stiff cases arising from continuousand discontinuous-Galerkin discretizations. Results obtained on relevant single-physics reference solutions, demonstrate the feasibility of the methods, paving the way for high-order solutions of fully coupled multi-physics problems.

关键词： Cauchy Stress Tensor multigrid algorithm Finite Element Solution Poisson's Ratio Navier Stokes Equations Fluid Structure Interaction Low Pressure Turbine NACA 0012 Computing Discontinuous Galerkin Method

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A Three Dimensional Multilevel Meshless Method: Point-Based Convergence Accelerator

A Three Dimensional Multilevel Meshless Method: Point-Based ...

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AIAA Science and Technology Forum and Exposition (AIAA SciTech)

作者： Rhee, Jae Sang Huh, Jin Young Kung, Tae Yoon Kim, Kyu Hong Seoul Natl Univ Dept Mech & Aerosp Engn Gwanak Ro 1KS013 Seoul 151744 South Korea Seoul Natl Univ Aerosp Inst Adv Aerosp Technol Gwanak Ro 1KS013 Seoul 151744 South Korea

ISBN: (数字)9781624105951

ISBN: (纸本)9781624105951

A Three dimensional convergence accelerator for meshless methods is presented. The widely known accelerator 'multigrid method' is modified to the meshless method. The accelerator is applied to 2 and 3-D Euler equations. For robust and accurate flux calculation, a spatial discretization was performed based on AUSMPW+. Then, LU-SGS, widely used implicit scheme, was selected for the time integration method. The performance of the accelerator was shown by comparing the results obtained from the single to the multilevel results. The comparison highlights the enhancement due to the multilevel method. Furthermore, an extension to 3-D space shows the power of meshless method in coarsening procedure.

关键词： Numerical Analysis Euler Equations Compressible Flow Unstructured Grid Finite Volume Method Total Energy Mach Number Pressure Coefficient multigrid algorithm Specific Heat

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