10,797 results on '"multigrid method"'
Search Results
2. A numerical solution of parabolic quasi-variational inequality nonlinear using Newton-multigrid method
- Author
-
M. Bahi, M. Beggas, N. Nesba, and A. Imtiaz
- Subjects
newton’s method ,multigrid method ,parabolic variational inequality ,finite element method ,hamilton–jacobi–bellman equation ,Applied mathematics. Quantitative methods ,T57-57.97 - Abstract
In this article, we apply three numerical methods to study the L∞-convergence of the Newton-multigrid method for parabolic quasi-variational inequalities with a nonlinear right-hand side. To discretize the problem, we utilize a finite element method for the operator and Euler scheme for the time. To obtain the system discretization of the problem, we reformulate the parabolic quasi-variational inequality as a Hamilton–Jacobi–Bellman equation. For linearizing the problem on the coarse grid, we employ Newton’s method as an external interior iteration of the Jacobian system. On the smooth grid, we apply the multigrid method as an interior iteration on the Jacobian system. Finally, we provide a proof for the L∞-convergence of the Newton-multigrid method for parabolic quasi-variational inequalities with a nonlinear right-hand, by giving a numerical example for this problem. more...
- Published
- 2024
- Full Text
- View/download PDF
Catalog
3. A reduced-form multigrid approach for ANN equivalent to classic multigrid expansion.
- Author
-
Seo, Jeong-Kweon
- Subjects
- *
ARTIFICIAL neural networks , *PARTIAL differential equations , *BOUNDARY value problems , *ARTIFICIAL intelligence , *LINEAR systems - Abstract
In this paper, we investigate the method of solving partial differential equations (PDEs) using artificial neural network (ANN) structures, which have been actively applied in artificial intelligence models. The ANN model for solving PDEs offers the advantage of providing explicit and continuous solutions. However, the ANN model for solving PDEs cannot construct a conventionally solvable linear system with known matrix solvers; thus, computational speed could be a significant concern. We study the implementation of the multigrid method, developing a general concept for a coarse-grid correction method to be integrated into the ANN-PDE architecture, with the goal of enhancing computational efficiency. By developing a reduced form of the multigrid method for ANN, we demonstrate that it can be interpreted as an equivalent representation of the classic multigrid expansion. We validated the applicability of the proposed method through rigorous experiments, which included analyzing loss decay and the number of iterations along with improvements in terms of accuracy, speed, and complexity. We accomplished this by employing the gradient descent method and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method to update the gradients while solving the given ANN systems of PDEs. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
- Full Text
- View/download PDF
4. A numerical solution of parabolic quasi-variational inequality nonlinear using Newton-multigrid method.
- Author
-
Bahi, M., Beggas, M., Nesba, N., and Imtiaz, A.
- Subjects
NONLINEAR analysis ,FINITE element method ,DISCRETIZATION methods ,PROBLEM solving ,STOCHASTIC convergence - Abstract
In this article, we apply three numerical methods to study the L∞-convergence of the Newton-multigrid method for parabolic quasi-variational inequalities with a nonlinear right-hand side. To discretize the problem, we utilize a finite element method for the ofaator and Euler scheme for the time. To obtain the system discretization of the problem, we reformulate the parabolic quasi-variational inequality as a Hamilton–Jacobi–Bellman equation. For linearizing the problem on the coarse grid, we employ Newton's method as an external interior iteration of the Jacobian system. On the smooth grid, we apply the multigrid method as an interior iteration on the Jacobian system. Finally, we provide a proof for the L∞-convergence of the Newton-multigrid method for parabolic quasi-variational inequalities with a nonlinear right-hand, by giving a numerical example for this problem. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
- Full Text
- View/download PDF
5. Homogeneous multigrid for HDG applied to the Stokes equation.
- Author
-
Lu, Peipei, Wang, Wei, Kanschat, Guido, and Rupp, Andreas
- Subjects
- *
LAGRANGE equations , *GALERKIN methods , *LINEAR equations , *LINEAR systems , *MATHEMATICS , *STOKES equations - Abstract
We propose a multigrid method to solve the linear system of equations arising from a hybrid discontinuous Galerkin (in particular, a single face hybridizable, a hybrid Raviart–Thomas, or a hybrid Brezzi–Douglas–Marini) discretization of a Stokes problem. Our analysis is centered around the augmented Lagrangian approach and we prove uniform convergence in this setting. Beyond this, we establish relations, which resemble those in Cockburn & Gopalakrishnan (2008, Error analysis of variable degree mixed methods for elliptic problems via hybridization. Math. Comput. , 74 , 1653–1677) for elliptic problems, between the approximates that are obtained by the single-face hybridizable, hybrid Raviart–Thomas and hybrid Brezzi–Douglas–Marini methods. Numerical experiments underline our analytical findings. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
- Full Text
- View/download PDF
6. 杂质颗粒对滚针轴承润滑特性影响研究.
- Author
-
宋琦, 王优强, 竺俊杰, 马金月, and 徐莹
- Abstract
Copyright of Lubrication Engineering (0254-0150) is the property of Editorial Office of LUBRICATION ENGINEERING and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) more...
- Published
- 2024
- Full Text
- View/download PDF
7. Heterogeneous Parallel Implementation of a Multigrid Method with Full Approximation in the Noisette Code.
- Author
-
Gorobets, A. V., Soukov, S. A., and Magomedov, A. R.
- Abstract
This article studies accelerating simulations of compressible flows on hybrid cluster systems using the full approximation scheme multigrid (FAS MG) method. The basic numerical algorithm has the following features: for spatial discretization, unstructured mixed-element meshes and schemes with the definition of mesh functions at mesh nodes are used; for temporal discretization, a fully implicit scheme is used. The aim of this study is to achieve acceleration of stationary simulations on both central and graphics processors without significant losses in parallel efficiency. We describe an approach to construct mesh levels; a technique for improving the quality of the mesh representation of the geometry of the modeled objects; and parallel implementation within the framework of complex parallelization, combining the message passing interface (MPI) for a distributed-memory parallel model, OpenMP for a shared-memory model, and OpenCL for computing on GPUs of various architectures. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
- Full Text
- View/download PDF
8. THE EFFECT OF APPROXIMATE COARSEST-LEVEL SOLVES ON THE CONVERGENCE OF MULTIGRID V-CYCLE METHODS.
- Author
-
VACEK, PETR, CARSON, ERIN, and SOODHALTER, KIRK M.
- Subjects
- *
LINEAR equations , *LINEAR systems , *USER experience , *PROBLEM solving , *KRYLOV subspace - Abstract
The multigrid V-cycle method is a popular method for solving systems of linear equations. It computes an approximate solution by using smoothing on fine levels and solving a system of linear equations on the coarsest level. Solving on the coarsest level depends on the size and difficulty of the problem. If the size permits, it is typical to use a direct method based on LU or Cholesky decomposition. In settings with large coarsest-level problems, approximate solvers such as iterative Krylov subspace methods, or direct methods based on low-rank approximation, are often used. The accuracy of the coarsest-level solver is typically determined based on the experience of the users with the concrete problems and methods. In this paper, we present an approach to analyzing the effects of approximate coarsest-level solves on the convergence of the V-cycle method for symmetric positive definite problems. Using these results, we derive coarsest-level stopping criterion through which we may control the difference between the approximation computed by a V-cycle method with approximate coarsest-level solver and the approximation which would be computed if the coarsestlevel problems were solved exactly. The coarsest-level stopping criterion may thus be set up such that the V-cycle method converges to a chosen finest-level accuracy in (nearly) the same number of V-cycle iterations as the V-cycle method with exact coarsest-level solver. We also utilize the theoretical results to discuss how the convergence of the V-cycle method may be affected by the choice of a tolerance in a coarsest-level stopping criterion based on the relative residual norm. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
- Full Text
- View/download PDF
9. 基于弹流润滑的滚子轴承参数研究.
- Author
-
黄嘉铭 and 顾春兴
- Abstract
Copyright of Lubrication Engineering (0254-0150) is the property of Editorial Office of LUBRICATION ENGINEERING and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) more...
- Published
- 2024
- Full Text
- View/download PDF
10. hp-Multigrid Preconditioner for a Divergence-Conforming HDG Scheme for the Incompressible Flow Problems.
- Author
-
Fu, Guosheng and Kuang, Wenzheng
- Abstract
In this study, we present an hp-multigrid preconditioner for a divergence-conforming HDG scheme for the generalized Stokes and the Navier–Stokes equations using an augmented Lagrangian formulation. Our method relies on conforming simplicial meshes in two- and three-dimensions. The hp-multigrid algorithm is a multiplicative auxiliary space preconditioner that employs the lowest-order space as the auxiliary space, and we develop a geometric multigrid method as the auxiliary space solver. For the generalized Stokes problem, the crucial ingredient of the geometric multigrid method is the equivalence between the condensed lowest-order divergence-conforming HDG scheme and a Crouzeix–Raviart discretization with a pressure-robust treatment as introduced in Linke and Merdon (Comput Methods Appl Mech Engrg 311:304–326, 2022), which allows for the direct application of geometric multigrid theory on the Crouzeix–Raviart discretization. The numerical experiments demonstrate the robustness of the proposed hp-multigrid preconditioner with respect to mesh size and augmented Lagrangian parameter, with iteration counts insensitivity to polynomial order increase. Inspired by the works by Benzi and Olshanskii (SIAM J Sci Comput 28:2095–2113, 2006) and Farrell et al. (SIAM J Sci Comput 41:A3073–A3096, 2019), we further test the proposed preconditioner on the divergence-conforming HDG scheme for the Navier–Stokes equations. Numerical experiments show a mild increase in the iteration counts of the preconditioned GMRes solver with the rise in Reynolds number up to 10 3 . [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
- Full Text
- View/download PDF
11. Multilevel local defect-correction method for the non-selfadjoint Steklov eigenvalue problems.
- Author
-
Xu, Fei, Wang, Bingyi, and Xie, Manting
- Abstract
In this paper, we design a multilevel local defect-correction method to solve the non-selfadjoint Steklov eigenvalue problems. Since the computation work needed for solving the non-selfadjoint Steklov eigenvalue problems increases exponentially as the scale of the problems increase, the main idea of our algorithm is to avoid solving large-scale equations especially large-scale Steklov eigenvalue problems directly. Firstly, we transform the non-selfadjoint Steklov eigenvalue problem into some symmetric boundary value problems defined in a multilevel finite element space sequence, and some small-scale non-selfadjoint Steklov eigenvalue problems defined in a low-dimensional auxiliary subspace. Next, the local defect-correction method is used to solve the symmetric boundary value problems, then the difficulty of solving these symmetric boundary value problems is further reduced by decomposing these large-scale problems into a series of small-scale subproblems. Overall, our algorithm can obtain the optimal error estimates with linear computational complexity, and the conclusions are proved by strict theoretical analysis which are different from the developed conclusions for equations with the Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
- Full Text
- View/download PDF
12. An Extensible Approach to Organizing Parallel Computations in the Software Package for the LS-STAG Simulation in Coupled Aerohydroelastic Problems
- Author
-
Marchevsky, Ilia, Puzikova, Valeria, Ghosh, Ashish, Editorial Board Member, Zhou, Lizhu, Editorial Board Member, Sokolinsky, Leonid, editor, Zymbler, Mikhail, editor, Voevodin, Vladimir, editor, and Dongarra, Jack, editor more...
- Published
- 2024
- Full Text
- View/download PDF
13. Topology Optimization of Multi-material Structures Based on Improved BESO Algorithm
- Author
-
He, Gang, Zhou, Yang, Cao, Zhaoyuan, Yang, Hu, Ceccarelli, Marco, Series Editor, Corves, Burkhard, Advisory Editor, Glazunov, Victor, Advisory Editor, Hernández, Alfonso, Advisory Editor, Huang, Tian, Advisory Editor, Jauregui Correa, Juan Carlos, Advisory Editor, Takeda, Yukio, Advisory Editor, Agrawal, Sunil K., Advisory Editor, Tan, Jianrong, editor, Liu, Yu, editor, Huang, Hong-Zhong, editor, Yu, Jingjun, editor, and Wang, Zequn, editor more...
- Published
- 2024
- Full Text
- View/download PDF
14. Homogeneous multigrid method for hybridizable interior penalty method
- Author
-
Lu, Peipei and Wang, Juan
- Published
- 2024
- Full Text
- View/download PDF
15. Local and parallel multigrid method for semilinear Neumann problem with nonlinear boundary condition.
- Author
-
Xu, Fei, Wang, Bingyi, and Xie, Manting
- Subjects
- *
NEUMANN problem , *NONLINEAR equations , *SEMILINEAR elliptic equations , *BOUNDARY value problems , *MULTIGRID methods (Numerical analysis) , *COMPUTATIONAL complexity - Abstract
A novel local and parallel multigrid method is proposed in this study for solving the semilinear Neumann problem with nonlinear boundary condition. Instead of solving the semilinear Neumann problem directly in the fine finite element space, we transform it into a linear boundary value problem defined in each level of a multigrid sequence and a small-scale semilinear Neumann problem defined in a low-dimensional correction subspace. Furthermore, the linear boundary value problem can be efficiently solved using local and parallel methods. The proposed process derives an optimal error estimate with linear computational complexity. Additionally, compared with existing multigrid methods for semilinear Neumann problems that require bounded second order derivatives of nonlinear terms, ours only needs bounded first order derivatives. A rigorous theoretical analysis is proposed in this paper, which differs from the maturely developed theories for equations with Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
- Full Text
- View/download PDF
16. 基于GPU加速的等几何拓扑优化高效多重网格 求解方法.
- Author
-
杨峰, 罗世杰, 杨江鸿, and 王英俊
- Abstract
Copyright of China Mechanical Engineering is the property of Editorial Board of China Mechanical Engineering and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) more...
- Published
- 2024
- Full Text
- View/download PDF
17. A fast proximal gradient method and convergence analysis for dynamic mean field planning.
- Author
-
Yu, Jiajia, Lai, Rongjie, Li, Wuchen, and Osher, Stanley
- Subjects
- *
ELLIPTIC equations , *MEAN field theory , *CONJUGATE gradient methods , *ALGORITHMS - Abstract
In this paper, we propose an efficient and flexible algorithm to solve dynamic mean-field planning problems based on an accelerated proximal gradient method. Besides an easy-to-implement gradient descent step in this algorithm, a crucial projection step becomes solving an elliptic equation whose solution can be obtained by conventional methods efficiently. By induction on iterations used in the algorithm, we theoretically show that the proposed discrete solution converges to the underlying continuous solution as the grid becomes finer. Furthermore, we generalize our algorithm to mean-field game problems and accelerate it using multilevel and multigrid strategies. We conduct comprehensive numerical experiments to confirm the convergence analysis of the proposed algorithm, to show its efficiency and mass preservation property by comparing it with state-of-the-art methods, and to illustrate its flexibility for handling various mean-field variational problems. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
- Full Text
- View/download PDF
18. MULTIGRID METHODS FOR NON COERCIVE VARIATIONAL INEQUALITIES.
- Author
-
NESBA, Nour El Houda and BEGGAS, Mohammed
- Subjects
- *
SUBDIFFERENTIALS , *MULTIGRID methods (Numerical analysis) , *ALGORITHMS - Abstract
In this study, our examination centers around the numerical resolution of non-coercive issues using a multi-grid approach. Our particular emphasis is directed towards employing multi-grid methodologies to tackle non-linear variational inequalities. Our primary goal involves confirming the consistent convergence of the multi-grid algorithm. To attain this objective, we make use of fundamental sub-differential calculus and glean insights from the convergence principles of non-linear multi-grid techniques. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
- Full Text
- View/download PDF
19. Kinetics of suspended particles with different shapes interacting in a fluid channel
- Author
-
Shahid, Muhammad and Usman, Kamran
- Published
- 2024
- Full Text
- View/download PDF
20. Compact Approximation of a Two-Dimensional Boundary Value Problem for Elliptic Equations of the Second Order with a Discontinuous Coefficient.
- Author
-
Gordin, V. A. and Shadrin, D. A.
- Abstract
For an elliptic equation of the second order with variable discontinuous coefficients and the right side, a scheme of the fourth order of accuracy is constructed. On the jump line, the docking conditions (Kirchhoff) are assumed to be satisfied. The use of Richardson's extrapolation, as the numerical experiments show, increases the order of accuracy to about the sixth order. It is shown that relaxation methods, including multigrid methods, are applicable to solve such systems of linear algebraic equations (SLAEs) corresponding to a compact finite-difference approximation of the problem. In comparison with the classical approximation, the accuracy increases by a factor of about 100 with the same complexity. Various variants of the equation and boundary conditions are considered, as well as the problem of determining the eigenvalues and functions for a piecewise constant coefficient of the equation. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
- View/download PDF
21. A scalable physically consistent particle method for high-viscous incompressible flows
- Author
-
Kondo, Masahiro, Matsumoto, Junichi, and Sawada, Tomohiro
- Published
- 2024
- Full Text
- View/download PDF
22. Tuning Soft Mutations of the Evolution Algorithm for Optimizing the Linear Solver Parameters.
- Author
-
Petrushov, A. A. and Krasnopolsky, B. I.
- Abstract
Automating the process of tuning the methods for solving systems of linear algebraic equations is an actual issue that can simplify the process of performing the calculations and improve their performance. This paper discusses ways of improving the hybrid evolution strategy proposed earlier by the authors to deal with this problem. Several aspects of evolution strategy mutation operators, including the locking of soft mutations for some parameters, optimizing the continuous parameters, adjusting the parameter search space, and tuning the parameter change stepping, are evaluated. The efficiency of the proposed modifications is investigated in detail, and the potential performance improvements in terms of the linear system optimal solution times and the optimization algorithm calculation times are outlined. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
- View/download PDF
23. Multigrid Methods for The Solution of Nonlinear Variational Inequalities.
- Author
-
El Houda, Nesba Nour, Mohammed, Beggas, Essaid, Belouafi Mohammed, Ahmad, Imtiaz, Ahmad, Hijaz, and Askar, Sameh
- Subjects
- *
SUBDIFFERENTIALS , *NONLINEAR theories , *MULTIGRID methods (Numerical analysis) - Abstract
In this research, we investigate the numerical solution of second member problems that depend on the solution obtained through a multigrid method. Specifically, we focus on the application of multigrid techniques for solving nonlinear variational inequalities. The main objective is to establish the uniform convergence of the multigrid algorithm. To achieve this, we employ elementary subdifferential calculus and draw insights from the convergence theory of nonlinear multigrid methods. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
- View/download PDF
24. Combination of Multigrid with Constraint Data for Inverse Problem of Nonlinear Diffusion Equation.
- Author
-
Liu, Tao, Ouyang, Di, Guo, Lianjun, Qiu, Ruofeng, Qi, Yunfei, Xie, Wu, Ma, Qiang, and Liu, Chao
- Subjects
- *
BURGERS' equation , *NONLINEAR equations , *TIKHONOV regularization , *POROUS materials - Abstract
This paper delves into a rapid and accurate numerical solution for the inverse problem of the nonlinear diffusion equation in the context of multiphase porous media flow. For the realization of this, the combination of the multigrid method with constraint data is utilized and investigated. Additionally, to address the ill-posedness of the inverse problem, the Tikhonov regularization is incorporated. Numerical results demonstrate the computational performance of this method. The proposed combination strategy displays remarkable capabilities in reducing noise, avoiding local minima, and accelerating convergence. Moreover, this combination method performs better than any one method used alone. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
- View/download PDF
25. An explicit cavitation multigrid algorithm for accelerating simulation of surface-textured bearings.
- Author
-
Xu, Wanjun, Zhao, Shanhui, Zhang, Mingjie, and Yang, Jiangang
- Abstract
The cavitation has a potential effect on the performance of surface-textured bearings. The traditional mass-conserving cavitation algorithms based on the Gauss-Seidel iteration method become slow as the grid increases. The computation time is unacceptable once the grid consists of a large array of textures. This study developed an explicit cavitation multigrid algorithm to accelerate the simulation of surface-textured bearings. The algorithm was based on the Ausas cavitation algorithm with the advantage of a matrix-free implementation. Novel numerical technologies were developed to overcome the difficulty of the floating free boundaries caused by the cavitation characteristics. An example of surface-dimpled thrust bearings was studied to analyze the algorithm efficiency. The algorithm is shown to be approximate optimal under reasonable solution parameters. It is an alternative to other cavitation algorithms. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
- View/download PDF
26. An Effective Multigrid Method for Solving Problems of High-Frequency Vibrational Convection.
- Author
-
Fedyushkin, A. I., Ivanov, K. A., and Puntus, A. A.
- Abstract
The paper describes an implemented algorithm for solving the problem of vibrational convection in a rectangular domain filled with an unevenly heated incompressible fluid. The mathematical model is based on the solution of the Simonenko–Zenkovskaya equations obtained by averaging the Navier-Stokes equations under the assumption that the volume of liquid executes high-frequency translational vibrations. To solve the Poisson equations, an algebraic multigrid method is implemented in combination with a highly efficient dynamic programming method (based on R. Bellman's optimal control principle) and fast Fourier transform. Mathematical software written in C/C++ has been developed. Examples of solving model problems with various directions of the heating flow of a square domain relative to the vibration vector are given. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
- View/download PDF
27. Smoothing analysis of two-color distributive relaxation for solving 2D Stokes flow by multigrid method
- Author
-
Xingwen Zhu and Lixiang Zhang
- Subjects
Smoothing analysis ,Local Fourier analysis ,Distributive relaxation ,Two-color relaxation ,Multigrid method ,Mathematics ,QA1-939 - Abstract
Smoothing properties of two-color distributive relaxation for solving a two-dimensional (2D) Stokes flow by multigrid method are theoretically investigated by using the local Fourier analysis (LFA) method. The governing equation of the 2D Stokes flow in consideration is discretized with the non-staggered grid and an added pressure stabilization term with stabilized parameters to be determined is introduced into the discretization system in order to enhance the smoothing effectiveness in the analysis. So, an important problem caused by the added pressure stabilization term is how to determine a suitable zone of parameters in the added term. To that end, theoretically, a two-color distributive relaxation, developed on the two-color Jacobi point relaxation, is established for the 2D Stokes flow. Firstly, a mathematical constitution based on the Fourier modes with various frequency components is constructed as a base of the two-color smoothing analysis, in which the related Fourier representation is presented by the form of two-color Jacobi point relaxation. Then, an optimal one-stage relaxation parameter and related smoothing factor for the two-color distributive relaxation are applied to the discretization system, and an analytical expression of the parameter zone on the added pressure stabilization term is established by LFA. The obtained analytical results show that numerical schemes for solving 2D Stokes flow by multigrid method on the two-color distributive relaxation have a specific convergence zone on the parameters of the added pressure stabilization term, and the property of convergence is independent of mesh size, but depends on the parameters of the pressure stabilization term. more...
- Published
- 2023
- Full Text
- View/download PDF
28. MULTILEVEL SPECTRAL DOMAIN DECOMPOSITION.
- Author
-
BASTIAN, PETER, SCHEICHL, ROBERT, SEELINGER, LINUS, and STREHLOW, ARNE
- Subjects
- *
POSITIVE systems , *GALERKIN methods , *LINEAR systems , *FINITE element method , *PROBLEM solving , *SCHWARZ function - Abstract
Highly heterogeneous, anisotropic coefficients, e.g., in the simulation of carbon-Fiber composite components, can lead to extremely challenging finite element systems. Direct solvers for the resulting large and sparse linear systems suffer from severe memory requirements and limited parallel scalability, while iterative solvers in general lack robustness. Two-level spectral domain decomposition methods can provide such robustness for symmetric positive definite linear systems by using coarse spaces based on independent generalized eigenproblems in the subdomains. Rigorous condition number bounds are independent of mesh size, number of subdomains, and coefficient contrast. However, their parallel scalability is still limited by the fact that (in order to guarantee robustness) the coarse problem is solved via a direct method. In this paper, we introduce a multilevel variant in the context of subspace correction methods and provide a general convergence theory for its robust convergence for abstract, elliptic variational problems. Assumptions of the theory are verified for conforming as well as for discontinuous Galerkin methods applied to a scalar diffusion problem. Numerical results illustrate the performance of the method for two- and three-dimensional problems and for various discretization schemes, in the context of scalar diffusion and linear elasticity. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
- View/download PDF
29. Multigrid reduced-order topology optimization scheme for structures subjected to stationary random excitations.
- Author
-
Wang, Bo, You, Haodong, Ma, Xiangtao, Shi, Yunfeng, Hao, Peng, and Zhang, Jiaxiong
- Abstract
Topology optimization methods for structures subjected to random excitations are difficult to widely apply in aeronautic and aerospace engineering, primarily due to the high computational cost of frequency response analysis for large-scale systems. Conventional methods are either unsuitable or inefficient for large-scale engineering structures, especially for structures consisting of multi-materials with non-proportional damping systems. To address this challenge, an accurate and highly efficient reduced-order method (ROM) based on the second-order Krylov subspace and the multigrid method is proposed in this paper, which is applicable to non-proportional damping systems. Moreover, a novel multigrid reduced-order topology optimization scheme for structures subjected to stationary random excitations is proposed based on the pseudo-excitation method (PEM). Two 3D numerical examples demonstrate the accuracy and efficiency of the proposed scheme for multi-material topology optimization. For a cantilever beam with about 6.7 × 10 5 degrees of freedom (DOFs), compared against the original reduced-order method, the efficiency of pseudo-harmonic analysis of the multigrid reduced-order method is improved by about 91% with sufficient accuracy, and the efficiency of the whole optimization process of the multigrid reduced-order method is improved by more than 71%. For a pedestal structure with about 3.5 × 10 5 DOFs, compared against the original reduced-order method, the efficiency of pseudo-harmonic analysis of the multigrid reduced-order method is improved by about 61%. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
- View/download PDF
30. A simplified new multigrid algorithm of lattice Boltzmann method for steady states.
- Author
-
An, Bo, Bergadà, J.M., and Sang, W.M.
- Subjects
- *
LATTICE Boltzmann methods , *MULTIGRID methods (Numerical analysis) , *ALGORITHMS - Abstract
In the present paper, a new strategy of multigrid method is introduced to accelerate the convergence speed of numerical simulations via lattice Boltzmann method. Based on the popular V-cycle multigrid algorithm, a simplified multigrid algorithm is presented and validated through the simulations of the classic lid-driven cavity flow for steady states. The novelty of the present algorithm resides in the construction of the information transferring process, in which, for a full cycle, the numerical simulation starts on the coarse mesh, transferring to medium mesh and then streams to the fine mesh through a prolongation operator. Afterwards, instead of using a restriction operator, the fluid information jumps back directly from the fine mesh to the coarse mesh via an assignment operator. The accuracy and efficiency of the simplified new algorithm are validated by comparing the results obtained when employing the classic V-cycle multigrid algorithm and the traditional lattice Boltzmann method with uniform Cartesian grid. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
- View/download PDF
31. Multigrid Methods for Hellan–Herrmann–Johnson Mixed Method of Kirchhoff Plate Bending Problems
- Author
-
Chen, Long, Hu, Jun, and Huang, Xuehai
- Subjects
Kirchhoff plate ,Hellan-Herrmann-Johnson mixed method ,Multigrid method ,Exact sequence ,Stable decomposition ,math.NA ,65N30 ,65N55 (Primary) ,74K20 ,Applied Mathematics ,Numerical and Computational Mathematics ,Computation Theory and Mathematics - Abstract
A V-cycle multigrid method for the Hellan–Herrmann–Johnson (HHJ) discretization of the Kirchhoff plate bending problems is developed in this paper. It is shown that the contraction number of the V-cycle multigrid HHJ mixed method is bounded away from one uniformly with respect to the mesh size. The uniform convergence is achieved for the V-cycle multigrid method with only one smoothing step and without full elliptic regularity assumption. The key is a stable decomposition of the kernel space which is derived from an exact sequence of the HHJ mixed method, and the strengthened Cauchy Schwarz inequality. Some numerical experiments are provided to confirm the proposed V-cycle multigrid method. The exact sequences of the HHJ mixed method and the corresponding commutative diagram is of some interest independent of the current context. more...
- Published
- 2018
32. Multigrid Method for Nonlinear Eigenvalue Problems Based on Newton Iteration.
- Author
-
Xu, Fei, Xie, Manting, and Yue, Meiling
- Abstract
In this paper, a novel multigrid method based on Newton iteration is proposed to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue λ and eigenfunction u separately, we treat the eigenpair (λ , u) as one element in a product space R × H 0 1 (Ω) . Then in the presented multigrid method, only one discrete linear boundary value problem needs to be solved for each level of the multigrid sequence. Because we avoid solving large-scale nonlinear eigenvalue problems directly, the overall efficiency is significantly improved. The optimal error estimate and linear computational complexity can be derived simultaneously. In addition, we also provide an improved multigrid method coupled with a mixing scheme to further guarantee the convergence and stability of the iteration scheme. More importantly, we prove convergence for the residuals after each iteration step. For nonlinear eigenvalue problems, such theoretical analysis is missing from the existing literatures on the mixing iteration scheme. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
- View/download PDF
33. Local grid refinement in multigrid method for point contact problems of polycrystalline anisotropic material under dry and lubricated conditions.
- Author
-
Zhang, Binbin, Vlogman, Tristan G., Andric, Predrag, Bor, Ton C., and Venner, Cornelis H.
- Subjects
SURFACE roughness ,ROLLER bearings ,PROBLEM solving ,ELECTRON diffraction ,ELASTOHYDRODYNAMIC lubrication - Abstract
Predicting rolling bearing fatigue life requires knowledge of the three-dimensional (3D) stress fields in the roller and raceway near the lubricated contact. Owing to the increasingly severe operating conditions, the effect of localized features such as surface roughness, subsurface inclusions, and even the crystallographic structure of the material becomes important. Achieving such detail requires (locally) extremely dense gridding in simulations, which in 3D is a major challenge. Multigrid techniques have been demonstrated to be capable of solving such problems. In this study, multigrid techniques are shown to further increase the efficiency of the solution by exploiting local grid refinement while maintaining the simplicity of a uniform discretization. This is achieved by employing increasingly finer grids only locally, where the highest resolution is required. Results are presented for dry contact and elastohydrodynamically lubricated contact cases, circular as well as elliptic, with varying crystallographic structure, and with surface roughness. The results show that the developed algorithm is very well suited for detailed analysis, with also excellent prospects for computational diagnostics involving actual material crystallographic structure from electron backscatter diffraction measurements. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
34. High performance computing for the discontinuous Galerkin methods
- Author
-
Mukhamedov, Farukh, Maischak, M., and Shaw, S.
- Subjects
515 ,High order polynomial approximation FEM ,C1 basis functions ,Schwarz method ,Multigrid method ,Parallel computing with OpenMP and MPI - Abstract
Discontinuous Galerkin methods form a class of numerical methods to find a solution of partial differential equations by combining features of finite element and finite volume methods. Methods are defined using a weak form of a particular model problem, allowing for discontinuities in the discrete trial and test spaces. Using a discontinuous discrete space mesh provides proper flexibility and a compact discretisation pattern, allowing a multidomain and multiphysics simulation. Discontinuous Galerkin methods with a higher approximation polynomial order, the socalled p-version, performs better in terms of convergence rate, compared with the low order h-version with smaller element sizes and bigger mesh. However, the condition number of the Galerkin system grows subsequently. This causes surge in the amount of required storage, computational complexity and in the time required for computation. We use the following three approaches to keep the advantages and eliminate the disadvantages. The first approach will be a specific choice of basis functions which we call C1 polynomials. These ensure that the majority of integrals over the edge of the mesh elements disappears. This reduces the total number of non-zero elements in the resulting system. This decreases the computational complexity without loss in precision. This approach does not affect the number of iterations required by chosen Conjugate Gradients method when compared to the other choice of basis functions. It actually decreases the total number of algebraic operations performed. The second approach is the introduction of suitable preconditioners. In our case, the Additive two-layer Schwarz method, developed in [4], for the iterative Conjugate Gradients method is considered. This directly affects the spectral condition number of the system matrix and decreases the number of iterations required for the computation. This approach, however, increases the total number of algebraic operations and might require more operational time. To tackle the rise in the number of algebraic operations, we introduced a modified Additive two-layer non-overlapping Schwarz method with a Multigrid process. This using a fixed low-order approximation polynomial degree on a coarse grid. We show that this approach is spectrally equivalent to the first preconditioner, and requires less time for computation. The third approach is a development of an efficient mathematical framework for distributed data structure. This allows a high performance, massively parallel, implementation of the discontinuous Galerkin method. We demonstrate that it is possible to exploit properties of the system matrix and C1 polynomials as basis functions to optimize the parallel structures. The previously mentioned parallel data structure allows us to parallelize at the same time both the matrix-vector multiplication routines for the Conjugate Gradients method, as well as the preconditioner routines on the solver level. This minimizes the transfer ratio amongst the distributed system. Finally, we combined all three approaches and created a framework, which allowed us to successfully implement all of the above. more...
- Published
- 2018
35. An efficient multigrid method with preconditioned smoother for two-dimensional anisotropic space-fractional diffusion equations.
- Author
-
Xu, Yuan, Lei, Siu-Long, and Sun, Hai-Wei
- Subjects
- *
LINEAR systems , *ANISOTROPY - Abstract
The anisotropic space-fractional diffusion equations in two dimensions are discretized by the Crank-Nicolson difference scheme with the weighted and shifted Grünwald formula, which is unconditionally stable and second-order convergence. The coefficient matrix of the discretized linear system possesses a two-level Toeplitz-like structure. Due to the anisotropy, the standard multigrid method converges slowly. By utilizing the GMRES method with a newly proposed tridiagonal preconditioner as a smoother, the convergence rate of the multigrid method can be accelerated significantly. The proposed tridiagonal preconditioner is shown to be invertible and a numerical experiment is given to demonstrate the efficiency of the proposed multigrid method with preconditioned smoother. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
36. An efficient multigrid method for semilinear interface problems.
- Author
-
Xu, Fei, Guo, Yasai, Huang, Qiumei, and Ma, Hongkun
- Subjects
- *
MULTIGRID methods (Numerical analysis) , *SEQUENCE spaces , *FINITE element method - Abstract
In this paper, we study an efficient multigrid method to solve the semilinear interface problems. We first give an optimal finite element error estimate for the semilinear interface problems under a weak condition for the nonlinear term compared with the existing conclusions. Then next based on the finite element error estimate, we design a novel multigrid method for semilinear elliptic problems. The proposed multigrid method only requires to solve a linear interface problem in each level of the multilevel space sequence and a small-scale semilinear interface problem in a correction space. The involved linear interface problem can be solved efficiently by the multigrid iteration. The dimension of the correction space is small and fixed, which is independent from the fine spaces. Thus the computational time of the correction step is negligible compared with that of the linear interface problems in the fine spaces. On the whole, the efficiency of the presented multigrid method is nearly the same as that of the multigrid method for linear interface problems. Additionally, unlike the existing finite element error estimates and the multigrid methods for semilinear interface problems, which always require the bounded second order derivatives of the nonlinear terms, all the analysis in our paper only requires a Lipschitz continuous condition. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
37. Multigrid Methods for a Mixed Finite Element Method of the Darcy–Forchheimer Model
- Author
-
Huang, Jian, Chen, Long, and Rui, Hongxing
- Subjects
Applied Mathematics ,Mathematical Sciences ,Darcy-Forchheimer model ,Multigrid method ,Peaceman-Rachford iteration ,Numerical and Computational Mathematics ,Computation Theory and Mathematics ,Applied mathematics ,Numerical and computational mathematics - Abstract
An efficient nonlinear multigrid method for a mixed finite element method of the Darcy-Forchheimer model is constructed in this paper. A Peaceman-Rachford type iteration is used as a smoother to decouple the nonlinearity from the divergence constraint. The nonlinear equation can be solved element-wise with a closed formulae. The linear saddle point system for the constraint is reduced into a symmetric positive definite system of Poisson type. Furthermore an empirical choice of the parameter used in the splitting is proposed and the resulting multigrid method is robust to the so-called Forchheimer number which controls the strength of the nonlinearity. By comparing the number of iterations and CPU time of different solvers in several numerical experiments, our multigrid method is shown to convergent with a rate independent of the mesh size and the Forchheimer number and with a nearly linear computational cost. more...
- Published
- 2018
38. Numerical approach to interpret the attributes of porous journal bearings using couple-stress fluid
- Author
-
Patil, Shalini M., Vinay, C.V., and P.A., Dinesh
- Published
- 2021
- Full Text
- View/download PDF
39. Shock-Capturing Exponential Multigrid Methods for Steady Compressible Flows.
- Author
-
Li, Shu-Jie
- Subjects
- *
TIME integration scheme , *RUNGE-Kutta formulas , *SHOCK waves , *COMPRESSIBLE flow - Abstract
In this paper, a robust and efficient exponential multigrid framework is proposed for computing steady compressible flows. The algorithm based on a global coupling, exponential time integration scheme can provide strong damping effects to accelerate the convergence towards the steady state, while high-frequency, high-order spatial error modes are smoothed out with a -stage preconditioned Runge–Kutta method. The resultant exponential multigrid framework is shown to be effective for smooth flows and can stabilize shock-capturing computations without limiting or adding artificial dissipation for medium-strength shock waves. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
40. Multigrid Method for Solving Inverse Problems for Heat Equation.
- Author
-
Al-Mahdawi, Hassan K. Ibrahim, Abotaleb, Mostafa, Alkattan, Hussein, Tareq, Al-Mahdawi Zena, Badr, Amr, and Kadi, Ammar
- Subjects
- *
INVERSE problems , *ALGEBRAIC equations , *INITIAL value problems , *MULTIGRID methods (Numerical analysis) , *INTEGRAL equations , *BOUNDARY value problems , *HEAT equation - Abstract
In this paper, the inverse problems for the boundary value and initial value in a heat equation are posed and solved. It is well known that those problems are ill posed. The problems are reformulated as integral equations of the first kind by using the separation-of-variables method. The discretization of the integral equation allowed us to reduce the integral equation to a system of linear algebraic equations or a linear operator equation of the first kind on Hilbert spaces. The Landweber-type iterative method was used in order to find an approximation solution. The V-cycle multigrid method is used to obtain more frequent and fast convergence for iteration. The numerical computation examples are presented to verify the accuracy and fast computing of the approximation solution. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
41. Calculation method for comprehensive damping of ball bearings based on multigrid method
- Author
-
Sun, Peng, Chen, Weifang, Shen, Yusu, and Wang, Dan
- Published
- 2020
- Full Text
- View/download PDF
42. Improved Numerical-Analytical Thermal Modeling Method of the PCB With Considering Radiation Heat Transfer and Calculation of Components’ Temperature
- Author
-
Yabin Zhang
- Subjects
Thermal modeling of the PCB ,radiation heat transfer ,Fourier-series analytical solution ,finite volume method ,multigrid method ,iterative method ,Electrical engineering. Electronics. Nuclear engineering ,TK1-9971 - Abstract
The previous work relating to the numerical-analytical coupling method for steady-state thermal analysis of the laminated PCB structure is first briefly reviewed. The Fourier-series analytical solution of temperature and the finite volume method were linked together for thermally modeling the PCB. For further modeling the PCB with components, thermal-resistance parameters of the components are then used for correlating components’ temperatures with the variable arrays in the coupling equations. For further considering radiation heat transfer between the PCB and the ambient, an iterative method is proposed. The radiation-equivalent heat transfer coefficient for each surface cell and each component can be updated during the iteration. Moreover, for improving the efficiency, the multigrid strategy is integrated in the coupling method for generating discrete cells of three levels in the metal layer and PCB surface region. To testify the iterative method, the model of a simple one-layer structure is compared with that built in COMSOL Multiphysics. The modeling results of the PCB of a phantom DC-DC power supply under radiation heat transfer are also given and discussed, and the modeling accuracy is approximately derived based on Richardson’s extrapolation. more...
- Published
- 2021
- Full Text
- View/download PDF
43. Numerical solution of thermal EHL line contact with bio-based oil as lubricant.
- Author
-
Awati, Vishwanath B., Kumar N, Mahesh, and Bujurke, N.M.
- Subjects
- *
ELASTOHYDRODYNAMIC lubrication , *REYNOLDS equations , *FINITE differences , *PETROLEUM , *TEMPERATURE distribution , *LUBRICATION & lubricants - Abstract
The paper presents the numerical solution of line contact thermal elastohydrodynamic lubrication (EHL) with bio-based lubricant. The model comprises Reynolds equation, film thickness, load balance and energy equations with appropriate boundary conditions by incorporating viscosity–pressure–temperature and density–pressure–temperature relations. Second-order finite difference scheme is used for the discretised form and their equation. The multigrid method with full approximation scheme is used to solve the Reynolds equation along with multilevel multi-integration method for film thickness equation. The pressure, film thickness and temperature distributions for two rolling velocities and various loads with a bio-based lubricant are presented in detail. The present findings yield a reduction in the minimum film thickness for high speed. Details of pressure spike as a function of relevant parameters are given. The results are compared with earlier findings based on different methods. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
44. A Multigrid Multilevel Monte Carlo Method for Stokes–Darcy Model with Random Hydraulic Conductivity and Beavers–Joseph Condition.
- Author
-
Yang, Zhipeng, Ming, Ju, Qiu, Changxin, Li, Maojun, and He, Xiaoming
- Abstract
A multigrid multilevel Monte Carlo (MGMLMC) method is developed for the stochastic Stokes–Darcy interface model with random hydraulic conductivity both in the porous media domain and on the interface. Three interface conditions with randomness are considered on the interface between Stokes and Darcy equations, especially the Beavers–Joesph interface condition with random hydraulic conductivity. Because the randomness through the interface affects the flow in the Stokes domain, we investigate the coupled stochastic Stokes–Darcy model to improve the fidelity. Under suitable assumptions on the random coefficient, we prove the existence and uniqueness of the weak solution of the variational form. To construct the numerical method, we first adopt the Monte Carlo (MC) method and finite element method, for the discretization in the probability space and physical space, respectively. In order to improve the efficiency of the classical single-level Monte Carlo (SLMC) method, we adopt the multilevel Monte Carlo (MLMC) method to dramatically reduce the computational cost in the probability space. A strategy is developed to calculate the number of samples needed in MLMC method for the stochastic Stokes–Darcy model. In order to accomplish the strategy for MLMC method, we also present a practical method to determine the variance convergence rate for the stochastic Stokes–Darcy model with Beavers–Joseph interface condition. Furthermore, MLMC method naturally provides the hierarchical grids and sufficient information on these grids for multigrid (MG) method, which can in turn improve the efficiency of MLMC method. In order to fully make use of the dynamical interaction between this two methods, we propose a multigrid multilevel Monte Carlo (MGMLMC) method with finite element discretization for more efficiently solving the stochastic model, while additional attention is paid to the interface and the random Beavers–Joesph interface condition. The computational cost of the proposed MGMLMC method is rigorously analyzed and compared with the SLMC method. Numerical examples are provided to verify and illustrate the proposed method and the theoretical conclusions. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
45. Independence of placement for local Fourier analysis.
- Subjects
- *
UNITARY transformations , *DISCRETIZATION methods , *DEGREES of freedom , *FOURIER analysis , *MULTIGRID methods (Numerical analysis) , *GENERALIZATION - Abstract
Local Fourier analysis (LFA) serves a prominent role in the prediction of the convergence factor of multigrid methods for discretizations of PDEs. However, few discussions about the implementation of LFA for complicated discretizations of PDEs exist, such as higher order finite elements, as staggered meshes could lead to complex LFA representations of the grid‐transfer operator. In this work, we prove that the LFA representation for d‐dimensional PDEs is independent of the placement of degrees of freedom (DoFs). Intuitively speaking, the seeding of the unknowns has no direct effect on the LFA presentation, instead, it serves as a unitary transformation between different representations. Thus, different LFA representations for a given PDE have the same spectrum and norm. Furthermore, we provide a uniform representation in terms of the location of the unknowns, named simple representation, where we allocate all of the DoFs at nodes, resulting in a simple and unified way to compute the symbols of the discrete operators, especially for the grid‐transfer operators. This simple representation can contribute to the generalization of the implementation of LFA for different types of discretizations and different problems, especially for higher order discretization methods. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
- View/download PDF
46. A multigrid preconditioner for tensor product spline smoothing.
- Author
-
Siebenborn, Martin and Wagner, Julian
- Subjects
- *
TENSOR products , *CONJUGATE gradient methods , *SPLINES , *COMPUTER systems , *COMPUTATIONAL complexity - Abstract
Penalized spline smoothing is a well-established, nonparametric regression method that is efficient for one and two covariates. Its extension to more than two covariates is straightforward but suffers from exponentially increasing memory demands and computational complexity, which brings the method to its numerical limit. Penalized spline smoothing with multiple covariates requires solving a large-scale, regularized least-squares problem where the occurring matrices do not fit into storage of common computer systems. To overcome this restriction, we introduce a matrix-free implementation of the conjugate gradient method. We further present a matrix-free implementation of a simple diagonal as well as more advanced geometric multigrid preconditioner to significantly speed up convergence of the conjugate gradient method. All algorithms require a negligible amount of memory and therefore allow for penalized spline smoothing with multiple covariates. Moreover, for arbitrary but fixed covariate dimension, we show grid independent convergence of the multigrid preconditioner which is fundamental to achieve algorithmic scalability. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
- View/download PDF
47. A multigrid based finite difference method for solving parabolic interface problem.
- Author
-
Feng, Hongsong and Zhao, Shan
- Subjects
- *
FINITE differences , *DISCRETIZATION methods , *CRANK-nicolson method , *SCHUR complement , *NUMERICAL analysis - Abstract
In this paper, a new Cartesian grid finite difference method is introduced to solve two-dimensional parabolic interface problems with second order accuracy achieved in both temporal and spatial discretization. Corrected central difference and the Matched Interface and Boundary (MIB) method are adopted to restore second order spatial accuracy across the interface, while the standard Crank-Nicolson scheme is employed for the implicit time stepping. In the proposed augmented MIB (AMIB) method, an augmented system is formulated with auxiliary variables introduced so that the central difference discretization of the Laplacian could be disassociated with the interface corrections. A simple geometric multigrid method is constructed to efficiently invert the discrete Laplacian in the Schur complement solution of the augmented system. This leads a significant improvement in computational efficiency in comparing with the original MIB method. Being free of a stability constraint, the implicit AMIB method could be asymptotically faster than explicit schemes. Extensive numerical results are carried out to validate the accuracy, efficiency, and stability of the proposed AMIB method. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
- View/download PDF
48. A-POSTERIORI-STEERED p-ROBUST MULTIGRID WITH OPTIMAL STEP-SIZES AND ADAPTIVE NUMBER OF SMOOTHING STEPS.
- Author
-
MIRAÇI, ANI, PAPEŽ, JAN, and VOHRALÍK, MARTIN
- Subjects
- *
POLYNOMIALS , *MULTIGRID methods (Numerical analysis) , *CONTRACTS - Abstract
We develop a multigrid solver steered by an a posteriori estimator of the algebraic error. We adopt the context of a second-order elliptic diffusion problem discretized by conforming finite elements of arbitrary polynomial degree p ≤ 1. Our solver employs zero pre- and one postsmoothing by the overlapping Schwarz (block-Jacobi) method and features an optimal choice of the step-sizes in the smoothing correction on each level by line search. This leads to a simple Pythagorean formula of the algebraic error in the next step in terms of the current error and levelwise and patchwise error reductions. We show the following two results and their equivalence: the solver contracts the algebraic error independently of the polynomial degree p; and the estimator represents a two-sided p-robust bound on the algebraic error. The p-robustness results are obtained by carefully applying the results of [J. Schöberl et al., IMA J. Numer. Anal., 28 (2008), pp. 1-24] for one mesh, combined with a multilevel stable decomposition for piecewise affine polynomials of [J. Xu, L. Chen, and R. H. Nochetto, Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids, in Multiscale, Nonlinear and Adaptive Approximation, Springer, Berlin, 2009, pp. 599-659]. We consider quasi-uniform or graded bisection simplicial meshes and prove at most linear dependence on the number of mesh levels for minimal H¹-regularity and complete independence for H²-regularity. We also present a simple and effective way for the solver to adaptively choose the number of postsmoothing steps necessary at each individual level, yielding a yet improved error reduction. Numerical tests confirm p-robustness and show the benefits of the adaptive number of smoothing steps. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
- View/download PDF
49. Transient elastohydrodynamic analysis of finite line contact under load impulse.
- Author
-
Chippa, SP and Borse, NV
- Abstract
Numerical analysis is performed to study the transient behavior of EHL finite line contact of a cylindrical roller and flat plane under load impulse. In the present work, effect of pressure on the density and viscosity of lubricant is considered. Finite difference method is used to discretize the governing equations. Multilevel Multi-integration method is used to calculate the elastic deformation. Moreover, Multigrid method is implemented to accelerate the convergence process. Uniqueness of this finite line contact analysis is that it provides an ability to determine the transient behavior of lubricated contact even at the edges of roller. Results show that the load impulse causes squeezing and separation movement within the contact that develops film dimple and pressure ripples at the inlet region, which propagate towards the exit region due to the entrainment motion. It is noticed that the time taken by oil film (t oil ) to travel the Hertzian contact width and the time period (t p) of load impulse decides the behavior of lubricated contacts. Firstly, under a relatively heavy load when the contact width is large enough so that t oil ≫ t p , then a significant rise in central film thickness (CFT), central minimum film thickness (CMFT) and minimum film thickness (MFT) occurs after the execution of load impulse. Further, under the light load generating a relatively small contact width such that t oil ≈ t p , then comparatively a small rise in CFT occurs right during the load impulse. Lastly, for a given load if the time period of impulse (t p) is large enough satisfying the condition t p ≫ t oil , then a considerable reduction in CFT, CMFT and MFT takes place during the application of load impulse. Moreover, as compare to other cases, for t oil ≫ t p the steady state condition is reestablished after a relatively more number of time cycles. It is observed that the maximum pressure and MFT occurs at the contact edges of roller which can be controlled by a proper choice of the radius of end profile (R y) . [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
- View/download PDF
50. An Optimal Multigrid Algorithm for the Combining P1-Q1 Finite Element Approximations of Interface Problems Based on Local Anisotropic Fitting Meshes.
- Author
-
Hu, Jun and Wang, Hua
- Abstract
A new finite element method is proposed for second order elliptic interface problems based on a local anisotropic fitting mixed mesh. The local anisotropic fitting mixed mesh is generated from an interface-unfitted mesh by simply connecting the intersected points of the interface and the underlying mesh successively. Optimal approximation capabilities on anisotropic elements are proved, the convergence rates are linear and quadratic in H 1 and L 2 norms, respectively. The discrete system is usually ill-conditioned due to anisotropic and small elements near the interface. Thereupon, a new multigrid method is presented to handle this issue. The convergence rate of the multigrid method is shown to be optimal with respect to both the coefficient jump ratio and mesh size. Numerical experiments are presented to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
- View/download PDF
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.