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Academic literature on the topic 'Adaptive multigrid algorithms'

Author: Grafiati

Published: 4 June 2021

Last updated: 7 February 2022

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Journal articles on the topic "Adaptive multigrid algorithms"

1

Bartsch, Guido, and Christian Wulf. "Adaptive Multigrid for Helmholtz Problems." Journal of Computational Acoustics 11, no. 03 (September 2003): 341–50. http://dx.doi.org/10.1142/s0218396x03001997.

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Solving Helmholtz problems for low frequency sound fields by a truncated modal basis approach is very efficient. The most time-consuming process is the calculation of the undamped modes. Using traditional FE solvers, the user has to provide a mesh which has at least six nodes per wavelength in each spatial direction to achieve acceptable results. Because the mesh size increases with the 3rd power of the highest frequency of interest, this uniform dense mesh approach is a very expensive way of creating a modal space. However, the number of modes and the accuracy of the modal basis directly influences the solution quality. It is well known that the representation of sound fields by modal basis functions φi is optimal with respect to the L2 error norm. This means that having a modal basis Φ := {φi, i = 1⋯n}, the distance between true and approximated sound field takes its minimum in the mean square. So, it is necessary to have a FE basis which also minimizes the discretization error when computing the modal basis. One can reach this goal by applying adaptive mesh refinements. Additionally, this yields the opportunity of using fast multigrid methods to solve discrete eigenvalue problems. In context of this presentation we will discuss the results of our adaptive multigrid algorithms.

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Carstensen, Carsten, and Jun Hu. "Hierarchical Argyris Finite Element Method for Adaptive and Multigrid Algorithms." Computational Methods in Applied Mathematics 21, no. 3 (June 1, 2021): 529–56. http://dx.doi.org/10.1515/cmam-2021-0083.

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Abstract The global arrangement of the degrees of freedom in a standard Argyris finite element method (FEM) enforces C 2 {C^{2}} at interior vertices, while solely global C 1 {C^{1}} continuity is required for the conformity in H 2 {H^{2}} . Since the Argyris finite element functions are not C 2 {C^{2}} at the midpoints of edges in general, the bisection of an edge for mesh-refinement leads to non-nestedness: the standard Argyris finite element space A ′ ⁢ ( 𝒯 ) {A^{\prime}(\mathcal{T})} associated to a triangulation 𝒯 {\mathcal{T}} with a refinement 𝒯 ^ {\widehat{\mathcal{T}}} is not a subspace of the standard Argyris finite element space A ′ ⁢ ( 𝒯 ^ ) {A^{\prime}(\widehat{\mathcal{T}})} associated to the refined triangulation 𝒯 ^ {\widehat{\mathcal{T}}} . This paper suggests an extension A ⁢ ( 𝒯 ) {A(\mathcal{T})} of A ′ ⁢ ( 𝒯 ) {A^{\prime}(\mathcal{T})} that allows for nestedness A ⁢ ( 𝒯 ) ⊂ A ⁢ ( 𝒯 ^ ) {A(\mathcal{T})\subset A(\widehat{\mathcal{T}})} under mesh-refinement. The extended Argyris finite element space A ⁢ ( 𝒯 ) {A(\mathcal{T})} is called hierarchical, but is still based on the concept of the Argyris finite element as a triple ( T , P 5 ⁢ ( T ) , ( Λ 1 , … , Λ 21 ) ) {(T,P_{5}(T),(\Lambda_{1},\dots,\Lambda_{21}))} in the sense of Ciarlet. The other main results of this paper are the optimal convergence rates of an adaptive mesh-refinement algorithm via the abstract framework of the axioms of adaptivity and uniform convergence of a local multigrid V-cycle algorithm for the effective solution of the discrete system.

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Stals, Linda. "Algorithm-based fault recovery of adaptively refined parallel multilevel grids." International Journal of High Performance Computing Applications 33, no. 1 (August 23, 2017): 189–211. http://dx.doi.org/10.1177/1094342017720801.

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On future extreme scale computers, it is expected that faults will become an increasingly serious problem as the number of individual components grows and failures become more frequent. This is driving the interest in designing algorithms with built-in fault tolerance that can continue to operate and that can replace data even if part of the computation is lost in a failure. For fault-free computations, the use of adaptive refinement techniques in combination with finite element methods is well established. Furthermore, iterative solution techniques that incorporate information about the grid structure, such as the parallel geometric multigrid method, have been shown to be an efficient approach to solving various types of partial different equations. In this article, we present an advanced parallel adaptive multigrid method that uses dynamic data structures to store a nested sequence of meshes and the iteratively evolving solution. After a fail-stop fault, the data residing on the faulty processor will be lost. However, with suitably designed data structures, the neighbouring processors contain enough information so that a consistent mesh can be reconstructed in the faulty domain with the goal of resuming the computation without having to restart from scratch. This recovery is based on a set of carefully designed distributed algorithms that build on the existing parallel adaptive refinement routines, but which must be carefully augmented and extended.

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Yang, Yidu, Yu Zhang, and Hai Bi. "Multigrid Discretization and Iterative Algorithm for Mixed Variational Formulation of the Eigenvalue Problem of Electric Field." Abstract and Applied Analysis 2012 (2012): 1–25. http://dx.doi.org/10.1155/2012/190768.

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This paper discusses highly finite element algorithms for the eigenvalue problem of electric field. Combining the mixed finite element method with the Rayleigh quotient iteration method, a new multi-grid discretization scheme and an adaptive algorithm are proposed and applied to the eigenvalue problem of electric field. Theoretical analysis and numerical results show that the computational schemes established in the paper have high efficiency.

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Bader, M., S. Schraufstetter, C. A. Vigh, and J. Behrens. "Memory efficient adaptive mesh generation and implementation of multigrid algorithms using Sierpinski curves." International Journal of Computational Science and Engineering 4, no. 1 (2008): 12. http://dx.doi.org/10.1504/ijcse.2008.021108.

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Yang, Feng Wei, Chandrasekhar Venkataraman, Vanessa Styles, and Anotida Madzvamuse. "A Robust and Efficient Adaptive Multigrid Solver for the Optimal Control of Phase Field Formulations of Geometric Evolution Laws." Communications in Computational Physics 21, no. 1 (December 5, 2016): 65–92. http://dx.doi.org/10.4208/cicp.240715.080716a.

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AbstractWe propose and investigate a novel solution strategy to efficiently and accurately compute approximate solutions to semilinear optimal control problems, focusing on the optimal control of phase field formulations of geometric evolution laws. The optimal control of geometric evolution laws arises in a number of applications in fields including material science, image processing, tumour growth and cell motility. Despite this, many open problems remain in the analysis and approximation of such problems. In the current work we focus on a phase field formulation of the optimal control problem, hence exploiting the well developed mathematical theory for the optimal control of semilinear parabolic partial differential equations. Approximation of the resulting optimal control problemis computationally challenging, requiring massive amounts of computational time and memory storage. The main focus of this work is to propose, derive, implement and test an efficient solution method for such problems. The solver for the discretised partial differential equations is based upon a geometric multigrid method incorporating advanced techniques to deal with the nonlinearities in the problem and utilising adaptive mesh refinement. An in-house two-grid solution strategy for the forward and adjoint problems, that significantly reduces memory requirements and CPU time, is proposed and investigated computationally. Furthermore, parallelisation as well as an adaptive-step gradient update for the control are employed to further improve efficiency. Along with a detailed description of our proposed solution method together with its implementation we present a number of computational results that demonstrate and evaluate our algorithms with respect to accuracy and efficiency. A highlight of the present work is simulation results on the optimal control of phase field formulations of geometric evolution laws in 3-D which would be computationally infeasible without the solution strategies proposed in the present work.

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Miraçi, Ani, Jan Papež, and Martin Vohralík. "Contractive Local Adaptive Smoothing Based on Dörfler’s Marking in A-Posteriori-Steered p-Robust Multigrid Solvers." Computational Methods in Applied Mathematics 21, no. 2 (February 5, 2021): 445–68. http://dx.doi.org/10.1515/cmam-2020-0024.

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Abstract In this work, we study a local adaptive smoothing algorithm for a-posteriori-steered p-robust multigrid methods. The solver tackles a linear system which is generated by the discretization of a second-order elliptic diffusion problem using conforming finite elements of polynomial order p ≥ 1 {p\geq 1} . After one V-cycle (“full-smoothing” substep) of the solver of [A. Miraçi, J. Papež, and M. Vohralík, A-posteriori-steered p-robust multigrid with optimal step-sizes and adaptive number of smoothing steps, SIAM J. Sci. Comput. 2021, 10.1137/20M1349503], we dispose of a reliable, efficient, and localized estimation of the algebraic error. We use this existing result to develop our new adaptive algorithm: thanks to the information of the estimator and based on a bulk-chasing criterion, cf. [W. Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 1996, 3, 1106–1124], we mark patches of elements with increased estimated error on all levels. Then, we proceed by a modified and cheaper V-cycle (“adaptive-smoothing” substep), which only applies smoothing in the marked regions. The proposed adaptive multigrid solver picks autonomously and adaptively the optimal step-size per level as in our previous work but also the type of smoothing per level (weighted restricted additive or additive Schwarz) and concentrates smoothing to marked regions with high error. We prove that, under a numerical condition that we verify in the algorithm, each substep (full and adaptive) contracts the error p-robustly, which is confirmed by numerical experiments. Moreover, the proposed algorithm behaves numerically robustly with respect to the number of levels as well as to the diffusion coefficient jump for a uniformly-refined hierarchy of meshes.

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Sun, J., and P. Monk. "An Adaptive Algebraic Multigrid Algorithm for Micromagnetism." IEEE Transactions on Magnetics 42, no. 6 (June 2006): 1643–47. http://dx.doi.org/10.1109/tmag.2006.872004.

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Ji, Hua, Fue-Sang Lien, and Eugene Yee. "Parallel Adaptive Mesh Refinement Combined with Additive Multigrid for the Efficient Solution of the Poisson Equation." ISRN Applied Mathematics 2012 (March 12, 2012): 1–24. http://dx.doi.org/10.5402/2012/246491.

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Three different speed-up methods (viz., additive multigrid method, adaptive mesh refinement (AMR), and parallelization) have been combined in order to provide a highly efficient parallel solver for the Poisson equation. Rather than using an ordinary tree data structure to organize the information on the adaptive Cartesian mesh, a modified form of the fully threaded tree (FTT) data structure is used. The Hilbert space-filling curve (SFC) approach has been adopted for dynamic grid partitioning (resulting in a partitioning that is near optimal with respect to load balancing on a parallel computational platform). Finally, an additive multigrid method (BPX preconditioner), which itself is parallelizable to a certain extent, has been used to solve the linear equation system arising from the discretization. Our numerical experiments show that the proposed parallel AMR algorithm based on the FTT data structure, Hilbert SFC for grid partitioning, and additive multigrid method is highly efficient.

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LOPEZ, S., and R. CASCIARO. "ALGORITHMIC ASPECTS OF ADAPTIVE MULTIGRID FINITE ELEMENT ANALYSIS." International Journal for Numerical Methods in Engineering 40, no. 5 (March 15, 1997): 919–36. http://dx.doi.org/10.1002/(sici)1097-0207(19970315)40:5<919::aid-nme95>3.0.co;2-u.

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Dissertations / Theses on the topic "Adaptive multigrid algorithms"

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Mayfield, Andrew James. "Adaptive mesh refinement." Thesis, University of Oxford, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.358687.

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Cakmak, Mehtap. "Development Of A Multigrid Accelerated Euler Solver On Adaptively Refined Two- And Three-dimensional Cartesian Grids." Master's thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/2/12610753/index.pdf.

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Cartesian grids offer a valuable option to simulate aerodynamic flows around complex geometries such as multi-element airfoils, aircrafts, and rockets. Therefore, an adaptively-refined Cartesian grid generator and Euler solver are developed. For the mesh generation part of the algorithm, dynamic data structures are used to determine connectivity information between cells and uniform mesh is created in the domain. Marching squares and cubes algorithms are used to form interfaces of cut and split cells. Geometry-based cell adaptation is applied in the mesh generation. After obtaining appropriate mesh around input geometry, the solution is obtained using either flux vector splitting method or Roe&rsquo
s approximate Riemann solver with cell-centered approach. Least squares reconstruction of flow variables within the cell is used to determine high gradient regions of flow. Solution based adaptation method is then applied to current mesh in order to refine these regions and also coarsened regions where unnecessary small cells exist. Multistage time stepping is used with local time steps to increase the convergence rate. Also FAS multigrid technique is used in order to increase the convergence rate. It is obvious that implementation of geometry and solution based adaptations are easier for Cartesian meshes than other types of meshes. Besides, presented numerical results show the accuracy and efficiency of the algorithm by especially using geometry and solution based adaptation. Finally, Euler solutions of Cartesian grids around airfoils, projectiles and wings are compared with the experimental and numerical data available in the literature and accuracy and efficiency of the solver are verified.

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3

Pathak, Harshavardhana Sunil. "Adaptive Mesh Redistribution for Hyperbolic Conservation Laws." Thesis, 2013. http://hdl.handle.net/2005/3281.

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An adaptive mesh redistribution method for efficient and accurate simulation of multi dimensional hyperbolic conservation laws is developed. The algorithm consists of two coupled steps; evolution of the governing PDE followed by a redistribution of the computational nodes. The second step, i.e. mesh redistribution is carried out at each time step iteratively with the primary aim of adapting the grid to the computed solution in order to maximize accuracy while minimizing the computational overheads. The governing hyperbolic conservation laws, originally defined on the physical domain, are transformed on to a simplified computational domain where the position of the nodes remains independent of time. The transformed governing hyperbolic equations are recast in a strong conservative form and are solved directly on the computational domain without the need for interpolation that is typically associated with standard mesh redistribution algorithms. Several standard test cases involving numerical solution of scalar and system of hyperbolic conservation laws in one and two dimensions are presented in order to demonstrate the accuracy and computational efficiency of the proposed technique.

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Books on the topic "Adaptive multigrid algorithms"

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Mavriplis, Dimitri J. Multigrid solution of the Euler equations on unstructured and adaptive meshes. Hampton, Va: ICASE, 1987.

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2

Stals, Linda. The solution of radiation transport equations with adaptive finite elements. Hampton, Va: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 2001.

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3

1948-, Hackbusch W., and Wittum Gabriel 1956-, eds. Adaptive methods--algorithms, theory and applications: Proceedings of the Ninth GAMM-Seminar, Kiel, January 22-24, 1993. Braunschweig: Vieweg, 1994.

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4

Lang, Jens. Adaptive multilevel solution of nonlinear parabolic PDE systems: Theory, algorithm, and applications. Berlin: Springer, 2001.

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5

The solution of radiation transport equations with adaptive finite elements. Hampton, Va: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 2001.

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6

K, Leaf G., Van Rosendale John R, and Institute for Computer Applications in Science and Engineering., eds. A dynamically adaptive multigrid algorithm for the incompressible Navier-Stokes equations: Validation and model problems. Hampton, Va: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1991.

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Book chapters on the topic "Adaptive multigrid algorithms"

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Botorog, George Horatiu, and Herbert Kuchen. "Algorithmic skeletons for adaptive multigrid methods." In Parallel Algorithms for Irregularly Structured Problems, 27–41. Berlin, Heidelberg: Springer Berlin Heidelberg, 1995. http://dx.doi.org/10.1007/3-540-60321-2_2.

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Douglas, Craig C., Jonathan Hu, Wolfgang Karl, Markus Kowarschik, Ulrich Rüde, and Christian Weiß. "Fixed and Adaptive Cache Aware Algorithms for Multigrid Methods." In Lecture Notes in Computational Science and Engineering, 87–93. Berlin, Heidelberg: Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/978-3-642-58312-4_11.

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3

Vasileva, Daniela. "On an Adaptive Semirefinement Multigrid Algorithm for Convection-Diffusion Problems." In Lecture Notes in Computer Science, 572–79. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-00464-3_67.

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Thompson, C. P. "A Parallel Adaptive Multigrid Algorithm for the Incompressible Navier-Stokes Equations." In Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters, 293–309. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-011-1810-1_19.

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Richert, Thomas. "Dynamic Load Balancing for Parallel Adaptive Multigrid Solvers with Algorithmic Skeletons." In Euro-Par 2000 Parallel Processing, 325–28. Berlin, Heidelberg: Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/3-540-44520-x_42.

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Li, Wei, and Yunqing Huang. "A Modified Adaptive Algebraic Multigrid Algorithm for Elliptic Obstacle Problems." In Series in Contemporary Applied Mathematics, 160–78. CO-PUBLISHED WITH HIGHER EDUCATION PRESS, 2006. http://dx.doi.org/10.1142/9789812774194_0008.

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Conference papers on the topic "Adaptive multigrid algorithms"

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Gilles, Luc, Brent L. Ellerbroek, and Curtis R. Vogel. "Layer-oriented multigrid wavefront reconstruction algorithms for multiconjugate adaptive optics." In Astronomical Telescopes and Instrumentation, edited by Peter L. Wizinowich and Domenico Bonaccini. SPIE, 2003. http://dx.doi.org/10.1117/12.459347.

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Irmisch, Stefan. "Adaptive Finite-Volume Solution of the Two-Dimensional Euler Equations on Unstructured Meshes." In ASME 1994 International Gas Turbine and Aeroengine Congress and Exposition. American Society of Mechanical Engineers, 1994. http://dx.doi.org/10.1115/94-gt-087.

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This paper presents a finite-volume method for solving the compressible, two-dimensional Euler equations using unstructured triangular meshes. The integration in time, to a steady-state solution, is performed using an explicit, multistage Runge-Kutta algorithm. A special treatment of the artificial viscosity along the boundaries reduces the production of numerical losses. Convergence acceleration is achieved by employing local time-stepping, implicit residual smoothing and a multigrid technique. The use of unstructured meshes, based on Delaunay triangulation, automatically adapted to the solution, allows arbitrary geometries and complex flow features to be treated easily. The employed refinement criterion does not only detect strong shocks, but also weak flow features. Solutions are presented for several subsonic and transonic standard test cases and cascade flows that illustrate the capability of the algorithm.

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Ebna Hai, Bhuiyan Shameem Mahmood, and Markus Bause. "Adaptive Multigrid Methods for Extended Fluid-Structure Interaction (eXFSI) Problem: Part I — Mathematical Modelling." In ASME 2015 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/imece2015-53265.

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This contribution is the first part of three papers on Adaptive Multigrid Methods for eXtended Fluid-Structure Interaction (eXFSI) Problem, where we introduce a monolithic variational formulation and solution techniques. In a monolithic nonlinear fluid-structure interaction (FSI), the fluid and structure models are formulated in different coordinate systems. This makes the FSI setup of a common variational description difficult and challenging. This article presents the state-of-the-art of recent developments in the finite element approximation of FSI problem based on monolithic variational formulation in the well-established arbitrary Lagrangian Eulerian (ALE) framework. This research will focus on the newly developed mathematical model of a new FSI problem which is called eXtended Fluid-Structure Interaction (eXFSI) problem in ALE framework. This model is used to design an on-live Structural Health Monitoring (SHM) system in order to determine the wave propagation in moving domains and optimum locations for SHM sensors. eXFSI is strongly coupled problem of typical FSI with a wave propagation problem on the fluid-structure interface, where wave propagation problems automatically adopted the boundary conditions from of the typical FSI problem at each time step. The ALE approach provides a simple, but powerful procedure to couple fluid flows with solid deformations by a monolithic solution algorithm. In such a setting, the fluid equations are transformed to a fixed reference configuration via the ALE mapping. The goal of this work is the development of concepts for the efficient numerical solution of eXFSI problem, the analysis of various fluid-mesh motion techniques and comparison of different second-order time-stepping schemes. This work consists of the investigation of different time stepping scheme formulations for a nonlinear FSI problem coupling the acoustic/elastic wave propagation on the fluid-structure interface. Temporal discretization is based on finite differences and is formulated as an one step-θ scheme; from which we can consider the following particular cases: the implicit Euler, Crank-Nicolson, shifted Crank-Nicolson and the Fractional-Step-θ schemes. The nonlinear problem is solved with Newton’s method whereas the spatial discretization is done with a Galerkin finite element scheme. To control computational costs we apply a simplified version of a posteriori error estimation using the dual weighted residual (DWR) method. This method is used for the mesh adaptation during the computation. The implementation is accomplished via the software library package DOpElib and deal.II for the computation of different eXFSI configurations.

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Ebna Hai, Bhuiyan Shameem Mahmood, Markus Bause, and Paul Kuberry. "Finite Element Approximation of the Extended Fluid-Structure Interaction (eXFSI) Problem." In ASME 2016 Fluids Engineering Division Summer Meeting collocated with the ASME 2016 Heat Transfer Summer Conference and the ASME 2016 14th International Conference on Nanochannels, Microchannels, and Minichannels. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/fedsm2016-7506.

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This contribution is the second part of three papers on Adaptive Multigrid Methods for the eXtended Fluid-Structure Interaction (eXFSI) Problem, where we introduce a monolithic variational formulation and solution techniques. To the best of our knowledge, such a model is new in the literature. This model is used to design an on-line structural health monitoring (SHM) system in order to determine the coupled acoustic and elastic wave propagation in moving domains and optimum locations for SHM sensors. In a monolithic nonlinear fluid-structure interaction (FSI), the fluid and structure models are formulated in different coordinate systems. This makes the FSI setup of a common variational description difficult and challenging. This article presents the state-of-the-art in the finite element approximation of FSI problem based on monolithic variational formulation in the well-established arbitrary Lagrangian Eulerian (ALE) framework. This research focuses on the newly developed mathematical model of a new FSI problem, which is referred to as extended Fluid-Structure Interaction (eXFSI) problem in the ALE framework. The eXFSI is a strongly coupled problem of typical FSI with a coupled wave propagation problem on the fluid-solid interface (WpFSI). The WpFSI is a strongly coupled problem of acoustic and elastic wave equations, where wave propagation problems automatically adopts the boundary conditions from the FSI problem at each time step. The ALE approach provides a simple but powerful procedure to couple solid deformations with fluid flows by a monolithic solution algorithm. In such a setting, the fluid problems are transformed to a fixed reference configuration by the ALE mapping. The goal of this work is the development of concepts for the efficient numerical solution of eXFSI problem, the analysis of various fluid-solid mesh motion techniques and comparison of different second-order time-stepping schemes. This work consists of the investigation of different time stepping scheme formulations for a nonlinear FSI problem coupling the acoustic/elastic wave propagation on the fluid-structure interface. Temporal discretization is based on finite differences and is formulated as a one step-θ scheme, from which we can consider the following particular cases: the implicit Euler, Crank-Nicolson, shifted Crank-Nicolson and the Fractional-Step-θ schemes. The nonlinear problem is solved with a Newton-like method where the discretization is done with a Galerkin finite element scheme. The implementation is accomplished via the software library package DOpElib based on the deal.II finite element library for the computation of different eXFSI configurations.

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