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Unstructured Anisotropic All Quad Mesh Adaptation Based On A Continuous Local Error Model For The Discontinuous Galerkin Method
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Book Synopsis Unstructured Anisotropic All-quad Mesh Adaptation Based on a Continuous Local Error Model for the Discontinuous Galerkin Method by : Keigan MacLean
Download or read book Unstructured Anisotropic All-quad Mesh Adaptation Based on a Continuous Local Error Model for the Discontinuous Galerkin Method written by Keigan MacLean and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: "This thesis will present a method of unstructured anisotropic adaptation for all-quad mesh suitable for use with high-order methods. The overall goal is to minimize the computational cost needed to perform highly accurate Computational Fluid Dynamics (CFD) simulations. With adaptive methods this can be accomplished through modifying the underlying spatial discretization of the domain. Here, this will be tailored to the particular problem at hand through a continuous local error model. In this form, the physical mesh geometry is replaced by the notion of a frame field, allowing us to extend previous ideas from metric-based adaptation methods. This will serve as a means to restate the discrete error minimization problem in order to determine the target element size and shape based on an optimal distribution of degrees of freedom.Furthermore, the goal of most CFD simulations is to measure some particular output of interest (e.g. such as lift or drag). Therefore, extensions to the method will be examined for goal-oriented adaptation. This involves using the adjoint problem solution and dual-weighted residuals to target mesh refinement in areas most crucial to the accuracy of these measurements.Finally, this work will also consider the implementation of a fully unstructured and anisotropic all-quad mesh generator capable of conforming to the aforementioned target elements. Here, the procedure is based on an Lp-CVT energy minimization problem used to determine updated node positions. Overall, this work will combine each of these aspects in order to establish a complete adaptive framework for use in solving high-order problems with the Discontinuous Galerkin method"--
Book Synopsis Anisotropic hp-Mesh Adaptation Methods by : Vít Dolejší
Download or read book Anisotropic hp-Mesh Adaptation Methods written by Vít Dolejší and published by Springer Nature. This book was released on 2022-06-06 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mesh adaptation methods can have a profound impact on the numerical solution of partial differential equations. If devised and implemented properly, adaptation significantly reduces the size of the algebraic systems resulting from the discretization, while ensuring that applicable error tolerances are met. In this monograph, drawing from many years of experience, the authors give a comprehensive presentation of metric-based anisotropic hp-mesh adaptation methods. A large part of this monograph is devoted to the derivation of computable interpolation error estimates on simplicial meshes, which take into account the geometry of mesh elements as well as the anisotropic features of the interpolated function. These estimates are then used for the optimization of corresponding finite element spaces in a variety of settings. Both steady and time dependent problems are treated, as well as goal-oriented adaptation. Practical aspects of implementation are also explored, including several algorithms. Many numerical experiments using the discontinuous Galerkin method are presented to illustrate the performance of the adaptive techniques. This monograph is intended for scientists and researchers, including doctoral and master-level students. Portions of the text can also be used as study material for advanced university lectures concerning a posteriori error analysis and mesh adaptation.
Book Synopsis Mesh Adaptation for Computational Fluid Dynamics, Volume 1 by : Alain Dervieux
Download or read book Mesh Adaptation for Computational Fluid Dynamics, Volume 1 written by Alain Dervieux and published by John Wiley & Sons. This book was released on 2022-09-21 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: Simulation technology, and computational fluid dynamics (CFD) in particular, is essential in the search for solutions to the modern challenges faced by humanity. Revolutions in CFD over the last decade include the use of unstructured meshes, permitting the modeling of any 3D geometry. New frontiers point to mesh adaptation, allowing not only seamless meshing (for the engineer) but also simulation certification for safer products and risk prediction. Mesh Adaptation for Computational Dynamics 1 is the first of two volumes and introduces basic methods such as feature-based and multiscale adaptation for steady models. Also covered is the continuous Riemannian metrics formulation which models the optimally adapted mesh problem into a pure partial differential statement. A number of mesh adaptative methods are defined based on a particular feature of the simulation solution. This book will be useful to anybody interested in mesh adaptation pertaining to CFD, especially researchers, teachers and students.
Book Synopsis Two-dimensional, Three-dimensional and Four-dimensional Anisotropic Mesh Adaptation for the Time-continuous Space-time Finite Element Method with Applications to the Incompressible Navier-Stokes Equations by : Pascal Tremblay
Download or read book Two-dimensional, Three-dimensional and Four-dimensional Anisotropic Mesh Adaptation for the Time-continuous Space-time Finite Element Method with Applications to the Incompressible Navier-Stokes Equations written by Pascal Tremblay and published by . This book was released on 2007 with total page 544 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes by : Andrea Cangiani
Download or read book hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes written by Andrea Cangiani and published by Springer. This book was released on 2017-11-27 with total page 133 pages. Available in PDF, EPUB and Kindle. Book excerpt: Over the last few decades discontinuous Galerkin finite element methods (DGFEMs) have been witnessed tremendous interest as a computational framework for the numerical solution of partial differential equations. Their success is due to their extreme versatility in the design of the underlying meshes and local basis functions, while retaining key features of both (classical) finite element and finite volume methods. Somewhat surprisingly, DGFEMs on general tessellations consisting of polygonal (in 2D) or polyhedral (in 3D) element shapes have received little attention within the literature, despite the potential computational advantages. This volume introduces the basic principles of hp-version (i.e., locally varying mesh-size and polynomial order) DGFEMs over meshes consisting of polygonal or polyhedral element shapes, presents their error analysis, and includes an extensive collection of numerical experiments. The extreme flexibility provided by the locally variable elemen t-shapes, element-sizes, and element-orders is shown to deliver substantial computational gains in several practical scenarios.
Book Synopsis A Parallel, Adaptive Discontinuous Galerkin Method for Hyperbolic Problems on Unstructured Meshes by : Andrew Giuliani
Download or read book A Parallel, Adaptive Discontinuous Galerkin Method for Hyperbolic Problems on Unstructured Meshes written by Andrew Giuliani and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis is concerned with the parallel, adaptive solution of hyperbolic conservation laws on unstructured meshes. First, we present novel algorithms for cell-based adaptive mesh refinement (AMR) on unstructured meshes of triangles on graphics processing units (GPUs). Our implementation makes use of improved memory management techniques and a coloring algorithm for avoiding race conditions. The algorithm is entirely implemented on the GPU, with negligible communication between device and host. We show that the overhead of the AMR subroutines is small compared to the high-order solver and that the proportion of total run time spent adaptively refining the mesh decreases with the order of approximation. We apply our code to a number of benchmarks as well as more recently proposed problems for the Euler equations that require extremely high resolution. We present the solution to a shock reflection problem that addresses the von Neumann triple point paradox. We also study the problem of shock disappearance and self-similar diffraction of weak shocks around thin films. Next, we analyze the stability and accuracy of second-order limiters for the discontinuous Galerkin method on unstructured triangular grids. We derive conditions for a limiter such that the numerical solution preserves second order accuracy and satisfies the local maximum principle. This leads to a new measure of cell size that is approximately twice as large as the radius of the inscribed circle. It is shown with numerical experiments that the resulting bound on the time step is tight. We also consider various combinations of limiting points and limiting neighborhoods and present numerical experiments comparing the accuracy, stability, and efficiency of the corresponding limiters. We show that the theory for strong stability preserving (SSP) time stepping methods employed with the method of lines-type discretizations of hyperbolic conservation laws may result in overly stringent time step restrictions. We analyze a fully discrete finite volume method with slope reconstruction and a second order SSP Runge-Kutta time integrator to show that the maximum stable time step can be increased over the SSP limit. Numerical examples show that this result extends to two-dimensional problems on triangular meshes. Finally, we propose a moment limiter for the discontinuous Galerkin method applied to hyperbolic conservation laws in two and three dimensions. The limiter works by finding directions in which the solution coefficients can be separated and limits them independently of one another by comparing to forward and backward reconstructed differences. The limiter has a precomputed stencil of constant size, which provides computational advantages in terms of implementation and runtime. We provide examples that demonstrate stability and second order accuracy of solutions.
Book Synopsis Residual-based Isotropic and Anisotropic Mesh Adaptation for Computational Fluid Dynamics by : Amir R. Baserinia
Download or read book Residual-based Isotropic and Anisotropic Mesh Adaptation for Computational Fluid Dynamics written by Amir R. Baserinia and published by . This book was released on 2008 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: The accuracy of a fluid flow simulation depends not only on the numerical method used for discretizing the governing equations, but also on the distribution and topology of the mesh elements. Mesh adaptation is a technique for automatically modifying the mesh in order to improve the simulation accuracy in an attempt to reduce the manual work required for mesh generation. The conventional approach to mesh adaptation is based on a feature-based criterion that identifies the distinctive features in the flow field such as shock waves and boundary layers. Although this approach has proved to be simple and effective in many CFD applications, its implementation may require a lot of trial and error for determining the appropriate criterion in certain applications. An alternative approach to mesh adaptation is the residual-based approach in which the discretization error of the fluid flow quantities across the mesh faces is used to construct an adaptation criterion. Although this approach provides a general framework for developing robust mesh adaptation criteria, its incorporation leads to significant computational overhead. The main objective of the thesis is to present a methodology for developing an appropriate mesh adaptation criterion for fluid flow problems that offers the simplicity of a feature-based criterion and the robustness of a residual-based criterion. This methodology is demonstrated in the context of a second-order accurate cell-centred finite volume method for simulating laminar steady incompressible flows of constant property fluids. In this methodology, the error of mass and momentum flows across the faces of each control volume are estimated with a Taylor series analysis. Then these face flow errors are used to construct the desired adaptation criteria for triangular isotropic meshes and quadrilateral anisotropic meshes. The adaptation results for the lid-driven cavity flow show that the solution error on the resulting adapted meshes is 80 to 90 percent lower than that of a uniform mesh with the same number of control volumes. The advantage of the proposed mesh adaptation method is the capability to produce meshes that lead to more accurate solutions compared to those of the conventional methods with approximately the same amount of computational effort.
Book Synopsis Time-accurate Anisotropic Mesh Adaptation for Three-dimensional Moving Mesh Problems by : Nicolas Barral
Download or read book Time-accurate Anisotropic Mesh Adaptation for Three-dimensional Moving Mesh Problems written by Nicolas Barral and published by . This book was released on 2015 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Time dependent simulations are still a challenge for industry, notably due to problems raised by moving boundaries, both in terms of CPU cost and accuracy. This thesis presents contributions to several aspects of simulations with moving meshes. A moving-mesh algorithm based on a large deformation time step and connectivity changes (swaps) is studied. An elasticity method and an Inverse Distance Weighted interpolation method are compared on many 3D examples, demonstrating the efficiency of the algorithm in handling large geometry displacement without remeshing. This algorithm is coupled with an Arbitrary-Lagrangian-Eulerian (ALE) solver, whose schemes and implementation in 3D are described in details. A linear interpolation scheme is used to handle swaps. Validation test cases showed that the use of swaps does not impact notably the accuracy of the solution, while several other complex 3D examples demonstrate the capabilities of the approach both with imposed motion and Fluid-Structure Interaction problems. Metric-based mesh adaptation has proved its efficiency in improving the accuracy of steady simulation at a reasonable cost. We consider the extension of these methods to unsteady problems, updating the previous fixed-point algorithm thanks to a new space-time error analysis based on the continuous mesh model. An efficient p-thread parallelization enables running 3D unsteady adaptative simulations with a new level of accuracy. This algorithm is extended to moving mesh problems, notably by correcting the optimal unsteady metric. Finally several 3D examples of adaptative moving mesh simulations are exhibited, that prove our concept by improving notably the accuracy of the solution for a reasonable time cost.
Book Synopsis Adjoint-based Error Estimation and Mesh Adaptation for Hybridized Discontinuous Galerkin Methods by : Georg May
Download or read book Adjoint-based Error Estimation and Mesh Adaptation for Hybridized Discontinuous Galerkin Methods written by Georg May and published by . This book was released on 2013 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Anisotropic Adaptation on Unstructured Grids by :
Download or read book Anisotropic Adaptation on Unstructured Grids written by and published by . This book was released on 2003 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The objective of the present research is to assess the use of grid adaptation to improve Computational Fluid Dynamics calculations. Many issues of the quality discretization of a flow domain are discussed. The representation of the highly directional features in a flow field, such as shocks and boundary layers, forms the focus of the analysis. Anisotropic adaptation, which uses stretched elements to resolve directional features, is more effective than isotropic adaptation. Anisotropic adaptation requires more degrees of freedom from the mesh and demands the use of unstructured grids in the adaptation. The size and orientation of an anisotropic element require a matrix-like local feature indicator. The Hessian, a matrix composed of the second derivatives of an appropriate flow variable, is defined and used as a feature indicator in the adaptation. The Hessian provides a metric that defines the length of an edge and the lengths of all edges are equal in the optimized mesh. The techniques to minimize the differences among edge lengths are discussed and those chosen include node enrichment, node removal, edge swapping and point smoothing. A unified procedure based on the advancing front method is implemented to reconstruct the local connectivity that has been removed in the node removal and edge swapping processes. The results indicate that the mesh in which the edge lengths are equalized is not correct for three major flow features one frequently encounters. The inflections existing near the wall in a boundary layer result in coarse grids there. A "wall" Hessian is defined to replace the second derivatives and give a more appropriate spacing for high Reynolds number flow modeling. Difficulties in the adaptation of discontinuities are addressed. These include the infinite refinement that tries to pull all the points close to the discontinuity and the deviation of the shock from the refinement region because it is too thin. Remedies proposed are to limit the minimum physical edge length and smooth the Hessian such that the refinement encompasses more layers of elements. The strength of a discontinuity is defined and methodology to refine the discontinuity equally is proposed. The invalidity of the Hessian in a free stream is corrected to give a reasonable grid size in that region. It is demonstrated that these suggested modifications improve the overall quality of the adapted mesh as well as the solution. The concepts involved in the extension of the length-based approach to three dimensions are addressed. The difference and difficulties in three-dimensional adaptation are discussed. Barriers exist which prevent the equidistribution of the edge lengths, the goal of the current approach.
Book Synopsis Anisotropic Mesh Adaptation Based on Hessian Recovery and A Posteriori Error Estimates by : Lennard Kamenski
Download or read book Anisotropic Mesh Adaptation Based on Hessian Recovery and A Posteriori Error Estimates written by Lennard Kamenski and published by . This book was released on 2009 with total page 113 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Optimesh: Anisotropic Mesh Adaptation with CAD Integrity for Verifiably Accurate CFD Solutions Over Complete Aircraft by : Wagdi G. Habashi
Download or read book Optimesh: Anisotropic Mesh Adaptation with CAD Integrity for Verifiably Accurate CFD Solutions Over Complete Aircraft written by Wagdi G. Habashi and published by . This book was released on 2003 with total page 7 pages. Available in PDF, EPUB and Kindle. Book excerpt: The code discussed here represents an advanced novel anisotropic mesh adaptation approach for unstructured grids that allows the customization of grids based on a specified error level. Equally important, it maintains the CAD integrity of the given geometry. Perhaps the best showcase of the abilities of this software is the adaptation of unstructured tetrahedral meshes over a complete Airbus 320 and a Boeing 747 with flow-through engine nacelles.
Book Synopsis Anisotropic Output-based Adaptation with Tetrahedral Cut Cells for Compressible Flows by : Michael Andrew Park
Download or read book Anisotropic Output-based Adaptation with Tetrahedral Cut Cells for Compressible Flows written by Michael Andrew Park and published by . This book was released on 2008 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: Anisotropic, adaptive meshing for flows around complex, three-dimensional bodies remains a barrier to increased automation in computational fluid dynamics. Two specific advances are introduced in this thesis. First, a finite-volume discretization for tetrahedral cut-cells is developed that makes possible robust, anisotropic adaptation on complex bodies. Through grid refinement studies on inviscid flows, this cut-cell discretization is shown to produce similar accuracy as boundary-conforming meshes with a small increase in the degrees of freedom. The cut-cell discretization is then combined with output-based error estimation and anisotropic adaptation such that the mesh size and shape are controlled by the output error estimate and the Hessian (i.e. second derivatives) of the Mach number, respectively. Using a parallel implementation, this output-based adaptive method is applied to a series of sonic boom test cases and the automated ability to correctly estimate pressure signatures at several body lengths is demonstrated starting with initial meshes of a few thousand control volumes. Second, a new framework for adaptation is introduced in which error estimates are directly controlled by removing the common intermediate step of specifying a desired mesh size and shape. As a result, output error control can be achieved without the adhoc selection of a specific field (such as Mach number) to control anisotropy, rather anisotropy in the mesh naturally results from both the primal and dual solutions. Furthermore, the direct error control extends naturally to higher-order discretizations for which the use of a Hessian is no longer appropriate to determine mesh shape. The direct error control adaptive method is demonstrated on a series of simple test cases to control interpolation error and discontinuous Galerkin finite element output error. This new direct method produces grids with less elements but the same accuracy as existing metric-based approaches.
Book Synopsis Adjoint-based error estimation and mesh adaption for hybridized discontinuous Galerkin methods by : Michael Woopen
Download or read book Adjoint-based error estimation and mesh adaption for hybridized discontinuous Galerkin methods written by Michael Woopen and published by . This book was released on 2013 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author :R. B. (R. Bruce) Simpson Publisher :Faculty of Mathematics, University of Waterloo ISBN 13 : Total Pages :18 pages Book Rating :4.:/5 (372 download)
Book Synopsis Anisotropic Mesh Transformations and Optimal Error by : R. B. (R. Bruce) Simpson
Download or read book Anisotropic Mesh Transformations and Optimal Error written by R. B. (R. Bruce) Simpson and published by Faculty of Mathematics, University of Waterloo. This book was released on 1992 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "Recently, research originating in several different applications has appeared on unstructured triangular meshes in which the vertex distribution is not locally uniform, i.e. anisotropic unstructured meshes. The techniques used have the common features that the distribution of triangle shapes for the mesh is controlled by specifying a symmetric tensor, and that the anisotropic mesh is the transform of an isotropic mesh. We discuss how these mechanisms arise in the theory of optimal error control, using simple model mesh generation problems, and review the related research in applications to computational fluid dynamics, surface triangulation, and semiconductor simulation."
Book Synopsis Curved and Anisotropic Unstructured Mesh Generation and Adaptivity Using the Winslow Equations by : Meire Fortunato
Download or read book Curved and Anisotropic Unstructured Mesh Generation and Adaptivity Using the Winslow Equations written by Meire Fortunato and published by . This book was released on 2016 with total page 79 pages. Available in PDF, EPUB and Kindle. Book excerpt: High-order methods are receiving considerable interest from the computational community because they can achieve higher accuracy with reduced computational cost compared to traditional low-order approaches. These methods generally require unstructured meshes of non-inverted curved elements, and the generation of high-order curved meshes in a robust and automatic way is an important and challenging open problem. We present a method to generate high-order unstructured curved meshes by solving the classical Winslow equations using a new continuous Galerkin finite element discretization. This formulation appears to produce high quality curved elements, which are highly resistant to inversion. In addition, the corresponding nonlinear equations can be solved efficiently using Picard iterations, even for highly stretched boundary layer meshes. Another challenge that mesh-based methods face is that the discretization of the domain is usually generated before the solution is known, which can lead to large numerical errors or non-convergent schemes. A tool that can be used to overcome this problem is mesh adaptivity. We use the Winslow variable diffusion equations -- which are a variation of the classical form -- to perform curved and anisotropic unstructured mesh adaptivity. We use a range of numerical examples to validate our models including complex geometries and stretched boundary layers. We demonstrate the high quality of the generated meshes and the performance of the nonlinear solver. Finally, we present an example of mesh adaptivity for shock capturing when solving the Euler equations of gas dynamics for supersonic flow.
Book Synopsis Discontinuous Galerkin Method by : Vít Dolejší
Download or read book Discontinuous Galerkin Method written by Vít Dolejší and published by Springer. This book was released on 2015-07-27 with total page 572 pages. Available in PDF, EPUB and Kindle. Book excerpt: The subject of the book is the mathematical theory of the discontinuous Galerkin method (DGM), which is a relatively new technique for the numerical solution of partial differential equations. The book is concerned with the DGM developed for elliptic and parabolic equations and its applications to the numerical simulation of compressible flow. It deals with the theoretical as well as practical aspects of the DGM and treats the basic concepts and ideas of the DGM, as well as the latest significant findings and achievements in this area. The main benefit for readers and the book’s uniqueness lie in the fact that it is sufficiently detailed, extensive and mathematically precise, while at the same time providing a comprehensible guide through a wide spectrum of discontinuous Galerkin techniques and a survey of the latest efficient, accurate and robust discontinuous Galerkin schemes for the solution of compressible flow.
