Provably Convergent Anisotropic Output-based Adaptation for Continuous Finite Element Discretizations PDF Download

Author: Hugh Alexander Carson
Publisher:
ISBN:
Category :
Languages : en
Pages : 131

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Book Description
The expansion of modern computing power has seen a commensurate rise in the reliance on numerical simulations for engineering and scientific purposes. Output error estimation combined with metric-based mesh adaptivity provides a powerful means of quantifiably controlling the error in these simulations, for output quantities of interest to engineers and scientists. The Mesh Optimization via Error Sampling and Synthesis (MOESS) algorithm, developed by Yano for Discontinuous Galerkin (DG) discretization, is a highly effective method of this class. This work begins with the extension of the MOESS algorithm to Continuous Galerkin (CG) discretization which requires fewer Degrees Of Freedom (DOF) on a given mesh compared to DG. The algorithm utilizes a vertex-based local error decomposition, and an edge-based local solve process in contrast to the element-centric construction of the original MOESS algorithm. Numerical results for linear problems in two and three dimensions demonstrate the improved DOF efficiency for CG compared to DG on adapted meshes. A proof of convergence for the new MOESS extension is then outlined, entailing the description of an abstract metric-conforming mesh generator. The framework of the proof is rooted in optimization, and its construction enables a proof of higher-order asymptotic rate of convergence irrespective of singularities. To the author’s knowledge, this is the first such proof for a Metric-based Adaptive Finite Element Method in the literature. A three dimensional Navier Stokes simulation of a delta wing is then used to compare the new formulation to the original MOESS algorithm. The required stabilization of the CG discretization is performed using a new stabilization technique: Variational Multi-Scale with Discontinuous sub-scales (VMSD). Numerical results confirm that VMSD adapted meshes require significantly fewer DOFs to achieve a given error level when compared to DG adapted meshes; these DOF savings are shown to translate into a reduction in overall CPU time and memory usage for a given accuracy

Author: Pascal Tremblay
Publisher:
ISBN:
Category : University of Ottawa theses
Languages : en
Pages : 544

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Author: Vít Dolejší
Publisher: Springer Nature
ISBN: 3031042794
Category : Mathematics
Languages : en
Pages : 258

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Book Description
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.

Author: Michael Andrew Park
Publisher:
ISBN:
Category :
Languages : en
Pages : 328

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Book Description
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.

Author: Thomas Apel
Publisher:
ISBN:
Category : Technology & Engineering
Languages : en
Pages : 268

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Author: Todd A. Oliver
Publisher:
ISBN:
Category :
Languages : en
Pages : 182

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This thesis presents high-order, discontinuous Galerkin (DG) discretizations of the Reynolds-Averaged Navier-Stokes (RANS) equations and an output-based error estimation and mesh adaptation algorithm for these discretizations. In particular, DG discretizations of the RANS equations with the Spalart-Allmaras (SA) turbulence model are examined. The dual consistency of multiple DG discretizations of the RANS-SA system is analyzed. The approach of simply weighting gradient dependent source terms by a test function and integrating is shown to be dual inconsistent. A dual consistency correction for this discretization is derived. The analysis also demonstrates that discretizations based on the popular mixed formulation, where dependence on the state gradient is handled by introducing additional state variables, are generally asymptotically dual consistent. Numerical results are presented to confirm the results of the analysis. The output error estimation and output-based adaptation algorithms used here are extensions of methods previously developed in the finite volume and finite element communities. In particular, the methods are extended for application on the curved, highly anisotropic meshes required for boundary conforming, high-order RANS simulations. Two methods for generating such curved meshes are demonstrated. One relies on a user-defined global mapping of the physical domain to a straight meshing domain. The other uses a linear elasticity node movement scheme to add curvature to an initially linear mesh. Finally, to improve the robustness of the adaptation process, an "unsteady" algorithm, where the mesh is adapted at each time step, is presented. The goal of the unsteady procedure is to allow mesh adaptation prior to converging a steady state solution, not to obtain a time-accurate solution of an unsteady problem. Thus, an estimate of the error due to spatial discretization in the output of interest averaged over the current time step is developed. This error estimate is then used to drive an h-adaptation algorithm. Adaptation results demonstrate that the high-order discretizations are more efficient than the second-order method in terms of degrees of freedom required to achieve a desired error tolerance. Furthermore, using the unsteady adaptation process, adaptive RANS simulations may be started from extremely coarse meshes, significantly decreasing the mesh generation burden to the user.

Author: Steffen Weißer
Publisher: Springer
ISBN: 303020961X
Category : Computers
Languages : en
Pages : 246

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Book Description
This book introduces readers to one of the first methods developed for the numerical treatment of boundary value problems on polygonal and polyhedral meshes, which it subsequently analyzes and applies in various scenarios. The BEM-based finite element approaches employs implicitly defined trial functions, which are treated locally by means of boundary integral equations. A detailed construction of high-order approximation spaces is discussed and applied to uniform, adaptive and anisotropic polytopal meshes. The main benefits of these general discretizations are the flexible handling they offer for meshes, and their natural incorporation of hanging nodes. This can especially be seen in adaptive finite element strategies and when anisotropic meshes are used. Moreover, this approach allows for problem-adapted approximation spaces as presented for convection-dominated diffusion equations. All theoretical results and considerations discussed in the book are verified and illustrated by several numerical examples and experiments. Given its scope, the book will be of interest to mathematicians in the field of boundary value problems, engineers with a (mathematical) background in finite element methods, and advanced graduate students.

Author: Gerd Kunert
Publisher: Logos Verlag Berlin
ISBN: 9783832504502
Category : Finite element method
Languages : en
Pages : 0

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Book Description
Certain classes of partial differential equations generically give rise to solutions with strong directional features, e.g. with boundary layers. Such solutions are called anisotropic. Their discretization by means of the finite element method (for example) can favourably employ so-called anisotropic meshes. These meshes are characterized by stretched, anisotropic finite elements with a (very) large stretching ratio. The widespread use of computer simulation leads to an increasing demand for semi- or fully automatic solution procedures. Within such self-adaptive algorithms, a posteriori error estimators form an indispensable ingredient for quality control. They are well understood for standard, isotropic discretizations. The knowledge about a posteriori error estimation on anisotropic meshes is much less mature. During the last decade the foundation and basic principles have been proposed, discussed and established, mostly for the Poisson problem. This monograph summarises some of the recent advances in anisotropic error estimation for more challenging problems. Emphasis is given to the contributions of the author. In Chapter 3 the investigation starts with singularly perturbed reaction diffusion problems which frequently lead to solutions with boundary layers. This problem class often arises when simplifying more complex models. Chapter 4 treats singularly perturbed convection diffusion problems, i.e. the convection is dominating. The solution structure is more intricate, and often features boundary layer and/or interior layer solutions. Chapter 5 is devoted to the Stokes equations. Flow problems generically give rise to anisotropic solutions (e.g. with edge singularities or containing layers). The Stokes equations often serve as a simplified or linearised model. In all three chapters, the main results consist in error estimators and corresponding error bounds that are robust with respect to the mesh anisotropy, as far as possible. Finally Chapter 6 addresses the robustness of a posteriori error estimation with respect to the mesh anisotropy.In particular the relation between anisotropic mesh construction and error estimation is investigated. This thesis presents the philosophy of anisotropic error estimation as well as the main results and the definitions required. Proofs and technical details are omitted; instead the key ideas are explained.The compact style of presentation aims at practitioners in particular by providing easily accessible error estimators and error bounds. Further insight is readily possible through the references.

Author: Nicolas Ringue
Publisher:
ISBN:
Category :
Languages : en
Pages :

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"Computational Fluid Dynamics (CFD) is an essential tool for scientists and engineers to understand and quantify aerodynamic flow. Numerous challenges however remain in order to improve computational efficiency, robustness and reliability of CFD methods. Mesh generation is one of the most time-consuming aspects of industrial CFD simulations since their accuracy heavily depends on mesh quality. In this context, mesh adaptation appears as one of the most promising approaches to improve CFD simulations. Since only an initial coarse grid is required, adaptive CFD algorithms enable a significant reduction in the amount of time spent on mesh generation as well as in the dependence of the accuracy of flow predictions on user input.This thesis presents a novel framework for hp-adaptation of high-order discontinuous finite element discretizations for compressible flow simulation. Using the sensitivities of a local adjoint-based error indicator, our method seeks element size, shape, and polynomial degree distributions which minimize a global error estimate for a specified number of degreesof freedom. This approach results in an optimized hp-mesh tailored to yield the most accurate prediction of an output quantity of interest, such as aerodynamic lift or drag coefficients, at a given computational cost. The proposed method features a reduced dependence on userdefined parameters compared to established fixed-fraction adjoint-based adaptive methods.It provides a unifying framework where adaptation decisions such as isotropic/anisotropic hp-refinement/coarsening do not only rely on local arbitrary measures of solution anisotropy and smoothness, but rather where a globally optimal distribution of degrees of freedom is sought to minimize the error in the chosen quantity of interest"--

Author: Krzysztof J. Fidkowski
Publisher:
ISBN:
Category :
Languages : en
Pages : 175

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Book Description
(Cont.) The compressible Navier-Stokes equations in both two and three dimensions are discretized using the discontinuous Galerkin (DG) finite element method. An anisotropic h-adaptation technique is presented for high-order (p> 1) discretizations, driven by an output-error estimate obtained from the solution of an adjoint problem. In two and three dimensions, algorithms are presented for intersecting the geometry with the background mesh and for constructing the resulting cut cells. In addition, a quadrature technique is proposed for accurately integrating high-order functions on arbitrarily-shaped cut cells and cut faces. Accuracy on cut-cell meshes is demonstrated by comparing solutions to those on standard, boundary-conforming meshes. In two dimensions, robustness of the cut-cell, adaptive technique is successfully tested for highly-anisotropic boundary-layer meshes representative of practical high-Re simulations. In three dimensions, robustness of cut cells is demonstrated for various representative curved geometries. Adaptation results show that for all test cases considered, p = 2 and p = 3 discretizations meet desired error tolerances using fewer degrees of freedom than p = 1.