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 PDF Download

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

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Author: Philip Claude Delhaye Caplan
Publisher:
ISBN:
Category :
Languages : en
Pages : 142

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Engineers and scientists are increasingly relying on high-fidelity numerical simulations. Within these simulations, mesh adaptation is useful for obtaining accurate predictions of an output of interest subject to a computational cost constraint. In the quest for accurately predicting outputs in problems with time-dependent solution features, a fully unstructured coupled spacetime approach has been shown to be useful in reducing the cost of the overall simulation. However, for the simulation of unsteady three-dimensional partial differential equations (PDEs), a four-dimensional mesh adaptation tool is needed. This work develops the first anisotropic metric-conforming four-dimensional mesh adaptation tool for performing adaptive numerical simulations of unsteady PDEs in three dimensions. The theory and implementation details behind our algorithm are first developed alongside an algorithm for constructing four-dimensional geometry representations. We then demonstrate our algorithm on three-dimensional benchmark cases and it appears to outperform existing implementations, both in metric-conformity and expected tetrahedra counts. We study the utility of the mesh adaptation components to justify the design of our algorithm. We then develop four-dimensional benchmark cases and demonstrate that metric-conformity and expected pentatope counts are also achieved. This is the first time anisotropic four-dimensional meshes have been presented in the literature. Next, the entire mesh adaptation framework, Mesh Optimization via Error Sampling and Synthesis (MOESS), is extended to the context of finding the optimal mesh to represent a function of four variables. The mesh size and aspect ratio distributions of the optimized meshes match the analytic ones, thus verifying our framework. Finally, we apply MOESS in conjunction with the mesh adaptation tool to perform the first four-dimensional anisotropic mesh adaptation for the solution of the advection-diffusion equation. The optimized meshes effectively refine the solution features corresponding to both a boundary layer solution as well as an expanding spherical wave.

Author:
Publisher:
ISBN:
Category : Aeronautics
Languages : en
Pages : 974

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Author: Nicolas Barral
Publisher:
ISBN:
Category :
Languages : en
Pages : 0

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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.

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

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

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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: Giovanni P. Galdi
Publisher: Birkhäuser
ISBN: 3034884249
Category : Mathematics
Languages : en
Pages : 300

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This volume consists of six articles, each treating an important topic in the theory ofthe Navier-Stokes equations, at the research level. Some of the articles are mainly expository, putting together, in a unified setting, the results of recent research papers and conference lectures. Several other articles are devoted mainly to new results, but present them within a wider context and with a fuller exposition than is usual for journals. The plan to publish these articles as a book began with the lecture notes for the short courses of G.P. Galdi and R. Rannacher, given at the beginning of the International Workshop on Theoretical and Numerical Fluid Dynamics, held in Vancouver, Canada, July 27 to August 2, 1996. A renewed energy for this project came with the founding of the Journal of Mathematical Fluid Mechanics, by G.P. Galdi, J. Heywood, and R. Rannacher, in 1998. At that time it was decided that this volume should be published in association with the journal, and expanded to include articles by J. Heywood and W. Nagata, J. Heywood and M. Padula, and P. Gervasio, A. Quarteroni and F. Saleri. The original lecture notes were also revised and updated.

Author: Daniel S.H. Lo
Publisher: CRC Press
ISBN: 041569048X
Category : Technology & Engineering
Languages : en
Pages : 676

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Highlights the Progression of Meshing Technologies and Their Applications Finite Element Mesh Generation provides a concise and comprehensive guide to the application of finite element mesh generation over 2D domains, curved surfaces, and 3D space. Organised according to the geometry and dimension of the problem domains, it develops from the basic meshing algorithms to the most advanced schemes to deal with problems with specific requirements such as boundary conformity, adaptive and anisotropic elements, shape qualities, and mesh optimization. It sets out the fundamentals of popular techniques, including: Delaunay triangulation Advancing-front (ADF) approach Quadtree/Octree techniques Refinement and optimization-based strategies From the geometrical and the topological aspects and their associated operations and inter-relationships, each approach is vividly described and illustrated with examples. Beyond the algorithms, the book also explores the practice of using metric tensor and surface curvatures for generating anisotropic meshes on parametric space. It presents results from research including 3D anisotropic meshing, mesh generation over unbounded domains, meshing by means of intersection, re-meshing by Delaunay-ADF approach, mesh refinement and optimization, generation of hexahedral meshes, and large scale and parallel meshing, along with innovative unpublished meshing methods. The author provides illustrations of major meshing algorithms, pseudo codes, and programming codes in C++ or FORTRAN. Geared toward research centers, universities, and engineering companies, Finite Element Mesh Generation describes mesh generation methods and fundamental techniques, and also serves as a valuable reference for laymen and experts alike.

Author: Institute for Computer Applications in Science and Engineering
Publisher:
ISBN:
Category :
Languages : en
Pages : 28

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An adaptive spectral element method has been developed for the efficient solution of time dependent partial differential equations. Adaptive mesh strategies that include resolution refinement and coarsening by three different methods are illustrated on solutions to the one-dimensional viscous Burgers equation and the two-dimensional Navier-Stokes equations for driven flow in a cavity. Sharp gradients, singularities and regions of poor resolution are resolved optimally as they develop in time using error estimators which indicate the choice of refinement to be used. The adaptive formulation presents significant increases in efficiency, flexibility and general capabilities for high order spectral methods.

Author: Paul Thomas Williams
Publisher:
ISBN:
Category :
Languages : en
Pages : 388

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