Download or read book The Development of a Robust Efficient Solver for Spectral and Spectral element Time Discretizations written by Nathan L. Mundis and published by . This book was released on 2014 with total page 156 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work examines alternative time discretizations for the Euler equations and methods for the robust and efficient solution of these discretizations. Specifically, the time-spectral method (TS), quasi-periodic time-spectral method (BDFTS), and spectral-element method in time (SEMT) are derived and examined in detail. For the two time-spectral based methods, focus is given to expanding these methods for more complicated problems than have been typically solved by other authors, including problems with spectral content in a large number of harmonics, gust response problems, and aeroelastic problems. To solve these more complicated problems, it was necessary to implement the flexible variant of the Generalized Minimal Residual method (FGMRES), utilizing the full second-order accurate spatial Jacobian, complete temporal coupling of the chosen time discretization, and fully-implicit coupling of the aeroelastic equations in the cases where they are needed. The FGMRES solver developed utilizes a block-colored Gauss-Seidel (BCGS) preconditioner augmented by a defect-correction process to increase its effectiveness. Exploration of more efficient preconditioners for the FGMRES solver is an anticipated topic for future work in this field. It was a logical extension to apply this already developed FGMRES solver to the spectral-element method in time, which has some advantages over the spectral methods already discussed. Unlike purely-spectral methods, SEMT allows for bothh - and p -refinement. This property could allow for element clustering around areas of sharp gradients and discontinuities, which in turn could make SEMT more efficient than TS for periodic problems that contain these sharp gradients and would require many time instances to produce a precise solution using the TS method. As such, a preliminary investigation of the SEMT method applied to the Euler equations is conducted and some areas for needed improvement in future work are identified. In this work, it is shown that many if not most periodic problems can be solved more quickly and more precisely (utilizing the current FGMRES solver) using the time-spectral method than the currently state-of-the-art second-order accurate time-implicit methods. Additionally, the potential efficiency gains of the quasi-periodic time-spectral method for strongly-periodic problems, over time-implicit methods for these same types of problems, is demonstrated. Problems with strong, moving shock waves, high reduced frequencies, and/or content in high harmonics (higher than the 20th harmonic) are particularly difficult to solve efficiently using TS, BDFTS, and SEMT, and future work should focus on solver improvements to address these types of problems in particular.
Download or read book Spectral Methods written by Claudio Canuto and published by Springer Science & Business Media. This book was released on 2007-06-30 with total page 616 pages. Available in PDF, EPUB and Kindle. Book excerpt: Following up the seminal Spectral Methods in Fluid Dynamics, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics contains an extensive survey of the essential algorithmic and theoretical aspects of spectral methods for complex geometries. These types of spectral methods were only just emerging at the time the earlier book was published. The discussion of spectral algorithms for linear and nonlinear fluid dynamics stability analyses is greatly expanded. The chapter on spectral algorithms for incompressible flow focuses on algorithms that have proven most useful in practice, has much greater coverage of algorithms for two or more non-periodic directions, and shows how to treat outflow boundaries. Material on spectral methods for compressible flow emphasizes boundary conditions for hyperbolic systems, algorithms for simulation of homogeneous turbulence, and improved methods for shock fitting. This book is a companion to Spectral Methods: Fundamentals in Single Domains.
Download or read book Implementing Spectral Methods for Partial Differential Equations written by David A. Kopriva and published by Springer Science & Business Media. This book was released on 2009-05-27 with total page 397 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explains how to solve partial differential equations numerically using single and multidomain spectral methods. It shows how only a few fundamental algorithms form the building blocks of any spectral code, even for problems with complex geometries.
Download or read book Efficient High Order Discretizations for Computational Fluid Dynamics written by Martin Kronbichler and published by Springer Nature. This book was released on 2021-01-04 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book introduces modern high-order methods for computational fluid dynamics. As compared to low order finite volumes predominant in today's production codes, higher order discretizations significantly reduce dispersion errors, the main source of error in long-time simulations of flow at higher Reynolds numbers. A major goal of this book is to teach the basics of the discontinuous Galerkin (DG) method in terms of its finite volume and finite element ingredients. It also discusses the computational efficiency of high-order methods versus state-of-the-art low order methods in the finite difference context, given that accuracy requirements in engineering are often not overly strict. The book mainly addresses researchers and doctoral students in engineering, applied mathematics, physics and high-performance computing with a strong interest in the interdisciplinary aspects of computational fluid dynamics. It is also well-suited for practicing computational engineers who would like to gain an overview of discontinuous Galerkin methods, modern algorithmic realizations, and high-performance implementations.
Download or read book Chebyshev and Fourier Spectral Methods written by John P. Boyd and published by Courier Corporation. This book was released on 2001-12-03 with total page 690 pages. Available in PDF, EPUB and Kindle. Book excerpt: Completely revised text focuses on use of spectral methods to solve boundary value, eigenvalue, and time-dependent problems, but also covers Hermite, Laguerre, rational Chebyshev, sinc, and spherical harmonic functions, as well as cardinal functions, linear eigenvalue problems, matrix-solving methods, coordinate transformations, methods for unbounded intervals, spherical and cylindrical geometry, and much more. 7 Appendices. Glossary. Bibliography. Index. Over 160 text figures.
Download or read book Spectral Methods written by Jie Shen and published by Springer Science & Business Media. This book was released on 2011-08-25 with total page 481 pages. Available in PDF, EPUB and Kindle. Book excerpt: Along with finite differences and finite elements, spectral methods are one of the three main methodologies for solving partial differential equations on computers. This book provides a detailed presentation of basic spectral algorithms, as well as a systematical presentation of basic convergence theory and error analysis for spectral methods. Readers of this book will be exposed to a unified framework for designing and analyzing spectral algorithms for a variety of problems, including in particular high-order differential equations and problems in unbounded domains. The book contains a large number of figures which are designed to illustrate various concepts stressed in the book. A set of basic matlab codes has been made available online to help the readers to develop their own spectral codes for their specific applications.
Download or read book Advances in Fluid Dynamics written by B. Rushi Kumar and published by Springer Nature. This book was released on 2020-07-10 with total page 1014 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book comprises selected peer-reviewed proceedings of the International Conference on Applications of Fluid Dynamics (ICAFD 2018) organized by the School of Advanced Sciences, Vellore Institute of Technology, India, in association with the University of Botswana and the Society for Industrial and Applied Mathematics (SIAM), USA. With an aim to identify the existing challenges in the area of applied mathematics and mechanics, the book emphasizes the importance of establishing new methods and algorithms to address these challenges. The topics covered include diverse applications of fluid dynamics in aerospace dynamics and propulsion, atmospheric sciences, compressible flow, environmental fluid dynamics, control structures, viscoelasticity and mechanics of composites. Given the contents, the book is a useful resource for students, researchers as well as practitioners.
Download or read book Higher Order Finite Element Methods written by Pavel Solin and published by CRC Press. This book was released on 2003-07-28 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: The finite element method has always been a mainstay for solving engineering problems numerically. The most recent developments in the field clearly indicate that its future lies in higher-order methods, particularly in higher-order hp-adaptive schemes. These techniques respond well to the increasing complexity of engineering simulations and
Download or read book A Spectral Multidomain Penalty Method Solver for Environmental Flow Processes written by Jorge Alberto Escobar-Vargas and published by . This book was released on 2012 with total page 286 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work presents the details behind each step in the development of a framework for two-dimensional quadrilateral discontinuous Spectral Multidomain Penalty Method (SMPM) solvers for environmental flow processes: a shallow water equation (SWE) solver and an incompressible Navier-Stokes equations (NSE) (under the Boussinesq approximation) solver, with additional emphasis given to the associated pressure solver. The potential for environmental flow simulations through spectral methods is very strong since these methods are exponentially accurate, non-dissipative and nondispersive. These characteristics translate into capturing the smallest resolved scales of the flow and the propagation of ocean/lake waves with minimum numerical error. In addition, the element-based capability of the method enables the appropriate resolution of the important scales of the processes being modeled, the localization of specific events, and the treatment of complex boundary conditions and geometries. Finally, the discontinuous character of the method add enhanced stability to the method for highlynonlinear under-resolved simulations, an intrinsic characteristic of environmental flow simulations. In the SWE solver, the SMPM is compared with a nodal discontinuous Galerkin method (DGM), where the equations are solved with an explicit SSP-RK34 method. The comparison is done by applying both methods to a suite of six commonly considered geophysical flow test cases; we also include results for a classical continuous Galerkin (i.e., spectral element) method for comparison. Both the analysis and numerical experiments show that the SMPM and DGM are essentially identical; both methods can be shown to be equivalent for very special choices of quadrature rules and Riemann solvers in the DGM along with special choices in the type of penalty term in the SMPM. In the NSE solver time is discretized with a high-order fractional step projection method, where the non-linear advection and forcing terms are advanced explicitly via a stiffly stable scheme. After that, an implicit solution of a Poisson pressure equation (PPE) is solved in order to introduce the incompressibility constraint. In the final fractional time-step linear viscosity forces are also solved implicitly by means of a modified Helmholtz equation. Stability of the numerical scheme for under-resolved simulations at high Reynolds numbers is ensured through use of penalty techniques, spectral filtering, dealiasing, and strong adaptive interfacial averaging. Special attention is given to the solution of the PPE linear system of equations, where the fundamental building blocks of the PPE solver presented here are a Kronecker (tensor) product-based computation of the left null singular value of the non-symmetric SMPM-discretized Laplacian matrix and a custom-designed two-level preconditioner. Both of these tools are essential towards ensuring existence and uniqueness of the solution of the discrete linear system of equations and enabling its efficient iterative calculation. Accuracy, efficiency, and stability of the multidomain model are assessed through the solution of the Taylor vortex, lid-driven cavity flow and double shear layer. The propagation of a non-linear internal wave of depression type is also presented to assess the potential of the solver for the study of environmental stratified flows. The availability of the quadrilateral SMPM solver allows the numerical investigation of a much broader range of environmental processes, namely those in streamwise, vertical non periodic domains with both horizontal and vertical localization.
Download or read book Mathematics of Wave Phenomena written by Willy Dörfler and published by Springer Nature. This book was released on 2020-10-01 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: Wave phenomena are ubiquitous in nature. Their mathematical modeling, simulation and analysis lead to fascinating and challenging problems in both analysis and numerical mathematics. These challenges and their impact on significant applications have inspired major results and methods about wave-type equations in both fields of mathematics. The Conference on Mathematics of Wave Phenomena 2018 held in Karlsruhe, Germany, was devoted to these topics and attracted internationally renowned experts from a broad range of fields. These conference proceedings present new ideas, results, and techniques from this exciting research area.
Download or read book Fractional Order Modeling of Dynamic Systems with Applications in Optimization Signal Processing and Control written by Ahmed G. Radwan and published by Academic Press. This book was released on 2021-10-22 with total page 530 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fractional-order Modelling of Dynamic Systems with Applications in Optimization, Signal Processing and Control introduces applications from a design perspective, helping readers plan and design their own applications. The book includes the different techniques employed to design fractional-order systems/devices comprehensively and straightforwardly. Furthermore, mathematics is available in the literature on how to solve fractional-order calculus for system applications. This book introduces the mathematics that has been employed explicitly for fractional-order systems. It will prove an excellent material for students and scholars who want to quickly understand the field of fractional-order systems and contribute to its different domains and applications. Fractional-order systems are believed to play an essential role in our day-to-day activities. Therefore, several researchers around the globe endeavor to work in the different domains of fractional-order systems. The efforts include developing the mathematics to solve fractional-order calculus/systems and to achieve the feasible designs for various applications of fractional-order systems. Presents a simple and comprehensive understanding of the field of fractional-order systems Offers practical knowledge on the design of fractional-order systems for different applications Exposes users to possible new applications for fractional-order systems
Download or read book Spectral and High order Methods with Applications written by Jie Shen and published by . This book was released on 2006 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: 中国科学院科学出版基金资助出版。
Download or read book Iterative Methods for Solving Linear Systems written by Anne Greenbaum and published by SIAM. This book was released on 1997-01-01 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Computing -- Numerical Analysis.
Download or read book Applied mechanics reviews written by and published by . This book was released on 1948 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book High Order Methods for Incompressible Fluid Flow written by M. O. Deville and published by Cambridge University Press. This book was released on 2002-08-15 with total page 532 pages. Available in PDF, EPUB and Kindle. Book excerpt: Publisher Description
Download or read book IUTAM Symposium on Discretization Methods in Structural Mechanics written by H.A. Mang and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 387 pages. Available in PDF, EPUB and Kindle. Book excerpt: The JUT AMlIACM Symposium on Discretization Methods in Structural Mechanics was nd th held in Vienna, Austria, from 2 to 6 June 1997. The site of the Symposium was the "Theatersaal" of the Austrian Academy of Sciences. The Symposium was attended by 71 persons from 23 countries. In addition, several Austrian graduate students and research associates participated in the meeting. In the 5-day Symposium a total of 48 papers were presented. All of them were invited and accorded equal weight in the programme. The following topics were covered: • Error-controlled adaptivity of finite element methods • Large deformations and buckling, including inelastic deformations • Inelastic brittle or ductile localization, phase transition and system failure, resulting from monotonic, cyclic or impact loading • Sensitivity analysis and inverse problems with special emphasis on identification of material parameters • Development of linear and nonlinear finite element methods for thin-walled structures and composites • Implicit integration schemes for nonlinear dynamics • Coupling of rigid and deformable structures; fluid-structures and acoustic-structure interaction • Competitive numerical methods (finite element methods, boundary element methods, coupling ofthese two methods) • Identification of material and structural data. Comments on details of the treatment of these topics are contained in the Concluding Remarks. The Editors would like to express their appreciation to E. Stein who has prepared these Concluding Remarks.
Download or read book Spectral Algorithms written by Ravindran Kannan and published by Now Publishers Inc. This book was released on 2009 with total page 153 pages. Available in PDF, EPUB and Kindle. Book excerpt: Spectral methods refer to the use of eigenvalues, eigenvectors, singular values and singular vectors. They are widely used in Engineering, Applied Mathematics and Statistics. More recently, spectral methods have found numerous applications in Computer Science to "discrete" as well as "continuous" problems. Spectral Algorithms describes modern applications of spectral methods, and novel algorithms for estimating spectral parameters. The first part of the book presents applications of spectral methods to problems from a variety of topics including combinatorial optimization, learning and clustering. The second part of the book is motivated by efficiency considerations. A feature of many modern applications is the massive amount of input data. While sophisticated algorithms for matrix computations have been developed over a century, a more recent development is algorithms based on "sampling on the fly" from massive matrices. Good estimates of singular values and low rank approximations of the whole matrix can be provably derived from a sample. The main emphasis in the second part of the book is to present these sampling methods with rigorous error bounds. It also presents recent extensions of spectral methods from matrices to tensors and their applications to some combinatorial optimization problems.
