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Numerical Solution Of The Poisson Interface Problem Using The Correction Function Method Framework
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Book Synopsis Numerical Solution of the Poisson Interface Problem Using the Correction Function Method Framework by : Philippe Blain
Download or read book Numerical Solution of the Poisson Interface Problem Using the Correction Function Method Framework written by Philippe Blain and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: "In this work we study the Poisson interface problem and a numerical method for its solution, the Correction Function Method. The Poisson interface problem complements the classical Poisson problem, a partial differential equation given in terms of the Laplacian operator, with jump conditions on the solution and its normal derivative along a co-dimension one interface inside the domain. Interface problems arise whenever materials with different properties come into contact, and thus have important applications in many fields of physics and engineering. The Correction Function Method (CFM) is a general framework to solve the Poisson interface problem in the wider context of finite difference methods. In essence, it modifies the finite difference stencils for nodes near the interface in order to take into account the jump conditions. However, these modifications are incorporated, using correction terms, to the right-hand side of the linear system deriving from the finite difference approximation of the partial differential equation, leaving the coefficient matrix identical to the one resulting from the regular Poisson problem. This is essential as many efficient linear solvers, termed "Fast Poisson solvers", are specifically designed to solve this system; by keeping the same matrix we can use them with no modification. In order to account for the multiple possible geometric configurations of the interface with respect to the finite difference stencil, we compute the correction terms locally using a functional minimization approach. Our specific implementation of the CFM is shown to be fourth-order accurate. We also present preliminary work towards a CFM-based Navier-Stokes solver for higher-order numerical simulation of multiphase flow." --
Book Synopsis Numerical Solution of Differential Equations by : Zhilin Li
Download or read book Numerical Solution of Differential Equations written by Zhilin Li and published by Cambridge University Press. This book was released on 2017-11-30 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: A practical and concise guide to finite difference and finite element methods. Well-tested MATLAB® codes are available online.
Book Synopsis Numerical Solution of Poisson Equation Using Finite Difference Method by : Bee Hong Tay
Download or read book Numerical Solution of Poisson Equation Using Finite Difference Method written by Bee Hong Tay and published by . This book was released on 2008 with total page 70 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Automated Solution of Differential Equations by the Finite Element Method by : Anders Logg
Download or read book Automated Solution of Differential Equations by the Finite Element Method written by Anders Logg and published by Springer Science & Business Media. This book was released on 2012-02-24 with total page 723 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Following are chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.
Book Synopsis Handbook of Numerical Methods for Hyperbolic Problems by : Remi Abgrall
Download or read book Handbook of Numerical Methods for Hyperbolic Problems written by Remi Abgrall and published by Elsevier. This book was released on 2016-11-17 with total page 668 pages. Available in PDF, EPUB and Kindle. Book excerpt: Handbook of Numerical Methods for Hyperbolic Problems explores the changes that have taken place in the past few decades regarding literature in the design, analysis and application of various numerical algorithms for solving hyperbolic equations. This volume provides concise summaries from experts in different types of algorithms, so that readers can find a variety of algorithms under different situations and readily understand their relative advantages and limitations. Provides detailed, cutting-edge background explanations of existing algorithms and their analysis Ideal for readers working on the theoretical aspects of algorithm development and its numerical analysis Presents a method of different algorithms for specific applications and the relative advantages and limitations of different algorithms for engineers or readers involved in applications Written by leading subject experts in each field who provide breadth and depth of content coverage
Book Synopsis Fast Solvers for 3D Poisson Equations Involving Interfaces in an Finite Or the Infinite Domain by :
Download or read book Fast Solvers for 3D Poisson Equations Involving Interfaces in an Finite Or the Infinite Domain written by and published by . This book was released on 2002 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, numerical methods are proposed for Poisson equations defined in a finite or in the infinite domain in three dimensions. In the domain, there can exists an interface across which the flux and the solution are discontinuous. To deal with the discontinuity in the source terms and in the flux, the original problem is transformed to a new one with a smooth solution. Such a transformation can he carried out easily through an extension of the jumps along the normal direction if the interface is expressed as the zero level set of a three dimensional function. An auxiliary sphere is used to separate the infinite region into an interior and exterior domain. The Kelvin's inversion is used to map the exterior domain into an interior domain. The two Poisson equations defined in the interior and the exterior written in spherical coordinates are solved simultaneously. By choosing the niesh size carefully and exploiting the fast Fourier transform, the resulting finite difference equations can he solved efficiently. The approach in dealing with the interface has also been used with the artificial boundary condition technique which truncates the infinite domain. Numerical results demonstrate second order accuracy of our algorithms.
Book Synopsis Solving PDEs in Python by : Hans Petter Langtangen
Download or read book Solving PDEs in Python written by Hans Petter Langtangen and published by Springer. This book was released on 2017-03-21 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a concise and gentle introduction to finite element programming in Python based on the popular FEniCS software library. Using a series of examples, including the Poisson equation, the equations of linear elasticity, the incompressible Navier–Stokes equations, and systems of nonlinear advection–diffusion–reaction equations, it guides readers through the essential steps to quickly solving a PDE in FEniCS, such as how to define a finite variational problem, how to set boundary conditions, how to solve linear and nonlinear systems, and how to visualize solutions and structure finite element Python programs. This book is open access under a CC BY license.
Book Synopsis Fast Enriched Finite Element Method for Poisson Equations Involving Interfaces by : Thanh Le Ngoc Huynh
Download or read book Fast Enriched Finite Element Method for Poisson Equations Involving Interfaces written by Thanh Le Ngoc Huynh and published by . This book was released on 2008 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: We develop a fast enriched finite element method for solving Poisson equations involving complex geometry interfaces by using regular Cartesian grids. The presence of interfaces is accounted for by developing suitable jump conditions. The immersed boundary method (IBM) and the immersed interface method (IIM) are successfully used to solve these problems when combined with a fast Fourier transform. However, the IBM and the IIM, which are developed from the finite difference method, have several disadvantages including the characterization of the null spaces and the inability to treat complex geometries accurately. We propose a solution to these difficulties by employing the finite element method. The continuous Galerkin solution approximations at the interface elements are modified using the enriched basis functions to make sure that the optimal convergence rates are obtained. Here, the FFT is applied in the fast Poisson solver to significantly accelerate the computational processes for solving the global matrix system. With reasonably small interfaces, the operational cost is almost linearly proportional to the number of the Cartesian grid points. The method is further extended to solve problems involving multi-materials while preserving the optimal accuracy. Several benchmark examples are shown to demonstrate the performance of the method.
Book Synopsis Numerical Solutions to the Poisson Equation in Media Surrounding Multiple Arbitrarily Shaped Bodies by : Zebadiah M. Smith
Download or read book Numerical Solutions to the Poisson Equation in Media Surrounding Multiple Arbitrarily Shaped Bodies written by Zebadiah M. Smith and published by . This book was released on 2010 with total page 121 pages. Available in PDF, EPUB and Kindle. Book excerpt: The dissertation describes a general, efficient, and parallelized approach to solving the Poisson equation in the volume surrounding multiple, arbitrarily shaped bodies. Green's function method is applied and a quadrature technique approximates the resulting integrals from which singularities are subtracted-yielding a large set of equations. The solution is implemented with a general, parallelized, functional program; the input of which are parametric equations defining the surfaces of multiple, arbitrarily shaped bodies and a number of possible boundary conditions. Taking full advantage of available computational resources, the method can be very efficient. The utility, generality, and efficiency of the approach is demonstrated by application to various problems in electrostatics and diffusion; both to validate its accuracy, and in specific cases, to elucidate the phenomena of interest. As the motivation for this research stems from aerosol science and specifically, the critical need for accurate source term modeling for gas-cooled, graphite-moderated nuclear reactors, discussion of the results is done in light of these topics. However, the approach is applicable to any phenomena governed by the Poisson equation-these phenomena are numerous and have implications in nearly all areas of science and engineering.
Book Synopsis Introduction to Numerical Methods for Variational Problems by : Hans Petter Langtangen
Download or read book Introduction to Numerical Methods for Variational Problems written by Hans Petter Langtangen and published by Springer Nature. This book was released on 2019-09-26 with total page 395 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook teaches finite element methods from a computational point of view. It focuses on how to develop flexible computer programs with Python, a programming language in which a combination of symbolic and numerical tools is used to achieve an explicit and practical derivation of finite element algorithms. The finite element library FEniCS is used throughout the book, but the content is provided in sufficient detail to ensure that students with less mathematical background or mixed programming-language experience will equally benefit. All program examples are available on the Internet.
Book Synopsis The Immersed Interface Method by : Zhilin Li
Download or read book The Immersed Interface Method written by Zhilin Li and published by SIAM. This book was released on 2006-01-01 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an introduction to the immersed interface method (IIM), a powerful numerical method for solving interface problems and problems defined on irregular domains for which analytic solutions are rarely available. This book gives a complete description of the IIM, discusses recent progress in the area, and describes numerical methods for a number of classic interface problems. It also contains many numerical examples that can be used as benchmark problems for numerical methods designed for interface problems on irregular domains.
Book Synopsis Encyclopedia of Two-Phase Heat Transfer and Flow IV by : John R Thome
Download or read book Encyclopedia of Two-Phase Heat Transfer and Flow IV written by John R Thome and published by World Scientific Publishing. This book was released on 2018-05-18 with total page 1372 pages. Available in PDF, EPUB and Kindle. Book excerpt: Set IV is a new addition to the previous Sets I, II and III. It contains 23 invited chapters from international specialists on the topics of numerical modeling of pulsating heat pipes and of slug flows with evaporation; lattice Boltzmann modeling of pool boiling; fundamentals of boiling in microchannels and microfin tubes, CO2 and nanofluids; testing and modeling of micro-two-phase cooling systems for electronics; and various special topics (flow separation in microfluidics, two-phase sensors, wetting of anisotropic surfaces, ultra-compact heat exchangers, etc.). The invited authors are leading university researchers and well-known engineers from leading corporate research laboratories (ABB, IBM, Nokia Bell Labs). Numerous 'must read' chapters are also included here for the two-phase community. Set IV constitutes a 'must have' engineering and research reference together with previous Sets I, II and III for thermal engineering researchers and practitioners.
Book Synopsis Numerical Methods in Geotechnical Engineering IX by : António S. Cardoso
Download or read book Numerical Methods in Geotechnical Engineering IX written by António S. Cardoso and published by CRC Press. This book was released on 2018-06-19 with total page 1656 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical Methods in Geotechnical Engineering IX contains 204 technical and scientific papers presented at the 9th European Conference on Numerical Methods in Geotechnical Engineering (NUMGE2018, Porto, Portugal, 25—27 June 2018). The papers cover a wide range of topics in the field of computational geotechnics, providing an overview of recent developments on scientific achievements, innovations and engineering applications related to or employing numerical methods. They deal with subjects from emerging research to engineering practice, and are grouped under the following themes: Constitutive modelling and numerical implementation Finite element, discrete element and other numerical methods. Coupling of diverse methods Reliability and probability analysis Large deformation – large strain analysis Artificial intelligence and neural networks Ground flow, thermal and coupled analysis Earthquake engineering, soil dynamics and soil-structure interactions Rock mechanics Application of numerical methods in the context of the Eurocodes Shallow and deep foundations Slopes and cuts Supported excavations and retaining walls Embankments and dams Tunnels and caverns (and pipelines) Ground improvement and reinforcement Offshore geotechnical engineering Propagation of vibrations Following the objectives of previous eight thematic conferences, (1986 Stuttgart, Germany; 1990 Santander, Spain; 1994 Manchester, United Kingdom; 1998 Udine, Italy; 2002 Paris, France; 2006 Graz, Austria; 2010 Trondheim, Norway; 2014 Delft, The Netherlands), Numerical Methods in Geotechnical Engineering IX updates the state-of-the-art regarding the application of numerical methods in geotechnics, both in a scientific perspective and in what concerns its application for solving practical boundary value problems. The book will be much of interest to engineers, academics and professionals involved or interested in Geotechnical Engineering.
Download or read book Physics Briefs written by and published by . This book was released on 1994 with total page 1146 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Numerical Methods in Geotechnical Engineering IX, Volume 1 by : Manuel de Matos Fernandes
Download or read book Numerical Methods in Geotechnical Engineering IX, Volume 1 written by Manuel de Matos Fernandes and published by CRC Press. This book was released on 2018-06-22 with total page 935 pages. Available in PDF, EPUB and Kindle. Book excerpt: NUMGE 2018 is the ninth in a series of conferences on Numerical Methods in Geotechnical Engineering organized by the ERTC7 under the auspices of the International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE). The first conference was held in 1986 in Stuttgart, Germany and the series continued every four years (1990 Santander, Spain; 1994 Manchester, United Kingdom; 1998 Udine, Italy; 2002 Paris, France; 2006 Graz, Austria; 2010 Trondheim, Norway; 2014 Delft, The Netherlands). The conference provides a forum for exchange of ideas and discussion on topics related to numerical modelling in geotechnical engineering. Both senior and young researchers, as well as scientists and engineers from Europe and overseas, are invited to attend this conference to share and exchange their knowledge and experiences. This work is the first volume of NUMGE 2018.
Book Synopsis Numerical Models for Differential Problems by : Alfio Quarteroni
Download or read book Numerical Models for Differential Problems written by Alfio Quarteroni and published by Springer Science & Business. This book was released on 2014-04-25 with total page 668 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this text, we introduce the basic concepts for the numerical modelling of partial differential equations. We consider the classical elliptic, parabolic and hyperbolic linear equations, but also the diffusion, transport, and Navier-Stokes equations, as well as equations representing conservation laws, saddle-point problems and optimal control problems. Furthermore, we provide numerous physical examples which underline such equations. We then analyze numerical solution methods based on finite elements, finite differences, finite volumes, spectral methods and domain decomposition methods, and reduced basis methods. In particular, we discuss the algorithmic and computer implementation aspects and provide a number of easy-to-use programs. The text does not require any previous advanced mathematical knowledge of partial differential equations: the absolutely essential concepts are reported in a preliminary chapter. It is therefore suitable for students of bachelor and master courses in scientific disciplines, and recommendable to those researchers in the academic and extra-academic domain who want to approach this interesting branch of applied mathematics.
Book Synopsis Numerical Methods for Elliptic and Parabolic Partial Differential Equations by : Peter Knabner
Download or read book Numerical Methods for Elliptic and Parabolic Partial Differential Equations written by Peter Knabner and published by Springer Science & Business Media. This book was released on 2006-05-26 with total page 437 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text provides an application oriented introduction to the numerical methods for partial differential equations. It covers finite difference, finite element, and finite volume methods, interweaving theory and applications throughout. The book examines modern topics such as adaptive methods, multilevel methods, and methods for convection-dominated problems and includes detailed illustrations and extensive exercises.
