Download or read book The Hybrid High Order Method for Polytopal Meshes written by Daniele Antonio Di Pietro and published by Springer Nature. This book was released on 2020-04-03 with total page 552 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph provides an introduction to the design and analysis of Hybrid High-Order methods for diffusive problems, along with a panel of applications to advanced models in computational mechanics. Hybrid High-Order methods are new-generation numerical methods for partial differential equations with features that set them apart from traditional ones. These include: the support of polytopal meshes, including non-star-shaped elements and hanging nodes; the possibility of having arbitrary approximation orders in any space dimension; an enhanced compliance with the physics; and a reduced computational cost thanks to compact stencil and static condensation. The first part of the monograph lays the foundations of the method, considering linear scalar second-order models, including scalar diffusion – possibly heterogeneous and anisotropic – and diffusion-advection-reaction. The second part addresses applications to more complex models from the engineering sciences: non-linear Leray-Lions problems, elasticity, and incompressible fluid flows. This book is primarily intended for graduate students and researchers in applied mathematics and numerical analysis, who will find here valuable analysis tools of general scope.

Download or read book Hybrid High Order Methods written by Matteo Cicuttin and published by Springer Nature. This book was released on 2021-11-11 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a comprehensive coverage of hybrid high-order methods for computational mechanics. The first three chapters offer a gentle introduction to the method and its mathematical foundations for the diffusion problem. The next four chapters address applications of increasing complexity in the field of computational mechanics: linear elasticity, hyperelasticity, wave propagation, contact, friction, and plasticity. The last chapter provides an overview of the main implementation aspects including some examples of Matlab code. The book is primarily intended for graduate students, researchers, and engineers working in related fields of application, and it can also be used as a support for graduate and doctoral lectures.

Download or read book Numerical Methods for PDEs written by Daniele Antonio Di Pietro and published by Springer. This book was released on 2018-10-12 with total page 323 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume gathers contributions from participants of the Introductory School and the IHP thematic quarter on Numerical Methods for PDE, held in 2016 in Cargese (Corsica) and Paris, providing an opportunity to disseminate the latest results and envisage fresh challenges in traditional and new application fields. Numerical analysis applied to the approximate solution of PDEs is a key discipline in applied mathematics, and over the last few years, several new paradigms have appeared, leading to entire new families of discretization methods and solution algorithms. This book is intended for researchers in the field.

Download or read book Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020 1 written by Jens M. Melenk and published by Springer Nature. This book was released on 2023-06-30 with total page 571 pages. Available in PDF, EPUB and Kindle. Book excerpt: The volume features high-quality papers based on the presentations at the ICOSAHOM 2020+1 on spectral and high order methods. The carefully reviewed articles cover state of the art topics in high order discretizations of partial differential equations. The volume presents a wide range of topics including the design and analysis of high order methods, the development of fast solvers on modern computer architecture, and the application of these methods in fluid and structural mechanics computations.

Download or read book Polyhedral Methods in Geosciences written by Daniele Antonio Di Pietro and published by Springer Nature. This book was released on 2021-06-14 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: The last few years have witnessed a surge in the development and usage of discretization methods supporting general meshes in geoscience applications. The need for general polyhedral meshes in this context can arise in several situations, including the modelling of petroleum reservoirs and basins, CO2 and nuclear storage sites, etc. In the above and other situations, classical discretization methods are either not viable or require ad hoc modifications that add to the implementation complexity. Discretization methods able to operate on polyhedral meshes and possibly delivering arbitrary-order approximations constitute in this context a veritable technological jump. The goal of this monograph is to establish a state-of-the-art reference on polyhedral methods for geoscience applications by gathering contributions from top-level research groups working on this topic. This book is addressed to graduate students and researchers wishing to deepen their knowledge of advanced numerical methods with a focus on geoscience applications, as well as practitioners of the field.

Download or read book Finite Volumes for Complex Applications IX Methods Theoretical Aspects Examples written by Robert Klöfkorn and published by Springer Nature. This book was released on 2020-06-09 with total page 727 pages. Available in PDF, EPUB and Kindle. Book excerpt: The proceedings of the 9th conference on "Finite Volumes for Complex Applications" (Bergen, June 2020) are structured in two volumes. The first volume collects the focused invited papers, as well as the reviewed contributions from internationally leading researchers in the field of analysis of finite volume and related methods. Topics covered include convergence and stability analysis, as well as investigations of these methods from the point of view of compatibility with physical principles. Altogether, a rather comprehensive overview is given on the state of the art in the field. The properties of the methods considered in the conference give them distinguished advantages for a number of applications. These include fluid dynamics, magnetohydrodynamics, structural analysis, nuclear physics, semiconductor theory, carbon capture utilization and storage, geothermal energy and further topics. The second volume covers reviewed contributions reporting successful applications of finite volume and related methods in these fields. The finite volume method in its various forms is a space discretization technique for partial differential equations based on the fundamental physical principle of conservation. Many finite volume methods preserve further qualitative or asymptotic properties, including maximum principles, dissipativity, monotone decay of free energy, and asymptotic stability, making the finite volume methods compatible discretization methods, which preserve qualitative properties of continuous problems at the discrete level. This structural approach to the discretization of partial differential equations becomes particularly important for multiphysics and multiscale applications. The book is a valuable resource for researchers, PhD and master’s level students in numerical analysis, scientific computing and related fields such as partial differential equations, as well as engineers working in numerical modeling and simulations.

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 SIAM International Meshing Roundtable 2023 written by Eloi Ruiz-Gironés and published by Springer Nature. This book was released on with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Error Control Adaptive Discretizations and Applications Part 1 written by and published by Elsevier. This book was released on 2024-06-29 with total page 446 pages. Available in PDF, EPUB and Kindle. Book excerpt: Error Control, Adaptive Discretizations, and Applications, Volume 58, Part One highlights new advances in the field, with this new volume presenting interesting chapters written by an international board of authors. Chapters in this release cover hp adaptive Discontinuous Galerkin strategies driven by a posteriori error estimation with application to aeronautical flow problems, An anisotropic mesh adaptation method based on gradient recovery and optimal shape elements, and Model reduction techniques for parametrized nonlinear partial differential equations. - Covers multi-scale modeling - Includes updates on data-driven modeling - Presents the latest information on large deformations of multi-scale materials

Download or read book PolyDPG written by Jaime David Mora Paz and published by . This book was released on 2020 with total page 506 pages. Available in PDF, EPUB and Kindle. Book excerpt: Over the last two decades, the computational mechanics community has witnessed a growing interest in the development of discretization of boundary value problems (BVPs) on meshes of polygons and polyhedra (i.e., polytopes) of a more general kind than the typical element shapes used in finite element (FE) methods. Among many problems addressed with polytopal discretization techniques, several authors have focused on simulating large deformation of elastomers with rigid inclusions. These works deal with the nonlinear elasticity equations in 2D and hyperelastic materials. Their results have demonstrated a larger capacity for stretching with polygonal meshes, in comparison with standard meshes (solving with a similar number of degrees of freedom). This is one of the scenarios where polytopal numerical methods have been claimed to deliver superior performance to the standard FE techniques. A specific practical problem, with similarities to the mentioned result, is a recent study by Brown and Long on the deformation under compressive loads of a silicone elastomer (the matrix material) with glass micro-balloons as inclusions, a type of composite material known as an elastomeric syntactic foam. The problem has been marked as a simulation challenge by Sandia National Laboratories (SNL). The positive experience with polygonal meshes in 2D suggests that, in 3D, methods that support polyhedral elements may turn out to be well-suited for this specific application. The Discontinuous Petrov-Galerkin (DPG) FE methodology has proven to be a variationally flexible discretization method, crafted with automatic discrete stability via the use of optimal test functions, and naturally coupled with adaptivity. DPG has been recently introduced into the family of polygonal methods by Vaziri, Fuentes, Mora and Demkowicz, who have labeled their proposed methodology as PolyDPG. In 2D, the extension of DPG to polygonal elements has been enabled by the ultraweak variational formulation and broken test spaces. With respect to the ultraweak formulation, PolyDPG is a conforming method in 2D and, in special cases, in 3D as well. For general polyhedral meshes, PolyDPG is actually a non-conforming discretization. Given the special properties of DPG and the ultraweak variational formulation, we believe that PolyDPG can develop into a tool capable of solving difficult linear and nonlinear BVPs. In this dissertation we have engaged in three main goals, all of which will ultimately take us closer to a solution of the presented challenge problem through PolyDPG. First, we analyze the mathematics of PolyDPG as a non-conforming DPG method for linear BVPs in 3D, which helps in understanding its convergence. As a second goal, we develop the computational tools that are required for the implementation of the three-dimensional version of PolyDPG as a geometrically flexible, multiphysics and high order approximation FE method. A series of numerical experiments for the Poisson equation and the linear elasticity equations illustrate the convergence behavior of PolyDPG for different types of meshes and approximation orders. The work of our third goal, which is embodied in Part II of this dissertation, has as its starting point, the study of the theory related to hyperelastic materials and the nonlinear problem of finite elasticity. Next, we cover the derivation of a novel ultraweak formulation of the corresponding linearized problem. Once we have a discrete formulation suitable for PolyDPG, we propose a model problem that resembles the elastomeric syntactic foam problem. By means of a nonlinear solver, we perform a set of numerical experiments to assess the performance of PolyDPG and the new formulation. The results provide evidence that PolyDPG can indeed be a useful tool to carry out simulations that successfully capture large deformations in hyperelastic materials in 3D

Download or read book The Virtual Element Method and its Applications written by Paola F. Antonietti and published by Springer Nature. This book was released on 2022-10-08 with total page 621 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this book is to present the current state of the art of the Virtual Element Method (VEM) by collecting contributions from many of the most active researchers in this field and covering a broad range of topics: from the mathematical foundation to real life computational applications. The book is naturally divided into three parts. The first part of the book presents recent advances in theoretical and computational aspects of VEMs, discussing the generality of the meshes suitable to the VEM, the implementation of the VEM for linear and nonlinear PDEs, and the construction of discrete hessian complexes. The second part of the volume discusses Virtual Element discretization of paradigmatic linear and non-linear partial differential problems from computational mechanics, fluid dynamics, and wave propagation phenomena. Finally, the third part contains challenging applications such as the modeling of materials with fractures, magneto-hydrodynamics phenomena and contact solid mechanics. The book is intended for graduate students and researchers in mathematics and engineering fields, interested in learning novel numerical techniques for the solution of partial differential equations. It may as well serve as useful reference material for numerical analysts practitioners of the field.

Download or read book Numerical Mathematics and Advanced Applications ENUMATH 2019 written by Fred J. Vermolen and published by Springer Nature. This book was released on 2021-04-30 with total page 1185 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gathers outstanding papers presented at the European Conference on Numerical Mathematics and Advanced Applications (ENUMATH 2019). The conference was organized by Delft University of Technology and was held in Egmond aan Zee, the Netherlands, from September 30 to October 4, 2019. Leading experts in the field presented the latest results and ideas regarding the design, implementation and analysis of numerical algorithms, as well as their applications to relevant societal problems. ENUMATH is a series of conferences held every two years to provide a forum for discussing basic aspects and new trends in numerical mathematics and scientific and industrial applications, all examined at the highest level of international expertise. The first ENUMATH was held in Paris in 1995, with successive installments at various sites across Europe, including Heidelberg (1997), Jyvaskyla (1999), lschia Porto (2001), Prague (2003), Santiago de Compostela (2005), Graz (2007), Uppsala (2009), Leicester (2011), Lausanne (2013), Ankara (2015) and Bergen (2017).

Download or read book The Gradient Discretisation Method written by Jérôme Droniou and published by Springer. This book was released on 2018-07-31 with total page 501 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph presents the Gradient Discretisation Method (GDM), which is a unified convergence analysis framework for numerical methods for elliptic and parabolic partial differential equations. The results obtained by the GDM cover both stationary and transient models; error estimates are provided for linear (and some non-linear) equations, and convergence is established for a wide range of fully non-linear models (e.g. Leray–Lions equations and degenerate parabolic equations such as the Stefan or Richards models). The GDM applies to a diverse range of methods, both classical (conforming, non-conforming, mixed finite elements, discontinuous Galerkin) and modern (mimetic finite differences, hybrid and mixed finite volume, MPFA-O finite volume), some of which can be built on very general meshes.span style="" ms="" mincho";mso-bidi-font-family:="" the="" core="" properties="" and="" analytical="" tools="" required="" to="" work="" within="" gdm="" are="" stressed,="" it="" is="" shown="" that="" scheme="" convergence="" can="" often="" be="" established="" by="" verifying="" a="" small="" number="" of="" properties.="" scope="" some="" featured="" techniques="" results,="" such="" as="" time-space="" compactness="" theorems="" (discrete="" aubin–simon,="" discontinuous="" ascoli–arzela),="" goes="" beyond="" gdm,="" making="" them="" potentially="" applicable="" numerical="" schemes="" not="" (yet)="" known="" fit="" into="" this="" framework.span style="font-family:" ms="" mincho";mso-bidi-font-family:="" this="" monograph="" is="" intended="" for="" graduate="" students,="" researchers="" and="" experts="" in="" the="" field="" of="" numerical="" analysis="" partial="" differential="" equations./ppiiiiibr/i/i/i/i/i/p

Download or read book BEM based Finite Element Approaches on Polytopal Meshes written by Steffen Weißer and published by Springer. This book was released on 2019-07-18 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Numerical Mathematics and Advanced Applications ENUMATH 2017 written by Florin Adrian Radu and published by Springer. This book was released on 2019-01-05 with total page 1070 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book collects many of the presented papers, as plenary presentations, mini-symposia invited presentations, or contributed talks, from the European Conference on Numerical Mathematics and Advanced Applications (ENUMATH) 2017. The conference was organized by the University of Bergen, Norway from September 25 to 29, 2017. Leading experts in the field presented the latest results and ideas in the designing, implementation, and analysis of numerical algorithms as well as their applications to relevant, societal problems. ENUMATH is a series of conferences held every two years to provide a forum for discussing basic aspects and new trends in numerical mathematics and scientific and industrial applications. These discussions are upheld at the highest level of international expertise. The first ENUMATH conference was held in Paris in 1995 with successive conferences being held at various locations across Europe, including Heidelberg (1997), Jyvaskyla (1999), lschia Porto (2001), Prague (2003), Santiago de Compostela (2005), Graz (2007), Uppsala (2009), Leicester (2011), Lausanne (2013), and Ankara (2015).

Download or read book Mathematical Aspects of Discontinuous Galerkin Methods written by Daniele Antonio Di Pietro and published by Springer Science & Business Media. This book was released on 2011-11-03 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces the basic ideas to build discontinuous Galerkin methods and, at the same time, incorporates several recent mathematical developments. The presentation is to a large extent self-contained and is intended for graduate students and researchers in numerical analysis. The material covers a wide range of model problems, both steady and unsteady, elaborating from advection-reaction and diffusion problems up to the Navier-Stokes equations and Friedrichs' systems. Both finite element and finite volume viewpoints are exploited to convey the main ideas underlying the design of the approximation. The analysis is presented in a rigorous mathematical setting where discrete counterparts of the key properties of the continuous problem are identified. The framework encompasses fairly general meshes regarding element shapes and hanging nodes. Salient implementation issues are also addressed.

Download or read book The Scaled Boundary Finite Element Method written by Chongmin Song and published by John Wiley & Sons. This book was released on 2018-06-19 with total page 775 pages. Available in PDF, EPUB and Kindle. Book excerpt: An informative look at the theory, computer implementation, and application of the scaled boundary finite element method This reliable resource, complete with MATLAB, is an easy-to-understand introduction to the fundamental principles of the scaled boundary finite element method. It establishes the theory of the scaled boundary finite element method systematically as a general numerical procedure, providing the reader with a sound knowledge to expand the applications of this method to a broader scope. The book also presents the applications of the scaled boundary finite element to illustrate its salient features and potentials. The Scaled Boundary Finite Element Method: Introduction to Theory and Implementation covers the static and dynamic stress analysis of solids in two and three dimensions. The relevant concepts, theory and modelling issues of the scaled boundary finite element method are discussed and the unique features of the method are highlighted. The applications in computational fracture mechanics are detailed with numerical examples. A unified mesh generation procedure based on quadtree/octree algorithm is described. It also presents examples of fully automatic stress analysis of geometric models in NURBS, STL and digital images. Written in lucid and easy to understand language by the co-inventor of the scaled boundary element method Provides MATLAB as an integral part of the book with the code cross-referenced in the text and the use of the code illustrated by examples Presents new developments in the scaled boundary finite element method with illustrative examples so that readers can appreciate the significant features and potentials of this novel method—especially in emerging technologies such as 3D printing, virtual reality, and digital image-based analysis The Scaled Boundary Finite Element Method: Introduction to Theory and Implementation is an ideal book for researchers, software developers, numerical analysts, and postgraduate students in many fields of engineering and science.