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Numerical Treatment of Partial Differential Equations
In 1988 we started work on the frst German edition of our book, which appeared in 1992. Our aim was to give students a textbook that contained the basic concepts and ideas behind most numerical methods for partial di?er- tial equations. The success of this frst edition and the second edition in 1994 encouraged us, ten years later, to write an almost completely new version, taking into account comments from colleagues and students and drawing on the enormous progress made in the numerical analysis of partial di?erential equations in recent times. The present English version slightly improves the third German edition of 2005: we have corrected some minor errors and added additional material and references.
Numerical solution of Variational Inequalities by Adaptive Finite Elements
Franz-Theo Suttmeier describes a general approach to a posteriori error estimation and adaptive mesh design for finite element models where the solution is subjected to inequality constraints. This is an extension to variational inequalities of the so-called Dual-Weighted-Residual method (DWR method) which is based on a variational formulation of the problem and uses global duality arguments for deriving weighted a posteriori error estimates with respect to arbitrary functionals of the error. In these estimates local residuals of the computed solution are multiplied by sensitivity factors which are obtained from a numerically computed dual solution. The resulting local error indicators are used in a feed-back process for generating economical meshes which are tailored according to the particular goal of the computation.
Numerical partial differential equations for environmental scientists and engineers : A first practical course
This book concerns the practical solution of Partial Differential Equations (PDEs). It reflects an interdisciplinary approach to problems occurring in natural environmental media: the hydrosphere, atmosphere, cryosphere, lithosphere, biosphere and ionosphere. It assumes the reader has gained some intuitive knowledge of PDE solution properties and now wants to solve some for real, in the context of practical problems arising in real situations. The practical aspect of this book is the infused focus on computation. It presents two major discretization methods - Finite Difference and Finite Element. The blend of theory, analysis, and implementation practicality supports solving and understanding complicated problems. It is divided into three parts. Part I is an overview of Finite Difference Methods. Part II focuses on Finite Element Methods, including an FEM tutorial. Part III deals with Inverse Methods, introducing formal approaches to practical problems which are ill-posed.
Numerical Mathematics and Advanced Applications ; Proceedings of ENUMATH 2007, the 7th European Conference on Numerical Mathematics and Advanced Applications, Graz, Austria, September 2007
The European Conference on Numerical Mathematics and Advanced Applications (ENUMATH) is a series of meetings held every two years to provide a forum for discussion on recent aspects of numerical mathematics and their applications. These proceedings contain a selection of invited plenary lectures, papers presented in minisymposia and contributed papers. Topics include theoretical aspects of new numerical techniques and algorithms as well as of applications in engineering and science. The book will be useful for a wide range of readers, giving them an excellent overview of the most modern methods, techniques, algorithms and results in numerical mathematics, scientific computing and their applications.
Numerical computation, data analysis and software in mathematics and engineering
Include the aspects of the meshless method, numerical simulation, mathematical models, deep learning and data analysis. Meshless methods, such as the improved element-free Galerkin method, the dimension-splitting, interpolating, moving, least-squares method, the dimension-splitting, generalized, interpolating, element-free Galerkin method and the improved interpolating, complex variable, element-free Galerkin method, are presented. Some complicated problems, such as tge cold roll-forming process, ceramsite compound insulation block, crack propagation and heavy-haul railway tunnel with defects, are numerically analyzed.
Numerical Approximation Methods for Elliptic Boundary Value Problems : Finite and Boundary Elements
Although the aim of this book is to give a unified introduction into finite and boundary element methods, the main focus is on the numerical analysis of boundary integral and boundary element methods. Starting from the variational formulation of elliptic boundary value problems boundary integral operators and associated boundary integral equations are introduced and analyzed. By using finite and boundary elements corresponding numerical approximation schemes are considered.
Nonlinear Finite Element Methods
Finite element methods have become ever more important to engineers as tools for design and optimization, now even for solving non-linear technological problems. However, several aspects must be considered for finite-element simulations which are specific for non-linear problems: These problems require the knowledge and the understanding of theoretical foundations and their finite-element discretization as well as algorithms for solving the non-linear equations. This book provides the reader with the required knowledge covering the complete field of finite element analyses in solid mechanics.
Nonlinear Elliptic and Parabolic Problems : A Special Tribute to the Work of Herbert Amann
The present volume is dedicated to celebrate the work of the renowned mathematician Herbert Amann, who had a significant and decisive influence in shaping Nonlinear Analysis. Most articles published in this book, which consists of 32 articles in total, written by highly distinguished researchers, are in one way or another related to the scientific works of Herbert Amann. The contributions cover a wide range of nonlinear elliptic and parabolic equations with applications to natural sciences and engineering. Special topics are fluid dynamics, reaction-diffusion systems, bifurcation theory, maximal regularity, evolution equations, and the theory of function spaces.
Nodal Discontinuous Galerkin Methods : Algorithms, Analysis, and Applications
This book discusses a family of computational methods, known as discontinuous Galerkin methods, for solving partial differential equations. While these methods have been known since the early 1970s, they have experienced an almost explosive growth interest during the last ten to fifteen years, leading both to substantial theoretical developments and the application of these methods to a broad range of problems. These methods are different in nature from standard methods such as finite element or finite difference methods, often presenting a challenge in the transition from theoretical developments to actual implementations and applications.
Multiscale Modeling in Epitaxial Growth
Epitaxy is a very active area of theoretical research since several years. It is experimentally well-explored and technologically relevant for thin film growth. Recently powerful numerical techniques in combination with a deep understanding of the physical and chemical phenomena during the growth process offer the possibility to link atomistic effects at the surface to the macroscopic morphology of the film. The goal of this book is to summarize recent developments in this field, with emphasis on multiscale approaches and numerical methods. It covers atomistic, step-flow, and continuum models and provides a compact overview of these approaches. It also serves as an introduction into this highly active interdisciplinary field of research for applied mathematicians, theoretical physicists and computational materials scientists.
Multifield Problems in Solid and Fluid Mechanics
This book summarizes the main scientific results of the Collaborative Research Center on Multifield Problems in Continuum Mechanics. The book is divided into three main sections: A: Volume-Coupled Problems, devoted to fields which are coupled inside the processing domain or volume, B: Boundary-Coupled Problems, here physical fields and processes are coupled via domain boundaries, C: Fundamental Methods, search into the mathematical concepts and backgrounds of multifield and multiscale modeling.
Modeling, Control and Implementation of Smart Structures : A FEM-State Space Approach
This monograph presents an introductory overview of smart structures, their concepts, their active involvement in the vibration control, their applications and the extensive research work done on it so far. The modelling of flexible beams using two types of beam theories, viz., the Euler-Bernoulli theory and the Timoshenko beam theory is presented, including a new concept of finite element modeling of the flexible structures using Timoshenko beam theory with the inclusion of the shear both in the piezo-patches as well as in the host structure. It presents the design of the periodic output feedback control system for smart structure systems, the design of the FOS controllers for active vibration control and the design of Discrete Sliding Mode controllers using multirate output feedback technique.
Modeling of metal forming and machining processes : By finite element and soft computing methods
The physics of metal forming and metal removing is normally expressed using non-linear partial differential equations which can be solved using the finite element method (FEM). However, when the process parameters are uncertain and/or the physics of the process is not well understood, soft computing techniques can be used with FEM or alone to model the process.
Modeling Excitable Tissue : The EMI Framework
This volume presents a novel computational framework for understanding how collections of excitable cells work. The key approach in the text is to model excitable tissue by representing the individual cells constituting the tissue. This is in stark contrast to the common approach where homogenization is used to develop models where the cells are not explicitly present. The approach allows for very detailed analysis of small collections of excitable cells, but computational challenges limit the applicability in the presence of large collections of cells.
Mixed Finite Elements, Compatibility Conditions, and Applications : Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy June 26–July 1, 2006
Since the early 70's, mixed finite elements have been the object of a wide and deep study by the mathematical and engineering communities. The fundamental role of this method for many application fields has been worldwide recognized and its use has been introduced in several commercial codes. An important feature of mixed finite elements is the interplay between theory and application. Discretization spaces for mixed schemes require suitable compatibilities, so that simple minded approximations generally do not work and the design of appropriate stabilizations gives rise to challenging mathematical problems.
Micro-Macro-interaction : In Structured media and Particle Systems
This book presents the development of new methods in this interdisciplinary field of macro- micro-interactions of different engineering branches like mechanical and process engineering, applied mathematics, theoretical, and computational physics. In particular, solids with microstructures and particle systems are considered.
Meshless Methods in Solid Mechanics
The main objective of this book is to provide a textbook for graduate courses on the computational analysis of continuum and solid mechanics based on meshless (also known as mesh free) methods. It can also be used as a reference book for engineers and scientists who are exploring the physical world through computer simulations. Emphasis of this book is given to the understanding of the physical and mathematical characteristics of the procedures of computational solid mechanics. It naturally brings the essence, advantages and challenging problems of meshless methods into the picture. The subjects in this book cover the fundamentals of continuum mechanics, the integral formulation methods of continuum problems, the basic concepts of finite element methods, and the methodologies, formulations, procedures, and applications of various meshless methods. It also provides general and detailed procedures of meshless analysis on elastostatics, elastodynamics, non-local continuum mechanics and plasticity with a large number of numerical examples. Some basic and important mathematical methods are included in the Appendixes. For the readers who want to gain knowledge through hands-on experience, the meshless programs for elastostatics and elastodynamics are also introduced in the book and included in the disc.
Meshfree Methods for Partial Differential Equations IV
The numerical treatment of partial differential equations with particle methods and meshfree discretization techniques is a very active research field both in the mathematics and engineering community. Due to their independence of a mesh, particle schemes and meshfree methods can deal with large geometric changes of the domain more easily than classical discretization techniques. Furthermore, meshfree methods offer a promising approach for the coupling of particle models to continuous models. This volume of LNCSE is a collection of the proceedings papers of the Fourth International Workshop on Meshfree Methods held in September 2007 in Bonn. The articles address the different meshfree methods (SPH, PUM, GFEM, EFGM, RKPM, etc.) and their application in applied mathematics, physics and engineering. The volume is intended to foster this very active and exciting area of interdisciplinary research and to present recent advances and results in this field.
Meshfree Methods for Partial Differential Equations III
Meshfree methods for the numerical solution of partial differential equations are becoming more and more mainstream in many areas of applications. Their flexiblity and wide applicability are attracting engineers, scientists, and mathematicians to this very dynamic research area. This volume represents the state of the art in meshfree methods. It consists of articles which address the different meshfree techniques, their mathematical properties and their application in applied mathematics, physics and engineering.
Meshfree Methods for Partial Differential Equations II
A Particle Strategy for Solving the Fokker-Planck Equation Modelling the Fiber Orientation Distribution in Steady Recirculating Flows Involving Short Fiber Suspensions.- Extended Meshfree Method for Elastic and Inelastic Media.- Meshfree Petrov-Galerkin Methods for the Incompressible Navier-Stokes Equations.- The ?-shape Based Natural Element Method in Solid and Fluid Mechanics.- A Particle-Partition of Unity Method Part VI: A p-robust Multilevel Solver.- Enriched Reproducing Kernel Approximation: Reproducing Functions with Discontinuous Derivatives.- Reproducing Kernel Element Interpolation: Globally Conforming I m/C n/P k Hierarchies.- Multi-scale Analysis Using Two Influence Radii in EFGM.- Solution of a Dynamic Main Crack Interaction with a System of Micro-Cracks by the Element Free Galerkin Method.- Finite Cover Method for Physically and Geometrically Nonlinear Problems.- A Numerical Scheme for Solving Incompressible and Low Mach Number Flows by the Finite Pointset Method.- SPH Renormalized Hybrid Methods for Conservation Laws: Applications to Free Surface Flows.- Discontinuous Radial Basis Function Approximations for Meshfree Methods.- Treating Moving Interfaces in Thermal Models with the C-NEM.- Bridging Scale Particle and Finite Element Methods.
