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The field of multiscale problems occurs in many fields of science, such as microstructures in materials, sharp-interface models, and others. Reporting on the mathematical developments in the DFG Priority programme, this book provides the state-of-the-art on the mathematical foundations of the modeling and the numerical treatment of such problems.

Mathematical analysis. --- Mathematical models. --- Models, Mathematical --- Simulation methods --- 517.1 Mathematical analysis --- Mathematical analysis --- Computer science --- Differential equations, partial. --- Numerical analysis. --- Computational Mathematics and Numerical Analysis. --- Partial Differential Equations. --- Numerical Analysis. --- Mathematics. --- Partial differential equations --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Mathematics --- Computer mathematics. --- Partial differential equations.

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Partial differential equations --- Numerical analysis --- Computer. Automation --- differentiaalvergelijkingen --- informatica --- wiskunde --- numerieke analyse

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This monograph provides both an introduction to and a thorough exposition of the theory of rate-independent systems, which the authors have worked on with a number of collaborators over many years. The focus is mostly on fully rate-independent systems, first on an abstract level with or without a linear structure, discussing various concepts of solutions with full mathematical rigor. The usefulness of the abstract concepts is then demonstrated on the level of various applications primarily in continuum mechanics of solids, including suitable approximation strategies with guaranteed numerical stability and convergence. Particular applications concern inelastic processes such as plasticity, damage, phase transformations, or adhesive-type contacts both at small strains and at finite strains. Other physical systems such as magnetic or ferroelectric materials, and couplings to rate-dependent thermodynamic models are also considered. Selected applications are accompanied by numerical simulations illustrating both the models and the efficiency of computational algorithms. This book presents the mathematical framework for a rigorous mathematical treatment of rate-independent systems in a comprehensive form for the first time. Researchers and graduate students in applied mathematics, engineering, and computational physics will find this timely and well-written book useful.

Differential equations, Partial --- Differential equations --- Banach spaces --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Differential equations, Partial. --- Differential equations. --- Banach spaces. --- 517.91 Differential equations --- Partial differential equations --- Mathematics. --- Partial differential equations. --- Physics. --- Continuum mechanics. --- Partial Differential Equations. --- Mathematical Methods in Physics. --- Continuum Mechanics and Mechanics of Materials. --- Mechanics of continua --- Elasticity --- Mechanics, Analytic --- Field theory (Physics) --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Math --- Science --- Functions of complex variables --- Generalized spaces --- Topology --- Differential equations, partial. --- Mathematical physics. --- Mechanics. --- Mechanics, Applied. --- Solid Mechanics. --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Quantum theory --- Physical mathematics

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Understanding the interaction between various processes is a pre-requisite for solving problems in natural and engineering sciences. Many phenomena can not be described by concentrating on them in isolation - therefore multifield models and concepts that include various kinds of field problems and processes are needed. This book summarizes the main scientific results of the Collaborative Research Center on Multifield Problems in Continuum Mechanics (Sonderforschungsbereich Mehrfeldprobleme in der Kontinuumsmechanik, SFB 404) funded by the German Research Foundation (DFG) from 1995-2006. 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.

Continuum mechanics --- Mechanics, Analytic --- Fluid mechanics --- Milieux continus, Mécanique des --- Mécanique analytique --- Mécanique des fluides --- Continuum mechanics. --- Fluid mechanics. --- Mechanics, Analytic. --- Applied Mathematics --- Civil Engineering --- Engineering & Applied Sciences --- Civil & Environmental Engineering --- Hydromechanics --- Analytical mechanics --- Kinetics --- Mechanics of continua --- Engineering. --- Computer mathematics. --- Mechanics. --- Computational intelligence. --- Mechanics, Applied. --- Theoretical and Applied Mechanics. --- Continuum Mechanics and Mechanics of Materials. --- Computational Intelligence. --- Computational Mathematics and Numerical Analysis. --- Elasticity --- Field theory (Physics) --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Intelligence, Computational --- Artificial intelligence --- Soft computing --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Construction --- Industrial arts --- Technology --- Mathematics --- Mechanics, applied. --- Computer science --- Solid Mechanics. --- Classical Mechanics. --- Mathematics. --- Solids. --- Engineering Mechanics. --- Data processing. --- Solid state physics --- Transparent solids

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This monograph provides both an introduction to and a thorough exposition of the theory of rate-independent systems, which the authors have worked on with a number of collaborators over many years. The focus is mostly on fully rate-independent systems, first on an abstract level with or without a linear structure, discussing various concepts of solutions with full mathematical rigor. The usefulness of the abstract concepts is then demonstrated on the level of various applications primarily in continuum mechanics of solids, including suitable approximation strategies with guaranteed numerical stability and convergence. Particular applications concern inelastic processes such as plasticity, damage, phase transformations, or adhesive-type contacts both at small strains and at finite strains. Other physical systems such as magnetic or ferroelectric materials, and couplings to rate-dependent thermodynamic models are also considered. Selected applications are accompanied by numerical simulations illustrating both the models and the efficiency of computational algorithms. This book presents the mathematical framework for a rigorous mathematical treatment of rate-independent systems in a comprehensive form for the first time. Researchers and graduate students in applied mathematics, engineering, and computational physics will find this timely and well-written book useful.

Partial differential equations --- Mathematical physics --- Classical mechanics. Field theory --- Solid state physics --- Physics --- Applied physical engineering --- differentiaalvergelijkingen --- toegepaste mechanica --- wiskunde --- fysica --- mechanica