This book is an introduction to numerical analysis in geomechanics and is intended for advanced undergraduate and beginning graduate study of the mechanics of porous, jointed rocks and soils. Although familiarity with the concepts of stress, strain and so on is assumed, a review of the fundamentals of solid mechanics including concepts of physical laws, kinematics and material laws is presented in an appendix. Emphasis is on the popular finite element method but brief explanations of the boundary element method, the distinct element method (also known as the discrete element method) and discontinuous deformation analysis are included. Familiarity with a computer programming language such as Fortran, C++ or Python is not required, although programming excerpts in Fortran are presented at the end of some chapters. This work begins with an intuitive approach to interpolation over a triangular element and thus avoids making the simple complex by not doing energy minimization via a calculus of variations approach so often found in reference books on the finite element method. The presentation then proceeds to a principal of virtual work via the well-known divergence theorem to obtain element equilibrium and then global equilibrium, both expressed as stiffness equations relating force to displacement. Solution methods for the finite element approach including elimination and iteration methods are discussed. Hydro-mechanical coupling is described and extension of the finite element method to accommodate fluid flow in porous geological media is made. Example problems illustrate important concepts throughout the text. Additional problems for a 15-week course of study are presented in an appendix; solutions are given in another appendix.

Advances in science and technology are driven by the development of rigorous mathematical foundations for the study of both theoretical and experimental models. With certain methodological variations, this type of study always comes down to the application of analytic or computational integration procedures, making such tools indispensible. With a wealth of cutting-edge research in the field, Integral Methods in Science and Engineering: Progress in Numerical and Analytic Techniques provides a detailed portrait of both the construction of theoretical integral techniques and their application to specific problems in science and engineering. The chapters in this volume are based on talks given by well-known researchers at the Twelfth International Conference on Integral Methods in Science and Engineering, July 23-27, 2012, in Porto Alegre, Brazil. They address a broad range of topics, from problems of existence and uniqueness for singular integral equations on domain boundaries to numerical integration via finite and boundary elements, conservation laws, hybrid methods, and other quadrature-related approaches. The contributing authors bring their expertise to bear on a number of topical problems that have to date resisted solution, thereby offering help and guidance to fellow professionals worldwide. Integral Methods in Science and Engineering: Progress in Numerical and Analytic Techniques will be a valuable resource for researchers in applied mathematics, physics, and mechanical and electrical engineering, for graduate students in these disciplines, and for various other professionals who use integration as an essential tool in their work.

This easy-to-follow textbook provides a student-friendly introduction to programming and algorithms. Emphasis is placed on the threshold concepts that present barriers to learning, including the questions that students are often too embarrassed to ask. The book promotes an active learning style in which a deeper understanding is gained from evaluating, questioning, and discussing the material, and practised in hands-on exercises. Although R is used as the language of choice for all programs, strict assumptions are avoided in the explanations in order for these to remain applicable to other programming languages. Features: provides exercises at the end of each chapter; includes three mini projects in the final chapter; presents a list of titles for further reading at the end of the book; discusses the key aspects of loops, recursions, program and algorithm efficiency and accuracy, sorting, linear systems of equations, and file processing; requires no prior background knowledge in this area.

This is a unique monograph on numerical conformal mapping that gives a comprehensive account of the theoretical, computational and application aspects of the problems of determining conformal modules of quadrilaterals and of mapping conformally onto a rectangle. It contains a detailed study of the theory and application of a domain decomposition method for computing the modules and associated conformal mappings of elongated quadrilaterals, of the type that occur in engineering applications. The reader will find a highly useful and up-to-date survey of available numerical methods and associated computer software for conformal mapping. The book also highlights the crucial role that function theory plays in the development of numerical conformal mapping methods, and illustrates the theoretical insight that can be gained from the results of numerical experiments. This is a valuable resource for mathematicians, who are interested in numerical conformal mapping and wish to study some of the recent developments in the subject, and for engineers and scientists who use, or would like to use, conformal transformations and wish to find out more about the capabilities of modern numerical conformal mapping. Contents: Standard Conformal Mappings and Software Conformal Modules of Quadrilaterals Domain Decomposition for Special Quadrilaterals Domain Decomposition for General Quadrilaterals Readership: Mathematicians, engineers and scientists interested in the theory and application of conformal mappings; advanced undergraduate and graduate students in mathematics.

Iterative methods use successive approximations to obtain more accurate solutions. Iterative Methods and Preconditioners for Systems of Linear Equations presents historical background, derives complete convergence estimates for all methods, illustrates and provides Matlab codes for all methods, and studies and tests all preconditioners first as stationary iterative solvers. This textbook is appropriate for undergraduate and graduate students in need of an overview or of deeper knowledge about iterative methods. It can be used in courses on Advanced Numerical Analysis, Special Topics on Numerical Analysis, Topics on Data Science, Topics on Numerical Optimization, and Topics on Approximation Theory. Scientists and engineers interested in new topics and applications will also find the text useful.

Data evaluation and data combination require the use of a wide range of probability theory concepts and tools, from deductive statistics mainly concerning frequencies and sample tallies to inductive inference for assimilating non-frequency data and a priori knowledge. Computational Methods for Data Evaluation and Assimilation presents interdisciplinary methods for integrating experimental and computational information. This self-contained book shows how the methods can be applied in many scientific and engineering areas. After presenting the fundamentals underlying the evaluation of experimental data, the book explains how to estimate covariances and confidence intervals from experimental data. It then describes algorithms for both unconstrained and constrained minimization of large-scale systems, such as time-dependent variational data assimilation in weather prediction and similar applications in the geophysical sciences. The book also discusses several basic principles of four-dimensional variational assimilation (4D VAR) and highlights specific difficulties in applying 4D VAR to large-scale operational numerical weather prediction models.

This workbook and solutions manual is intended for advanced undergraduate or beginning graduate students as a supplement to a traditional course in numerical mathematics and as preparation for independent research involving numerical mathematics. The solutions manual provides complete MATLAB code and numerical results for each of the exercises in the workbook and will be especially useful for those students without previous MATLAB programming experience. It is also valuable for classroom instructors to help pinpoint the author's intent in each exercise and to provide a model for graders. Upon completion of this material, students will have a working knowledge of MATLAB programming, they will have themselves programmed algorithms encountered in classwork and textbooks, and they will know how to check and verify their own programs against hand calculations and by reference to theoretical results, special polynomial solutions and other specialized solutions. No previous programming experience with MATLAB is necessary.

Meshless, or meshfree methods, which overcome many of the limitations of the finite element method, have achieved significant progress in numerical computations of a wide range of engineering problems. A comprehensive introduction to meshless methods, Meshless Methods and Their Numerical Properties gives complete mathematical formulations for the most important and classical methods, as well as several methods recently developed by the authors. This book also offers a rigorous mathematical treatment of their numerical properties-including consistency, convergence, stability, and adaptivity-to help you choose the method that is best suited for your needs. Get Guidance for Developing and Testing Meshless Methods Developing a broad framework to study the numerical computational characteristics of meshless methods, the book presents consistency, convergence, stability, and adaptive analyses to offer guidance for developing and testing a particular meshless method. The authors demonstrate the numerical properties by solving several differential equations, which offer a clearer understanding of the concepts. They also explain the difference between the finite element and meshless methods. Explore Engineering Applications of Meshless Methods The book examines how meshless methods can be used to solve complex engineering problems with lower computational cost, higher accuracy, easier construction of higher-order shape functions, and easier handling of large deformation and nonlinear problems. The numerical examples include engineering problems such as the CAD design of MEMS devices, nonlinear fluid-structure analysis of near-bed submarine pipelines, and two-dimensional multiphysics simulation of pH-sensitive hydrogels. Appendices supply useful template functions, flowcharts, and data structures to assist you in implementing meshless methods. Choose the Best Method for a Particular Problem Providing insight into the special features and intricacies of meshless methods, this is a valuable reference for anyone developing new high-performance numerical methods or working on the modelling and simulation of practical engineering problems. It guides you in comparing and verifying meshless methods so that you can more confidently select the best method to solve a particular problem.

This book gives a compact exposition of the fundamentals of the theory of locally convex topological vector spaces. Furthermore it contains a survey of the most important results of a more subtle nature, which cannot be regarded as basic, but knowledge which is useful for understanding applications. Finally, the book explores some of such applications connected with differential calculus and measure theory in infinite-dimensional spaces. These applications are a central aspect of the book, which is why it is different from the wide range of existing texts on topological vector spaces. Overall, this book develops differential and integral calculus on infinite-dimensional locally convex spaces by using methods and techniques of the theory of locally convex spaces. The target readership includes mathematicians and physicists whose research is related to infinite-dimensional analysis.

Multi-grid methods are the most efficient tools for solving elliptic boundary value problems. The reader finds here an elementary introduction to multi-grid algorithms as well as a comprehensive convergence analysis. One section describes special applications (convection-diffusion equations, singular perturbation problems, eigenvalue problems, etc.). The book also contains a complete presentation of the multi-grid method of the second kind, which has important applications to integral equations (e.g. the "panel method") and to numerous other problems. Readers with a practical interest in multi-grid methods will benefit from this book as well as readers with a more theoretical interest.

This volume includes contributions from diverse disciplines including electrical engineering, biomedical engineering, industrial engineering, and medicine, bridging a vital gap between the mathematical sciences and neuroscience research. Covering a wide range of research topics, this volume demonstrates how various methods from data mining, signal processing, optimization and cutting-edge medical techniques can be used to tackle the most challenging problems in modern neuroscience.

This book gathers outstanding papers on numerical modeling in Mechanical Engineering (Volume 2) as part of the 2-volume proceedings of the 4th International Conference on Numerical Modeling in Engineering (NME 2021), which was held in Ghent, Belgium, on 24-25 August 2021. The overall objective of the conference was to bring together international scientists and engineers in academia and industry from fields related to advanced numerical techniques, such as the finite element method (FEM), boundary element method (BEM), isogeometric analysis (IGA), etc., and their applications to a wide range of engineering disciplines. This book addresses numerical simulations of various mechanical and materials engineering industrial applications such as aerospace applications, acoustic analysis, bio-mechanical applications, contact problems and wear, heat transfer analysis, vibration and dynamics, transient analysis, nonlinear analysis, composite materials, polymers, metal alloys, fracture mechanics, fatigue of materials, creep, mechanical behavior, micro-structure, phase transformation, and crystal plasticity.

In this monograph the authors develop a theory for the robust control of discrete-time stochastic systems, subjected to both independent random perturbations and to Markov chains. Such systems are widely used to provide mathematical models for real processes in fields such as aerospace engineering, communications, manufacturing, finance and economy. The theory is a continuation of the authors work presented in their previous book entitled "Mathematical Methods in Robust Control of Linear Stochastic Systems" published by Springer in 2006.

Key features:

- Provides a common unifying framework for discrete-time stochastic systems corrupted with both independent random perturbations and with Markovian jumps which are usually treated separately in the control literature;

- Covers preliminary material on probability theory, independent random variables, conditional expectation and Markov chains;

- Proposes new numerical algorithms to solve coupled matrix algebraic Riccati equations;

- Leads the reader in a natural way to the original results through a systematic presentation;

- Presents new theoretical results with detailed numerical examples.

The monograph is geared to researchers and graduate students in advanced control engineering, applied mathematics, mathematical systems theory and finance. It is also accessible to undergraduate students with a fundamental knowledge in the theory of stochastic systems."

The book teaches a student to model a scientific problem and write a computer program in C language to solve that problem. To do that, the book first introduces the student to the basics of C language, dealing with all syntactical aspects, but without the pedantic content of a typical programming language manual. Then the book describes and discusses many algorithms commonly used in scientific applications (e.g. searching, graphs, statistics, equation solving, Monte Carlo methods etc.).This important book fills a gap in current available bibliography. There are many manuals for programming in C, but they never explain programming technicalities to solve a given problem. This book illustrates many relevant algorithms and shows how to translate them in a working computer program.

While the finite element method (FEM) has become the standard technique used to solve static and dynamic problems associated with structures and machines, ANSYS software has developed into the engineer's software of choice to model and numerically solve those problems.

An invaluable tool to help engineers master and optimize analysis, The Finite Element Method for Mechanics of Solids with ANSYS Applications explains the foundations of FEM in detail, enabling engineers to use it properly to analyze stress and interpret the output of a finite element computer program such as ANSYS.

Illustrating presented theory with a wealth of practical examples, this book covers topics including:

• Essential background on solid mechanics (including small- and large-deformation elasticity, plasticity, and viscoelasticity) and mathematics
• Advanced finite element theory and associated fundamentals, with examples
• Use of ANSYS to derive solutions for problems that deal with vibration, wave propagation, fracture mechanics, plates and shells, and contact

Totally self-contained, this text presents step-by-step instructions on how to use ANSYS Parametric Design Language (APDL) and the ANSYS Workbench to solve problems involving static/dynamic structural analysis (both linear and non-linear) and heat transfer, among other areas. It will quickly become a welcome addition to any engineering library, equally useful to students and experienced engineers alike.

Essential Numerical Methods for Electromagnetics presents key contributions selected from the volume in the Handbook of Numerical Analysis: Numerical Methods for Electromagnetics Vol. 13 (2005).

This reference is an accessible resource on the basics of modeling. It is designed to assist professionals in the development of electromagnetic designs for electronic components and devices. It provides essential numerical methods and applications necessary for the development of technologies and simulation modeling. Numerical methods are a key ingredient in a simulation environment where researchers create virtually simulated experiments versus physical experiments. This book serves as a useful guide for scientists, engineers, and researchers providing a quick reference of commonly used numerical methods to help solve a variety of problems in the electronic industry.

The basics of modeling aspects provide an accessible resource; Numerical solution procedures for quick reference; Special numerical techniques are presented to assist in specialization; Most commonly used methods and applications to create simulation experiments;

Classical and Modern Numerical Analysis: Theory, Methods and Practice provides a sound foundation in numerical analysis for more specialized topics, such as finite element theory, advanced numerical linear algebra, and optimization. It prepares graduate students for taking doctoral examinations in numerical analysis.

The text covers the main areas of introductory numerical analysis, including the solution of nonlinear equations, numerical linear algebra, ordinary differential equations, approximation theory, numerical integration, and boundary value problems. Focusing on interval computing in numerical analysis, it explains interval arithmetic, interval computation, and interval algorithms. The authors illustrate the concepts with many examples as well as analytical and computational exercises at the end of each chapter.

This advanced, graduate-level introduction to the theory and methods of numerical analysis supplies the necessary background in numerical methods so that students can apply the techniques and understand the mathematical literature in this area. Although the book is independent of a specific computer program, MATLAB(r) code is available on the authors' website to illustrate various concepts.

Numerical and Analytical Methods with MATLAB? presents extensive coverage of the MATLAB programming language for engineers. It demonstrates how the built-in functions of MATLAB can be used to solve systems of linear equations, ODEs, roots of transcendental equations, statistical problems, optimization problems, control systems problems, and stress analysis problems. These built-in functions are essentially black boxes to students. By combining MATLAB with basic numerical and analytical techniques, the mystery of what these black boxes might contain is somewhat alleviated.

This classroom-tested text first reviews the essentials involved in writing computer programs as well as fundamental aspects of MATLAB. It next explains how matrices can solve problems of linear equations, how to obtain the roots of algebraic and transcendental equations, how to evaluate integrals, and how to solve various ODEs. After exploring the features of Simulink, the book discusses curve fitting, optimization problems, and PDE problems, such as the vibrating string, unsteady heat conduction, and sound waves. The focus then shifts to the solution of engineering problems via iteration procedures, differential equations via Laplace transforms, and stress analysis problems via the finite element method. The final chapter examines control systems theory, including the design of single-input single-output (SISO) systems.

Two Courses in One Textbook
The first six chapters are appropriate for a lower level course at the sophomore level. The remaining chapters are ideal for a course at the senior undergraduate or first-year graduate level. Most of the chapters contain projects that require students to write a computer program in MATLAB that produces tables, graphs, or both. Many sample MATLAB programs (scripts) in the text provide guidance on completing these projects.

A systematic introduction to partial differential
equations and modern finite element methods for their efficient numerical solution
Partial Differential Equations and the Finite Element Method provides a much-needed, clear, and systematic introduction to modern theory of partial differential equations (PDEs) and finite element methods (FEM). Both nodal and hierachic concepts of the FEM are examined. Reflecting the growing complexity and multiscale nature of current engineering and scientific problems, the author emphasizes higher-order finite element methods such as the spectral or hp-FEM.
A solid introduction to the theory of PDEs and FEM contained in Chapters 1-4 serves as the core and foundation of the publication. Chapter 5 is devoted to modern higher-order methods for the numerical solution of ordinary differential equations (ODEs) that arise in the semidiscretization of time-dependent PDEs by the Method of Lines (MOL). Chapter 6 discusses fourth-order PDEs rooted in the bending of elastic beams and plates and approximates their solution by means of higher-order Hermite and Argyris elements. Finally, Chapter 7 introduces the reader to various PDEs governing computational electromagnetics and describes their finite element approximation, including modern higher-order edge elements for Maxwell's equations.
The understanding of many theoretical and practical aspects of both PDEs and FEM requires a solid knowledge of linear algebra and elementary functional analysis, such as functions and linear operators in the Lebesgue, Hilbert, and Sobolev spaces. These topics are discussed with the help of many illustrative examples in Appendix A, which is provided as a service for those readers who need to gain the necessary background or require a refresher tutorial. Appendix B presents several finite element computations rooted in practical engineering problems and demonstrates the benefits of using higher-order FEM.
Numerous finite element algorithms are written out in detail alongside implementation discussions. Exercises, including many that involve programming the FEM, are designed to assist the reader in solving typical problems in engineering and science.
Specifically designed as a coursebook, this student-tested publication is geared to upper-level undergraduates and graduate students in all disciplines of computational engineeringand science. It is also a practical problem-solving reference for researchers, engineers, and physicists.

Nonlinearity plays a major role in the understanding of most physical, chemical, biological, and engineering sciences. Nonlinear problems fascinate scientists and engineers, but often elude exact treatment. However elusive they may be, the solutions do exist-if only one perseveres in seeking them out. Self-Similarity and Beyond presents a myriad of approaches to finding exact solutions for a diversity of nonlinear problems. These include group-theoretic methods, the direct method of Clarkson and Kruskal, traveling waves, hodograph methods, balancing arguments, embedding special solutions into a more general class, and the infinite series approach. The author's approach is entirely constructive. Numerical solutions either motivate the analysis or confirm it, therefore they are treated alongside the analysis whenever possible. Many examples drawn from real physical situations-primarily fluid mechanics and nonlinear diffusion-illustrate and emphasize the central points presented. Accessible to a broad base of readers, Self-Similarity and Beyond illuminates a variety of productive methods for meeting the challenges of nonlinearity. Researchers and graduate students in nonlinearity, partial differential equations, and fluid mechanics, along with mathematical physicists and numerical analysts, will re-discover the importance of exact solutions and find valuable additions to their mathematical toolkits.

This important textbook provides an introduction to the concepts of the newly developed extended finite element method (XFEM) for fracture analysis of structures, as well as for other related engineering applications.
One of the main advantages of the method is that it avoids any need for remeshing or geometric crack modelling in numerical simulation, while generating discontinuous fields along a crack and around its tip. The second major advantage of the method is that by a small increase in number of degrees of freedom, far more accurate solutions can be obtained. The method has recently been extended to nonlinear materials and other disciplines such as modelling contact and interface, simulation of inclusions and holes, moving and changing phase problems, and even to multiscale analyses.
The book is self contained, with summaries of both classical and modern computational techniques. The main chapters include a comprehensive range of numerical examples describing various features of XFEM.

Explore a Unified Treatment of the Finite Element Method The finite element method has matured to the point that it can accurately and reliably be used, by a careful analyst, for an amazingly wide range of applications. With expanded coverage and an increase in fully solved examples, the second edition of Finite Element Analysis: Thermomechanics of Solids presents a unified treatment of the finite element method in theremomechanics, from the basics to advanced concepts. An Integrated Presentation of Critical Technology As in the first edition, the author presents and explicates topics in a way that demonstrates the highly unified structure of the finite element method. The presentation integrates continuum mechanics and relevant mathematics with persistent reliance on variational and incremental-variational foundations. The author exploits matrix-vector formalisms and Kronecker product algebra to provide transparent and consistent notation throughout the text. Nearly twice as long as the first edition, this second edition features: Greater integration and balance between introductory and advanced material Increased number of fully solved examples Selected developments in numerical methods, detailing accelerating computations in eigenstructure extraction, time integration, and stiffness matrix triangularization More extensive coverage of the arc length method for nonlinear problems Expanded and enhanced treatment of rotating bodies and buckling Provides Sophisticated Understanding of Capabilities and Limitations This new edition of a popular text includes significant illustrative examples and applications, modeling strategies, and explores a range

This book addresses the modelling of mechanical waves by asking the right questions about them and trying to find suitable answers. The questions follow the analytical sequence from elementary understandings to complicated cases, following a step-by-step path towards increased knowledge. The focus is on waves in elastic solids, although some examples also concern non-conservative cases for the sake of completeness. Special attention is paid to the understanding of the influence of microstructure, nonlinearity and internal variables in continua. With the help of many mathematical models for describing waves, physical phenomena concerning wave dispersion, nonlinear effects, emergence of solitary waves, scales and hierarchies of waves as well as the governing physical parameters are analysed. Also, the energy balance in waves and non-conservative models with energy influx are discussed. Finally, all answers are interwoven into the canvas of complexity.

The book presents, in a systematic manner, the optimal controls under different mathematical models in fermentation processes. Variant mathematical models - i.e., those for multistage systems; switched autonomous systems; time-dependent and state-dependent switched systems; multistage time-delay systems and switched time-delay systems - for fed-batch fermentation processes are proposed and the theories and algorithms of their optimal control problems are studied and discussed. By putting forward novel methods and innovative tools, the book provides a state-of-the-art and comprehensive systematic treatment of optimal control problems arising in fermentation processes. It not only develops nonlinear dynamical system, optimal control theory and optimization algorithms, but can also help to increase productivity and provide valuable reference material on commercial fermentation processes.

This book provides a clear summary of the work of the author on the construction of nonstandard finite difference schemes for the numerical integration of differential equations. The major thrust of the book is to show that discrete models of differential equations exist such that the elementary types of numerical instabilities do not occur. A consequence of this result is that in general bigger step-sizes can often be used in actual calculations and/or finite difference schemes can be constructed that are conditionally stable in many instances whereas in using standard techniques no such schemes exist. The theoretical basis of this work is centered on the concepts of "exact" and "best" finite difference schemes. In addition, a set of rules is given for the discrete modeling of derivatives and nonlinear expressions that occur in differential equations. These rules often lead to a unique nonstandard finite difference model for a given differential equation.