Dynamic relaxation (DR) is a widely spread numerical method used in fields as structural dynamics (particularly in form-finding), geomechanics or biomechanics, among others. When using DR in numerical simulations, researchers usually follow two different paths in the choice of a damping strategy for the oscillations: some use a viscous damping while others use a kinetic damping. However, it is difficult to find comparisons between both methods to help deciding whether one is better than the other for a particular application. Focused in the field of form-finding of thin-walled structures, the main objective of this book is to make a contribution to the development of DR methods and a review of some of the existing DR methods in order to compare them. Also, as an application for the studied methods, a contribution to the modelling of inflatable lifejackets will be presented: the aim of this part of the work is to present some contributions to the creation of a numerical tool permitting to test the functioning of commercial inflatable lifejackets by means of Finite Elements calculations.
This book is designed as an advanced guide for numerical methods in the science. It covers many topics of practical numerical methods applied in the science: solutions of equations with one variable: bisection, secant, false rule, Newton-Raphson, fixed point, etc., solution of systems of equations: Gauss, Gauss-Jordan, Cramer, Inverse Matrix, Jacobi, Gauss-Seidel, Gauss-Seidel with relaxation, etc., polynomial interpolation: Lagrange interpolation, Newton interpolation, interpolation with equidistant spaces, etc., the method of the least square method for a polynomial fit (regression analysis), etc., numerical derivatives, finite differential discretization of the derivative, numerical integrations: trapeze method, Simpson 1/3, Simpson 3/8, differential equations: Euler, Runge-Kutta, differential equations with boundary values, etc. It is included the deduction of many formulas in order to clear the concepts of the numerical methods applied in Science. It is hoped that this book fills all needs of the students to get the fundaments of the numerical methods and to achieve the interest and motivation of the students for this topic.
Numerical Simulation in Micropolar Fluid Dynamics presents a detailed study of various multi-physical flow problems of micropolar non-Newtonian fluids. A comprehensive overview of modern micropolar fluid mechanics and applications is provided. A number of powerful numerical methods including finite element methods, homotopy analysis methods, network simulation and differential transform methods are then applied to specific steady flow scenarios. Magnetohydrodynamic, double-diffusive convection, magneto-tribological, stagnation point, chemically-reacting and enclosure flows are all studied. Extensive details of numerical analysis are provided. It is hoped that this monograph will stimulate further research in related fields.
Numerical methods are powerful problem-solving tools, Techniques of these methods are capable of handing large system of equation, some problem in physics and engineering which are impossible to be solved analytically. The main object in this thesis is to study and reformulate some numerical methods for solving system of retarded delay differential equations. We get good results in each presented methods for solving system of retarded delay differential equations. This is done by writing a computer Maple program version 13. For the purpose of comparison we compared exact results with approximate solution which is obtained by the above methods.
Precise numerical analysis may be defined as the study of computer methods for solving mathematical problems either exactly or to prescribed accuracy. This book explains how precise numerical analysis is constructed. It includes a CD-ROM which contains executable Windows XP programs for the PC and which demonstrates how these programs can be used to solve typical problems of elementary numerical analysis with precision. The book also provides exercises which illustrate points from the text and references for the methods presented.A· Clearer, simpler descriptions and explanations ofthe various numerical methodsA· Windows based softwareA· Two new types of numerical problems; accurately solving partial differential equations with the included software and computing line integrals in the complex plane.
This book contains Concept of Numerical Analysis, The Sources and Propagation of Error, Computer Representation of Numbers, Sources of Error, Propagation of Errors, Propagation of Error in Sum, Stability in Numerical Analysis, Root Finding For Nonlinear Equations, Simple Enclosure Methods, Secant Method, Newton’s Method, General Theory for One Point Iteration Methods, Error Tests. Numerical Evaluation of Multiple Roots, Roots of Polynomials, Mullers Method, Non-Linear Systems of Equations, Newton’s Method for Non- Linear Systems, Interpolation Theory, Polynomial Interpolation Theory, Newton’s Divided Differences, Finite Difference and Table Oriented Interpolation formulas, Forward-Differences, Hermite Interpolation, Spline Functions,Approximation of Functions, Numerical Integration,Numerical Methods for Differential Equation. This book is useful for post graduate students.
Dynamic loads can cause severe damage to bridges and lead to malfunction of transportation networks. A comprehensive understanding of the nature of the dynamic loads and the structural response of bridges can prevent undesired failures while keeping the cost-safety balance. Dissimilar to the static behavior, the dynamic response of bridges depends on several structural parameters such as material properties, damping and mode shapes. Furthermore, dynamic load characteristics can significantly change the structural response. In most cases, complexity and involvement of numerous parameters require the designer to investigate the bridge response via a massive numerical study. This study targets three main dynamic loads applicable to railway and highway bridges, and explores particular issues related to each classification: seismic loads, vehicular dynamic loads, and high-speed train loads.
This work presents some numerical methods for options valuation. Chapter one is the introduction. Chapter two is the literature review which covers key literature findings and theoretical underpinning on a number of real options pricing approaches. Chapter three discusses numerical methods for options valuation. Chapter four presents numerical implementation and examples. Chapter five is the summary and conclusion.
Computer oriented numerical methods have revolutionized the study of fluid mechanics. The problems of fluid mechanics which left previously due to non availability computing facilities are amenable for solution now-a-days due to computing facilities. The classical method of polynomial interpolation is replaced by computer oriented numerical methods. The method of solving algebraic and transcendental equations, ordinary partial differential equation has been modified so as to provide facilities for computation in digital computers. The numerical methods for the solutions of initial and boundary value problems presented in this book will be helpful to the various researchers and investigators in the field of Fluidmechanics.
In this study, several numerical experiments are performed through the use of ReConAn Finite Element Analysis (FEA) software. These experiments involve the numerical assessment of the nonlinear behavior of reinforced concrete (RC) structures under limit state loading. The main objective of this research work, is the development of an object-oriented FEA code (ReConAn FEA), capable of easily incorporating advanced numerical techniques and modeling methods for the analysis of RC structures through the use of state-of-the-art programming techniques. An excessive literate and numerical research is presented on different methodologies and numerical methods used for the analysis of RC structures. This research reveals the advantages and disadvantages that each numerical method has, through which the proposed Hybrid modeling (HYMOD) approach derived. The HYMOD foresees the combination of beam and hexahedral elements so as to discretize the geometry of any RC building. This is performed in order to decrease the computational demands of the resulted numerical model but at the same time maintain the required numerical accuracy during the nonlinear solution procedure.
Numerical methods for solving ordinary and partial differential equations have always been important in scientific investigations. With the advent of computers, the use of numerical methods has been popularized, and more importantly, people are now able to tackle those problems which are fundamental to our understanding of scientific phenomena, but were so much more difficult to study in the past. Spectral methods is the name given to a numerical approach for the solution of differential, integral and integro equations. Our intension in this book is to develop formulas which are new to the best of our knowledge; for the generation of higher order pseudospectral integration matrices. This is used for solving integral and ordinary differential equation applying the proposed formulas transforms the linear integral equation into a system of linear equations which can be solved using any of the well-known numerical methods.
This book examines the time vibration of the displacement of a structure due to the internal forces, with no damping or external forcing. Practically, vibrations decay with time but in theory these vibrations do not actually decay. For vibrations due to purely internal forces, the dynamic systems are referred to as conservative systems. The methods of solution adopted for solving non-linear single-degree-of-freedom problems may be extended to multi-degree-of-freedom problems. There are many studies in literature on the application of these methods of solution to linear problems and yet so few have been applied to the non-linear problems. Dynamic vibration equations are of great importance not only for understanding the dynamic motion of structures, but also for providing knowledge of differential equations to mathematicians. Several attempts have been made to study dynamic vibration equations. This book gives a nicer approach to numerical solutions to second order ordinary differential equations. It also gives good examples for learners easy understanding of the techniques used.
The matrix functions computations have an important area of research in numerical analysis for introducing numerical solutions of some types of differential equations and linear systems of equations. The main objective of this book, which consists of four chapters, is to introduce an analytical and numerical treatment for computing matrix functions based on five definitions of matrix functions for some types of matrices. Namely: matrix functions for square matrices having pure complex or mixed eigenvalues using Vandermonde matrix, Lagrange-Sylvester interpolation and mixed interpolation methods. Also for square matrices having mixed or repeated real eigenvalues using extension of Sylvester's definition and Newton's divided difference. The analytical analysis and numerical treatment of the proposed methods and techniques is studied. The accuracy of these proposed methods and techniques is demonstrated by several test problems where the obtained numerical results are compared with the exact values for some types of matrix functions and in other time compared with previous methods.
The work is devoted to the construction of efficient parallel algorithms of the integration step control in for simulation of dynamic objects. To select the optimum step size the paper proposes several parallel algorithms that are based on well-known, specially restructured methods of solving systems of ordinary differential equations. For these purposes, the parallel nested methods, explicit and implicit extrapolation schemes of variable order are used. As a criterion for the selection of the numerical scheme the inequalities which control accuracy and stability are used. When solving stiff problems, this allows at each step to select the optimum in terms of computational cost numerical scheme. Also the new difference block methods with the possibility of adapting the step, addressed at parallel implementation, are proposed. The basic idea, on which the design of block methods was based, is to obtain simultaneous approximations of the exact solution at points forming a block. On the basis of the proposed step size control algorithms test problems are implemented, the numerical solution of which provides the required accuracy with the maximum possible integration step.
Finite Elements Analysis in the setting of Fracture Mechanics is a topic that engineers, mathematics and physics frequently face. In this book we describe crack-microcrack interactions in theoretical and numerical way proposing a multi-field scheme for elastic microcracked bodies and its discretization in presence of macroscopic cracks. We start reviewing the basic laws of crack propagation in Cauchy's Continuua and show numerical schemes based on Finite Element Methods (FEM) and eXtended Finite Element Methods (XFEM). The reading of this book is suggested for students and researchers who are interested in mathematical modelling and numerical computation of cracking phenomena. The content of this book was awarded during the European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2004, Jyvaskyla, Finland, 24 - 28 July 2004.