Showing papers on "Nonlinear system published in 1989"

A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle.

Abstract: This paper proposes a very tractable approach to modeling changes in regime. The parameters of an autoregression are viewed as the outcome of a discrete-state Markov process. For example, the mean growth rate of a nonstationary series may be subject to occasional, discrete shifts. The econometrician is presumed not to observe these shifts directly, but instead must draw probabilistic inference about whether and when they may have occurred based on the observed behavior of the series. The paper presents an algorithm for drawing such probabilistic inference in the form of a nonlinear iterative filter

Energy function analysis for power system stability

Abstract: 1 Power System Stability in Single Machine System.- 1.1 Introduction.- 1.2 Statement of the Stability Problem.- 1.3 Mathematical Formulation of the Problem.- 1.4 Modeling Issues.- 1.5 Motivation Through Single Machine Infinite Bus System.- 1.6 Chapter Outline.- 2 Energy Functions for Classical Models.- 2.1 Introduction.- 2.2 Internal Node Representation.- 2.3 Energy Functions for Internal Node Models.- 2.4 Individual Machine and other Energy Functions.- 2.5 Structure Preserving Energy Functions.- 2.6 Alternative Form of the Structure Preserving Energy Function.- 2.7 Positive Definiteness of the Energy Integral.- 2.8 Tsolas-Araposthasis-Varaiya Model.- 3 Reduced Order Energy Functions.- 3.1 Introduction.- 3.2 Individual Machine and Group Energy Function.- 3.3 Simplified Form of the Individual Machine Energy Function.- 3.4 Cutset Energy Function.- 3.5 Example of Cutset Energy Function.- 3.6 Extended Equal Area Criterion (EEAC).- 3.7 The Quasi Unstable Equilibrium Point (QUEP) Method.- 3.8 Decomposition-Aggregation Method.- 3.9 Time Scale Energies.- 4 Energy Functions with Detailed Models of Synchronous Machines and Its Control.- 4.1 Introduction.- 4.2 Single Machine System With Flux Decay Model.- 4.3 Multi-Machine Systems With Flux Decay Model (Method of Parameter Variations).- 4.4 Lyapunov Functions for Multi-Machine Systems With Flux Decay Model.- 4.5 Multi-Machine Systems With Flux Decay Models and AVR.- 4.6 Energy Functions With Detailed Models.- 4.7 Lyapunov Function for Multi-Machine Systems With Flux Decay and Nonlinear Voltage Dependent Loads.- 5 Region of Stability in Power Systems.- 5.1 Introduction.- 5.2 Characterization of the Stability Boundary.- 5.3 Region of Stability.- 5.4 Method of Hyperplanes and Hypersurfaces.- 5.5 Potential Energy Boundary Surface (PEBS) Method.- 5.6 Hybrid Method Using the Gradient System.- 6 Practical Applications of the Energy Function Method.- 6.1 Introduction.- 6.2 The Controlling u.e.p. Method.- 6.3 Modifications to the Controlling u.e.p. Method.- 6.4 Potential Energy Boundary Surface (PEBS) Method.- 6.5 Mode of Instability (MOI) Method.- 6.6 Dynamic Security Assessment.- 7 Future Research Issues.- Appendix A 10 Machine 39 Bus System Data.- References.

Parallel and distributed computation

Abstract: This book focuses on numerical algorithms suited for parallelization for solving systems of equations and optimization problems Emphasis on relaxation methods of the Jacobi and Gauss-Seidel type, and issues of communication and synchronization Topics covered include: Algorithms for systems of linear equations and matrix inversion; Herative methods for nonlinear problems; and Shortest paths and dynamic programming

Dynamics of Solitons in Nearly Integrable Systems

Abstract: A detailed survey of the technique of perturbation theory for nearly integrable systems, based upon the inverse scattering transform, and a minute account of results obtained by means of that technique and alternative methods are given. Attention is focused on four classical nonlinear equations: the Korteweg-de Vries, nonlinear Schr\"odinger, sine-Gordon, and Landau-Lifshitz equations perturbed by various Hamiltonian and/or dissipative terms; a comprehensive list of physical applications of these perturbed equations is compiled. Systems of weakly coupled equations, which become exactly integrable when decoupled, are also considered in detail. Adiabatic and radiative effects in dynamics of one and several solitons (both simple and compound) are analyzed. Generalizations of the perturbation theory to quasi-one-dimensional and quantum (semiclassical) solitons, as well as to nonsoliton nonlinear wave packets, are also considered.

Adaptive control of linearizable systems

Abstract: The authors give some initial results on the adaptive control of minimum-phase nonlinear systems which are exactly input-output linearizable by state feedback. Parameter adaptation is used as a technique to make robust the exact cancellation of nonlinear terms, which is called for in the linearization technique. The application of the adaptive technique to control of robot manipulators is discussed. Only the continuous-time case is considered; extensions to the discrete-time and sampled-data cases are not obvious. >

Solitons: An introduction

Abstract: This textbook is an introduction to the theory of solitons and its diverse applications to nonlinear systems that arise in the physical sciences. The authors explain the generation and properties of solitons, introducing the mathematical technique known as the Inverse Scattering Transform. Their aim is to present the essence of inverse scattering clearly, rather than rigorously or completely. Thus, the prerequisites (i.e., partial differential equations, calculus of variations, Fourier integrals, linear waves and Sturm–Liouville theory), and more advanced material is explained in the text with useful references to further reading given at the end of each chapter. Worked examples are frequently used to help the reader follow the various ideas, and the exercises at the end of each chapter not only contain applications but also test understanding. Answers, or hints to the solution, are given at the end of the book. Sections and exercises that contain more difficult material are indicated by asterisks.

Numerical methods and software

Abstract: 1. Introduction. 2. Computer Arithmetic and Computational Errors. 3. Linear systems of Equations. 4. Interpolation. 5. Numerical Quadrature. 6. Linear Least-Square Data Fitting. 7. Solution of Nonlinear Equations. 8. Ordinary Diernetial Equations. 9. Optimization and Nonlinear Least Squares. 10. Simulation and Random Numbers. 11. Trigonometirc Approximation and the Fast Fourier Transorm. Bibliography.

Representations of non-linear systems: the NARMAX model

Abstract: Input-output representations of non-linear discrete-time systems are discussed. It is shown that the NARMAX (Non-linear AutoRegressive Moving Average with eXogenous inputs) model is a general and natural representation of non-linear systems and contains, as special cases, several existing non-linear models. The problem of approximating non-linear input-output systems is also addressed and several properties of non-linear models are highlighted using simple examples.

Continuum Mechanics of Electromagnetic Solids

Abstract: 1. Essential Properties of Electromagnetic Solids. 2. Elements of Continuum Mechanics. 3. General Equations of Nonlinear Electromagnetic Continua. 4. Elastic Dielectrics and Piezoelectricity. 5. Elastic Conductors. 6. Elastic Ferromagnets. 7. Elastic Ionic Crystals, Ferroelectrics and Ceramics. Appendices. Index.

Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials

Abstract: 1 Formulation of Elementary Boundary Value Problems- 1 The Concept of the Classical Formulation of a Boundary Value Problem for Equations with Discontinuous Coefficients- 2 The Concept of Generalized Solution- 3 Generalized Formulations of Problems for the Basic Equations of Mathematical Physics- 2 The Concept of Asymptotic Expansion A Model Example to Illustrate the Averaging Method- 1 Asymptotic expansion A Formal Asymptotic Solution- 2 Asymptotic Expansion of a Solution of the Equation u = 1 + ?u3- 3 Asymptotic Expansion of a Solution of the Equation (K(x/?)u?)?= f(x) by the Averaging Method- 4 Generalization of the Averaging Method in the Case of a Piecewise Smooth Coefficient- 5 Averaging the System of Differential Equations- 3 Averaging Processes in Layered Media- 1 Problem of Small Longitudinal Vibrations of a Rod- 2 Nonstationary Problem of Heat Conduction- 3 Averaging Maxwell Equations- 4 Averaging Equations of a Viscoelastic Medium- 5 Media with Slowly Changing Geometric Characteristics- 6 Heat Transfer Through a System of Screens- 7 Averaging a Nonlinear Problem of the Elasticity Theory in an Inhomogeneous Rod- 8 The System of Equations of Elasticity Theory in a Layered Medium- 9 Considerations Permitting Reduction of Calculations in Constructing Averaged Equations- 10 Nonstationary Nonlinear Problems- 11 Averaging Equations with Rapidly Oscillating Nonperiodic Coefficients- 12 Problems of Plasticity and Dynamics of Viscous Fluid as Described by Functions Depending on Fast Variables- 4 Averaging Basic Equations of Mathematical Physics- 1 Averaging Stationary Thermal Fields in a Composite- 2 Asymptotic Expansion of Solution of the Stationary Heat Conduction Problem- 3 Stationary Thermal Field in a Porous Medium- 4 Averaging a Stationary System of Equations of Elasticity Theory in Composite and Porous Materials- 5 Nonstationary Systems of Equations of Elasticity and Diffusion Theory- 6 Averaging Nonstationary Nonlinear System of Equations of Elasticity Theory- 7 Averaging Stokes and Navier-Stokes Equations The Derivation of the Percolation Law for a Porous Medium (Darcy's Law)- 8 Averaging in case of Short-Wave Propagation- 9 Averaging the Transition Equation for a Periodic Medium- 10 Eigenvalue Problems- 5 General Formal Averaging Procedure- 1 Averaging Nonlinear Equations- 2 Averaged Equations of Infinite Order for a Linear Periodic Medium and for the Equation of Moment Theory- 3 A Method of Describing Multi-Dimensional Periodic Media that does not Involve Separating Fast and Slow Variables- 6 Properties of Effective Coefficients Relationship Among Local and Averaged Characteristics of a Solution- 1 Maintaining the Properties of Convexity and Symmetry of the Minimized Functional in Averaging- 2 On the Principle of Equivalent Homogeneity- 3 The Symmetry Properties of Effective Coefficients and Reduction of Periodic Problems to Boundary Value Problems- 4 Agreement Between Theoretically Predicted Values of Effective Coefficients and Those Determined by an Ideal Experiment- 7 Composite Materials Containing High-Modulus Reinforcement- 1 The Stationary Field in a Layered Material- 2 Composite Materials with Grains for Reinforcement- 3 Dissipation of Waves in Layered Media- 4 High-Modulus 3D Composite Materials- 5 The Splitting Principle for the Averaged Operator for 3D High-Modulus Composites- 8 Averaging of Processes in Skeletal Structures- 1 An Example of Averaging a Problem on the Simplest Framework- 2 A Geometric Model of a Framework- 3 The Splitting Principle for the Averaged Operator for a Periodic Framework- 4 The Splitting Principle for the Averaged Operator for Trusses and Thin-walled Structures- 5 On Refining the Splitting Principle for the Averaged Operator- 2 The Concept of Generalized Solution- 3 Generalized Formulations of Problems for the Basic Equations of Mathematical Physics- 2 The Concept of Asymptotic Expansion A Model Example to Illustrate the Averaging Method- 1 Asymptotic expansion A Formal Asymptotic Solution- 2 Asymptotic Expansion of a Solution of the Equation u = 1 + ?u3- 3 Asymptotic Expansion of a Solution of the Equation (K(x/?)u?)?= f(x) by the Averaging Method- 4 Generalization of the Averaging Method in the Case of a Piecewise Smooth Coefficient- 5 Averaging the System of Differential Equations- 3 Averaging Processes in Layered Media- 1 Problem of Small Longitudinal Vibrations of a Rod- 2 Nonstationary Problem of Heat Conduction- 3 Averaging Maxwell Equations- 4 Averaging Equations of a Viscoelastic Medium- 5 Media with Slowly Changing Geometric Characteristics- 6 Heat Transfer Through a System of Screens- 7 Averaging a Nonlinear Problem of the Elasticity Theory in an Inhomogeneous Rod- 8 The System of Equations of Elasticity Theory in a Layered Medium- 9 Considerations Permitting Reduction of Calculations in Constructing Averaged Equations- 10 Nonstationary Nonlinear Problems- 11 Averaging Equations with Rapidly Oscillating Nonperiodic Coefficients- 12 Problems of Plasticity and Dynamics of Viscous Fluid as Described by Functions Depending on Fast Variables- 4 Averaging Basic Equations of Mathematical Physics- 1 Averaging Stationary Thermal Fields in a Composite- 2 Asymptotic Expansion of Solution of the Stationary Heat Conduction Problem- 3 Stationary Thermal Field in a Porous Medium- 4 Averaging a Stationary System of Equations of Elasticity Theory in Composite and Porous Materials- 5 Nonstationary Systems of Equations of Elasticity and Diffusion Theory- 6 Averaging Nonstationary Nonlinear System of Equations of Elasticity Theory- 7 Averaging Stokes and Navier-Stokes Equations The Derivation of the Percolation Law for a Porous Medium (Darcy's Law)- 8 Averaging in case of Short-Wave Propagation- 9 Averaging the Transition Equation for a Periodic Medium- 10 Eigenvalue Problems- 5 General Formal Averaging Procedure- 1 Averaging Nonlinear Equations- 2 Averaged Equations of Infinite Order for a Linear Periodic Medium and for the Equation of Moment Theory- 3 A Method of Describing Multi-Dimensional Periodic Media that does not Involve Separating Fast and Slow Variables- 6 Properties of Effective Coefficients Relationship Among Local and Averaged Characteristics of a Solution- 1 Maintaining the Properties of Convexity and Symmetry of the Minimized Functional in Averaging- 2 On the Principle of Equivalent Homogeneity- 3 The Symmetry Properties of Effective Coefficients and Reduction of Periodic Problems to Boundary Value Problems- 4 Agreement Between Theoretically Predicted Values of Effective Coefficients and Those Determined by an Ideal Experiment- 7 Composite Materials Containing High-Modulus Reinforcement- 1 The Stationary Field in a Layered Material- 2 Composite Materials with Grains for Reinforcement- 3 Dissipation of Waves in Layered Media- 4 High-Modulus 3D Composite Materials- 5 The Splitting Principle for the Averaged Operator for 3D High-Modulus Composites- 8 Averaging of Processes in Skeletal Structures- 1 An Example of Averaging a Problem on the Simplest Framework- 2 A Geometric Model of a Framework- 3 The Splitting Principle for the Averaged Operator for a Periodic Framework- 4 The Splitting Principle for the Averaged Operator for Trusses and Thin-walled Structures- 5 On Refining the Splitting Principle for the Averaged Operator- 6 Asymptotic Expansion of a Solution of a Linear Equation in Partial Derivatives for a Rectangular Framework- 7 Skeletal Structures with Random Properties- 9 Mathematics of Boundary-Layer Theory in Composite Materials- 1 Problem on the Contact of Two Layered Media- 2 The Boundary Layer for an Elliptic Equation Defined on a Half-Plane- 3 The Boundary Layer Near the Interface of Two Periodic Structures- 4 Problem on the Contact of Two Media Divided by a Thin Interlayer- 5 The Boundary Layer for the Nonstationary System of Equations of Elasticity Theory- 6 On the Ultimate Strength of a Composite- 7 Boundary Conditions of Other Types- 8 On the Averaging of Fields in Layer Media with Layers of Composite Materials- 9 The Time Boundary Layer for the Cauchy Parabolic Problem- Supplement: Existence and Uniqueness Theorems for the Problem on a Cell

Response‐Surface Approach for Reliability Analysis

Abstract: The present paper introduces and discusses a stochastic finite-element method. It can be used for the analysis of structural and mechanical systems whose geometrical and material properties have spatial random variability. The method utilizes a polynomial expansion of the numerical nonlinear structural operator (for which actual analytical form is unknown). The expansion is made according to a response-surface approximation in terms of spatial averages of the design variables. The polynomial form is then modified by suitable error factors, one for each geometrical or mechanical property. Each error factor is due to the deviations, of the single property, from its spatial average in the different finite elements. The method demands an accurate design of the experiments to be conducted in order to identify the model parameters. A numerical example has been worked out. In this numerical example, the stresses and the strains in a light-water reactor pressurized vessel are computed by a stochastic three-dimensional finite element nonlinear analysis.

A class of high resolution explicit and implicit shock-capturing methods

Abstract: The development of shock-capturing finite difference methods for hyperbolic conservation laws has been a rapidly growing area for the last decade. Many of the fundamental concepts, state-of-the-art developments and applications to fluid dynamics problems can only be found in meeting proceedings, scientific journals and internal reports. This paper attempts to give a unified and generalized formulation of a class of high-resolution, explicit and implicit shock capturing methods, and to illustrate their versatility in various steady and unsteady complex shock waves, perfect gases, equilibrium real gases and nonequilibrium flow computations. These numerical methods are formulated for the purpose of ease and efficient implementation into a practical computer code. The various constructions of high-resolution shock-capturing methods fall nicely into the present framework and a computer code can be implemented with the various methods as separate modules. Included is a systematic overview of the basic design principle of the various related numerical methods. Special emphasis will be on the construction of the basic nonlinear, spatially second and third-order schemes for nonlinear scalar hyperbolic conservation laws and the methods of extending these nonlinear scalar schemes to nonlinear systems via the approximate Riemann solvers and flux-vector splitting approaches. Generalization of these methods to efficiently include real gases and large systems of nonequilibrium flows will be discussed. Some perbolic conservation laws to problems containing stiff source terms and terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for one-, two- and three-dimensional gas-dynamics problems. The use of the Lax-Friedrichs numerical flux to obtain high-resolution shock-capturing schemes is generalized. This method can be extended to nonlinear systems of equations without the use of Riemann solvers or flux-vector splitting approaches and thus provides a large savings for multidimensional, equilibrium real gases and nonequilibrium flow computations.

Stability analysis of nonlinear systems

Abstract: Investigates stability theory in terms of two different measures, treats the theory of a variety of inequalities, and demonstrates manifestations of the general Lyapunov method. Also covers the importance of utilizing different forms of nonlinear variation of parametric formulae, constructive methods generated by monotone iterative technique and th

Interior a priori estimates for solutions of fully non-linear equations

Abstract: On etend une theorie de perturbations aux solutions d'equations uniformement elliptiques d'ordre 2 totalement non lineaires

Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms

Abstract: Discretization of the Primitive Variable Formulation: A Primitive Variable Formulation. The Finite Element Problem and the Div-St abi lity Condition. Finite Element Spaces. Alternate Weak Forms, Boundary Conditions and Numerical Integration. Penalty Methods. Solution of the Discrete Equations: Newton's Method and Other Iterative Methods. Solving the Linear Systems. Solution Methods for Large Reynolds Numbers. Time Dependent Problems: A Weak Formulation and Spatial Discretizations. Time Discretizations. The Streamfunction-Vorticity Formulation: Algorithms for the Streamfunction-Vorticity Equations. Solution Techniques for Multiply Connected Domains. The Streamfunction Formulation: Algorithms for Determining Streamfunction Approximations. Eigenvalue Problems Connected with Stability Studies for Viscous Flows: Energy Stability Analysis of Viscous Flows. Linearized Stability Analysis of Stationary Viscous Flows. Exterior Problems: Truncated Domain-Artificial Boundary Condition Methods. Nonlinear Constitutive Relations: A Ladyzhenskaya Model and Algebraic Turbulence Models. Bingham Fluids. Electromagnetically or Thermally Coupled Flows: Flows of Liquid Metals. The Boussinesq Equations. Remarks on Some Topics That Have Not Been Considered: Problems, Formulations, Algorithms, and Other Issues That Have Not Been Considered. Bibliography. Glossary of Symbols. Index.

A critical evaluation of recent developments in hydrogeochemical transport models of reactive multichemical components

Abstract: Many hydrogeochemical models have appeared in recent years for simulating subsurface solute transport. The hydrological transport of solutes can be described by a set of linear partial differential equations, and the chemical equilibria are described by a set of nonlinear algebraic equations. Three approaches are currently used to formulate the problem: (1) the mixed differential and algebraic equation (DAE) approach, (2) the direct substitution approach (DSA), and (3) the sequential iteration approach (SIA). An extremely important consideration in any approach is the choice of primary dependent variables (PDVs). Six types of PDVs have been employed in the existing models: (1) concentrations of all species, (2) concentrations of all component species and precipitated species, (3) total analytical concentrations of aqueous components, (4) total dissolved concentrations of aqueous components, (5) concentrations of aqueous component species, and (6) hybrid concentrations. Because of many possible combinations of PDVs and approaches, many hydrogeochemical transport models for multicomponent systems have been developed. This paper critically evaluates and discusses these models. The discussion and evaluation are conducted in terms of (1) how severe can the constraints be that a model imposes on computer resources, (2) which equilibrium geochemical processes can a model include, and (3) how easily can a model be modified to deal with mixed kinetic and equilibrium reactions. The use of SIA models leads to the fewest constraints on computer resources in terms of central processing unit (CPU) memory and CPU time; both DAE and DSA models require excessive CPU memory and CPU time for realistic two- and three-dimensional problems. Only those models that use the first three types of PDVs can treat the full complement of equilibrium reactions simultaneously. DAE and SIA models can be modified with reasonable ease to handle mixed chemical kinetics and equilibria. DSA models require strenuous efforts to modify for treating mixed chemical kinetics and equilibria. Therefore SIA models using the third type of PDVs are recommended for their practicality and flexibility. DSA and DAE models should remain research tools for one-dimensional investigations.

Nonlinear Dynamical Economics and Chaotic Motion

Abstract: The behavior of nonlinear dynamical systems can differ completely from that of linear dynamical systems. Regular oscillatory motion as well as seemingly irregular motion can emerge in these systems without resort to exogenous influences or peculiar parameter values. Recent mathematical advances made in the theory of nonlinear dynamical systems allow the restrictive linear approach to dynamical phenomena in economics to be discarded and observable fluctuations of economic variables to be modelled in a simple and convenient way. This book attempts to familiarize the reader with the standard tools in nonlinear dynamical systems theory. The usage of these tools is demonstrated with simple examples from different fields like business cycle theory, optimal control theory, growth theory, and population dynamics. The presentation encompasses a short reminder of linear dynamical systems and traditional themes in nonlinear systems like the existence and uniqueness of limit cycles and closed orbits in predator-prey systems. The main part of the book deals with chaotic motion in economic systems. It is demonstrated, that irregular dynamical behavior can be generated in deterministic economic systems with relative ease, once the linear approach in modelling dynamic phenomena is abandoned. One-dimensional and higher-dimensional difference equations as well as higher-dimensional differential equation systems are discussed and illustrated with a variety of economic examples. Recently emerging techniques for discriminating stochastic and deterministic dynamical systems are presented, as well as the question of whether the nature of actual economic time series is chaotic or stochastic.

Chaos and Integrability in Nonlinear Dynamics: An Introduction

Abstract: The Dynamics of Differential Equations. Hamiltonian Dynamics. Classical Perturbation Theory. Chaos in Hamiltonian Systems and Area--Preserving Mappings. The Dynamics of Dissipative Systems. Chaos and Integrability in Semiclassical Mechanics. Nonlinear Evolution Equations and Solitons. Analytic Structure of Dynamical Systems. Index.

An Alternating Frequency/Time Domain Method for Calculating the Steady-State Response of Nonlinear Dynamic Systems

Abstract: A method is proposed for analyzing the steady-state response of nonlinear dynamic systems. The method iterates to obtain the discrete Fourier transform of the system response, returning to the time domain at each iteration to take advantage of the ease in evaluating nonlinearities there-rather than analytically describing the nonlinear terms in the frequency domain. The updated estimates of the nonlinear terms are transformed back into the frequency domain in order to continue iterating on the frequency spectrum of the steady-state response. The method is demonstrated by solving a problem with friction damping in which the excitation has multiple discrete frequencies.

Convergence of spectral methods for nonlinear conservation laws

Abstract: The convergence of the Fourier method for scalar nonlinear conservation laws which exhibit spontaneous shock discontinuities is discussed. Numerical tests indicate that the convergence may (and in fact in some cases must) fail, with or without post-processing of the numerical solution. Instead, a new kind of spectrally accurate vanishing viscosity is introduced to augment the Fourier approximation of such nonlinear conservation laws. Using compensated compactness arguments, it is shown that this spectral viscosity prevents oscillations, and convergence to the unique entropy solution follows.

The anionic group theory of the non-linear optical effect and its applications in the development of new high-quality NLO crystals in the borate series

Abstract: Starting from a general quantum-mechanical perturbation theory on the nonlinear optical (NLO) effect in crystals, this review gives a systematic presentation of the basic concepts and calculation methods of the ‘anionic group theory for the NLO effect of crystals’ and a brief discussion of the approximations involved. Calculations have been made for the second harmonic generation (SHG) coefficients of a few typical NLO crystals. Comparisions between these theoretical values and the experimental values made both on powdered crystals and on single crystals suffice to show the feasibility of the theoretical treatment and calculation methods. On this basis, borate ions of various structure types are classified and systematic calculations are carried out for the NLO susceptibilities of some typical borate crystals with good prospects of applications in opto-electronics. Through these calculations, a series of structural criteria serving as useful guidelines for searching and developing new NLO crystal...