Advances in scientific computing have made modelling and simulation an important part of the decision-making process in engineering, science, and public policy. This book provides a comprehensive and systematic development of the basic concepts, principles, and procedures for verification and validation of models and simulations. The emphasis is placed on models that are described by partial differential and integral equations and the simulations that result from their numerical solution. The methods described can be applied to a wide range of technical fields, from the physical sciences, engineering and technology and industry, through to environmental regulations and safety, product and plant safety, financial investing, and governmental regulations. This book will be genuinely welcomed by researchers, practitioners, and decision makers in a broad range of fields, who seek to improve the credibility and reliability of simulation results. It will also be appropriate either for university courses or for independent study.

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Verification and Validation in Scientific Computing

Precise dynamic models of processes are required for many applications, ranging from control engineering to the natural sciences and economics. Frequently, such precise models cannot be derived using theoretical considerations alone. Therefore, they must be determined experimentally. This book treats the determination of dynamic models based on measurements taken at the process, which is known as system identification or process identification. Both offline and online methods are presented, i.e. methods that post-process the measured data as well as methods that provide models during the measurement. The book is theory-oriented and application-oriented and most methods covered have been used successfully in practical applications for many different processes. Illustrative examples in this book with real measured data range from hydraulic and electric actuators up to combustion engines. Real experimental data is also provided on the Springer webpage, allowing readers to gather their first experience with the methods presented in this book. Among others, the book covers the following subjects: determination of the non-parametric frequency response, (fast) Fourier transform, correlation analysis, parameter estimation with a focus on the method of Least Squares and modifications, identification of time-variant processes, identification in closed-loop, identification of continuous time processes, and subspace methods. Some methods for nonlinear system identification are also considered, such as the Extended Kalman filter and neural networks. The different methods are compared by using a real three-mass oscillator process, a model of a drive train. For many identification methods, hints for the practical implementation and application are provided. The book is intended to meet the needs of students and practicing engineers working in research and development, design and manufacturing.

Identification of Dynamic Systems: An Introduction with Applications

Forecasting the ground water level fluctuations is an important requirement for planning conjunctive use in any basin. This paper reports a research study that investigates the potential of artificial neural network technique in forecasting the groundwater level fluctuations in an unconfined coastal aquifer in India. The most appropriate set of input variables to the model are selected through a combination of domain knowledge and statistical analysis of the available data series. Several ANN models are developed that forecasts the water level of two observation wells. The results suggest that the model predictions are reasonably accurate as evaluated by various statistical indices. An input sensitivity analysis suggested that exclusion of antecedent values of the water level time series may not help the model to capture the recharge time for the aquifer and may result in poorer performance of the models. In general, the results suggest that the ANN models are able to forecast the water levels up to 4 months in advance reasonably well. Such forecasts may be useful in conjunctive use planning of groundwater and surface water in the coastal areas that help maintain the natural water table gradient to protect seawater intrusion or water logging condition.

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Groundwater Level Forecasting in a Shallow Aquifer Using Artificial Neural Network Approach

Tutorial introduction Algorithms of CM, TM, and CTM Coupling techniques Boundary element methods Other kinds of boundary methods Comparisons Part I: Collocation Trefftz method 1 - Basic algorithms and theory Notations and preliminaries Approximation problems Error estimates Debye-Huckel equation Stability analysis 2 - Motz's problem and its variants Introduction Basic algorithms of CTM Error bounds and integration approximation Variants of Motz's problem Concluding remarks 3 - Coupling techniques Introduction Description of generalized TMs Penalty TMs Simplified hybrid TMs Penalty plus hybrid TMs Lagrange multiplier TM Effective condition number Numerical experiments 4 - Biharmonic equations with singularities Introduction The Green formulas of A2u The collocation Trefftz methods Error bounds Part II: Collocation methods 5 - Collocation methods Introduction Description of collocation methods Error analysis Robin boundary conditions Inverse inequalities Final remarks 6 - Combinations of collocation and finite element methods Introduction Combinations of FEMs Linear algebraic equations of combination of FEM and CM Uniformly Vh0-elliptic inequality Uniformly Vh0-elliptic inequality involving integration approximation Final remarks 7 - Radial basis function collocation methods Introduction Radial basis functions Description of radial basis function collocation methods Inverse estimates for radial basis functions Numerical experiments Comparisons and conclusions Part III: Advanced topics 8 - Combinations with high-order FEMs Introduction Combinations of TM and Lagrange FEMs Global superconvergence Adini's elements 9 - Eigenvalue problems Introduction New numerical algorithms for eigenvalue problems Error bounds of eigenvalues Error bounds of eigenfunctions Computational models and numerical experiments Eigenvalues for the singularity problem Summaries and discussions 10 - The Helmholtz equation Introduction The Trefftz method Error analysis Summaries and discussions 11 - Explicit harmonic solutions of Laplace's equation Introduction Harmonic functions Harmonic solutions involving Neumann conditions Extensions and analysis on singularity New models of singularities for Laplace's equation Concluding remarks Appendix - Historic review of boundary methods Potential theory Existence and uniqueness Reduction in dimensions and Green's formula Integral equations Extended Green's formula Pre-electronic computer era Electronic computer era Boundary integral equation and boundary element methods

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Trefftz and Collocation Methods

Background/purpose: Various analyses have been performed to identify the mechanical properties of the human skin tissue in vivo. They generally use different approaches and hypotheses (behavior laws as well as mechanical tests) and the obtained results are consequently difficult to analyze and compare. In this paper, an inverse method that can be adapted to any kind of mechanical tests and behavior laws is presented.



Method: A suction deformation performed on the volar aspect of the forearm of a subject is considered. This test is modeled with the finite element method to compare the experimental and simulated curves using an inverse method that allows the skin mechanical parameters identification. This process is based on two optimization algorithms, Kalman's filter and Gauss–Newton's methods. To account for the nonlinear behavior of the skin, a specific nonlinear elastic law, which is then compared with standard linear elastic and neo-Hookean's mechanical behaviors, was developed.



Results: The obtained results first prove that neither linear elasticity nor neo-Hookean's laws can be used to model the skin. On the contrary, the nonlinear elastic model presents a relevant fit of the experimental curves. The skin thickness is also proved to be another key point to be taken into consideration.



Conclusions: The obtained results are successfully compared with literature and the reliability of the proposed method is underlined with the identification of 300 additional experimental curves. The different works we are currently focusing on are finally introduced.

A nonlinear elastic behavior to identify the mechanical parameters of human skin in vivo.

Modelling and Systems Parameter Estimation for Dynamic Systems presents a detailed examination of the estimation techniques and modeling problems. The theory is furnished with several illustrations and computer programs to promote better understanding of system modeling and parameter estimation. The material is presented in a way that makes for easy reading and enables the user to implement and execute the programs himself to gain first hand experience of the estimation process.

Modelling and Parameter Estimation of Dynamic Systems

In this chapter, we discuss the least squares/equation error techniques for parameter estimation, which are used for aiding the parameter estimation of dynamic systems (including algebraic systems), in general, and the aerodynamic derivatives of aerospace vehicles from the flight data, in particular. In the first few sections, some basic concepts and techniques of the least squares approach are discussed with a view to elucidating the more involved methods and procedures in the later chapters. Since our approach is model-based, we need to define a mathematical model of the dynamic (or static) system.

Least squares methods

The type of (linear or nonlinear) mathematical model, and the presence of process or measurement noise in data or both mainly drive the choice of the estimation method and the intended use of results. The equation error method has a cost function that is linear in parameters. It is simple and easy to implement. The output error method is more complex and requires the nonlinear optimisation technique (Gauss-Newton method) to estimate model parameters. The iterative nature of the approach makes it a little more computer intensive. The third approach is the filter error method which is the most general approach to parameter estimation problem accounting for both process and measurement noise. Being a combination of the Kalman filter and output error method, it is the most complex of the three techniques with high computational requirements. The output error method is perhaps the most widely used approach for aircraft parameter estimation and is discussed in this chapter, after discussing the concepts of maximum likelihood. The Gaussian least squares differential correction method is also an output error method, but it is not based on the maximum likelihood principle.

Output error method

For the on-line/real-time parameter estimation problem, several aspects are important: i) the estimation algorithm should be robust; ii) it should converge to an estimate close to the true value; iii) its computational requirements should be moderately low or very low; and iv) the algorithm should be numerically reliable and stable so that condition (i) is assured. It is possible to apply on-line techniques to an industrial process as long as transient responses prevail, since when these responses die out or subside, there is no activity and all input-output signals of the process (for identification) have attained the steady state and hence these signals are not useful at all for parameter estimation. Only the steady state gain of the plant/system can be determined.

Real-time parameter estimation

Research in the area of artificial neural networks has advanced at a rapid pace in recent times. The artificial neural network possesses a good ability to learn adaptively. The decision process in an artificial neural network is based on certain nonlinear operations. Such nonlinearities are useful: i) in improving the convergence speed (of the algorithm); ii) to provide more general nonlinear mapping between input-output signals; and iii) to reduce the effect of outliers in the measurements. One of the most successful artificial neural networks is the so-called feed forward neural network. In this chapter first the description of the feed forward neural network and its training algorithms is given. Next, parameter estimation using this approach is discussed. The presentation of training algorithms is such that it facilitates MATLAB implementation. Subsequently, recurrent neural networks are described. Several schemes based on recurrent neural networks are presented for parameter estimation of dynamical systems. Subsequently, the genetic algorithm is described and its application to parameter estimation considered.

J. R. Raol

Papers

Modelling and Parameter Estimation of Dynamic Systems

Least squares methods

Output error method

Real-time parameter estimation

Parameter estimation using artificial neural networks and genetic