Engineering Analysis – Fall 2009Class organization Lecture 1MotivationMathematicaMapleMatlabModelsComputer simulationMathematical ModelsMathematical Model (cont’d)Mathematical Model (cont’d)Mathematical Model (cont’d)Analytical versus numerical methods for model solvingExample: the analytical modelExample – the analytical solutionExample: numerical solutionExample: numerical resultsThe solution of the analytical modelEngineering Analysis – Fall 2009Dan C. MarinescuOffice: HEC 439 BOffice hours: Tu-Th 11:00-12:00Lecture 12Class organization z Class webpage:z www.cs.ucf.edu/~dcm/Teaching/EngineeringAnalysisz Textbook:z "Applied Numerical Methods with Matlab" (Second Edition) by S. C. Chapra. Publisher Mc. Graw Hill 2008. ISBN 978-0-07-313290-7z Class Notes.Lecture 13GradeWeight of different activitiesLecture 14Lecture 15z The textbook covers five categories of numerical methods:Lecture 16Lecture 1z Motivation for the use of mathematical software packages z From Models to Analytical and to Numerical Simulationz ExampleLecture 17Motivationz Science and engineering demand a quantitative analysisof physical phenomena. Such an analysis requires a sophisticated mathematical apparatus. z Computers are very helpful; several software packages for mathematical software exist. z Specialized packages such as Ellpack for solving elliptic boundary value problems. z General-purpose systems are: z (i) Mathematica of Wolfram Research; z (ii) Maple of Maplesoft;z (iii) Matlab of Mathworks); and z (iv) IDL.Lecture 18Mathematicaz All-purpose mathematical software package. z It integrates z swift and accurate symbolic and numerical calculation, z all-purpose graphics, and z a powerful programming language. z It has a sophisticated ``notebook interface'' for documenting and displaying work. It can save individual graphics in several graphics format. z Its functional programming language (as opposed to procedural) makes it possible to do complex programming using very short concise commands; it does, however, allow the use of basic procedural programming constructs like Do and For. z Drawbacks: steeper learning curve for beginners used to procedural languages; more expensive.Lecture 19Maplez Powerful analytical and mathematical software.z Does the same sorts of things that Mathematica does, with similar high quality. z Maple's programming language is procedural (like C or Fortran or Basic) although it has a few functional programming constructs. z Drawbacks: Worksheet interface/typesetting not as developed as Mathematica's, but it is less expensive.Lecture 110Matlabz Combines efficient computation, visualization and programming for linear-algebraic technical work and other mathematical areas. z Widely used in the Engineering schools.z Drawbacks: Does not support analytical/symbolic math.Lecture 111Modelsz Abstractions of physical, social, economical, systems or phenomena.z Design to allow us to understand complex systems or phenomena. z A model captures only aspects of the original system relevant for the type of analysis being conducted.z Example: the study of the liftoff properties of a wing in a wind tunnel.Lecture 112Computer simulationz Theoretical studies, experiment and computer simulation are three exploratory methods in science and engineering.z In this class we are only concerned with computer models of physical systems.Lecture 113Mathematical Modelsz A formulation or equation that expresses the essential features of a physical system or process in mathematical terms.z Models can be represented by a functional relationship between:z dependent variables, z independent variables, z parameters, and z forcing functions.Dependentvariable= findependentvariables, parameters, forcingfunctions⎛ ⎝ ⎜ ⎞ ⎠ ⎟Lecture 114Mathematical Model (cont’d)•Dependent variable Î a characteristic that usually reflects the behavior or state of the system•Independent variables Î dimensions, such as time and space, along which the system’s behavior is being determined•Parameters Î constants reflective of the system’s properties or composition•Forcing functions Î external influences acting upon the systemLecture 115Mathematical Model (cont’d)z Conservation laws provide the foundation for many model functions. Examples of such laws:z Conservation of massz Conservation of momentumz Conservation of chargez Conservation of energyz Some system models will be given as implicit functions or as differential equations - these can be solved either using analytical methods or numerical methods.Lecture 116Mathematical Model (cont’d)•Dependent variable Î a characteristic that usually reflects the behavior or state of the system•Independent variables Î dimensions, such as time and space, along which the system’s behavior is being determined•Parameters Î constants reflective of the system’s properties or composition•Forcing functions Î external influences acting upon the systemLecture 117Analytical versus numerical methods for model solvingz Once a mathematical model is constructed one could use z Analytical methodsz Numerical methodsz Analytical methodsz Produce exact solutionsz Not always feasiblez May require mathematical sophysticationz Numerical methodsz Produce an approximate solutionz The time to solve a numerical problem is a function of the desired accuracy of the approximation.Lecture 118Example: the analytical modeldvdt= g−cdmv2Consider a bungee jumper in midair. The model for itsvelocity is given by the differential equation:Dependent variable - velocity vIndependent variables - time tParameters - mass m, drag coefficient cdForcing function - gravitational acceleration gThe change in velocity is affected by: the gravitational force which pulls it down and are opposed by the drag forceLecture 119Example – the analytical solutionvt()=gmcdtanhgcdmt⎛ ⎝ ⎜ ⎞ ⎠ ⎟ zThe model can be used to generate a graph. Example: the velocity of a 68.1 kg jumper, assuming a drag coefficient of 0.25 kg/mLecture 120Example: numerical solutionz For the numerical solution we observe that the time rate of change of velocity can be approximated as:dvdt≈ΔvΔt=vti+1()−vti()ti+1− tiLecture 121Example: numerical resultsz The efficiency and accuracy of numerical methods depend upon how the method is applied.z Applying the previous method in 2 s intervals yields:Lecture 122The solution of the analytical modelz Presented on the white