Engineering Analysis Fall 2009 Dan C Marinescu Office HEC 439 B Office hours Tu Th 11 00 12 00 Class organization z Class webpage z z www cs ucf edu dcm Teaching EngineeringAnalysis Textbook z Applied Numerical Methods with Matlab Second Edition by S C Chapra Publisher Mc Graw Hill 2008 ISBN 978 0 07 313290 7 z Class Notes Lecture 1 2 Grade Weight of different activities Lecture 1 3 Lecture 1 4 z The textbook covers five categories of numerical methods Lecture 1 5 Lecture 1 z Motivation for the use of mathematical software packages z From Models to Analytical and to Numerical Simulation z Example Lecture 1 6 Motivation z z Science and engineering demand a quantitative analysis of physical phenomena Such an analysis requires a sophisticated mathematical apparatus Computers are very helpful several software packages for mathematical software exist z z Specialized packages such as Ellpack for solving elliptic boundary value problems General purpose systems are z z z z i Mathematica of Wolfram Research ii Maple of Maplesoft iii Matlab of Mathworks and iv IDL Lecture 1 7 Mathematica z z All purpose mathematical software package It integrates z z z z z z swift and accurate symbolic and numerical calculation all purpose graphics and a powerful programming language It has a sophisticated notebook interface for documenting and displaying work It can save individual graphics in several graphics format 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 Drawbacks steeper learning curve for beginners used to procedural languages more expensive Lecture 1 8 Maple z z z z Powerful analytical and mathematical software Does the same sorts of things that Mathematica does with similar high quality Maple s programming language is procedural like C or Fortran or Basic although it has a few functional programming constructs Drawbacks Worksheet interface typesetting not as developed as Mathematica s but it is less expensive Lecture 1 9 Matlab z z z Combines efficient computation visualization and programming for linear algebraic technical work and other mathematical areas Widely used in the Engineering schools Drawbacks Does not support analytical symbolic math Lecture 1 10 Models z z z z Abstractions of physical social economical systems or phenomena Design to allow us to understand complex systems or phenomena A model captures only aspects of the original system relevant for the type of analysis being conducted Example the study of the liftoff properties of a wing in a wind tunnel Lecture 1 11 Computer simulation z z Theoretical studies experiment and computer simulation are three exploratory methods in science and engineering In this class we are only concerned with computer models of physical systems Lecture 1 12 Mathematical Models z z A formulation or equation that expresses the essential features of a physical system or process in mathematical terms Models can be represented by a functional relationship between z z z z dependent variables independent variables parameters and forcing functions independent Dependent forcing f parameters variable functions variables Lecture 1 13 Mathematical Model cont d Dependent variable a characteristic that usually reflects the behavior or state of the system Independent variables dimensions such as time and space along which the system s behavior is being determined Parameters constants reflective of the system s properties or composition Forcing functions external influences acting upon the system Lecture 1 14 Mathematical Model cont d z Conservation laws provide the foundation for many model functions Examples of such laws z z z z z Conservation of mass Conservation of momentum Conservation of charge Conservation of energy 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 1 15 Mathematical Model cont d Dependent variable a characteristic that usually reflects the behavior or state of the system Independent variables dimensions such as time and space along which the system s behavior is being determined Parameters constants reflective of the system s properties or composition Forcing functions external influences acting upon the system Lecture 1 16 Analytical versus numerical methods for model solving z Once a mathematical model is constructed one could use z z z Analytical methods z z z z Analytical methods Numerical methods Produce exact solutions Not always feasible May require mathematical sophystication Numerical methods z z Produce an approximate solution The time to solve a numerical problem is a function of the desired accuracy of the approximation Lecture 1 17 Example the analytical model Consider a bungee jumper in midair The model for its velocity is given by the differential equation dv cd 2 g v m dt The change in velocity is affected by the gravitational force which pulls it down and are opposed by the drag force Dependent variable velocity v Independent variables time t Parameters mass m drag coefficient cd Forcing function gravitational acceleration g Lecture 1 18 Example the analytical solution v t gc gm d tanh t cd m 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 m Lecture 1 19 Example numerical solution z For the numerical solution we observe that the time rate of change of velocity can be approximated as dv v v ti 1 v ti ti 1 ti dt t Lecture 1 20 Example numerical results z z The efficiency and accuracy of numerical methods depend upon how the method is applied Applying the previous method in 2 s intervals yields Lecture 1 21 The solution of the analytical model z Presented on the white board Lecture 1 22