Engineering Computation Lecture 1 Presentation E. T. S. Ingeniera - PowerPoint PPT Presentation

Engineering Computation Lecture 1 Presentation E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 1

Introduction Objectives 1. Introduction (quite ambitious!) to numerical methods for engineering as a general and fundamental tool for all engineering disciplines. We plan to review some main topics of Algebra (matrix calculations, equations,…) and Calculus (functions, integration, differential equations,…) with the perspective given by the availability of a computer. 2. Computer tools and programming will be important; we will use commercial software widely used in science and engineering: MATLAB. 3. We will illustrate and discuss how numerical methods are used in practice. We will consider examples from Engineering. E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 2

Introduction ENGCOMP Course overview 1. Approximation, errors. 2. Taylor series. Numerical derivatives. 3. Numerical methods for ODE’s. 4. Introduction to numerical solutions of PDE’s. 5. Interpolation, Curve-fitting. 6. Numerical integration. 7. Solution of nonlinear equations and systems. 8. Simultaneous linear equations: Gaussian elimination. Factorization. Norms. Iterative methods. E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 3

Introduction Why are Numerical Methods so widely used in Engineering? • Engineers use mathematical (equations and data) and physical modeling to describe and predict the behavior of systems. • Closed-form (analytical) solutions are only possible and complete for simple problems (geometry, properties, etc.). • Computers are widely available, powerful, and (relatively) cheap. • Powerful software packages are available (special or general purpose). E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 4

Introduction A few applications of Numerical Methods in Engineering: • Structural/mechanical analysis, design, and behavior. • Communication/power Network simulation Train and traffic networks • Computational Fluid Dynamics (CFD): Flow circulation Groundwater & pollutant movement Weather prediction E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 5

Introduction Why study Numerical Methods? Numerical Analysis is a Discipline : • Need to understand concepts and theory - Know what problems can be solved. - Know what problems cannot be solved, or when problems will be troublesome. • Need to understand methods and techniques - Know why methods work, or judge when they are working. - Be able to create or modify tools (software) as needed. - Evaluate errors, convergence, and stability of arithmetic approximations. E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 6

Introduction Why study Numerical Methods? (continued) Use of Numerical Methods is an Art : • Numerical methods are approximate. • The most appropriate method(s) is not always obvious. • Evaluating precision and accuracy is an essential part of the process. E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 7

Introduction Instructors: Prof. Amparo Gil Dept de Matemática Aplicada y CC. de la Computación E.T.S. de Ingeniería de Caminos (1st floor) Universidad de Cantabria e-mail: amparo.gil@unican.es http://personales.unican.es/gila/UC-Cornell2016.pdf Office Hours: by appointment via e-mail. Prof. Jaime Puig-Pey Dept de Matemática Aplicada y CC. de la Computación E.T.S. de Ingeniería de Caminos (1st floor) Universidad de Cantabria e-mail: puigpeyj@unican.es Office Hours: by appointment via e-mail and Monday-Thursday (16:30 to 18:30 h) E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 8

Introduction Course computing framework: Software environment: MATLAB Electronic Communication by e-mail: • Computer assignments will be submitted as attachments via e-mail: amparo.gil@unican.es , puigpeyj@unican.es • Text files, MATLAB documents as attachments. • documents will be distributed directly or via web. E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 9

Introduction ENGCOMP Course Materials Required Textbook and Notes: • Chapra & Canale, Numerical Methods for Engineers, 7th Ed., 2015 Computer sessions (recommended texts): • Palm, Introduction to MATLAB for Engineers . • The MathWorks, The Student Edition of MATLAB. • Pratap, Getting Started with MATLAB. Additional material will be available at the course website, e.g.: "Matlab_primer.pdf " , "Matlab_capabilities.pdf “ , … E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 10

Selected bibliography . Atkinson, K.E. "An introduction to Numerical Analysis" . John Wiley & Sons, New York, 2nd Edition, 1989. . Burden, R. L., Faires J.D. “ Numerical Analysis ”, 9th ed. 2010, Brooks/Cole Ed. . Fish, J., Belytschko, T. " A First Course in Finite Elements ". John Wiley & Sons. 2007 . Gil, A., Segura, J., Temme, NM, “Numerical Methods for Special Functions” , 2007, SIAM. . Gockenbach, M.S. " Partial Differential Equations: Analytical and Numerical Methods ". SIAM. 2002. . Lambert, J.D., “Numerical Methods for Ordinary Differential Equations ”, 1973, John Wiley & Sons. . Mitchell, A.R., Griffiths, D.F., “The Finite Difference Method in Partial Differential Equations”, 1980, Wiley, London. . Quarteroni A., Saleri F. "Cálculo científico con MATLAB y Octave". Springer Verlag. 2006 E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 11

Introduction Periodic Assignments • Problem Sets (PS) - teams of 2/3; work together, learn from each other - teams to be formed at the end of September. • Computer Assignments (CA) - teams of 2/3; work together, learn from each other - submit electronically • Assignment submissions must follows the standards described on the course web page. E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 12

Introduction Schedule: Monday: 11:00-13:00 h. Tuesday: 13:00-14:00 h. Thursday: 13:00-14:00 h. E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 13

Contributions to the final grade: For 2 Prelims 40% Final Exam 20% Computer Assignments 20% Problem Sets and active participation 20% E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 14

• Preliminary exams : Prelim 1: To be announced (November) Prelim 2: To be announced. • Final exam To be announced E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 15

Introduction • Numerical analysis is a part of mathematics, but it works on questions that are strongly related to the use of computers and to applications from Science and Engineering. • Using numerical analysis we will be able, for instance, to handle large systems of equations, non-linearities, complicated geometries and solving engineering problems which have no analytical solution. E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 16

Introduction Roots of equations : • – We will be interested in = x \ f ( x ) 0 . 0 0 methods for solving – These methods are very useful in engineering projects, because in many occasions it is not possible to solve the design equations analytically. E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 17

Introduction Systems of linear equations : • + = a x a x b 11 1 12 2 1 + = a x a x b 21 1 22 2 2 – We will study methods for computing the set of values that simultaneously satisfy a system of algebraic equations. – Applications: calculus of structures, electric circuits, supply networks, fit of curves, etc. E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 18

Introduction Optimization : • – Determine the value x 0 leading to the optimal value of f(x). – These problems can be subject to constraints. E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 19

Introduction Fitting curves . Fitting techniques can be divided into two groups: • – Regression . It is used when one has errors in the experimental data. One looks for the curve showing the trend of the data. – Interpolation . It is used to fit tabulated data and predict intermediate values or extrapolated data. E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 20

Introduction b ∫∫ Ω • Integration : ∫ = g(x, y)dxdy I f ( x ) dx a - Determine the area below a given curve, the f(x) volume under a surface. - It has many applications Integral in engineering. Calculation of centers of gravity, areas, volumes, etc. - It can also be used to a b x solve differential equations. E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 21

Introduction • Ordinary differential equations : dy dt = f ( t; y) – ODE’s are very important because many problems can be stated in terms of variations and not in terms of magnitudes. – There are two types of problems: Initial value problems, and boundary value problems. E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 22

Introduction • Partial differential equations : 2 2 ∂ ∂ u u + = f ( x , y ) 2 2 ∂ ∂ x y y – Used for characterizing engineering problems where the behavior of the physical magnitude can be expressed in terms of speed change with respect to two or more variables. x – Approximation by finite differences or the finite element method. E. T. S. Ingeniería de Caminos, Canales y Puertos. Santander 23