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

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Lecture 1 Presentation

Engineering Computation

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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.

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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.

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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).

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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

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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.

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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.

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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)

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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.

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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 “ , …

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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

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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.

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Introduction

Schedule: Monday: 11:00-13:00 h. Tuesday: 13:00-14:00 h. Thursday: 13:00-14:00 h.

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Contributions to the final grade:

For 2 Prelims 40% Final Exam 20% Computer Assignments 20% Problem Sets and active participation 20%

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Preliminary exams:

Prelim 1: To be announced (November) Prelim 2: To be announced.

Final exam

To be announced

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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.

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Introduction

Roots of equations :

– We will be interested in methods for solving – These methods are very useful in engineering projects, because in many

ccasions it is not possible

to solve the design equations analytically.

. ) x ( f \ x =

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Introduction

Systems of linear equations:

– 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.

2 2 22 1 21 1 2 12 1 11

b x a x a b x a x a = + = +

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Introduction

Optimization:

– Determine the value x0 leading to the optimal value of f(x). – These problems can be subject to constraints.

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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.

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Introduction

Integration:
Determine the area below

a given curve, the volume under a surface.

It has many applications

in engineering. Calculation of centers of gravity, areas, volumes, etc.

It can also be used to

solve differential equations.

∫

=

b a

dx ) x ( f I f(x) x Integral a b

∫∫Ω

y)dxdy g(x,

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Introduction

Ordinary differential

equations : – ODE’s are very important because many problems can be stated in terms of variations and not in terms

f magnitudes.

– There are two types of problems: Initial value problems, and boundary value problems.

dy dt = f ( t; y)

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Introduction

Partial differential

equations: – 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. – Approximation by finite differences or the finite element method.

) y , x ( f y u x u

2 2 2 2

= ∂ ∂ + ∂ ∂ y x

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Mathematical Models

CONSERVATION LAWS -- BALANCE

F

x =

∑ F

y =

∑

At each node, Balance= Force Equilibrium P V1 H1 V2 H2 b1 b2 V1 H1 Fb1

6 equations - 6 unknowns

Flow in = 80 Flow in = 100 Flow out = ? (60) Flow out = 120 Incompressible fluid flow, Each junction: Flow Balance , flow in = flow out Increases = Decreases

Single elements – Systems – Networks

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Mathematical Models

Physical Laws
Newton Law, dynamic equilibrium
Analytic solution (t=0, v=0):
Approximate solution (“Euler method (forward)”):

D

F

U

F

m·a; = F

) s / kgm ( F

2

) kg ( m ) s / m ( a

2

; m F a = ; m F dt dv = ; m (Up) F (Down) F dt dv

U D

+ = ; m cv mg dt dv − = ) s / Kg ( c

) t

(t

) v(t m c

g

) v(t ) v(t m c·v

m·g

dt ) dv(t

i 1 i i i 1 i t t t t i

i i

+ + = =

      + = =      

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Mathematical Models

To solve the problem numerically, one replaces the derivative by

a finite difference, thus transforming the problem into a very simple one containing only simple algebraic operations:

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Numerical Differentiation

Forward: Centered:

How big a step size should we select?
One- or two-sided formula:

What are the advantages of each?

How is optimal step size affected by:
precision of numerical calculations?
precision with which f is computed?
choice of formula?

( ) ( ) ( )

i 1 i i

f x f x f x h

+

− ′ =

( ) ( ) ( )

i 1 i 1 i

f x f x f x 2h

+ −

− ′ =

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Mathematical Models

Comparing solutions:

2 4 6 8 10 12 14 16 18 20 10 20 30 40 50 60

Numerical solution, ∆t=1sec Numerical solution, ∆t=2seg T (sec) V (m/sec) Exact solution

Approximate Approximat (∆t=2s.) (∆t=1s.) 2 16,422 19,62 17,819339 4 27,798 32,037357 29,697439 6 35,678 39,896213 37,615198 8 41,137 44,870026 42,893056 10 44,919 48,017917 46,411195 12 47,539 50,010194 48,756333 14 49,353 51,271092 50,319566 16 50,611 52,069105 51,361594 18 51,481 52,574162 52,056193 20 52,085 52,893809 52,519203 Exact t(sec)