ME 261: Numerical Analysis

Transcription

1 Dr. A.B.M. Toufique Hasan Associate Professor Department of Mechanical Engineering, BUET Term: July August 016 Lecture-1: Lift, F L Some mechanical engineering problems??? Flapping wing Thrust, T Drag, F D Weight, W Fig. Aerodynamic forces in a flight vehicle Fig. Flying of birds Fig. Unmanned Aerial Vehicle (UAV) 1

2 Analysis of Engineering problems involve- 1. The development of a mathematical model (theoretical Model) to represent all the important characteristics of the physical system.. The derivation of the governing equations of the model by applying physical laws such as Newton s law of motion, conservation of mass, conservation of momentum and conservation of energy. 3. Solution of the governing equations (mathematical problem); 4. Interpretation of the solution. Depending on the system being used, the governing equations may be a set of linear and non linear algebraic equations, a set of transcendental equations, a set of ordinary or partial differential equations, or an equation involving integrals or derivatives. 3 Governing equations for aerodynamic problems are the Navier-Stokes equations which consist of the following laws - 1. Conservation of mass. Conservation of Momentum mass : momentum : ρ ( ρui ) + = 0 t i ( ρui ) ( ρuiu j ) p τ + = + t j i i j Symbols have their usual meaning. j Solution of such system of equations is complicated and usual mathematical methods are failed in most cases. Mostly, we have to look for some alternatives and numerical methods (analysis) plays a great role in solving such complicated equations. 4

3 Solution of governing equations may or may not be able to find in analytical form. Analytical solutions denote exact solutions that can be used to study the behavior of the system with varying properties. Unfortunately very few practical systems lead to analytical solutions, and analytical solutions are of limited use. Numerical solutions are those that can not be expressed in the form of complete mathematical expressions. For example, result of the following integration has no closed form solution: π 1+ cos x dx 0 No analytic solution. Can only be evaluated numerically. Numerical solution (analysis) is a must. 5 To solve complicated mechanics not only in fluid mechanics but also in thermal science and applied mechanics or other physical science- the approach is known as computational mechanics. Some well defined divisions are- 1. Computational Fluid Dynamics (CFD). Computational Heat Transfer (CHT) 3. Computational Structural Dynamics (CSD) Further, there are situations where multi disciplinary physics are involved. Efficient ad proper applications of numerical techniques can be used to analyse such problems. One of such problems is known as Fluid Structure Interaction (FSI) where fluid dynamics interact with structural dynamics. For example, biomedical fluid flow problem: Heart valve movement analysis using FSI 6 3

4 Numerical analysis is the process by which mathematical problems are formulated so that they can be solved with arithmetic operations such as addition (+), subtraction (-), multiplication ( ), division ( ), and comparison. Since these operations are exactly those that computers can do, numerical analysis and computers are intimately related. Numerical analysis has a rich store of methods to find the answer by purely arithmetical operations. WHY NUMERICAL ANALYSIS - Numerical analysis can solve problems where analytical solutions are not available (using mathematical approach). - Numerical methods are capable of handling large systems of equations, different degrees of nonlinearities which are common in engineering practice. - Numerical methods can handle any complicated physical geometries which are often impossible to solve analytically. - The intelligent use of computer programs is often predicted on knowledge of basic theory underlying the methods. - Many problems cannot be approached using commercially available computer programs (using ANSYS, COMSOL, Star CCM, ADINA, FASTRAN etc), so if your expert in programming, you can design your own programs to solve problems. 7 Another difference between a numerical result and the analytical answer is that the former is always an approximation. Analytical methods are usually give the result in terms of mathematical functions that can be evaluated for a specific instance. This also has the advantage that the behavior and properties of the function are often apparent; this is not the case for a numerical answer. Parabolic velocity profile V = V(r) V max Analytic form of solution Numerical solution While the numerical result is an approximation, this can usually be as accurate as needed. Different order of accuracy (1 st order accurate, nd order accurate,.. ) can be obtained through numerical techniques. 8 4

5 In this course (ME 61) On hand practice of numerical problems using calculators. WITHOUT SCIENTIFIC CALCULATOR, NO ONE NEED TO ATTEND THE LECTURE. Use of Mathematical package-matlab (in Lab class-me 6) To be patient in solving problems. Lots of iterations, simple calculations etc. 9 Approximations Approximations and errors are an integral part of human life. They are everywhere and unavoidable. This is more so in the life of a computational scientist. Errors come in a variety of forms and sizes; some are avoidable and some are not. For example, data conversion and round off errors cannot be avoided, but a human error can be eliminated. Although certain errors cannot be eliminated completely, we must at least know the bounds of these errors to make use of our final solution. Exact numbers are those which can be expressed completely by using finite quantities of numbers or symbols. Example- 1,, 61, 3,. 1/, 1/3, 3/4, 4/5. π, e,

6 Approximations 11 Approximations 1 6

7 Course Content: 1. and motivation. Approximations and errors in numerical computations 3. Roots of equations (Polynomials and Transcendental) 4. Solution of set of non-linear equation 5. Solution of set of linear algebraic equations 6. Curve fitting 7. Interpolation methods 8. Numerical differentiation 9. Numerical integration 10. Solution of differential equations 11. Solution of set of homogeneous equations Text Book: Numerical Methods for Engineers (6 th or latest edition) -Steven C. Chapra and Raymond P. Canale 13 7