1. PRESENTED BY M.PRAVEEN M.YOGESH K.VISHAL S.SURYA R.VENGATESH GUIDED BY ASHOK (AP/S&H)

2. SUMMARY NUMERICAL METHODS IN ENGINEERING NUMERICAL METHODS IN CRIME DETECTION NUMERICAL METHODS IN SCIENTIFIC COMPUTING NUMERICAL METHODS IN FINDING ROOTS NUMERICAL METHODS IN HEAT EQUQTION 3. Introduction Why are Numerical Methods so widely used in Engineering? Engineers use mathematical modeling (equations and data) 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). 4. 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): Weather prediction Groundwater & pollutant movement 5. Numerical Methods in Engineering 6. Topic 1 Overview of the Modelling Process 7. Why numerical modelling? To reduce the number of prototypes constructed. To use as the forward solver in an optimization routine to gain the best possible design. 8. What are the alternatives? Design and build lots of physical models Time consuming Expensive Measurement difficulties (cables, calibration, mechanical precision, interference, mutual coupling, field confinement, etc) Analytical solutions Limited to simple models because of the computational complexity 9. Numerical Modelling Procedure Select/write your numerical modelling code. Find a simple analytical model and divide into segments/pixels/voxels. These MUST have every dimension less than l/10 where l is the wavelength in the material being modelled. (Note that this is NOT the free space wavelength). 10. Lets do the calculation of l The complex propagation coefficient g is given by the equation where s is the conductivity, e is the absolute permittivity, m is the absolute magnetic permeability, w is the angular radiation frequency, and j is the imaginary number, a is the attenuation constant b is the phase constant )( weswmbag jjj 11. The wavelength in the material b l 2 At this frequency we have The wavelength depths on the electromagnetic properties of the material. Remember NO segment/pixel/voxel can be larger than in any dimension l/10 . This can make life very difficult when you have electrically large conductive materials. 12. Solution MoM and FDTD both use perfectly conducting materials. To introduce finite conductivity, we can use lumped impedance elements. 13. Maths and crime: Deblurring a number plate A short crime story Burglar robs a bank Escapes in a getaway car Pursued by police 14. GOOD NEWS Police take a photo BAD NEWS Photo is blurred 15. SOLUTION Find a model of the blurring process Blurring function g Original image f Blurring formula Inverting the formula we can get rid the blur BUT need to know the blurring function g ydygyxfxh 2 )()()( 16. 2/)/( 222 dxgdexhdeef xixiyi Inversion formula h(x) f(x) An example of Image Processing 17. Scientific computing Design and analysis of algorithms for numerically solving mathematical problems in science and engineering Considers the effect of approximations and performs error analysis modern simulations of engineering applications 18. Computational problems: attack strategy Develop mathematical model (usually requires a combination of math skills and some a priori knowledge of the system) Come up with numerical algorithm (numerical analysis skills) Implement the algorithm (software skills) Run, debug, test the software Visualize the results Interpret and validate the results Mathematical modeling 19. Computational problems: well-posedness The problem is well-posed, if (a) solution exists (b) it is unique (c) it depends continuously on problem data Simplification strategies: Infinite finite Nonlinear linear High-order low-order 20. What computers cant do Solve (by reasoning) general mathematical problems they can only repetitively apply arithmetic primitives to input. Solve problems exactly. Represent all numbers. Only a finite subset of the numbers between 0 and 1 can be represented. 21. Finding roots / solving equations General solution exists for equations such as ax2 + bx + c = 0 The quadratic formula provides a quick answer to all quadratic equations. However, no exact general solution (formula) exists for equations with exponents greater than 4. 22. Finding roots Even if exact procedures existed, we are stuck with the problem that a computer can only represent a finite number of values thus, we cannot validate our answer because it will not come out exactly However we can say how accurate our solution is as compared to the exact solution 23. Finding roots, continued Transcendental equations: involving geometric functions (sin, cos), log, exp. These equations cannot be reduced to solution of a polynomial. Convergence: we might imagine a reasonable procedure for finding solutions, but can we guarantee it terminates? 24. Square root program If the input c < 1, the root lies between c and 1. Else, the root lies between 1 and c. The (positive) square root function is continuous and has a single solution. c = x2 F(x) = x2 - c Example: F(x) = x2 - 4 -6 -4 -2 0 2 4 6 0 0.5 1 1.5 2 2.5 3 25. double Sqrt(double c, double tol) { double a,b, mid, f; // set initial boundaries of interval if (c < 1) { a = c; b = 1} else { a = 1; b = c} do { mid = ( a + b ) / 2.0; f = mid * mid - c; if ( f < 0 ) a = mid; else b = mid; } while( fabs( f ) > tol ); return mid; } -6 -4 -2 0 2 4 6 0 0.5 1 1.5 2 2.5 3 26. Heat equations The heat capacity c of a material gives the amount of energy needed to raise the temperature T one kilogram of the material by one degree K (Kelvin). The thermal conductivity k of a material indicates the amount of energy transmitted trough a plate with thickness 1 m and 1sq.m area if the temperatures at the two sides differ by 1 K. A(x) c ( T (t, x)/t)=/x(k A(x)(T (t, x)/x))+A(x)f(t,x) 27. The flux of thermal energy is a vector indicates the direction of the flow and the amount of thermal energy flowing per second and square meter Fouriers law of heat conduction can be stated as = k T 28. One Dimensional Heat Equation If a temperature T over a solid (with constant cross section A) is known to depend on one coordinate x only, then the change of temperature T measured over a distance x will lead to a flow of thermal energy Q. If the time difference is t then Q/t = k A (T/x) 29. Two Dimensional Heat Equation If the domain G R2 with boundary curve C describes a thin plate with constant thickness h then we may assume that the temperature will depend on t, x and y only and not on z. E (t) =