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CVEN302-502

CVEN302-502. Computer Applications in Engineering and Construction. Modeling is the development of a mathematical representation of a physical/biological/chemical/ economic/etc. system Putting our understanding of a system/problem into math Numerical methods are one means by which

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CVEN302-502

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  1. CVEN302-502 Computer Applications in Engineering and Construction

  2. Modeling is the development of a mathematical representation of a physical/biological/chemical/ economic/etc. system Putting our understanding of a system/problem into math Numerical methods are one means by which mathematical models are solved

  3. Example: Falling Parachutist F=ma =Fdown +Fup =mg-cv (gravity minus air resistance) Where does mg come from? Observations. Where does -cv come from? More observations!

  4. Now we have fundamental physical laws, so we combine those with observations to model system. A lot of what you will do is “canned” but need to know how to make use of observations. How have computers changed problem solving in engineering? Allow us to focus more on the correct description of the problem at hand, rather than worry about how to solve it.

  5. Example: Finite elements and structural analysis Complex truss Simple truss - force balance Instead of limiting our analysis to simple cases, numerics allows us to work on realistic cases.

  6. What are the fundamental laws we use in modeling? Conservation of mass - i.e. traffic flow estimation Conservation of momentum -i.e. force balance in structures Conservation of energy - i.e. redox chemistry in water treatment plant

  7. Issues to be considered in modeling and numeric methods 1.Nonlinear vs. Linear 2.Large vs. Small systems 3.Nonideal vs. Ideal 4.Sensitivity analysis 5.Design

  8. Back to our example: the falling parachutist F=ma=mg-cv dv m = mg - cv dt dv mg - cv = dt m Analytic solution (from calculus) gm ( ) ( ) ( ) - / c m t v t = 1 - e c

  9. Numerical solution discretize original equation ( ) ( ) dv D v v t - v t i + 1 i @ = dt D t t - t i + 1 i ( ) ( ) v t - v t c ( ) i + 1 i = g - v t i t - t m i + 1 i c é ù ( ( ) ( ( ) ) ) v t = v t + g - v t t - t ê ú i + 1 i i i = 1 i m ë û

  10. Finally, combining analytical and numerical techniques Catenary cable (power lines) From force balances, displacement can be modeled by a differential equation 2 2 d y w dy æ ö = 1 + ç ÷ 2 dx T dx è ø a

  11. Forces acting on catenary 12 Tb 10 8 6 W=ws Ta 4 2 0 -6 -4 -2 0 2 4 6 8 10 12

  12. Can solve by integration æ ö T T w ç ÷ a a y = x + y - cosh ç ÷ 0 w T w è ø a Where 1 ( ) x x - cosh x = e + e 2 This equation is not analytically solvable for w or Ta

  13. Say we are given w, y0 and the value of y at an x. Can solve for Ta using numerical methods æ ö T T w ç ÷ a a y = x + y - cosh ç ÷ 0 w T w è ø a Becomes Try different values of Ta until lhs is 0

  14. We will use Matlab as the computer language of choice for this course • Anything you can do in Fortran or C you can do in Matlab • Easier debugging system • Built-in graphics • Many, many functions already exist • Excellent help capabilities Matlab Truss example – nice animation 14

  15. In short, you will use numeric methods throughout your career - even if you don’t write programs - even if you go into management If we didn’t have numerical methods, we might as well be...

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