#### Transcript Introduction - Texas A&M University

CVEN302-501 Computer Applications in Engineering and Construction Dr. Jun Zhang Teaching Assistants Mr. TBA (Section 501 and computer lab) Office: Room Phone: Email: Office Hour: Lab Hours: Mon & Thur 7:00-9:00pm Course Objectives To develop the ability to solve engineering problems numerically To evaluate numerical solution methods (knowing the advantages and limitations of numerical methods) To design, implement, and test computer programs To improve the skills of writing MATLAB codes Objective: “Numerical methods” Numerical methods give solutions to math problems written as algebraic statements that computers can execute We will learn to formulate the solutions and evaluate their applicability and performance. Objective: “Design” Writing the solutions as a series of steps a computer can execute flow chart and pseudo-code Objective: “Implementation” Converting the pseudo-code solution into a computer program. Objective: “Testing” Checking that the computer program actually solves the equations you mean to solve Evaluating the success of the numerical solution chosen Numerical accuracy, stability, and efficiency Objective: “Evaluation” Critical evaluation of the solution the program gives for the actual engineering problem. This requires all your engineering and computer skills Objective: “Presentation” Communication of the results of a computer program to the people who need to know the answer Clients Bosses Regulators Contractors Chapter 1 Mathematical Modeling and Engineering Problem Solving Engineering Problems Empirical observation and experiment certain aspects of empirical studies occur repeatedly such general behavior can be expressed as fundamental laws that essentially embody the cumulative wisdom of past experience Theoretical / Numerical formulation of fundamental laws F ma ; E Algebraic ODE PDE dv d2x F m m 2 dt dt 2T 2T 2 q 2 x y Mathematical Models Modeling is the development of a mathematical representation of a physical/biological/chemical/ economic/etc. system Putting our understanding of a system into math Problem Solving Tools: Analytic solutions, statistics, numerical methods, graphics, etc. Numerical methods are one means by which mathematical models are solved Mathematical Modeling The process of solving an engineering or physical problem. Engineering or Physical problems (Description) Mathematical Modeling Approximation & Assumption Formulation or Governing Equations Analytical & Numerical Methods Solutions Applications Common features operation Bungee Jumper You are asked to predict the velocity of a bungee jumper as a function of time during the free-fall part of the jump Use the information to determine the length and required strength of the bungee cord for jumpers of different mass The same analysis can be applied to a falling parachutist or a rain drop Bungee Jumper / Falling Parachutist Newton’s Second Law F = ma = Fdown - Fup = mg - cdv2 (gravity minus air resistance) Observations / Experiments Where does mg come from? Where does -cdv2 come from? Now we have fundamental physical laws, so we combine those with observations to model the 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 worrying about how to solve it. Exact (Analytic) Solution Newton’s Second Law dv m mg c d v 2 dt cd 2 dv g v dt m Exact Solution v( t ) gc d mg tanh t cd m Numerical Method What if cd = cd (v) const? Solve the ODE numerically! dv v lim dt t 0 t v v ( t i 1 ) v ( t i ) t ti1 ti Assume constant slope (i.e, constant drag force) over t Numerical (Approximate) Solution Finite difference (Euler’s) method dv v v( t i 1 ) v( t i ) dt t ti1 ti v( t i 1 ) v( t i ) cd g v( t i ) 2 ti1 ti m Numerical Solution cd 2 v( t i 1 ) v( t i ) g v( t i ) ( t i 1 t i ) m Example 1.2 Hand Calculations A stationary bungee jumper with m = 68.1 kg leaps from a stationary hot air balloon. Use the Euler’s method with a time increment of 2 s to compute the velocity for the first 12 s of free fall. Assume a drag coefficient of 0.25 kg/m. cd 2 v( t i 1 ) v( t i ) g v( t i ) ( t i 1 t i ) m t0 0; v( t0 ) 0 Explicit time-marching scheme m = 68.1 kg, g = 9.81 m/s2, cd = 0.25 kg/m Euler’s Method Use a constant time increment t = 2 s Step 1 Step 2 0.25 v 0 9.81 (0 ) 2 ( 2 0 ) 19.6200m / s 68.1 0.25 t 4 s; v 19.6200 9.81 ( 19.6200) 2 ( 4 2 ) 36.4317m / s 68.1 t 2 s; Step 3 t 6 s; Step 4 t 8 s; Step 5 Step 6 0.25 v 36.4137 9.81 ( 36.4137) 2 (6 4 ) 46.2983m / s 68.1 0.25 v 46.2983 9.81 ( 46.2983) 2 ( 8 6 ) 50.1802m / s 68.1 0.25 t 10s; v 50.1802 9.81 ( 50.1802) 2 ( 10 8 ) 51.3123m / s 68.1 0.25 t 12s; v 51.3123 9.81 ( 51.3123) 2 ( 12 10) 51.6008m / s 68.1 The solution accuracy depends on time increment Example: Bungee Jumper Olympic 10-m Platform Diving Air : a dv 2 m mg c da v mg dt w dv Water : m mg cdw v v mg dt Buoyant Force cda cdw Example: Finite Elements and Structural Analysis Simple truss - force balance Complex truss Instead of limiting our analysis to simple cases, numerical method allows us to work on realistic cases.