Prepared by Asif Ali Shaikh/Sania Qureshi Assistant Professors Department of BSRS Mehran, UET, Jamshoro
Note: Detailed discussion of following topics is being carried out in the classes. It is highly recommended that you may follow textbooks suggested by your mentors. These handouts can never ever replace textbooks.
MEHRAN UNIVERSITY OF ENGINEERING AND TECHNOLOGY, JAMSHORO Odd Semester 2015 15 Batch 1st Semester M.E. in Structural Engineering Subject: Computer Applications in Structural Engineering From 24/03/2015 to 12/07/2015 LECTURE 01 Conducted on: 25/03/2015 Timings: 3pm to 4.30pm
Computer Applications in Structural Engineering
Scientific Computing Computational Science Scientific Computation Numerical Analysis (Derivation of algorithms, sensitivity, consistency, order, convergence and their error analysis) Numerical Methods (Use of developed algorithms) Iterative Methods Indirect Methods Approximate Methods Without mathematics, Numerical Methods and Numerical Analysis there's nothing you can do. Analytical Methods Everything around you is Exact Methods mathematics. Everything Direct Methods around you is numbers. Non-Iterative Methods by What is Iteration? Shakuntala Devi
Repetition of certain series of steps. Need and Importance of Scientific Computing
Computational Science (also scientific computing or scientific computation) is concerned with constructing mathematical models and quantitative analysis techniques and using computers to analyze and solve scientific problems. In practical use, it is typically the application of computer simulation and other forms of computation from numerical analysis and theoretical computer science to problems in various scientific disciplines. Simply, Numerical Analysis is the study of algorithms that use numerical approximations for the problems of mathematical analysis. The field is different from theory and laboratory experiment which are the traditional forms of science and engineering. The scientific computing approach is to gain understanding, mainly through the analysis of mathematical models implemented on computers.
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Prepared by Asif Ali Shaikh/Sania Qureshi Assistant Professors Department of BSRS Mehran, UET, Jamshoro
Note: Detailed discussion of following topics is being carried out in the classes. It is highly recommended that you may follow textbooks suggested by your mentors. These handouts can never ever replace textbooks.
Main objective of Scientific Computing is to design and critically analyze the techniques used to obtain approximate but accurate solution to complicated mathematical and engineering problems. Direct or Analytical Methods compute the solution to a problem in a finite number of steps. These methods would give the precise answer if they were performed in infinite precision arithmetic. Such methods are: Gauss-Elimination and Jordan Methods, QR Factorization Method and Simplex technique. On the other hand, iterative methods used in Scientific Computing are not expected to terminate the process of computation in a finite number of steps. In practice, finite precision is used and the result is an approximation of the true solution (assuming stability). While using an iterative technique, solution to a problem starts with an initial guess that converge to the exact solution only in finite time limit. Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps (in general). Examples include Newton's method, the bisection method, and Jacobi iteration. Iterative methods are more common than direct methods in numerical analysis. Some methods are direct in principle but are usually used as though they were not, e.g. GMRES and the conjugate gradient method. For these methods the number of steps needed to obtain the exact solution is so large that an approximation is accepted in the same manner as for an iterative method. Furthermore, continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem; this process is called discretization. For example, the solution of a differential equation is a function. This function must be represented by a finite amount of data, for instance by its value at a finite number of points at its domain, even though this domain is a continuum. Numerical vs. Analytic Solutions Suppose you have a mathematical model and you want to understand its behavior. That is, you want to find a solution to the set of equations. The best is when you can use calculus, trigonometry, and other math techniques to write down the solution. Now you know absolutely how the model will behave under any circumstances. This is called the analytic solution, because you used analysis to figure it out. It is also referred to as a closed form solution. But this tends to work only for simple models. For more complex models, the math becomes quite complicated. Then you turn to numerical methods of solving the equations, such as the Runge-Kutta method. For a differential equation that describes behavior over time, the numerical method starts with the initial values of the variables, and then uses the equations to figure out the changes in these variables over a very brief time period. It’s only an approximation, but it can be a very good approximation under certain circumstances. A computer must be used to perform the thousands of repetitive calculations involved. The result is a long list of numbers, not an equation. This long list of numbers can be used to drive an animated simulation. There is also a middle ground between these two methods. There are many important nonlinear equations for which it is not possible to find an analytic solution. However, there are techniques where you can find approximate analytic solutions that are close to the true solution, at least within a certain range. One such method is called the perturbation method. The advantage over a numerical solution is that you wind up with an equation (instead of just a long list of numbers) which you can gain some insight from. There are number of Mathematical Models of considerable importance which are not solvable by analytical methods available in literature of Mathematics. Such as:
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Prepared by Asif Ali Shaikh/Sania Qureshi Assistant Professors Department of BSRS Mehran, UET, Jamshoro
Note: Detailed discussion of following topics is being carried out in the classes. It is highly recommended that you may follow textbooks suggested by your mentors. These handouts can never ever replace textbooks.
Large Linear Algebraic Systems arising from study of Electrical Networks, Mass-Spring Systems, Ordinary Differential and Partial Differential Equations are required to be solved using iterative schemes rather than using Gauss-Elimination or Cramer’s Rules. Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations. Analysis of Mass-Spring Damper Model in Mechanical Engineering and Subject of Structural Engineering give rise to an Eigenvalue Problems which are required to be solved by numerical techniques (Power’s Method, Shifted Power’s Method, Inverse Method and Rayleigh Quotient Iteration). Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving partial differential equations numerically. A Mathematical relationship (model) can be established using experimental data and then intermediate and future values are predicted using tools of Scientific Computing (Model Fitting and Data Analysis) Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. Historically, such algorithms were developed within the overlapping field of operations research. Insurance companies use numerical programs for actuarial analysis. Numerical Analysis finds applications nearly in all fields of engineering and physical sciences but in the 21st century, fields like Life Sciences and Arts have also adopted elements of scientific computations. Ordinary Differential Equations appear in Celestial Mechanics (planets, stars and galaxies), Numerical Linear Algebra is important for Data Analysis; Stochastic Differential Equations and Markov Chains are essential in simulating living cells for Medicines and Biology. Much of the Numerical Analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors. When using numerical algorithms and computing finite precision; errors of approximation or rounding and truncation errors are introduced. It is highly important to have a notion of their nature and order. A newly developed algorithm is worthless without discussion of error analysis. Neither does it make sense to use methods which introduce errors with magnitude larger than the effects to be measured or simulated. On the other hand, using a method with very high accuracy might be computationally too expensive to justify the gain in accuracy.
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Prepared by Asif Ali Shaikh/Sania Qureshi Assistant Professors Department of BSRS Mehran, UET, Jamshoro
Note: Detailed discussion of following topics is being carried out in the classes. It is highly recommended that you may follow textbooks suggested by your mentors. These handouts can never ever replace textbooks.
Engineering fields for Scientific Computing: Civil Engineering
Structural Analysis Traffic and Environmental Simulations Geographic Information
Mechanical and Aerospace Engineering
Heat Flow Fluid Dynamics Structural Optimization
Electrical and Computer Engineering
Network Analysis Signal Processing Electromagnetic Fields Load Flow Analysis
Chemical and Pharmaceutical Engineering
Molecular Modeling System Simulations Biomechanical and Biomedical Engineering
Numerical Simulations of certain systems using Scientific Computing
Astrophysics: Life cycle of galaxies
Climate Research: Gulf Stream, Greenhouse Effect and etc
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Prepared by Asif Ali Shaikh/Sania Qureshi Assistant Professors Department of BSRS Mehran, UET, Jamshoro
Note: Detailed discussion of following topics is being carried out in the classes. It is highly recommended that you may follow textbooks suggested by your mentors. These handouts can never ever replace textbooks.
Weather Forecast: Tornadoes – where, when, and how strong? Simulation of Stock-markets or predicting economic effects
Security of nuclear power plants, tests for nuclear weapons Statics: Stability of buildings
Propagation of harmful substances
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Prepared by Asif Ali Shaikh/Sania Qureshi Assistant Professors Department of BSRS Mehran, UET, Jamshoro
Note: Detailed discussion of following topics is being carried out in the classes. It is highly recommended that you may follow textbooks suggested by your mentors. These handouts can never ever replace textbooks.
Aerodynamics and Turbulence: objects in a wind tunnel and so on.
Car Industry: Crash tests
Analysis and study of Proteins
Molecular dynamics: crystal structure and macromolecules 6
Prepared by Asif Ali Shaikh/Sania Qureshi Assistant Professors Department of BSRS Mehran, UET, Jamshoro
Note: Detailed discussion of following topics is being carried out in the classes. It is highly recommended that you may follow textbooks suggested by your mentors. These handouts can never ever replace textbooks.
Example. Find length of y sin x from
0 to ?
Solution: From Calculus
b
2 Arc Length 1 y dx a
Arc Length 1 cos2 x dx 0
This integral cannot be evaluated by techniques so far learnt. Therefore, an attempt is made to solve the integral using MATLAB which produced the following result MATLAB Commands >> syms x >> int(sqrt(1+cos(x)^2),0,pi) >>ans =2*2^(1/2)*ellipticE(1/2) >> vpa(ans) >>ans =3.8201977890277120179047620821714 Thus, result of integral is not elementary mathematical function and its approximate answer is 3.8202 which can be obtained using a good technique of numerical integration. Accuracy refers closeness of computed solution (using an iterative method) to true (exact) solution of the given problem. Precision refers to how closely computed values (using an iterative method) agree with each other.
Accuracy vs Precision 7
