Mathematical Modeling in Civil Engineering Problems PDF

MATHEMATICAL MODELING IN CIVIL ENGINEERING PROBLEMS. LEARNING OBJECTIVES At the end of the lesson, you shall be able to 1. Gain a fundamental understanding of the importance of numerical approximations, estimates, and prediction for mathematical modeling to civil engineering problems. I...

MATHEMATICAL MODELING IN CIVIL ENGINEERING PROBLEMS. LEARNING OBJECTIVES At the end of the lesson, you shall be able to 1. Gain a fundamental understanding of the importance of numerical approximations, estimates, and prediction for mathematical modeling to civil engineering problems. Introduction Knowledge and understanding are prerequisites for the effective implementation of any tool. This is particularly true when using computers to solve civil engineering problems. Although they have great potential utility, computers are practically useless without a fundamental understanding of how engineering systems work. This understanding is initially gained by empirical means- that is, by observation and experiment. MATHEMATICAL MODELING Mathematical Models A mathematical model can be broadly defined as a formulation or equation that expresses the essential features of a physical system or process in mathematical terms. In very general sense, it can be represented as a functional relationship of the form Dependent variable = f (independent variable = parameters forcing functions) Dependent Variable = reflecting characteristics of a system Independent Variables = dimensions of system\'s behavior Parameters = system\'s properties or composition Forcing Functions = external influences in the system Solutions to a Mathematical Model The general solution is called an analytical, or exact, solution because it exactly satisfies the original differential equation. Unfortunately, there are many mathematical models that cannot be solved exactly. In many of these cases, the only alternative is to develop a numerical solution that approximates the exact solution. Numerical methods are those in which the mathematical problem is reformulated so it can be solved by arithmetic operations. CONSERVATION OF LAWS AND ENGINEERING Conservation Laws in Engineering Among the most important are the conservation laws. Although they form the basis for a variety of complicated and powerful mathematical models, the great conservation laws of science and engineering are conceptually easy to understand. They all boil down to: Change = Increases - Decreases Although simple, the above equation embodies one of the most fundamental ways in which conservation laws are used in engineering that is, to predict changes with respect to time. Hence, this gives the special name time variable (or transient) computations. Conservation Laws in Engineering Aside from predicting changes, another way in which conservation laws are applied is for cases where change is nonexistent. If change is zero, the equation becomes: Change = 0 = Increases - Decreases Increase = Decrease Thus, if no change occurs, the increases and decreases must be in balance. This case, which is also given a special name steady-state computation has many applications in engineering. Computer Programs Computer programs are merely a set of instructions that direct the computer to perform a certain task. Most engineers merely require the ability to perform engineering-oriented numerical calculations. This can be narrowed down the complexity to a few programming topics: 1\. Simple Information Representations 2\. Advanced Information Representations 3\. Mathematical Formulas 4\. Input/output 5\. Logical Representation 6\. Modular Programming Role of Computers in Numerical Analysis Numerical methods are techniques by which mathematical problems are formulated so that they can be solved with arithmetic operations. Although there are many kinds of numerical methods, they have one common characteristic: they invariably involve large numbers of tedious arithmetic calculations. It is little wonder that with the development of fast, efficient digital computers, the role of numerical methods in engineering problem solving has increased dramatically in recent years. Role of Computers in Numerical Analysis Solutions were derived for some problems using analytical, or exact, methods. However, these solutions can be derived for only a limited class of problems. These include those that can be approximated with linear models and those that have simple geometry and low dimensionality. ☐ Graphical solutions were used to characterize the behavior of systems. Although graphical techniques can often be used to solve complex problems, the results are not very precise. ☐ Graphical solutions were used to characterize the behavior of systems. Although graphical techniques can often be used to solve complex problems, the results are not very precise. Practical Issue: Nonlinear vs. Linear Linear Much of classical engineering depends on linearization to permit analytical solutions. Although this is often appropriate, expanded insight can often be gained if nonlinear problems are examined. Practical Issue: Small vs. Large Systems Without a computer, it is often not feasible to examine systems with over three interacting components. With computers and numerical methods, more realistic multicomponent systems can be examined. Practical Issue: Non-Ideal vs. Ideal Idealized laws abound in engineering. Often there are nonidealized alternatives that are more realistic but more computationally demanding. Approximate numerical approaches can facilitate the application of these nonideal relationships. Practical Issue: Sensitivity Analysis Because they are SO involved, many manual calculations require a great deal of time and effort for successful implementation. This sometimes discourages the analyst from implementing the multiple computations that are necessary to examine how a system responds under different conditions. FNUMERICAL TERMINOLOGIES Practical Issue: Design It is often a straightforward proposition to determine the performance of a system as a function of its parameters. It is usually more difficult to solve the inverse problem that is, determining the parameters when the required performance is specified. Numerical methods and computers often permit this task to be implemented in an efficient manner. Pre-Numerical Backgrounds Roots of Equations. These problems are concerned with the value of a variable or a parameter that satisfies a single nonlinear equation. These problems are especially valuable in engineering design contexts where it is often impossible to explicitly solve design equations for parameters. System of Linear Algebraic Equations. These problems are similar in spirit to roots of equations in the sense that they are concerned with values that satisfy equations. However, in contrast to satisfying a single equation, a set of values is sought that simultaneously satisfies a set of linear algebraic equations. Pre-Numerical Backgrounds Optimization. These problems involve determining a value or values of an independent variable that correspond to a \"best\" or optimal value of a function. Thus, optimization involves identifying maxima and minima. Curve Fitting. You will often have occasion to fit curves to data points. The techniques developed for this purpose can be divided into two general categories: regression and interpolation. Regression is employed where there is a significant degree of error associated with the data. Interpolation is used where the objective is to determine intermediate values between relatively error-free data. Pre-Numerical Backgrounds Integration. As depicted, a physical interpretation of numerical integration is the determination of the area under a curve. Ordinary Differential Equations. Ordinary differential equations are of great significance in engineering practice. This is because many physical laws are couched in terms of the rate of change of a quantity rather than the magnitude of the quantity itself. Partial Differential Equations. These are used to characterize engineering systems where the behavior of a physical quantity is couched in terms of its rate of change with respect to two or more independent variables.