CE 421 - Civil Engineering Systems

Unit Objectives

 

  

UNIT 1: Introduction to Systems Analysis and Problem Solving

UNIT 2: Basic Theory of Probability and Statistics

UNIT 3: Correlative Models

UNIT 4: Introduction to Numerical Analysis

UNIT 5: Introduction to Optimization

 
 
 
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UNIT 1: Introduction to Systems Analysis and Problem Solving

  1. To be able to define the following terms:
    1. Systems Analysis
    2. Engineering System
  2. To be able to identify the four characteristics of an engineering system.
  3. To be able to diagram and identify the different components of the systems approach methodology.
  4. To be able to identify the 8 principles of problem definition.
  5. To be able to identify and apply the 7 strategies for generating solutions. 
  6. To be able to define the following terms:
    1. mathematical model
    2. economic model
    3. descriptive model
    4. prescriptive model
  7. To be able to construct a flowchart of the mathematical modeling process.
  8. To be able to describe the four types of solution evaluation.

UNIT 2: Basic Theory of Probability and Statistics

  1. To be able to construct:
    1. Histogram
    2. Frequency Diagram
    3. Cumulative frequency diagram
  2. To be able to determine:
    1. Mode, median, mean
    2. Standard Deviation, coefficient of variation, skewness
  3. To be able to describe the two different ways to define probability
    1. frequency approach
    2. set approach
  4. To be able to understand and apply the basic postulates and theorems of probability.
  5. To be able to construct a histogram and associated frequency diagram for a given set of data.
  6. To be able to explain the following concepts:
    1. random variable
    2. probability mass function - discrete
    3. probability density function - continuous
    4. To be able to understand and use standard discrete and continuous probability functions.
    5. To be able to construct a cumulative distribution function from a given set of data.
    6. To be able to select and fit a continuous probability density function for a given set of data.
    7. To understand the concept of monte carlo simulation and to be able to construct and apply a monte carlo simulation model.

UNIT 3: Correlative Models

  1. Regression Models:
    1. To be able to define a correlative model and give an example of one.
    2. To be able to identify when it is appropriate to use a regression model.
    3. To be able to determine the parameters of the "best" regression model.
    4. To be able to define and calculate the following:
      1. Standard error of the Estimate.
      2. Coefficient of Determination.
    5. To be able to identify the two ways to develop nonlinear regression models.
      1. Linear transformation method.
      2. Polynomial regression method.
    6. To be able to develop either a linear transformation model or a polynomial regression model.
    7. To be able to know when to apply multiple linear regression and general non-linear regression.
  2. Interpolation Models:
    1. To understand when to use regression and when to use interpolation.
    2. To understand the problems associated with using higher order polynomial interpolation.
    3. To know when to apply Newton Polynomial Interpolation.
    4. To know how to determine the coefficients associated with applying a second order polynomial interpolation using the Newton Divided Difference Method.
    5. To know when to use Lagrange Polynomial Interpolation.
    6. To know how to determine coefficients associated with applying a second order polynomial interpolation using the Lagrange Polynomial Method.
    7. To know what a spline is and to know the advantages and disadvantages of the different types of splines.
    8. To be able to apply linear, quadratic and cubic splines.

UNIT 4: Introduction to Numerical Analysis

  1. Linear Systems Analysis: (Chapra and Canale chapters 7-10)
    1. To be able to write the system of state equations in matrix form for a given engineering system.
    2. To understand the basic rules of matrix algebra.
    3. To be able to perform the following matrix algebra operation:
      1. Add and multiply two matrices.
      2. Determine the transpose of a matrix.
      3. Determine the determinant of a matrix.
      4. Determine the inverse of a matrix.
    4. To be able to determine the solution of two linear equations:
      1. Graphically.
      2. Using Cramer's Rule.
      3. Using Direct Substitution.
    5. To be able to explain the graphical significance of a matrix determinant:
      1. equal to zero.
      2. approximately equal to zero.
    6. To be able to solve a system of linear equations using Gauss Elimination.
    7. To be able to identify the advantages and disadvantages of Gauss Elimination.
    8. To be able to identify strategies for dealing with the disadvantages of the Gauss Elimination method.
    9. To be able to identify the uses of a matrix inverse.
      1. Sensitivity analysis.
      2. Solution of problems with multiple loading vectors.
    10. To be able to identify advantages and disadvantages of Iterative Solution Methods.
    11. To be able to characterize a diagonally dominant system.
    12. To be able to identify the differences between the following:
      1. The Gauss-Seidel Method.
      2. The Jacobi Method.
    13. To be able to solve a system of equations with an iterative method.
    14. To be able to improve the convergence characteristics of a weakly diagonally dominant system of equations.
    15. To be able to identify the advantages and limitations of the LU Decomposition Method.
    16. To be able to decompose a matrix into its [U] and [L] triangular matrices using
      1. Gauss Elimination
      2. Crout Decomposition
    17. To be able to solve a system of equations using [U] and [L].
    18. To be able to apply the Thomas Algorithm to tri-diagonal systems of equations.
  2. Nonlinear Analysis: (Chapra and Canale chapters 4-6)
    1. To be able to recast a nonlinear equation into the form of a root determination problem.
    2. To be able to identify the major difference between open and closed methods.
    3. To be able to identify a significant advantage and limitation of the following methods:
      1. Bisection Method.
      2. False Position Method.
      3. Newton's Method.
      4. Secant Method.
    4. To be able to solve a nonlinear equation using the above methods.
    5. To be understand and be able to apply the Taylor's series expansion.
  3. Numerical Calculus: (Chapra and Canale chapters 15-18)
    1. To be able to write and apply the following first order difference equations:
      1. forward difference
      2. backward difference
      3. central difference
    2. To be able to derive the difference equations from the Taylor Series and know the order of accuracy of each equation and why.
    3. To be able to write and apply the following integration equations:
      1. Trapezoidal Rule
      2. Simpson's 1/3 Rule
      3. Simpson's 3/8 Rule
    4. To recognize the special significance of both Simpson's Rules.
    5. To be able to identify the limitations of the Newton Coates equations.
    6. To be able to write and apply the following integration equations:
      1. Romberg Integration
      2. Gauss Quadrature Integration
    7. To be able to identify the advantages and limitations of the following:
      1. Romberg Integration
      2. Gauss Quadrature Integration
  4. Differential Equations: (Chapra and Canale chapters 19-20)
    1. To understand the graphical relationship between a differential equation and its solution.
    2. To be able to apply the following methods and know the difference between them:
      1. Euler's Method
      2. Heun's Method
      3. Runge-Kutta Method(s).
      4. To be able solve a system of ordinary differential equations by the methods listed in D.2.

UNIT 5: Introduction to Optimization

  1. Unconstrained Optimization
    1. To be able to evaluate the necessary and sufficiency conditions for an unconstrained function. (Optimality Conditions)
    2. To be able to identify and apply the four ways to determine the optimal solution to a univariate unconstrained function (Unconstrained Optimization):
      1. Exact Explicit Method
      2. Exact Implicit Method (Region Elimination)
      3. Exact Implicit Method (Gradient Minimization)
      4. Approximate Explicit
    3. To be able to understand and apply the two analytical methods for determining the optimal solution to a multivariate unconstrained function:
      1. Explicit Method
      2. Newton's Method
    4. To be able to understand and apply the three major gradient methods for the determining the optimal solution to a multivariate unconstrained function:
      1. Steepest Descent Method (Cauchy)
      2. Conjugate Gradient Method (Fletcher-Reeves)
      3. Variable Metric Methods (Davidon-Fletcher-Powell)
  2. Constrained Optimization
    1. To be able to identify the analytical methods for solving constrained optimization methods (Constrained Optimization):
      1. Variable Substitution
      2. Lagrange Multiplier Method
      3. equality constraints
      4. inequality constraints
    2. To be able to solve a nonlinear optimization problem using the Lagrange multiplier method:
      1. with equality constraints
      2. with inequality constraints
    3. To understand the geometric significance of negative, positive, and zero values of the generalized Lagrange multiplier.
    4. To be able to identify the numerical methods for solving constrained optimization problems and know the advantages and disadvantages of each:
      1. Gradient Based Methods (Penalty Function Method)
      2. General Search Methods ( Box Complex Method)
    5. To be able to formulate a nonlinear optimization problem.
  3. Linear Programming
    1. To recognize the characteristics of a linear programming problem
      1. Linear objective function
      2. Linear Constraints
      3. Non-negative decision variables
    2. To be able to formulate a linear programming problem
    3. To be able to convert linear programming constraints into a system of equations.
    4. To be able to solve a linear programming problem.
      1. graphically
      2. By enumeration
    5. To be able to formulate and solve a problem using EXCEL.
    6. To be able to formulate and solve an linear programming problem using EXCEL.