# Lecture 1*: Mathematical and Numerical Modeling

Introduction to the course:

Why do we need numerical methods?

Why is computer essential for numerical methods?

# General steps followed in numerical analysis

In order to obtain a numerical solution for a problem in civil engineering or any other field of science and engineering, one should follow the following general six steps:

1-     Problem definition:

State the problem verbally including the process behavior, geometry, dimensions, known & unknown variables, and simplifying assumptions.

2-    Mathematical formulation:

Represent the information in step 1 by functional relationship(s) usually algebraic or differential equation(s) using fundamental laws and principals such as equilibrium.

3-    Numerical formulation:

Approximate the above mathematical formulation and develop a numerical procedure such that the problem can be solved by simple repeated arithmetic operations.

4-    Computer implementation:

The above numerical procedure is implemented in a computer code to carry out the repeated calculations. Note that for certain problems a computer package can be used where step 3 becomes unnecessary.

5-    Verification of the accuracy of the results:

The numerical results are checked against the results of other analytical or numerical methods, if available. The convergence of the solution can be checked using space or/and time mesh refinement.

6-    Interpretation of computer output:

The obtained numerical output is used to arrive at a decision. Sometimes this could be the hardest part of solving the problem because it requires full understanding of the physical problems.

The following simple example illustrates the use of the above four steps.

Example 1: Flow of water in open channels

Flow rate = q

Velocity of flow = v

Slope of channel= s                                                                       h

Roughness coefficient= n

b

Given q = 5 m3/s, b = 20 m, n= 0.03, s = 0.0002, determine the height of water in the channel.

1-Problem definition:

It is required to determine the value of h corresponding to the following parameters: q = 5 m3/s, b = 20 m, n= 0.03, s = 0.0002

2-Mathematical Formulation:

We need to derive a mathematical relation between h and all other variables. From hydraulics, we know that the continuity equation is given by:

(1)

The velocity is related to the cross-section geometry and slope through Manning equation:

where                                                    (2)

Using (2) in (1), we get the required relation:

or

Using the given data, the above equation becomes:

(3)

3-Numerical Formulation & 4-Computer Implementation:

The root of eq. (3) is the required value of h. We will learn in the first part of this course how to obtain roots of all kinds of equations. The result for this equation is h = 0.7023 m.

5- Verification of the accuracy of the results:

Using h = 0.7023 in (3), we get 0 = 0       O.K.

# Example2: Deflection of a beam

Determine the value and location of the maximum deflection for the beam loaded as shown.

W0

L

Assume that the beam is made of ASTM-A36 steel and has a w360x262 section.

1-Problem definition:

The load will deflect due to its own weight and the given triangular load. The shape of central line of the beam after deflection is called the elastic curve y(x). It is required to determine the value of x at which y(x) becomes maximum.

The following data is given:

the intensity of the triangular load, w0 = 2 kN/m

the length of the beam, L=4.5 m

From Tables:

the elastic modulus, E=200 N/mm2 for ASTM-A36 steel

the moment of the inertia about the axis of bending (z-axix), Iz =849x106mm4 for w360x262.

Furthermore let us assume that the self-weight of the beam is very small as compared to w0.

2-Mathematical Formulation:

From CE203 we have learnt how to derive the equation of the elastic curve (which is mainly based on equilibrium):

where w0, E, Iz and L are as given above.

In order to determine the location of maximum function y(x), we can set dy/dx=0 and determine the roots of the resulted polynomial, i.e.

One of the roots of the above equation corresponds to the required value of x.

3-Numerical Formulation & 4-Computer Implementation:

There are several numerical methods for computing the roots of polynomials that we are going to study in this course. After doing the numerical analysis and computer implementation, we obtain the following roots:

x1 = -4.50 m, x2 = 4.50 m, x3 = -2.01 m and x4 = 2.01 m.

5-Interpretation of the Computer Output:

The two negative values are disregarded because x has to be a positive value between 0 and 4.5 Furthermore the value of 4.5 is disregarded because the deflection of the beam is zero at x=4.5 m (The beam is fixed at the right end). We are left with x=2.01 m which is the location of the maximum deflection (ymax). Therefore, ymax = y(2.01) = -0.049 m.

* Important: This handout is only a summery of the lecture. The student is expected to take detailed notes during the class and refer to the textbook for more examples and discussion.