Why computing?¶

by Xiaofeng Liu, Ph.D., P.E. Associate Professor

Department of Civil and Environmental Engineering

Institute of Computational and Data Sciences

Penn State University

223B Sackett Building, University Park, PA 16802

Web: http://water.engr.psu.edu/liu/

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This chapter will cover the basics of programming and Python, in preparation for the rest of the book. This chapter will cover the following topics:

• Why study computing methods
• Basics about computer hardware and software
• A primer on programming
• Python programming language
• Plotting and scientific visualization with Python
• Numerical error analysis

This notebook will cover the first part, which is why computing is important for civil and environmental engineers.

Why study computing methods?¶

Civil and environmental engineering is a very old engineering field, which probably has existed since the beginning of human civilization (note the "civil" part of this word). In public eyes, a civil or environmental engineer might not have too much to do with high technological advancement in fields such as mordern computing. This is indeed an unfortunate misconception of not only the public but also many people in this very field. As you will see in this book and many other places, computing is everywhere in civil and environmental engineering. Computing has weaved itself in almost all aspects of not only CEE but also the whole society. As a result, we often take computing for granted, a.k.a, without recognizing its existance, letting along its importance. In the 21th century, CEE practioners should master computing skill as other essentials such as surveying and drafting.

Before we go further, a proper definition of "computing methods" is in order. Any computing method in this book is defined as the use of computer to solve a mathematical problem or perform a calculation task. A more traditional (and narrower) name for "computing methods" is "numerical methods". However, the defintion of computing has evolved and the name "computing methods" suggest a much broader definition. Any task which can be automated with a computer can be call "computing". There are many examples in the profession of CEE, which will be introduced later on. One simple example is the work flow of retrieving some data (either download from a website such as USGS or read the result file from a computer software), processing the data (e.g., cleaning, grouping, and computation), and making a plot to visualize the results. A computing work flow like this can be automated with some a customized code, which greatly increases productivity and efficiency, and reduce the chance of human errors when done manually.

The topic of this book is computing methods. Importance of computing has been emphsized. However, the intent is definitely not to discount the critical role of manual calculation. Across the U.S. and around the globe, most of the CEE program curriculums probably emphasize more on hand calculations. In-class exercises, examples, homework, and projects are mostly done with a piece of paper and a pen. Hand calculation is an essential step in CEE education because it is probably the most effective way to learn a new concept, tool, or method. To some degree, most CEE curriculums also contain some courses and trainings on computing methods. However, based on the author's observation and experience, what is missing is the link among theories, methods, hand calculation, and computing techniques. Many CEE students are required to take some computing methods and programming classes, mostly very early during college years. At that time, students are not much exposed to technical courses in CEE and thus can not make the link between computing and CEE subjects. On the other hand, in the delivery of technical courses, instructors are usually constrained by time and the need to cover a broad spectrum of topics. They usually focus much more on theory and hand calculation, without too much devotion to the use of computers and computing. It is also disappointing to note that CEE professional license exams have almost no coverage on computing methods or technique, despite the fact that civil and environmental engineers use computer to perform computing tasks almost everyday. Based on the status quo of current CEE education, a book such as this and corresonding computing methods course offering are needed.

Computational tools are widely used in all engineering professions, including CEE. The following is a list of non-exclusive examples:

• Design and calculations. Formulas and equations are used in many CEE design processes, whether it is to use the Manning's equation to calculate channel flow capacity or to calculate the displacement of a beam under various loading. These formulas and equations can be programmed or solved using computer code for use.
• Data processing management. In CEE practice, engineers and designers deal with all kinds of data, such as survey data, traffic data, bridge data, rainfall data, river discharge, and wind load data. These data usually need operations such as collection, processing, conversion, and further calculation. All these operations can be coded and computerized. For example, to calculate the wind load on a structure, wind data and their statistics (mean, standard deviation, etc.) are needed. A Python code can easily perform a task like this.
• Many engineering software have been developed which uses computing methods. For example, HEC-RAS for open channel flows solves the governing equations of fluid flow in channels. The Abaqus suite uses the finite-element method to solve multiphysics problems such as the deformation of a structure under loading. The AutoCAD package provdies many functionalities which of course need very complex geometric computation. Even Microsoft Excel, one of the most used office software, provides programming capability through VBA.

So no matter you realize or not, computing is everywhere in our life. Before you begin the reading and learning journey along with this book, it will be benefitial for you to think about the following two questions.

• Any examples of computing you can think of in your daily life?
• Any examples of computing in your previous course work, internship, or jobs?

Your answer to these questions hopefully makes you more motivated to dive deep into the fascinating world of computing. The mastering of computing techniques can make an engineer more efficient and competitive in this ever faster changing world. In your career, you will likely spend hours with a computer everyday.

Computational software and code¶

The implementation of computing methods will result in computational software or code. Some examples in CEE have been mentioned previously. Like anything else, there are some pros and cons associated with computational tools. These need to be always kept in mind when using computer codes.

The pros include:

• They can greatly improve the efficiency of engineering design (in comparison to hand calculations).
• They are less prone to human errors if the code is bug free and it is used correctly.

The cons include:

• Most of time computational software are used as a blackbox (unless it is open source). Engineers are usually not involved in the the development of computational software. They are mostly likely the end user. Although most software come with user and technical manuals, engineers may not have the time or willingness to learn all the details.
• They can be misused and fail if the user does not know what is behind the scene and how to properly use the computed results.

In this book, the more concrete definition of computing methods is a numerical analysis which uses computer to solve mathematical models in the form of equations. These mathematical model equations, either algebraic or differential, are used to describe biological, geological, chemical, and many other physical processes encounted in CEE. The computed results, if they are computed correctly, are always only an approximation to the true solutions. In contrast to computing methods, analytical methods seek to solve the equations analytically (exactly). Unfortunately, analytical solutions are very rare and only exist when the situation is greatly simplified and many assumptions have to be made. For most real world problems, there is no analytical solution because of complexity. In addition to computing methods and analytical methods, we can also use experimental methods to solve problems, for example to use experimetns to test a design or study an engineering process.

Figure. Computing methods, analytical methods, and experimental methods.

In comparison with other two methods, computing methods and computational modeling can be used in the following scenarios:

• when it is very time consuming to obtain analytical solutions. In this case, even if analytical solutions may be found for certain cases, it may take a mathematician a long time to do the derivation. The resulted analytical solutions may also be cumbersome to use (imagine an analytical solution on a couple of pages).
• when it is impossible to obtain analytical solutions. As mentioned before, most problems do not have analytical solutions. An example is given below. The following equation is a 1D advection-diffusion-reaction equation with a 2nd-order reaction, which can be used to model the transport and transformation of contaminats in rivers.

\begin{equation} \frac{\partial C}{\partial t} + U\frac{\partial C}{\partial x} = D \frac{\partial^2 C}{\partial x^2} + k C^2 \end{equation}

Here, $C$ is contaminant concentration, $U$ is flow velocity, $D$ is a dispersion coefficient, $k$ is a reaction constant, $t$ is time, and $x$ is the coordiante along the river. Due to the nonlinear term $k C^2$, there is no analytical solution. However, with computing methods, it is relatively easy to get an approximate solution if boundary and initial conditions are set properly.

• when it is time consuming and expensive to perform experiments. In general, performing experiments take time for preparation, designing and manufacturing of speciman, conducting experiments, and analyzing measurement data. It is also expensive. When the physical experiments are replaced by virtual experiments on computers, it may take much less time and money to see the results.

There are many other practical engineering issues which limit either analytical approach or experiments:

• nonlinear vs. linear problems. As mentioned before, nonlinear problems usually do not have analytical solutions. Nonlinearity can emerge in almost all aspects of the problems encounted in CEE. For example the 2nd-order reaction term above. Another example is the nonlinear properties of material.
• Large vs. small systems. If a problem is defined on a small system, it may be possible to replicate in the lab with reasonable cost and time. But for large systems, such as the whole sewer system of a metropolitan area or the design of the next tallest building in the world, it is extremely hard, if not impossible, to study them as a whole in the lab. On the other hand, a computer model does not care about the size of the system as long as the computer has enough memory and computing power.
• Uncertainties vs. certainties. A good engineering design always needs to take into consideration of uncertainties because our knowledge of the world is limited. A common way to quantify uncertainties is to vary the uncertain system input (such as load of structures or upstream flood inflow) and check the system response. That means if we perform experiments, we need to repeat the experiments many times (e.g., in thousands) with varying input. This is almost impossible to do in the lab. However, with computer and parallel computing, uncerntainties quantification is routinely performed with computer models.
• Design alternatives. In a computer model, design alternative can be easily accommdated by changing the design, e.g., the change of road layout and signal, the size of supporting beams, or the depth of foundation. As you can imagine, these changes will be difficult to accommendate in the lab, at least not that easy to change many times.
• Repetitive work. In real world, many of the tasks are repetive. Computers are extremely good at repeating the same or similar tasks. A well written computer code can be used over and over again. In contrast, a physical model built in the lab may have to be removed or demolished after the project is over.

A quick example of computing method¶

As a preview of a typical computing method, this part shows a simple example. This example shows a simplified version of a spherical particle settling in quiescent fluid. Particle settling is important for water treatment and river water quality. The motion of the particle is governed by the Newton's second law, which is a mathematical model: \begin{equation} m \frac{dU_z}{dt} = F_z \end{equation} where $m=\rho_p V$ is the particle mass, $\rho_p$ is particle density, $V$ is particle volume, $U_z$ is the particle velocity in $z$ (vertical) direction, $F_z$ is the total force on the particle in $z$, and $t$ is time. For simplicity, we assume the particle only settles in the vertical direction and there is not motion in other directions. We also define the downward direction is positive.

The total vertical force is made of three parts:

• the gravity force $mg = \rho_p V g$, where $g$ is gravity acceleration constant
• the buoyancy foce $\rho_w V g$, where $\rho_w$ is the density of water
• the drag force exerted by water. The drag force $F_D$ is pointing upward because it is a drag. There are formulas to calculate the drag force, which can be found in any elementary fluid mechanics book. For simplicity, we assume the drag force is propportional to the velocity square, i.e.,

\begin{equation} F_D = - C_D U_z^2 \end{equation}

where $C_D$ is a constant.

Thus, the governing equation becomes \begin{equation} \rho_p V \frac{dU_z}{dt} = \rho_p V g - \rho_w V g - C_D U_z^2 \end{equation}

Divide both sides by the particle mass $\rho_p V$, we get Thus, the governing equation becomes \begin{equation} \frac{dU_z}{dt} = \frac{\rho_p-\rho_w}{\rho_p} g - \frac{C_D}{\rho_p V} U_z^2 \end{equation} which is a first-order, nonlinear ordinary differential equation for $U_z(t)$. In order to get an solution for this problem, we need an initial condition, i.e., what is the particle velocity at the beging. For example, if initially the particle is not moving, then the initial condition is \begin{equation} U_z(t=0) = 0 \end{equation}

To solve this equation with computer, one can use different methods. The simplest one is probably the finite difference method, which we will cover in later chapters. In essence, the continous the differential equation is "discretized" or "approximated" at discrete time, let's say time $t_{i}$ and $t_{i+1}$. We can also approximate the right hand side of the equation with values at time $t_i$. The discrete form the time derivative is then \begin{equation} \frac{dU_z}{dt} \approx \frac{U_z(t_{i+1})-U_z(t_{i})}{t_{i+1}-t_i} = \frac{\rho_p-\rho_w}{\rho_p} g - \frac{C_D}{\rho_p V} U_z(t_i)^2 \end{equation}

Starting with $i$ = 0, i.e., $t_0$ = 0, we can marching forward using the above approximation because the only unknown is $U_z(t_{i+1})$ and everything eles is either constant or already known at previous time step $t_i$. The method described here is only one of many which we will cover in the ODE chapter.

In the following, to show the result with less distraction to the numerical details, we will demonstrate the numerical solution of the problem by calling functions in a Python library. You do not need to pay attention to all the details of how the functions are called and the plot is made.

In the Python code below, the "odeint" function within the "scipy" library is called to the integrate the ODE from $t$ = 0 s to 0.5 s. Parameter values are specified in the code. Then, a plot is made to show the time history of the particle velocity. You can observe that the particle start from zero velocity and then increases its velocity (due to gravity) until it asymptotically approaches its terminal velocity at about 0.52 m/s.

Important things to keep in mind when utilizing computing tools¶

The above part of the chapter may give the reader the impression that it mostly praises computing methods and computational tools for their importance and usefulness. In fact, it is only part of the message. To make it clear that computing methods are only tools and they should be used properly, the following is a list of famous quotes in computational science which probably have been used in all computing-related courses and should be memorized by all who uses computer to solve engineering problems.

• "All models are wrong, but some models are useful" by George P.E. Box, who was a British statistician. All computer models have assumptions and simplifications which may not be correct or accurate in reality. No model can consider everything happening in the real world. That is why all models are wrong. However, as long as a model can correctly capture the processes of interest to us, they are useful.
• "The purpose of computing is insigh not numbers" by C. Hasting. To get the results (numbers) is not the final goal of computing. The goal is to get insigh and make the results useful for your purpose. Many times, people are obsessed with the colorful visualization of computing results, in the form of either beautiful figures or impressive animations. While these visualizations are useful, we have to always ask the question "So what?". Do we learn anything new or do we find a better solution/design, or do we found the cause of a problem? Etc.
• "Garbage in, garbage out". All computer models needs user input, such as model parameters or designs, such that it can compute the result corresponding to those input. If the input is garbage (not reasonable values or totally wrong values), the output is garbage too.