Learning Scenario - Laplace's Equation (Excel)

Basic Model:

Description

This model illustrates and animates the movement of an oscillating surface, such as a drumhead being struck, using the Laplace Relaxation Method. The center four cells are the source of the oscillation, and as they vibrate up and down, the surrounding cells dampen the effect. A 3-D animation with clear axes labels shows the movement of the cells. The user can speed up or slow down the animation as it runs to better visualize the movement.

Background Information

Laplace's Equation is used in determining heat conduction, electrostatic potential, and also has many other applications in the scientific world. In an object with boundary conditions, the Laplace Equation can be used to determine a particular value (for example electrostatic potential) of a location in space if that value is known for the cells adjacent to that space. The Laplace Relaxation Method is a method for estimating these values iteratively.

Science/Math

The fundamental principle behind this model is HAVE = HAD + CHANGE. The value of each cell in this model (expressed by location and color) depends on the values of the cells around it.

The oscillation in this model is controlled by the center four cells whose values increase and decrease based on the equation: 10*sin(iteration), where iteration step size is determined by the user-defined variable "step".

The values of the surrounding cells are determined by the equation, "I am the average of my neighbors." This can be written as the following: cell value = (cell[right] value + cell[left] value + cell[up] value + cell[down] value)/4

Teaching Strategies

Lead a discussion about Laplace's Equation. At the end of the discussion, students should be able to answer the following questions:

1. What is the essential equation behind Laplace's Equation?
2. How is Laplace's equation useful for estimating electrostatic potential?
3. In what other fields is Laplace's equation useful? How and why?
4. What is the Laplace Relaxation Method and why is it useful? How is it calculated?

Implementation:

How to use the Model

This model has one parameter that can be manipulated to produce different results:

• The parameter "step" determines how far the model progresses every time step. The default value is .1, but the value can range from .001 or smaller to 1, depending on the speed of the animation the user wants.

To begin the animation, the user must change the value of "start" on the left side of the spreadsheet to "1" and either hit and hold down "F9" on a PC or [command][=] on a Mac. As the model runs, the variable "iterations" on the left side of the spreadsheet tracks the number of time steps that have passed since the model began.

To reset the animation, change "start" from 1 back to 0 and hold down F9 or [command][=]. It may take a few seconds for the model to completely return to its original state. **Note: Make sure that under Excel-> preferences-> calculation: be sure to select calculate sheets "Manually", to check the box marked "Limit Iteration" and set "Maximum Iterations" = 1. For more information on Excel, reference the Excel tutorial at:

http://shodor.org/tutorials/excel/IntroToExcel

Learning Objectives:

1. Understand the function of the Laplace Relaxation Method in the animation.
2. Consider the role of ideal models in determining behavior in less ideal circumstances.

Objective 1

To accomplish this objective, have the students run the model, changing the step as needed, to try and understand how it calculates the values for the animation. Make sure they look at the equations in the charts. Ask the following questions:

1. Take a look at the equations in the lower and upper charts. How does the model determine the values for the cells surrounding the pulse points? Does this make sense? Why or why not? Each cell's value is the average of the values of the cells adjacent to it. It makes sense.
2. Look at the equations of the cells on the edges of the model. What are the boundary conditions in this model? Are they homogeneous? Explain.
3. Does this model have any consistent maxima or minima? Why or why not?
4. How is this model an application of the Laplace Relaxation Method? Which cells are direct applications of this? Why? Explain.

Objective 2

Have students run the model, changing the step as needed, to try and visualize the animation as a representation of a specific application (ex. electrostatic potentials). Ask the following questions:

1. Is this model ideal? Why or why not?
2. What elements of the real world is this model missing? Think of the model in terms of both the macro world (stick hitting drumhead) and the micro world (electrostatic potentials). Friction, pressure, etc.
3. How would those elements be incorporated into the model? How would that change the animation and the values in the model?
4. Is this model's basis on a grid of squares realistic to any real life situations? Why or why not? How could it be made more realistic?
5. Why is it useful to determine these behaviors in ideal environments before moving to less ideal environments?
6. What is the purpose of the Laplace Equation in science?

Extensions:

1. What other phenomena could this be used to model?
2. What would change in the model if the source of the oscillations were not in the center?

Extension 1:

Encourage students to discuss different phenomena in the environment and/or science and technology that this could be used to model. Ask the following questions:

1. What are other examples of oscillating surfaces? What physical things could this be used to model? Ex. Movement of a drumhead after being hit by a stick.
2. Consider the model in terms of temperature. In what situation would there be fluctuations in heat in an object? Ex. Hot iron taken out of the oven and put into cold water.
3. What are ways the model could be modified to represent different phenomena? Explain.
4. How are Laplace's equation and the Laplace Relaxation Method useful in representing these different phenomena? Explain?

Extension 2

Have students consider the effects of moving the source of oscillation in the model. Ask the following questions:

1. What would happen if the source of oscillation were still in the center, but were smaller? Or larger? How would this affect the surrounding cells?
2. What if the oscillation were moved to the corners of the model? How would the boundary conditions affect the movement of the cells? Because the cells are averages of their immediate neighbors, not including those at a diagonal, the boundary conditions would prevent any other cells from oscillating in response to the source.
3. What if the source of oscillation replaced the boundary conditions on one side of the model? What affect would that have on the animation?
4. What would these changes mean in terms of different applications for the model?

Related models

SimSurface

http://www.shodor.org/master/simsurface/software/

This is a model of simulated annealing, a common method for finding global maxima and minima of complicated multi-dimensional functions. The model uses the Relaxation Method to calculate electrostatic potential of the electrons in the model.

Basic Diffusion

http://www.shodor.org/refdesk/BioPortal/model/VSdiffusion

This is a model of basic diffusion created using the Vensim software. Students should discuss the ways in which the Laplace Relaxation Method is an application of diffusion (I am the average of my neighbors).