### Archive

Posts Tagged ‘spatial’

## Modeling a Watershed with Arrays

This is the second installment of a four-part series.  The other three parts can be accessed by clicking on the links below.
Methods for Using Arrays Effectively

Modeling Customers Switching Between Brands
Modeling Customers Switching Between Brands – The General Case

This is the second installment of a multipart series.  The first part can be found by clicking here. Part 3 is available here.

In the first post of this series, I showed how to conditionally pull information from an array.  In this post, I will extend this concept to show how to route information through an arrayed model.  This is especially useful in spatial modeling applications.

Routing Water Through a Watershed

A common ecology application is the modeling of a watershed.  Part of such a model will necessarily involve a network of stream or river segments – called reaches – which feed each other.  It is desirable to implement this in a way that makes it easy to modify the reach network.  Using an explicit stock-flow network makes this very difficult.  However, it is relatively straightforward to use arrays of stocks and flows to build an easily configurable network.

Imagine a small watershed broken down into reaches as shown below:

For our purposes, a new reach will need to be created at every junction point.  Therefore, in this example and from a topological point-of-view, it is not strictly necessary to treat reach 4 separately from reach 2 nor reach 5 separately from reach 3, but reaches 2 and 3 must be separate from reach 1.  There are, of course, other reasons to separate reach 4 from 2 and reach 5 from 3, for example, slope, channel width, length, etc.

Every reach flows into exactly one other reach at its head, but many reaches can flow into the head of the same reach.  This requires a many-to-one representation of the reach network.  This is accomplished quite easily with a routing map which, for each reach, contains the number of the reach that this reach flows into.  We also need someway to signify the outlet.  Since reach numbers start at one, we can use zero to signify the outlet.  Using these rules, the above network is completely represented in the following routing map:

 Reach Flows into 1 0 2 1 3 1 4 2 5 3

The nice thing about this representation is that it fits nicely into a one-dimensional array where the array index is the reach number and the reach it flows into is the value stored in that array element.

The model itself uses one stock to represent each reach.   That stock has one inflow for water entering the reach and one outflow for water leaving the reach: (Download the zipped STELLA model here)

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## Version 9.1.3 Updates Key Features

June 16th, 2009 1 comment

We just released another update to STELLA and iThink — Version 9.1.3.  This particular release has a number of updates specific to key features.  For example, if you’re a Macintosh user and rely on the data import/export functionality, you’ll definitely appreciate the updates we made to make sure the software is compatible with the latest versions of Excel for Mac.

Updates to the ARRAYRANK builtin include an optional parameter to specify a secondary sort field for variables with the same value.  You can now also use the ARRAYRANK builtin  in a non-arrayed variable.

My personal favorite in this release, though, is an enhancement we made to the Spatial Map utility. In addition to color configurations, you can now assign an image to a range of values.  This really expands your ability to create  interesting visual representations of spatial data.

Using images to visualize simulations

This simple model of trees burning in a forest illustrates how images can be used in a spatial map configuration.

The model is set up so that initially there is one tree in the middle of the forest that is burning.  All of the other trees in the forest are healthy living trees.

When you run the simulation, the images in Spatial Map allow you to see the fire spreading to adjacent trees. It really adds to the visual effect of what’s happening in the model. Eventually the burning trees die out and you’re left with a forest mostly full of dead trees — now that’s a visual!

If you want to learn more about Spatial Map, check out Karim Chichakly’s series on spatial modeling in iThink and STELLA .

A full list of the features and fixes in Version 9.1.3 is available on our web site.  If your Technical Support Contract is current, you can go ahead and download the update now from your My Software page.

Hope you have as much fun with the spatial mapping as I did!

Categories: STELLA & iThink Tags:

## Spatial Modeling with isee Spatial Map

Editor’s Note: This is part 3 of a 3-part series on spatial modeling in iThink and STELLA. Part 1 is available here. Part 2 is available here.

Last time, we explored a two dimensional diffusion problem by looking at a metal plate with constant heat applied to the center. The model is available here: 2d-diffusion. The results, using isee Spatial Map, of the start (left) and end (right) of a six-minute simulation are shown below.

I am frequently asked how to set up Spatial Map. isee Spatial Map is a simple program that can be used to display any dataset as a two dimensional grid with specific colors assigned to data ranges. Since it is stand-alone, iThink and STELLA communicate with it through the Export Data functionality. If you wish to plot simulation results in Spatial Map, you must first set up a persistent link to a CSV file. This persistent link is always going to be from a table that contains just one element of the array you wish to view in Spatial Map.

In this example, a table named “Temp Export Table” was created to export the temperature data. The first element, temperature[1, 1], was placed in the table. There is a subtlety here that cannot be overlooked. I wish to plot the values of the stock T as it varies over time. Yet I export a different variable named “temperature”. Why is this?

This is necessary because although stocks can be exported in the format Spatial Map expects, the export settings that are compatible with Spatial Map only export their initial values no matter where the simulation is. If we export T, we will only ever see the initial conditions in Spatial Map. Thus, when displaying a stock in Spatial Map, and we almost always do display stocks, it is necessary to create a converter that is set identically equal to the stock. The converter will export its current values, and since it is equal to the stock, the stock’s current values will be exported. The converter used for this purpose in this sample model is named “temperature”.

Next it is necessary to set up the persistent link. Choose Export Data… from the Edit menu. The Export Type should already be set to Persistent and Dynamic. Under Export Data Source, select “Export variables in table” and choose the table with the array element in it from the pop-up menu. In this case, that table is called “Temp Export Table”. Also select “One set of values” under Interval. This forces the data to be export in the format required by Spatial Map. These settings are shown below.

To finish setting up the export, choose the CSV file to export to and press OK. For this model, the file is named “2D Diffusion.csv”. Note that all of this has already been set up in the attached sample, so you will not be able to set it up again. You can examine the settings, though, by choosing Manage Persistent Links in the Edit menu and then pressing the Edit link at the end of the “Temp Export Data” line in the Export block.

The value of “temperature” will now be exported once at the start of each run and once at the end. If you wish to see the simulation unfold in Spatial Map, it will be necessary to set a Pause interval, as dynamic links are also exported every time the simulation pauses. Under Runs Specs… in the Run menu, you can see that I have set the Pause Interval to 20. This forces the Spatial Map to update every 20 seconds during the simulation run. This also forces the user to keep pressing Run to advance the simulation.

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## Spatial Modeling in Two Dimensions

Editor’s Note:  This is part 2 of a 3-part series on spatial modeling in iThink and STELLA.  Part 1 is available here.  Part 3 is available here.

Last time, we explored spatial modeling using the one-dimensional diffusion problem as an example.  Many spatial applications, however, require two dimensional formulations.  As an extension, we will now explore the two-dimensional diffusion problem.  Instead of a one-meter metal bar with constant heat applied at its ends, the two-dimensional diffusion problem looks at the response of a one-meter by one-meter metal plate with constant heat applied to its center.  We then watch the heat diffuse across the plate.

At first blush, one might think the two-dimensional case is much more difficult than the one-dimensional case.  In particular, if a grid is superimposed over the plate, each finite element on the plate has eight neighbors, as shown below.  It is tempting, therefore, to consider radiating heat in each of these eight directions.

However, without looking at the two-dimensional diffusion equations, if we consider just the physical layout of this system, the four corners of the finite element only touch the four corner neighbors (1, 3, 5, and 7) at one point.  In contrast, the four sides of the finite element are shared with each of its four immediate neighbors (2, 4, 6, and 8).  This suggests that heat only radiates to (and from) these four neighbors, not all eight.  In fact, if we examine the two-dimensional diffusion equation, we find that there are only component contributions in the x– and the y-directions.  There are no contributions on the diagonal (which would appear in the equation as ∂2u/∂xy and ∂2u/∂yx terms).

Intuitively, then, we have a finite element that is very similar to the one-dimensional case.  We only need to add corresponding flows in the y-direction.  This leads to the following model with the individual finite elements arrayed.

The array T is now two-dimensional, in x and in y.  In addition, dx can differ from dy, so the diffusion constant C must be broken down into its constituent parts Cx = k/dx2 and Cy = k/dy2.  This leads to the following set of equations for the radiant flows through the plate:

in left = Cx*T[X – 1, Y]                               in top = Cy*T[X, Y – 1]
out left
= Cx*T[X, Y]                                  out top = Cy*T[X, Y]
out right
= Cx*T[X, Y]                              out bottom = Cy*T[X, Y]
in right
= Cx*T[X + 1, Y]                          in right = Cy*T[X, Y + 1]

X and Y are dimension names for the elements in the x– and ­y-directions, respectively.

Using isee Spatial Map, it is possible to view the results of this diffusion across two dimensions.  Spatial Map displays an array as a one-dimensional or two-dimensional grid (depending on the array).  Each cell in the grid is filled with a color corresponding to the value in the corresponding cell of the array.  Below are two spatial maps.  The one on the left shows the initial conditions of the metal plate.  Note that heat only appears in the center of the plate, where it is being externally applied.  The map on the right shows the distribution of heat across the plate at the end of a six-minute simulation.

The model is available here:  2d-diffusion.  It is already configured to use isee Spatial Map.  In the final installment of this 3-part series, I will describe how to set up isee Spatial Map.

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## Spatial Modeling in iThink and STELLA

STELLA and iThink provide capabilities to model spatial problems.  Version 9.1.2 expands these capabilities to allow more intuitive equation formulations.

Spatial modeling is concerned with modeling behavior over space as well as time.  Common applications include modeling flows in lakes and streams or modeling development in an urban landscape.  To demonstrate the new capabilities in version 9.1.2, the one-dimensional diffusion problem will be modeled in STELLA.

Consider the diffusion of heat through a one meter metal bar when a constant temperature is applied to both ends.  This problem is typically introduced as the following partial differential equation for temperature function u(x, t) with the given boundary conditions:

u/∂tk2u/∂x2 = 0
u(x, 0) = 0, 0 < x < 1
u(0, t) = u0, t ≥ 0
u
(1, t) = u0, t ≥ 0

Here, k is the diffusion coefficient (based on thermal conductivity, density, and heat capacity) and u0 is the constant temperature applied to both ends of the bar.  The finite difference solution to this problem (with the above initial conditions) is:

u(x, t + dt) = u(x, t) + ∂u/∂t
u/∂t = k (u(x + dx, t) – 2u(x, t) + u(xdx, t))/dx2

It is also possible to derive a closed-form solution.  Unless you are an astrophysicist, this solution probably does little to help you understand what is happening.  It is far more intuitive to develop a physical model of the underlying mechanics than to wrestle with the mathematics.

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