Skip to content

Making Connections

isee systems blog

Making Connections

  • About
  • More Connections
Search

Spatial Modeling in Two Dimensions

Updated: August 25, 2009April 7, 2009Filed under: Modeling Tips28 Comments

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.

2d-radiation

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/∂x∂y and ∂2u/∂y∂x 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.

2d-element-model

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.

2d-element-results

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.

Modeling Tips
  • 2D array
  • arrays
  • diffusion
  • iThink/STELLA
  • spatial
  • Version 9.1.2

Post navigation

Previous Post:

Published Papers that Feature Models

Next Post:

Spatial Modeling with isee Spatial Map

About the Author

Karim Chichakly

Co-President

All posts byKarim ChichaklyWebsite

Hide

Categories

  • Education (4)
  • isee NetSim (3)
  • isee.NET Framework (1)
  • Modeling Tips (32)
  • News & Announcements (15)
  • STELLA & iThink (14)
  • Stories from the Field (5)
  • Systems Thinking (8)
  • Training (6)

Archives

Browse by keyword

2D array archetypes arrays Barry Richmond Bass diffusion builtins calibration Causal Loop CLD command line conferences crisis data diffusion Education environment export game graphical function h1n1 healthcare housing import iThink/STELLA market dynamics MODSIM modules mortgage netsim optimization Physics policy price releases scholarship software spatial Stella storytelling System Dynamics Society Systems Thinking Version 9.1.2 video webinar workshop

Recent Posts

  • COVID-19: Modeling Distributions of Incubation and Recovery Times April 1, 2020
  • Multiobjective Optimization January 9, 2018
  • Optimizing Model Performance December 22, 2017
  • Calibration in Stella® December 15, 2017
  • Drifting Goals March 9, 2016

RSS System Dynamics Forum

Recent Comments

  • best apps review on About
  • digital software on Modeling the Economic Crisis
  • Mishawaka Indiana on What are “Mental Models”?
  • La Paz Indiana on XMILE – An open standard for system dynamics models
  • Bristol Indiana on Modeling the Economic Crisis

Meta

  • Log in
  • Entries feed
  • Comments feed
  • WordPress.org
Products
Software (v1.3)
  • Stella Architect
  • Stella Professional
  • Stella Designer
  • Stella Simulator
  • iThink
  • Feature Updates
  • Policies
  • Publishing Options
  • License Agreement
Free Software
  • Stella Online
  • isee Player
  • Stella Architect Trial
Solutions
Consulting
  • Systems Innovation Practice
Common Applications
  • Business
  • Education
  • Research
  • Government
  • Energy
  • Health
  • Agriculture
  • Manufacturing
  • Conservation
Quick Links
About
  • isee systems
  • Systems Thinking
  • Barry Richmond Scholarship
Resources
  • Frequently Asked Questions
  • Product Help
  • Examples
  • Request Support
  • Request Quote
  • Systems in Focus
  • Quick Tips
  • Legacy Tutorials
News and Events
Upcoming Workshops
  • Introduction to Dynamic Modeling
  • Whole Systems Partnership
Newsletter
  • The Connector
Recent Webinars
  • Model Mysteries
Recent Training
  • Systems Thinking Practice
Press Release
  • Stella Architect Release

  Phone: (603) 448-4990   Email: info@iseesystems.com

   Monday - Friday: 9:00 am - 5:00 pm EDT | Saturday - Sunday: Closed
Wheelock Office Park | 31 Old Etna Rd, Suite 7N | Lebanon, NH 03766 | US

isee systems inc. holds registered trademark rights over the following: iThink®, STELLA®, Stella®, isee systems® and claims the following trademarks; isee NetSim™, Stella Live™, Causal Lens™ and Stella Online™.

Terms of Use

© 2017. isee systems inc . All rights reserved.