1. CSE351/ IT351 Modeling and Simulation
Mesh Models
Dr. Jim Holten

2. Mesh Models Background – Representing data from the real world
Mesh Types
Model Implementation via Matrices
Model Implementation using SRF

3. Background How do we represent a continuum?
How do we represent a deformable object?
How do we represent disjoint parts?

4. Representing a Continuum
Field equations (analytical)
Mesh points (discrete locations)
Mesh “cells” (discrete objects)
Combinations

5. Continuum Field Equations
Gravitation
Electric charges
Magnetic fields
Evenly propagated point source radiated energy

6. Continuum Mesh Points Attributed values at each point
Fixed grid points
Mobile grid points
Arbitrary (strategically placed) points
Multi-mesh points

7. Continuum Mesh Combinations
Objects, particles, and fields
Radiation transport and collisions
Protein models
Satellite orbits
Integrated systems
Boundary transports

8. Continuum Mesh Cells Point locations and attributes
Attribute values over line, surface, or volume regions (1-D, 2-D, or 3-D)
May form a hierarchy of cell types
Node (Point)
Edge (Line between points)
Face (Surface bounded by lines)
Zone (Volume bounded by surfaces)

9. Mesh Cell Behavior Types
Fixed positions (Eulerian Mesh)
Adaptive positions (Lagrangian Mesh)
Crushing collision surface
Wavefront deformation
Adaptive refinement for finer localized representation
Assemblies of parts
Separate “independent objects”
Boundaries

10. Implementing a Mesh Model
Choose a model object representation
Represent the state variables (properties of the represented objects)
Organize the model calculations (state variable calculations for each time step)
Decide how to view the time step values

11. State Variables
State variables are EVERY variable that changes over time and must be carried from time step to time step.
State variables are mostly “attributes” (properties) of the elements.
State variables generally are in vectors whose value entries correspond to the cells in a single cell set.

12. State Variable Representations
Each has its own data type, depending on what it represents.
Commonly float or double, may also be integer, enumerated value, vector, string, or ...
Generally loosely interpreted as meaning ANY attribute of an element or of the model itself.

13. State Variable Calculations
Inputs to calculations:
State variable values (one or many)
Constants and coefficients
Element associations
Outputs
New state variable values

14. State Variable Calculations
Temporary variables may or may not be considered state variables.
Calculations may be iterated many times over for each of many inputs (A differential equation solver or integrator) or be simple expression evaluation as in (a = b).
A state variable could be the state of a lesser (limited scope) Finite State Machine.

15. State Variable Calculations
Generally best to organize the calculations to iterate through the output values, calculating each before going on.
Each mesh cell has its own state variables, and it often is best to compute all for one cell at a time, but some algorithms require one attribute for all cells before the next attribute can be calculated.

16. State Variable Calculations
Selecting the order to do calculations among attributes can be a precedence-based scheduling problem.
Work to keep it as simple as possible!!
Use comments to inform source code readers why you have used a specific ordering or technique!

17. Choose a Model Object Representation
Finite Elements? (Pick from a zoo of predefined cell element types)
Regular Polyhedral Mesh? (Homogeneous mesh cells)
General Polyhedral Mesh? (Hierarchy of generalized cells)
Any combination of the above?

18. Model Cell Criteria Cell shapes? Cell properties (attributes)?
Cell associations with neighbor cells?
Cell associations with other cell set members?
Complexities of developing supporting code?

19. Finite Element Cell Types
1-D: Line, interpolated line, spline
2-D: Triangle, rectangle, trapezoid, circle, ellipse, interpolated shape variations
3-D: Tetrahedral, hexahedral, spherical, ellipsoidal
Special types: Springs, shock absorbers, circuit components, other custom variations

20. Regular Polyhedral Cell Types
Limited to “regular” shapes that will cover a “region”.
1-D: No problem.
2-D: Triangles, quadrangles, and hexagons only.
3-D: Hexahedrals only.
Does not cover irregularly shaped model objects/parts.

21. General Polyhedral Cell Types
A hierarchy of cells (nodes, edges, faces, and zones).
0-D: node (point) has a location (usually)
1-D: edge (line) connects two end points (nodes).
2-D: face is surrounded by edges.
3-D: zone is surrounded by faces.
Fully generalized polyhedral shapes, allowing extreme shape representation.

22. Matrix Representations of Mesh Models
Vectors
Each is over a single cell set (which is always made up of a single cell type)
Each contains values for a single property for each cell in the cell set
Matrices
Each matrix represents an association between values for a pair of cell sets
Each may be a “flag” (1 or 0) or may have an association value (constant coefficient)
Some are used as transformations to “relate” vector values in linear equations for the time step calculations
Commonly used for sets of homogeneous cell types

23. Matrix Mesh Model Vectors
Vectors of cell set member attributes (property values)
One value for each cell in the cell set
May be scalars, vectors, matrices, tensors, strings, enumerated values (hot/warm/cold, on/off, ...), or ?
Examples
Volume temperature, material makeup, mass, density, pressure, ...
Point coordinates (1-D, 2-D, or 3-D usually)
Each entry a vector of flow through a surface

24. Matrix Mesh Model Matrices
Represent relations which indicate
cell adjacency within a set of cells
cell associations between cell sets
May contain values representing
Cell interaction properties (weights, coefficients)
Linear state value propagator coefficients
Stochastic covariance values for property values between cell pairs

25. Matrix Mesh Model Characteristics
Most model matrices are sparse
Matrix math does NOT handle nonlinearities well, so nonlinearities must be handled as separate expressions for each transformation.
Matrix representations do NOT clearly indicate organization at higher levels of abstraction.
It is easy to get lost in the code and data relationships at all levels.
Parallel partitioning can be awkward.

26. Viewing Time Step Values
Generally state variable combinations are valid only between (not during) time steps. That is when to view them (except for debugging purposes).
Sometimes auxiliary state variables are created JUST so they can be viewed!!

27. Viewing Time Step Values
Print the values.
Plot the values.
Use the values to adjust visual “meters” or other indicators.
Use the values to adjust images on virtual displays.
Save the values for a later animation.
Postprocess the values for a custom view.