1. Soft Computing Applied to Finite Element Tasks
Dan Givoli
Dept. of Aerospace Eng., Technion
Akram Bitar & Larry Manevitz
IBM - R&D Labs, UH Dept. of Computer, UH
A. Bitar

2. Finite Element Method (FEM)
What is it ?
Arguably the most effective numerical techniques for solving various partial differential equations (PDEs) arising from mathematical physics and engineering
A. Bitar

3. Finite Element Method (FEM)
How does it work?
Mesh generation:
Divide up the PDE’s domain into
finite number of elements
Solution representation:
On each element represent solution as a
combination of simple basis functions with
unknown coefficients
FEM Mesh
Solve:
Solution found by linear algebra techniques
A. Bitar

4. Sub-Problem
Soft Computing Technique
Comments
Choice of Kinds of Elements, Topology
Expert System, Computational Geometry
Assigning Resources to Sub-Bodies
Genetic Algorithms; Automated Negotiations
Numbering of Nodes
M-G-Margi;
NP complete
Adaptive Meshing
Feed Forward NN
Time Series Prediction; M-G- Bitar
Load Balancing
Automated Negotiations, Genetic Algorithms
Kinds of Approximation on Elements
Expert System
Mesh Placement; Assigning Geometry to Topology
Self Organizing NNs
M-G-Yousef
Visualization
NNs
Avoid Interpolation

5. Time Dependent Partial Differential Equations
Hyperbolic
Wave Equations
Parabolic
Heat Equations

6. FEM and Time Dependent PDEs
For time dependent PDEs critical regions should be subject to local mesh refinement.
The critical regions are identified as those regions with large gradients (error indicators).
This regions change dynamically.

7. Mesh Adaptations Problem
Current methodology is to use indicators (e.g. gradients) from the solution at the current time step to identify where the mesh should be refined at the next time step.
The defect of this method is that one is always operating one step behind

8. Mesh Adaptation Problem
Refine
We miss the action

9. Our Method
To predict the “area of interest” at the next time stage and refine the mesh accordingly
Time Series Prediction via Neural Network methodology is used in order to predict the “area of interest” at the next time step
The Neural Network receives, as input, the gradient values at the current time and predicts the gradient values at the next time step

10. Neural Networks (NN) What is it?
A biologically inspired model, which tries to simulate the human nervous system
Consists of elements (neurons) and connections between them (weights)
Can be trained to perform complex functions (e.g. classifications) by adjusting the value of the weights.

11. Feed-Forward Networks
Step 2: Feed the Input Signal forward
Input Layer
Hidden Layers
Output Layer
Input Signals
Output Signals
Train the net over an input set until a convergence occurs
Step1:
Initialize
Weights
Step3:
Compute the
Error Signal
(difference between the NN output and the desired Output)
Step4: Feed the Error Signal backward and update the weights
(in order to minimize the error)

12. One Dimension Wave Equation
Analytic Solution
PDE

15. Two Dimension Wave Equation
Analytic Solution
PDE

16. Neural Network Predictor “Standard” Gradient Indicator
Analytic Solution
FEM Solution
Time=0.4

18. Summary
The Finite Element Method involves various tasks that need automating. Soft Computing Methods are appropriate for some of them.
Previously we have used Expert Systems, SONN, and Feed-forward NNs to automate three different tasks with good success.
In this talk we showed how to PREDICT gradients using NNs and use this to substantially improve adaptive meshing for time dependent PDEs.