Soft Computing Applied to Finite Element Tasks

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

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

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 FEMMesh • Solve: Solution found by linear algebra techniques

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

6. FEM and Time Dependent PDEs • For time dependentPDEs 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. Refine We miss the action Mesh Adaptation Problem

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. Step 2:Feed the Input Signal forward Input Signals Output Signals Input Layer Hidden Layers Output Layer Step4:Feed the Error Signal backward and update the weights (in order to minimize the error) Feed-Forward Networks 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)

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 Analytic Solution FEM Solution FEM Solution Time=0.4 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.