Exploiting the Potential of Bayesian Inference in Combination with CAE

9th December 2013 by Mike Heskitt

This post has been contributed by my colleague, Ichiro Shibata, Director at Altair ProductDesign Japan

“SHISAKU-LESS” has been a typical motto in Japanese manufacturing for the past few years. “SHISAKU” means a trial production, so by representing the virtual prototyping with CAD/CAE, we can reduce the number of trial productions by conducting all related simulations in the finite element (FE) models. If we are going to effectively use these “virtual prototypes” to improve design and reduce weight though, we must also consider real world effects in the simulations.

In the real world, there is uncertainty about the exact values of any parameter in a physical model because every design object has both controllable variables and uncontrollable errors. finite element analysis (FEA) though, returns just one response to an input to a particular system with defined variables.  Consequently, there exists a critical gap between the deterministic CAE result and the stochastic distribution in the real world.

How to deal with uncertainty in FEA has been considered by forward-thinking researchers for a long time.  However, best practices have yet to be determined.


Critical gap between CAE result and the real world


You may think that we can utilize stochastic FEA or even run tens of thousands of general FEA with Monte Carlo simulation, but it may not be practicable in our daily work with limited computer resources and time.  Also, the solution for the robust design cannot be provided by just running FEA.  We eventually need statistical knowledge to consider uncertainties as probability density functions and build a statistical model through a CAE-based stochastic study.

Regression analysis is one of the practical methods to build a statistical model and there are 2 different statistical methods to use for prediction as shown below;  Frequentist statistics that is based on the objective probability and Bayesian statistics that refers to the subjective probability.  Frequentist uses the MLE, Maximum Likelihood Estimation, to determine parameters as constant numbers, while Bayesian uses MCMC, Markov Chain Monte Carlo methods, to estimate parameters as stochastic distributions.  Bayesian is widely used to deal with big data because it’s an attractive technique to observe the propagation of stochastic uncertainty in unknown events.

Shown below is a comparison study between an ordinary regression and Bayesian regression conducted with a simple fictional dataset.  The horizontal and the vertical axis represent the observed and the predicted data, respectively. The exact match between the observed value and the predicted value is indicated by the diagonal black line.  Since the dataset has some stochastic errors, Bayesian inference is better in prediction accuracy than the ordinary one through the effect of hierarchical inference model shown above.  Bayesian result also shows the probabilistic distribution by vertical short lines.


Difference between the Frequentist and Bayesian in regression model


This simple example is suggesting a possible beneficial effect of Bayesian inference for CAE-based robust design.  As we are not data scientists but engineers, we just need to use statistics carefully, whether it includes Frequestist statistics, Bayesian inference, PRML, or whatever works for your solution.

Altair ProductDesign Japan is also focusing on the simplification of vehicle crash models.  The final objective of the research is to combine the simplified FE model with MCMC solver to conduct multi-objective robust design with Bayesian inference.  We have high expectations for the future solutions we will develop for our customers.


Comparison between an ordinary regression and Bayesian regression