Bayesian optimisation for the selection of representative load paths in computational material testing (best UK Ph.D. in Comp. Mech. for Dr Olivier Goury)

The aim of computational material testing is to obtain the relationship between forces and extensions of a complex material through simulations, given some information about its microstructure. This is conceptually similar to pulling on a specimen experimentally, and reporting the force that needs to be applied to obtain an overall deformation of the material. Of course, replacing this costly experimental setup by simulations allows practitioners to investigate the use of new materials before manufacturing them, to play with the microstructure (e.g. varying the fiber content in advanced fiber-reinforced concrete) in order to design a material that fits particular engineering needs, or to test and control the reaction of the material when used for the design of a complex, multiscale engineering system.

However, computational material testing (or computational homogenisation), is very demanding in terms of computational resources. Typically, the material model needs to be solved at every material point of the engineering system of interest, which can be arbitrarily large. 

The solution that we have been investigating to alleviate this issue is to develop efficient Model order Reduction technique. In an offline stage, the material is tested using pre-defined loading scenarios, which leads to a set of particular mechanical states (the snapshot). Then, when a response of the material is required by the engineer for design purposes (i.e. online), it is obtained by performing an optimal interpolation in the space spanned by these pre-computed states, which reduces the overall computational cost by orders of magnitude.

Figure 1: Model order reduction for computational material testing 

The choice of the pre-defined load scenarios is key to the success of this approach. All the states that will need to be predicted should be sufficiently well represented by the states contained in the snapshot. At the present stage of our research, we consider that all potential load cases are equally likely. Although very general, this framework leads to an immense space of mechanical states to explore and represent.

Figure 2: random load-path generation, in 2D.
In 3D the paths belong to an hypercube of dimension 6

One first approach to explore this space of likely states is to choose random load paths (see Figure 2). In the case of damage mechanics, we constrain the random generation to follows states of strictly increasing energy dissipation, thereby constraining the generator to explore non-trivial (i.e. damaged) mechanical states. The reduced order models obtained by applying this simple idea are surprisingly efficient and reliable in our numerical test cases (simple elastic damageable multiphase representative volume element).

Our second approach is more advanced and aims at providing a quasi-optimal family of load-cases. The idea is to locate the load case that is the most incorrectly predicted when using previously generated mechanical states, and iterate until the accuracy of the reduced order model is acceptable. This data-driven approach is very appealing but requires a measure of goodness of prediction over all potential load-cases, which is difficult to (i) define and (ii) obtain at reasonable numerical costs. We have tackled the affordability issue by adopting:

1. A hierarchical description of the load paths using adaptive shape functions in time: start with proportional loading, then introduce an increasing number of kinks at arbitrary location on the load path (see figure 3)



Figure 3: Automatic path-generation through probabilistic worst-case
scenario detection (left: proportional loading, right:
space of load-paths containing a unique kink).

2. A Bayesian optimisation algorithm to detect the case of worst prediction for a given approximate description of potential load-cases. Precisely, we compute the residual of the governing equations at points chosen quasi-optimally using Gaussian process interpolation and the associated notion of maximum probability of improvement. Subsequently, we establish a relationship between a measure of this residual and the goodness of prediction, through a probabilistic regression technique.

This new approach to load paths selection provides some confidence on the accuracy of the constructed reduced model (we do not leave unexplored regions in the space of potential mechanical states), but it is significantly more expensive than the random generation approach: we must pay for reliability, in particular in the context of complex, nonlinear problems. This overhead may decrease when considering more complex material behaviours (visco-plasticity, anisotropy), owing to the fact that the random generator may require more time to detect small outliers in the space of potential mechanical states. 

More details about this research can be found in the Ph.D. thesis of Dr Olivier Goury.

Goury, O., Amsallem, D., Bordas, S.P.A. et al. Comput Mech (2016) 58: 213. https://doi.org/10.1007/s00466-016-1290-2