Incorporating sufficient physical information into artificial neural networks: a guaranteed improvement via physics-based Rao-Blackwellization

The concept of Rao-Blackwellization is employed to improve predictions of artificial neural networks by physical information. The error norm and the proof of improvement are transferred from the original statistical concept to a deterministic one, using sufficient information on physics-based conditions. The proposed strategy is applied to material modeling and illustrated by examples of the identification of a yield function, elasto-plastic steel simulations, the identification of driving forces for quasi-brittle damage and rubber experiments. Sufficient physical information is employed, e.g., in the form of invariants, parameters of a minimization problem, dimensional analysis, isotropy and differentiability. It is proven how intuitive accretion of information can yield improvement if it is physically sufficient, but also how insufficient or superfluous information can cause impairment. Opportunities for the improvement of artificial neural networks are explored in terms of the training data set, the networks' structure and output filters. Even crude initial predictions are remarkably improved by reducing noise, overfitting and data requirements

Optimizing artificial neural networks for mechanical problems by physics-based Rao-Blackwellization: example of a hyperelastic microsphere model

The Rao-Blackwell scheme provides an algorithm on how to implement sufficient information into statistical models and is adopted here to deterministic material modeling. Even crude initial predictions are improved significantly by Rao-Blackwellization, which is proven by an error inequality. This is first illustrated by an analytical example of hyperelasticity utilizing knowledge on principal stretches. Rao-Blackwellization improves a 1-d uniaxial strain-energy relation into a 3-d relation that resembles the classical micro-sphere approach. The presented scheme is moreover ideal for data-based approaches, because it supplements existing predictions with additional physical information. A second example hence illustrates the application of Rao-Blackwellization to an artificial neural network to improve its prediction on load paths, which were absent in the original training process

A thermodynamically consistent modelling framework for strongly time-dependent bainitic phase transitions

In this work, a thermodynamically consistent constitutive framework is introduced that is capable of reproducing the significant time-dependent behaviour of austenite-to-bainite phase transformations. In particular, the aim is to incorporate the effect of these diffusion-controlled processes by plasticity-like evolution equations instead of incorporating related global diffusion equations. To this end, a variational principle for inelastic solids is adopted and enhanced by an additional term. This term essentially contributes to the evolution equations for the phase volume fractions of several crystallography-based bainite variants. Due to the specific modifications, special attention has to be paid with respect to the fulfilment of thermodynamical consistency, which can be shown to be unconditionally satisfied for the newly proposed modelling framework. The phase transformation model itself is based on the convexification of a multi-well energy density landscape in order to provide the effective material response for possible phase mixtures. Several material parameters are determined via parameter identification based on available experimental results for 51CrV4, which also allow the quantitative evaluation of the predicted results