FEM, the Finite Element Method is a widely-used method for solving differential equations in engineering and mathematical modeling. Applications span geology, reservoir modeling, engineering construction and fracking. LR (formerly Lloyds Register), through its Alan Turing Institute, reports a ‘radically redesigned’ approach to FEM, developed by researchers at the Universities of Cambridge and Western Australia.
The researchers observe that FEM results often do not match empirical evidence, revealing mismatches in a model that are amenable to a statistical approach. Such model ‘mis-specification’ is addressed by introducing ‘stochastic forcing’ to the partial differential equations before updating the FEM. Stochastic forcing* is introduced through a random function within the governing equations.
You may wonder why an apparently abstruse mathematical technique is being brought to our attention. The answer is that, for LR, the approach ‘lays the theoretical foundations and methodologies by which digital twins can be realized’. Researcher Mark Girolami said, ‘Digital Twins, the pairing of the physical and virtual world, are of significant current interest to the broader engineering community. By integrating data with FEMs, this new work provides the mathematical foundations of the Digital Twin revolution’.
The study is reported in the Proceedings of the National Academy of Sciences with a use case of modeling ‘solitons’ (ocean waves) which are said to be a threat to critical offshore infrastructure such as wind turbines.
* It would appear that (at least) some facets of stochastic forcing are available in software from Cossan.
We quizzed Cossan project lead Edoardo Patelli as to whether there was
a degree of hyperbola in the LR announcement. He told us ‘Yes, you see
a rebranding [of FME ]with cool terms like digital twins instead of
stochastic finite elements and perhaps with a pinch of machine learning
instead of data analysis. Maybe this is a better way to generate
commercial interest! But are people going to use your code without
understanding what is going on underneath?’ We challenged the paper’s
lead author Colin Duffin, suggesting there may be some ‘prior art’
here. He responded, ‘Stochastic finite elements are similar but to my
knowledge they are made for uncertainty quantification of
parameters/quantities of interest. Whereas in our work we are more
interested in updating our FEM solution with data and seeing the
resultant posterior measure’.
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