The criterion of subscale sufficiency and its application to the relationship between static capillary pressure, saturation and interfacial areas

Abstract

Modern imaging techniques, increased simulation capabilities and extended theoretical frameworks, naturally drive the development of multiscale modelling by the question: which new information should be considered? Given the need for concise constitutive relationships and efficient data evaluation; however, one important question is often neglected: which information is sufficient? For this reason, this work introduces the formalized criterion of subscale sufficiency. This criterion states whether a chosen constitutive relationship transfers all necessary information from micro to macroscale within a multiscale framework. It further provides a scheme to improve constitutive relationships. Direct application to static capillary pressure demonstrates usefulness and conditions for subscale sufficiency of saturation and interfacial areas.

Improving constitutive equations in multiscale modelling by means of the sufficiency criterion using the example of nano wire contraction

“…If we fix the values for E, γ and 1/w + 1/h, we automatically also fix the value for ε. Knowing the sufficient set, we can now improve the initial relation ship by a conditional average [2]. One possible interpretation of this average is averaging ε ini over constant values s of S, because these must be isolines as constant sufficient parameters imply a constant value of the dependent parameter (black lines in Fig.…”

“…Yet, it can be adapted to purely mechanical problems involving information transfer accross scales. For instance, the constitutive equation takes the role of the estimator, the macroscopic parameters replace the random variables and an error norm supersedes the expected deviation [2]. We will demonstrate how this concept can be adapted to non-statistical, multiscale problems using the example of contraction in relaxed nano wires.…”

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

“…The following analytical and numerical ANN examples aim at providing a simple system to illustrate the strategy, its benefits and open questions. A more detailed background can be found in [14] for application to deterministic multiscale problems.…”