Abstract
The use of constitutive equations calibrated from data has been implemented into standard numerical solvers for successfully addressing a variety problems encountered in simulation-based engineering sciences (SBES). However, the complexity remains constantly increasing due to the need of increasingly detailed models as well as the use of engineered materials. Data-Driven simulation constitutes a potential change of paradigm in SBES. Standard simulation in computational mechanics is based on the use of two very different types of equations. The first one, of axiomatic character, is related to balance laws (momentum, mass, energy, $$\ldots $$ ), whereas the second one consists of models that scientists have extracted from collected, either natural or synthetic, data. Data-driven (or data-intensive) simulation consists of directly linking experimental data to computers in order to perform numerical simulations. These simulations will employ laws, universally recognized as epistemic, while minimizing the need of explicit, often phenomenological, models. The main drawback of such an approach is the large amount of required data, some of them inaccessible from the nowadays testing facilities. Such difficulty can be circumvented in many cases, and in any case alleviated, by considering complex tests, collecting as many data as possible and then using a data-driven inverse approach in order to generate the whole constitutive manifold from few complex experimental tests, as discussed in the present work.
Highlights
Machine and manifold learning techniques, and nonlinear dimensionality reduction, as for example locally linear embedding (LLE), kernel-PCA, referred as k-PCA, local-PCA, among many other choices, allows us to remove correlations in data [10,17,19,20,21]
When dealing with machines, these intellectual needs are not inherent to their nature, and decisions can be made from a new kind of intelligence that, more than based on mathematical expressions, are based on data via data mining and data analytics
As soon as the uncorrelated parameters are extracted, two main options have been considered to date: (1) when a new case, not included in the data, must be analyzed, its solution is interpolated on the manifold from its closest neighbors [12] so that decisions can be taken in real time; and (2) an explicit parametric solution could be constructed by using the just extracted uncorrelated parameters so that it could be particularized in real-time [6,7]
Summary
Machine and manifold learning techniques, and nonlinear dimensionality reduction, as for example locally linear embedding (LLE), kernel-PCA (the nonlinear counterpart of principal component analysis—PCA), referred as k-PCA, local-PCA, among many other choices, allows us to remove correlations in data [10,17,19,20,21]. If instead of performing simple tests, we consider one involving complex and evolving loads applying on a quite complex geometry In this way numerous mechanical states will coexist in the part, and having access for example to the strain in a region of the specimen, we could by using an inverse identification strategy, identifying a large part of the constitutive manifold. Such a procedure has as main drawback the fact of using the elastic tensor as main mechanical variable as well as its complexity in the case of nonlinear behaviors, as discussed later Another appealing possibility consists of constructing a polynomial approximation of the elastic energy, whose second derivative results in the elastic tensor, and whose identification from collected data seems to be more robust.
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