Model-free data-driven methods in mechanics: material data identification and solvers

Abstract

This paper presents an integrated model-free data-driven approach to solid mechanics, allowing to perform numerical simulations on structures on the basis of measures of displacement fields on representative samples, without postulating a specific constitutive model. A material data identification procedure, allowing to infer strain-stress pairs from displacement fields and boundary conditions, is used to build a material database from a set of mutiaxial tests on a nonconventional sample. This database is in turn used by a data-driven solver, based on an algorithm minimizing the distance between manifolds of compatible and balanced mechanical states and the given database, to predict the response of structures of the same material, with arbitrary geometry and boundary conditions. Examples illustrate this modelling cycle and demonstrate how the datadriven identification method allows importance sampling of the material state space, yielding faster convergence of simulation results with increasing database size, when compared to synthetic material databases with regular sampling patterns.

Data-driven reduced homogenization for transient diffusion problems with emergent history effects

“…Spatial discretization of the domain Ω is performed by finite elements containing m = 1, 2, ..., M material (integration) points and i = 1, 2, ..., N nodes. The notations and terminologies adopted here, follow from [16,28].…”

Data‐driven solvers for strongly nonlinear material response

“…This approach was introduced in the seminal work of Kirchdoerfer and Ortiz 14 and was then further developed and analyzed for a range of problems arising in computational mechanics. [15][16][17][18][19][20][21] For the case of EM simulations, a comparable formulation based on the variational principle was already proposed by Rikabi et al 22 in 1988. The main idea behind the data-driven computing paradigm is the reformulation of the initial/boundary value problem (IBVP) describing the physical phenomenon under investigation, such that the material modeling step is bypassed altogether, thus eliminating the attached epistemic uncertainty.…”

“…A data-driven solver integrating the material response identification approach has also been developed, showing significant accuracy and efficiency improvements. 21 Approaches based on manifold learning are also available. 25,26 Thereby, the data-driven solution is not calculated using the original measurement data, but along a reduced-order manifold derived after the data set.…”

“…An alternative and fundamentally different paradigm regarding the incorporation of material data into numerical computations, bearing the name “data‐driven computing,” emerged recently from the field of computational mechanics. This approach was introduced in the seminal work of Kirchdoerfer and Ortiz 14 and was then further developed and analyzed for a range of problems arising in computational mechanics 15‐21 . For the case of EM simulations, a comparable formulation based on the variational principle was already proposed by Rikabi et al 22 in 1988.…”

“…The data‐driven computing paradigm has been further extended for the purpose of material response identification, 20 in which case the data‐driven solver is effectively “inverted” as to locate states that sample the mechanical response of elastic materials. A data‐driven solver integrating the material response identification approach has also been developed, showing significant accuracy and efficiency improvements 21 . Approaches based on manifold learning are also available 25,26 .…”