We develop a new computing paradigm, which we refer to as data-driven computing, according to which calculations are carried out directly from experimental material data and pertinent constraints and conservation laws, such as compatibility and equilibrium, thus bypassing the empirical material modeling step of conventional computing altogether. Data-driven solvers seek to assign to each material point the state from a prespecified data set that is closest to satisfying the conservation laws. Equivalently, data-driven solvers aim to find the state satisfying the conservation laws that is closest to the data set. The resulting data-driven problem thus consists of the minimization of a distance function to the data set in phase space subject to constraints introduced by the conservation laws. We motivate the data-driven paradigm and investigate the performance of data-driven solvers by means of two examples of application, namely, the static equilibrium of nonlinear three-dimensional trusses and linear elasticity. In these tests, the data-driven solvers exhibit good convergence properties both with respect to the number of data points and with regard to local data assignment. The variational structure of the data-driven problem also renders it amenable to analysis. We show that, as the data set approximates increasingly closely a classical material law in phase space, the data-driven solutions converge to the classical solution. We also illustrate the robustness of data-driven solvers with respect to spatial discretization. In particular, we show that the data-driven solutions of finite-element discretizations of linear elasticity converge jointly with respect to mesh size and approximation by the data set.

We formulate a Data Driven Computing paradigm, termed maxent Data Driven Computing, that generalizes distance-minimizing Data Driven Computing and is robust with respect to outliers. Robustness is achieved by means of clustering analysis. Specifically, we assign data points a variable relevance depending on distance to the solution and on maximum-entropy estimation. The resulting scheme consists of the minimization of a suitably-defined free energy over phase space subject to compatibility and equilibrium constraints. Distanceminimizing Data Driven schemes are recovered in the limit of zero temperature. We present selected numerical tests that establish the convergence properties of the max-ent Data Driven solvers and solutions.

We extend the Data-Driven formulation of problems in elasticity of Kirchdoerfer and Ortiz [1] to inelasticity. This extension differs fundamentally from Data-Driven problems in elasticity in that the material data set evolves in time as a consequence of the history dependence of the material. We investigate three representational paradigms for the evolving material data sets: i) materials with memory, i. e., conditioning the material data set to the past history of deformation; ii) differential materials, i. e., conditioning the material data set to short histories of stress and strain; and iii) history variables, i. e., conditioning the material data set to ad hoc variables encoding partial information about the history of stress and strain. We also consider combinations of the three paradigms thereof and investigate their ability to represent the evolving data sets of different classes of inelastic materials, including viscoelasticity, viscoplasticity and plasticity. We present selected numerical examples that demonstrate the range and scope of Data-Driven inelasticity and the numerical performance of implementations thereof.

SUMMARYWe formulate extensions to Data Driven Computing for both distance minimizing and entropy maximizing schemes to incorporate time integration. Previous works focused on formulating both types of solvers in the presence of static equilibrium constraints. Here formulations assign data points a variable relevance depending on distance to the solution and on maximum-entropy weighting, with distance minimizing schemes discussed as a special case. The resulting schemes consist of the minimization of a suitablydefined free energy over phase space subject to compatibility and a time-discretized momentum conservation constraint. We present selected numerical tests that establish the convergence properties of both types of Data Driven solvers and solutions.

Ballistic tests were performed on single-yarn, single-layer and ten-layer targets of Kevlar® KM2 (600 and 850 denier), Dyneema® SK-65 and PBO® (500 denier). The objective was to develop data for validation of numerical models so, multiple diagnostic techniques were used: (1) ultra-high speed photography, (2) high-speed video and (3) nickel-chromium wire technique. These techniques allowed thorough validation of the numerical models through five different paths. The first validation set was at the yarn level, where the transverse wave propagation obtained with analytical and numerical simulations was compared to that obtained in the experiments. The second validation path was at the single-layer level: the propagation of the pyramidal wave observed with the high speed camera was compared to the numerical simulations. The third validation consisted of comparing, for the targets with ten layers, the pyramid apex and diagonal positions from tests and simulations. The fourth validation, which is probably the most relevant, consisted of comparing the numerical and experimental ballistic limits. Finally for the fifth validation set, nickel-chromium wires were used to record electronically the waves propagating in the fabrics. It is shown that for the three materials the waves recorded during the tests match well the waves predicted by the numerical model.

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