Data-driven computational mechanics.

*(English)*Zbl 1425.74503Summary: 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.

##### MSC:

74S30 | Other numerical methods in solid mechanics (MSC2010) |

65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |

74B05 | Classical linear elasticity |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

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\textit{T. Kirchdoerfer} and \textit{M. Ortiz}, Comput. Methods Appl. Mech. Eng. 304, 81--101 (2016; Zbl 1425.74503)

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