  Data-Driven Computational Mechanics is a novel computing paradigm that
enables the transition from standard data-starved approaches to modern
data-rich approaches. At this early stage of development, one can distinguish
two mainstream directions. The first one relies on a discrete-continuous
optimization problem and seeks to assign to each material point a point in the
phase space that satisfies compatibility and equilibrium, while being closest
to the data set provided. The second one is a data driven inverse approach that
seeks to reconstruct a constitutive manifold from data sets by manifold
learning techniques, relying on a well-defined functional structure of the
underlying constitutive law. In this work, we propose a third route that
combines the strengths of the two existing directions and mitigates some of
their weaknesses. This is achieved by the formulation of an approximate
nonlinear optimization problem, which can be robustly solved, is
computationally efficient, and does not rely on any special functional
structure of the reconstructed constitutive manifold. Additional benefits
include the natural incorporation of kinematic constraints and the possibility
to operate with implicitly defined stress-strain relations. We discuss
important mathematical aspects of our approach for a data-driven truss element
and investigate its key numerical behavior for a data-driven beam element that
makes use of all components of our methodology.
