TY - JOUR

T1 - A framework for data-driven structural analysis in general elasticity based on nonlinear optimization

T2 - The static case

AU - Gebhardt, Cristian Guillermo

AU - Schillinger, Dominik

AU - Steinbach, Marc Christian

AU - Rolfes, Raimund

N1 - Funding information: C. G. Gebhardt and R. Rolfes gratefully acknowledge the financial support of the Lower Saxony Ministry of Science and Culture (research project ventus efficiens, FKZ ZN3024) and the German Research Foundation (research project ENERGIZE, GE 2773/3-1 – RO 706/20-1) that enabled this work. D. Schillinger acknowledges support from the German Research Foundation through the DFG Emmy Noether Award (SCH 1249/2-1), and from the European Research Council via the ERC Starting Grant “ImageToSim” (Action No. 759001).

PY - 2020/3/25

Y1 - 2020/3/25

N2 - 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, which can be classified as a direct approach, 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, which can be classified as an inverse approach, 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 hybrid approach 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.

AB - 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, which can be classified as a direct approach, 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, which can be classified as an inverse approach, 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 hybrid approach 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.

KW - Approximate nonlinear optimization problem

KW - Constitutive manifold

KW - Data-Driven Structural Analysis

KW - Finite element method

KW - General elasticity

KW - Geometrically exact beam

UR - http://www.scopus.com/inward/record.url?scp=85082109925&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2020.112993

DO - 10.1016/j.cma.2020.112993

M3 - Article

AN - SCOPUS:85082109925

VL - 365

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

M1 - 112993

ER -