On the modal analysis of nonlinear beam and shell structures with singular mass and stiffness matrices

Hente, Christian; Gebhardt, Cristian Guillermo; Pache, Dorian; Rolfes, Raimund

doi:10.1016/j.tws.2019.106310

Cited by 11 publications

(10 citation statements)

References 26 publications

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Section: Specialization Of the Proposed Approachmentioning

confidence: 99%

“…Since the formulation of constraints related to usual boundary conditions, for example, rigid support, simple support, movable support inter alia, are represented by linear equations in the nodal variables and thus, their treatment is straightforward, we omit their systematic presentation. For further details, see References 32,34. Finally, having at hand all vectors and matrices indicated within this subsection, the construction of the equations corresponding to the optimality conditions and their derivatives is straightforward.…”

Section: Data-driven Geometrically Exact Beam (Dynamic Setting)mentioning

confidence: 99%

See 1 more Smart Citation

A framework for data‐driven structural analysis in general elasticity based on nonlinear optimization: The dynamic case

Gebhardt

Steinbach

Schillinger

et al. 2020

Numerical Meth Engineering

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Summary In this article, we present an extension of the formulation recently developed by the authors to the structural dynamics setting. Inspired by a structure‐preserving family of variational integrators, our new formulation relies on a discrete balance equation that establishes the dynamic equilibrium. From this point of departure, we first derive an “exact” discrete‐continuous nonlinear optimization problem that works directly with data sets. We then develop this formulation further into an “approximate” nonlinear optimization problem that relies on a general constitutive model. This underlying model can be identified from a data set in an offline phase. To showcase the advantages of our framework, we specialize our methodology to the case of a geometrically exact beam formulation that makes use of all elements of our approach. We investigate three numerical examples of increasing difficulty that demonstrate the excellent computational behavior of the proposed framework and motivate future research in this direction.

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Section: Specialization Of the Proposed Approachmentioning

confidence: 99%

Section: Data-driven Geometrically Exact Beam (Dynamic Setting)mentioning

confidence: 99%

A framework for data‐driven structural analysis in general elasticity based on nonlinear optimization: The dynamic case

Gebhardt

Steinbach

Schillinger

et al. 2020

Numerical Meth Engineering

Self Cite

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show abstract

“…Details can be found in standard textbooks on numerical linear algebra or on optimization, such as [18,19]. In the context of slender structures, the reader may refer to [20,21] and references therein for the efficient calculation of the null-space basis.…”

Section: Optimization Problems For Data-driven Computational Mechanicsmentioning

confidence: 99%

“…The formulation of constraints that can be related to the usual types of boundary conditions (rigid support, simple support, movable support inter alia), can be achieved by linear equations in the nodal variables and is thus straightforward. In the scope of this work, we therefore omit their systematic presentation and refer for further details to [21,27]. With all vectors and matrices derived in this subsection at hand, the construction of the equations corresponding to the optimality conditions and their derivatives is straightforward.…”

Section: Data-driven Geometrically Exact Beam Elementmentioning

confidence: 99%

A Framework for Data-Driven Computational Mechanics Based on Nonlinear Optimization

Gebhardt,

Schillinger,

Steinbach

et al. 2019

Preprint

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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 datadriven 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.

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“…Since the formulation of constraints related to usual boundary conditions, e.g., rigid support, simple support, movable support inter alia, are represented by linear equations in the nodal variables and thus, their treatment is straightforward, we omit their systematic presentation. For further details, see [30,32].…”

Section: Data-driven Geometrically Exact Beam(dynamic Setting)mentioning

confidence: 99%

A Framework for Data-Driven Computational Dynamics Based on Nonlinear Optimization

Gebhardt,

Steinbach,

Schillinger

et al. 2019

Preprint

Self Cite

View full text Add to dashboard Cite

In this article, we present an extension of the formulation recently developed by the authors (A Framework for Data-Driven Computational Mechanics Based on Nonlinear Optimization, arXiv:1910.12736 [math.NA]) to the structural dynamics setting. Inspired by a structure-preserving family of variational integrators, our new formulation relies on a discrete balance equation that establishes the dynamic equilibrium. From this point of departure, we first derive an "exact" discrete-continuous nonlinear optimization problem that works directly with data sets. We then develop this formulation further into an "approximate" nonlinear optimization problem that relies on a general constitutive model. This underlying model can be identified from a data set in an offline phase.To showcase the advantages of our framework, we specialize our methodology to the case of a geometrically exact beam formulation that makes use of all elements of our approach. We investigate three numerical examples of increasing difficulty that demonstrate the excellent computational behavior of the proposed framework and motivate future research in this direction.

show abstract

On the modal analysis of nonlinear beam and shell structures with singular mass and stiffness matrices

Cited by 11 publications

References 26 publications

A framework for data‐driven structural analysis in general elasticity based on nonlinear optimization: The dynamic case

A framework for data‐driven structural analysis in general elasticity based on nonlinear optimization: The dynamic case

A Framework for Data-Driven Computational Mechanics Based on Nonlinear Optimization

A Framework for Data-Driven Computational Dynamics Based on Nonlinear Optimization

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