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Non-Eulerian Newmark Methods: A Powerful Tool for Free-Boundary Continuum Mechanics Problems

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Abstract

In this paper we introduce a new procedure to solve free-surface problems based on applying the Newmark family of time integration schemes to non-Eulerian formulations of the problem (i.e., in non-current domains). The methods obtained within this framework present important advantages over other methods in literature, as for instance, that the computational domain is independent of the current unknowns, the convective term disappears, and modelling and tracking of the free surface is straightforward. Moreover, the Newmark algorithm is convenient to obtain accurate and stable methods for solving continuum mechanics models which can be written in terms of either displacement, velocity or acceleration. We consider a viscous Newtonian fluid in a time dependent domain which may undergo large deformations along the time but not topological changes of interfaces. A unified formulation providing the general framework of the proposed schemes is stated in this context. They are combined with finite elements methods for space discretization. In particular, the Newmark family of pure Lagrange–Galerkin methods is stated. The non-Eulerian formulations are advantageous to obtain linear methods. More precisely, linearized versions of the standard (non-linear) Newmark schemes that present good approximation properties are also proposed. Moreover, we deal with problems associated with high mesh distortion by proposing a reinitialization method to be applied in the non-Eulerian Newmark framework that preserves the order of convergence. In order to assess the performance of the proposed numerical methods, we solve different problems in two space dimensions.

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Acknowledgements

This work was partially funded by FEDER and Xunta de Galicia (Spain) under Grant 2017 GRC GI-1563 and research Project ED431G/01, and by FEDER and the Spanish Ministry of Economy and Competitiveness under research Project MTM2017-86459-R. We would also like to thank the referee for his/her valuable comments.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidade da Coruña, Centro de Investigación CITIC, Elviña s/n, 15071, A Coruña, Spain

    M. Benítez

  2. Departamento de Matemática Aplicada and Instituto de Matemáticas, Universidade de Santiago de Compostela, c/ Lope Gómez de Marzoa s/n, 15782, Santiago de Compostela, Spain

    A. Bermúdez

  3. Instituto Tecnológico de Matemática Industrial (ITMATI), Rúa Constantino Candeira s/n, 15782, Santiago de Compostela, Spain

    A. Bermúdez

  4. REPSOL Technology Lab, Autovía de Extremadura s/n, 28935, Móstoles, Madrid, Spain

    P. Fontán

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  1. M. Benítez
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  3. P. Fontán
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Correspondence to M. Benítez.

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Benítez, M., Bermúdez, A. & Fontán, P. Non-Eulerian Newmark Methods: A Powerful Tool for Free-Boundary Continuum Mechanics Problems. J Sci Comput 83, 44 (2020). https://doi.org/10.1007/s10915-020-01207-y

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  • DOI: https://doi.org/10.1007/s10915-020-01207-y

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