Published in Int. J. Numer. Meth. Fluids Vol. 54 (6-8), pp. 639-664, 2007
doi:10.1002/fld.1480

Abstract

The Navier–Stokes equations written in Laplace form are often the starting point of many numerical methods for the simulation of viscous flows. Imposing the natural boundary conditions of the Laplace form or neglecting the viscous contributions on free surfaces are traditionally considered reasonable and harmless assumptions. With these boundary conditions any formulation derived from integral methods (like finite elements or finite volumes) recovers the pure Laplacian aspect of the strong form of the equations. This approach has also the advantage of being convenient in terms of computational effort and, as a consequence, it is used extensively. However, we have recently discovered that these resulting Laplacian formulations violate a basic axiom of continuum mechanics: the principle of objectivity. In the present article we give an accurate account about these topics. We also show that unexpected differences may sometimes arise between Laplace discretizations and divergence discretizations.

M. Dabrowski, M. Krotkiewski, D. Schmid. MILAMIN: MATLAB-based finite element method solver for large problems. Geochem. Geophys. Geosyst. 9(4) (2008) DOI 10.1029/2007gc001719

R. Rossi, A. Larese, P. Dadvand, E. Oñate. An efficient edge-based level set finite element method for free surface flow problems. Int. J. Numer. Meth. Fluids 71(6) (2012) DOI 10.1002/fld.3680

E. Oñate, A. Franci, J. Carbonell. Lagrangian formulation for finite element analysis of quasi-incompressible fluids with reduced mass losses. Int. J. Numer. Meth. Fluids 74(10) (2014) DOI 10.1002/fld.3870

P. Ryzhakov, J. Cotela, R. Rossi, E. Oñate. A two-step monolithic method for the efficient simulation of incompressible flows. Int. J. Numer. Meth. Fluids 74(12) (2014) DOI 10.1002/fld.3881

P. Ryzhakov, E. Oñate, R. Rossi, S. Idelsohn. Improving mass conservation in simulation of incompressible flows. Int. J. Numer. Meth. Engng 90(12) (2012) DOI 10.1002/nme.3370

M. Huber, U. Rüde, C. Waluga, B. Wohlmuth. Surface Couplings for Subdomain-Wise Isoviscous Gradient Based Stokes Finite Element Discretizations. J Sci Comput 74(2) (2017) DOI 10.1007/s10915-017-0470-3

S. Idelsohn, E. Oñate. The challenge of mass conservation in the solution of free-surface flows with the fractional-step method: Problems and solutions. Int. J. Numer. Meth. Biomed. Engng. 26(10) (2008) DOI 10.1002/cnm.1216

A. Franci. Unified Stabilized Formulation for Quasi-incompressible Materials. (2016) DOI 10.1007/978-3-319-45662-1_3

E. Oñate, A. Franci, J. Carbonell. A Particle Finite Element Method (PFEM) for Coupled Thermal Analysis of Quasi and Fully Incompressible Flows and Fluid-Structure Interaction Problems. (2014) DOI 10.1007/978-3-319-06136-8_6

P. Ryzhakov, R. Rossi, S. Idelsohn, E. Oñate. A monolithic Lagrangian approach for fluid–structure interaction problems. Comput Mech 46(6) (2010) DOI 10.1007/s00466-010-0522-0

E. Oñate, A. Franci, J. Carbonell. A particle finite element method for analysis of industrial forming processes. Comput Mech 54(1) (2014) DOI 10.1007/s00466-014-1016-2