Comput. Mech. (2006) 38: 294–309 DOI 10.1007/s00466-006-0058-5 ORIGINAL PAPER R. Aubry · S.R. Idelsohn · E. Oñate Fractional step like schemes for free surface problems with thermal coupling using the Lagrangian PFEM Received: 29 December 2005 / Accepted: 10 February 2006 / Published online: 25 April 2006 © Springer-Verlag 2006 Abstract The method presented in Aubry et al. (Comput Struc 83:1459–1475, 2005) for the solution of an incom- pressible viscous fluid flow with heat transfer using a fully Lagrangian description of motion is extended to three dimen- sions (3D) with particular emphasis on mass conservation. A modified fractional step (FS) based on the pressure Schur complement (Turek 1999), and related to the class of alge- braic splittings Quarteroni et al. (Comput Methods Appl Mech Eng 188:505–526, 2000), is used and a new advantage of the splittings of the equations compared with the classical FS is highlighted for free surface problems. The temperature is semi-coupled with the displacement, which is the main var- iable in a Lagrangian description. Comparisons for various mesh Reynolds numbers are performed with the classical FS, an algebraic splitting and a monolithic solution, in order to illustrate the behaviour of the Uzawa operator and the mass conservation. As the classical fractional step is equivalent to one iteration of the Uzawa algorithm performed with a stan- dard Laplacian as a preconditioner, it will behave well only in a Reynold mesh number domain where the preconditioner is efficient. Numerical results are provided to assess the supe- riority of the modified algebraic splitting to the classical FS. Keywords Lagrangian description · Mixed incompressible element · Coupled thermo mechanical analysis · Pressure schur complement · Generalized stokes solver 1 Introduction The Navier–Stokes equations have been traditionally associ- ated with an Eulerian description of motion, where the veloc- R. Aubry (B ) · E. Oñate Universidad Politécnica de Cataluña, International Center for Numerical Methods in Engineering (CIMNE), Barcelona, Spain E-mail: romain@cimne.upc.edu E-mail: onate@cimne.upc.edu S.R. Idelsohn ICREA Research Professor at CIMNE E-mail: sergio@cimne.upc.edu ity is known on each spatial point of the problem domain. The Lagrangian formulation offers a different point of view, as each particle knows its velocity. At the continuum level, both descriptions are strictly equivalent but give rise to differ- ent implementations and difficulties. A major drawback with a Lagrangian approach is perhaps the necessity to remesh frequently, which was not affordable until very recently. As mesh generation has undergone amazing progresses these last twenty years [31, 32, 40, 41, 63], this has made the use of the Lagrangian formulation now possible. At the computational level, a Lagrangian approach offers many advantages, as no convective term appears in the time derivative which means, amongst other things: – no stabilization of the convective term is necessary [38]. – the matrices to be solved are symmetric which provides minimization properties with short term recurrences for iterative solvers [3, 56]. – it almost halves storage as matrices are symmetric. – optimal preconditioners are available for the generalized Stokes problem but not for the Navier-Stokes equations [17, 29, 43]. – for free surface problems, it provides an explicit descrip- tion of the free surface and no additional transport or reini- tialization equations (Level set [46], VOF [36], pseudo concentration) need to be solved. It is furthermore far less diffusive [46] in this context. Finally, the solution of the non linear problem is the final solution whereas the solution with a level set-like method is most of the time explicitely convected due to instabilities with elements that are fluid then gaz inside the non linear process. – only the domain filled by the fluid is meshed. – boundary conditions (temperature, pressure, heat flux) on free surfaces are straightforward to impose. In this paper, the particle finite element method (PFEM) [26, 39, 45] (E. Oñate et al. 2004, submitted) is used with vari- ous mixed elements in order to solve thermal convection for incompressible fluid flows. Mass conservation constitutes an important problem in fluid mechanics. As depicted in [48], mass conservation is usually only partially weakly verified.