Lagrangian approach

Lagrangian approach naturally observes movement, and thus advection of flow. Points that describe continuum move along streamlines and transport their properties.

Eulerian approach statically observes some locations while the medium flows through them, which requires the addition of non-linear operator to transport equations (u · ∇).

RIP Courant–Friedrichs–Lewy

The classical CFL condition does not apply to Lagrangian methods, which may have large time-steps with stable advection.

Meshless framework

Due to freedom of movement, the Lagrangian method is ideal to be described by meshless (or mesh-free) framework. Novel 2nd order consistent Lagrangian Differencing (LD) spatial operators are introduced to accurately solve Partial Differential Equation (PDEs).

Incompressible flows

Navier-Stokes equations (NSE) are solved using the split-step scheme that decouples velocity and pressure. The scheme is thus fast and achieves 2nd order accuracy. The mesh-free nodes are explicitly moving each time step and adjusting to boundaries. The volume conservation is unconditionally stable and handles potentially compressible situations (as shown on an exaggerated example).

Fluid-structure interaction

Users do not have to deal with volumetric meshing, but they also do not have to deal with boundary surfaces meshing or conversion to meshless points. The solver implements novel treatment of boundary conditions, which are directly applied on imported triangulated geometry.

This makes transfer of loads and coupling with some FEM solver straightforward, in order to simulate FSI with large deformations and movements.

High-performance computing

The solver is completely parallelized, i.e. runs on multi-core and many-core devices. The only bottleneck is when the solver needs to transfer data (due to outputting results to disk, or waiting for some coupled solver).

As GPUs have many parallel execution units, they are ideal candidate for solving repetitive tasks in CFD. Rhoxyz simulations may run more than 20x faster on the modern GPU, than on the modern CPU.

Graphical user interface

The solver comes with a desktop application for setting up a simulation, controlling the simulation running process, and real-time visualisation of the results while the simulation is running.

The simulation results may be exported and loaded for more complex post-processing, e.g. using ParaView.


Bašić J., Degiuli N., Blagojević B., Ban D. Lagrangian differencing dynamics for incompressible flows. Journal of Computational Physics, 462 (2022), 111198

Bašić M., Blagojević B., Peng C., Basic J. Lagrangian Differencing Dynamics for Time-Independent Non-Newtonian Materials. Materials, 14 (2021), 6210

Bašić M., Blagojević B., Klarin B., Bašić J. Coupling of non-Newtonian Meshless Flow with Structural Solvers. VII International Conference on Particle-Based Methods (PARTICLES 2021). Volume IS17 – Particle Methods for Fluid-Structure Interactions. Hamburg, Germany (2021)

Bašić J., Blagojević B., Bašić M., Klarin B. Lagrangian Differencing Dynamics for microgravity sloshing. 6th International Conference on Smart and Sustainable Technologies (SpliTech), Bol, Croatia (2021).

Bašić J., Blagojević B., Bašić M., Sikora M. Parallelism and Iterative bi-Lanczos Solvers. 6th International Conference on Smart and Sustainable Technologies (SpliTech), Bol, Croatia (2021)

Peng C., Bašić M., Blagojević B., Bašić J., Wu W. A Lagrangian differencing dynamics method for granular flow modeling.
Computers and Geotechnics
, 137 (2021), 104297

Joubert J.C., Wilke D.N., Govender N., Pizette P., Bašić J., Abriak N-E. Boundary condition enforcement for renormalised weakly compressible meshless Lagrangian methods. Engineering Analysis with Boundary Elements, 130 (2021), 332-351

Andrun M., Bašić J., Blagojević B., Klarin B. Simulating hydroelastic slamming by coupled Lagrangian-FDM and FEM. in Proceedings of the 12th Symposium on High Speed Marine Vehicles, Naples, Italy (2020)

Bašić J., Degiuli N., Malenica Š., Ban D. Lagrangian finite-difference method for predicting green water loadings. Ocean Engineering, 209 (2020), 107533.

Bašić J. Development of numerical model for green water loading by coupling the mesh based flow models with the meshless models. PhD dissertation, University of Zagreb (2019).

Bašić J., Blagojević, B., Andrun, M., Degiuli, N. A Lagrangian Finite Difference Method for Sloshing: Simulations and Comparison with Experiments. in Proceedings of the Twenty-ninth International Ocean and Polar Engineering Conference 3412–3418 (2019).

Bašić J., Blagojevic B., Ban D., Ljubenkov B., Degiuli N. Lagrangian Finite Difference Method for Violent Fluid–Structure Interaction. in Proceedings of the 32nd Symposium on Naval Hydrodynamics 14 (2018).

Bašić J., Degiuli, N. & Ban, D. A class of renormalised meshless Laplacians for boundary value problems. Journal of Computational Physics. 354, 269–287 (2018).

Bašić J., Degiuli N., Ban D. Renormalised Lagrangian method for water entry impact simulation. in Proceedings of the 11th Symposium on High Speed Marine Vehicles 8 (2017).