Medusa: coordinate free implementation of meshless methods

Our team started the development of Medusa library in 2015 to support our research in the field of numerical analysis and to ease implementation of applied projects. Over time, the interface grew and matured, putting emphasis on modularity, extensibility and reusability. Similarly to many open-source FEM libraries, it relies heavily on the C++ template system and allows the programs to be written independently of the number of spatial dimensions with negligible run-time and memory overhead. Special care is also taken to increase expressiveness and to be able to explicitly translate mathematical notation into program source code. However, source code is still standard compliant C++, which allows the user to use entirety of the C++ ecosystem. The open-source nature of the library is a novelty compared to the other libraries. We already utilized Medusa library for solving broad spectra of problems ranging from pure academic experiments like high order solutions of Poisson’s equation in 2D, 3D, and 4D, to applied thermo-fluid and thermo-elastic simulations of real life problems.

Few examples are presented below: (1) The Pennes' bioheat equation on realistic geometry of human brain (2) Scattering from a triple dielectric step in 1D (3) adaptive solution of 3D Bousinesq’s problem (4) High order solution of Poisson’s equation in 4D (4) thermo-fluid natural convection case

Code repository, examples, API reference and project main page

Homepage
Homepage

The basic information about Medusa, including live demo, description of main features, download links, installation guide, and more, can be found on dedicated Medusa page.

Code
Code

Medusa library is open source and you can freely clone it from its git repository, where all the development happens. The code on repository is tested upon each change by executing unit and end-to-end tests using Continuous Integration provided by GitLab.

Documentation
Documentation

The complete C++ API of the library is available through the doxygen generated pages. The code is divided into modules and each class is documented and equipped with a short usage example.

Wiki
Wiki

A tool is useless if you don't know how to use it, and while we provide detailed documentation we understand that using a library for a first time user may be a very daunting task. In this spirit we created numerous examples to aid the process of learning. Our wiki page is a more frequently changing body of content, that has numerous examples, tutorials, experiments and other notes in somewhat relaxed form.

ResearchGate
ResearchGate

Follow Medusa publications at the project site on ResearchGate

Related projects

Multi-analysis of fretting fatigue using physical and virtual experiments
DTR in natural convection regime
Inertial effects on fluid flow in complex porous media
Advanced modelling of radio channels using ray-optical and numerical meshless methods
Core research programme P2-0095

P-Lab team


Publications

F. Strniša, M. Jančič, G. Kosec; Numerical analysis of small-strain elasto-plastic deformation using local radial basis function approximation with Picard iteration, Applied mathematical modelling, vol. 137, 2025 [DOI: 10.1016/j.apm.2024.115714][COBISS: 208700163] ::
M. Rot, G. Kosec; Refined radial basis function-generated finite difference analysis of non-Newtonian natural convection, Physics of fluids, vol. 37, 2025 [DOI: 10.1063/5.0257896][COBISS: 230006531] ::
A. Kolar-Požun, M. Jančič, M. Rot, G. Kosec; Some observations regarding the RBF-FD approximation accuracy dependence on stencil size, Journal of computational science, vol. 79, 2024 [DOI: 10.1016/j.jocs.2024.102284][COBISS: 194478851] ::
M. Jančič, G. Kosec; Strong form mesh-free hp-adaptive solution of linear elasticity problem, Engineering with computers, vol. 40, 2024 [DOI: 10.1007/s00366-023-01843-6][COBISS: 153678339] ::
M. Jančič, M. Založnik, G. Kosec; Meshless interface tracking for the simulation of dendrite envelope growth, Journal of computational physics, vol. 507, 2024 [DOI: 10.1016/j.jcp.2024.112973][COBISS: 191242243] ::
U. Duh, V. Shankar, G. Kosec; Discretization of non-uniform rational B-spline (NURBS) models for meshless isogeometric analysis, Journal of scientific computing, 2024 [DOI: 10.1007/s10915-024-02597-z][COBISS: 201006595] ::
M. Rot, M. Jančič, G. Kosec; Spatially dependent node regularity in meshless approximation of partial differential equations, Journal of computational science, vol. 79, 2024 [DOI: 10.1016/j.jocs.2024.102306][COBISS: 195505923] ::
M. Depolli, J. Slak, G. Kosec; Parallel domain discretization algorithm for RBF-FD and other meshless numerical methods for solving PDEs, Computers & Structures, 2022 [DOI: 10.1016/j.compstruc.2022.106773][COBISS: 98533123] ::
U. Duh, G. Kosec, J. Slak; Fast variable density node generation on parametric surfaces with application to mesh-free methods, SIAM journal on scientific computing, vol. 43, 2021 [DOI: 10.1137/20M1325642][COBISS: 56730115] ::
M. Jančič, J. Slak, G. Kosec; Monomial augmentation guidelines for RBF-FD from accuracy versus computational time perspective, Journal of scientific computing, vol. 87, 2021 [DOI: 10.1007/s10915-020-01401-y][COBISS: 52715011] ::
J. Močnik Berljavac, P. Mishra, J. Slak, G. Kosec; RBF-FD analysis of 2D time-domain acoustic wave propagation in heterogeneous media, Computers & Geosciences, 2021 [DOI: 10.1016/j.cageo.2021.104796][COBISS: 62417923] ::
J. Slak, G. Kosec; Medusa : A C++ library for solving PDEs using strong form 1 mesh-free methods, ACM transactions on mathematical software, vol. 47, 2021 [DOI: 10.1145/3450966][COBISS: 69777155] ::
G. Kosec, J. Slak; Radial basis function-generated finite differences solution of natural convection problem in 3D, International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2019, 23-28 September 2019, Rhodes, Greece, AIP conference proceedings, vol. 2293, 2020 [DOI: 10.1063/5.0027289][COBISS: 40426499]
M. Jančič, J. Slak, G. Kosec; GPU accelerated RBF-FD solution of poisson's equation, MIPRO 2020 : 43nd International Convention, September 28 - October 2, 2020, Opatija, Croatia, 2020 [DOI: 10.23919/MIPRO48935.2020.9245221][COBISS: 36556291]
U. Duh, M. Depolli, J. Slak, G. Kosec; Parallel point sampling for 3D bodies, MIPRO 2020 : 43nd International Convention, September 28 - October 2, 2020, Opatija, Croatia, 2020 [DOI: 10.23919/MIPRO48935.2020.9245354][COBISS: 36558851]
J. Slak, G. Kosec; Adaptive RBF-FD method : PhD thesis, 2020 [COBISS: 31592707] ::
M. Rot, M. Horvat, G. Kosec; Modalna dekompozicija naravne konvekcije v nenewtonskih tekočinah : magistrsko delo, 2020 [COBISS: 31313155] ::
J. Slak, G. Kosec; Refined meshless local strong form solution of Cauchy-Navier equation on an irregular domain, Engineering analysis with boundary elements, vol. 100, 2019 [DOI: 10.1016/j.enganabound.2018.01.001][COBISS: 31107623] ::
B. Stojanovič, J. Slak, G. Kosec; RBF-FD Solution of Electromagnetic ScatteringProblem, MIPRO 2019 : 42nd International Convention, May 20 -24, 2019, Opatija, Croatia, MIPRO ..., 2019 [DOI: 10.23919/MIPRO.2019.8756943][COBISS: 32388391]
G. Kosec, J. Slak, M. Depolli, R. Trobec, K. Pereira, S. Tomar, T. Jacquemin, S. Bordas, M. Abdel Wahab; Weak and strong from meshless methods for linear elastic problem under frettingcontact conditions, Tribology international, vol. 138, 2019 [DOI: 10.1016/j.triboint.2019.05.041][COBISS: 32424999] ::
J. Slak, B. Stojanovič, G. Kosec; High order RBF-FD approximations with application to a scattering problem, 4th International Conference on Smart and Sustainable Technologies, [SpliTech 2019], 18-21 June 2019, Split & Bol, Croatia, 2019 [COBISS: 33271591]
G. Kosec, J. Slak; Parallel RBF-FD solution of the Boussinesq's problem, Proceedings of the the Sixth International Conference on Parallel, Distributed, GPU and Cloud Computing for Engineering, PARENG 2019, 4-5 June 2019, Pécs, Hungary, 2019 [COBISS: 32414247]
J. Močnik Berljavac, J. Slak, G. Kosec; Parallel simulation of time-domain acoustic wavepropagation, MIPRO 2019 : 42nd International Convention, May 20 -24, 2019, Opatija, Croatia, MIPRO ..., 2019 [DOI: 10.23919/MIPRO.2019.8756946][COBISS: 32388647]
J. Slak, G. Kosec; Adaptive radial basis function-generated finite differences method for contact problems, International journal for numerical methods in engineering, vol. 119, 2019 [DOI: 10.1002/nme.6067][COBISS: 32230439] ::
J. Slak, G. Kosec; Fast generation of variable density node distributions for mesh-free methods, Boundary Elements and other Mesh Reduction Methods XXXXI, 2019 [COBISS: 32052519]
M. Maksić, V. Djurica, A. Souvent, J. Slak, M. Depolli, G. Kosec; Cooling of overhead power lines due to the natural convection, International journal of electrical power & energy systems, 2019 [DOI: 10.1016/j.ijepes.2019.05.005][COBISS: 32388135] ::
J. Slak, G. Kosec; On generation of node distributions for meshless PDE discretizations, SIAM journal on scientific computing, vol. 41, 2019 [DOI: 10.1137/18M1231456][COBISS: 32782887] ::
G. Kosec; A local numerical solution of a fluid-flow problem on an irregular domain, Advances in engineering software, vol. 120, 2018 [DOI: 10.1016/j.advengsoft.2016.05.010][COBISS: 29512743] ::
G. Kosec, J. Slak; Numerical simulation of natural convection from a heated cylinder, Proceedings of the International Conference on Computational Methods, ICCM2018, 6th - 10th August 2018, Rome, Italy, 2018 [COBISS: 31642919]
J. Slak, G. Kosec; Parallel coordinate free implementation of local meshless method, MIPRO 2018 : 41st International Convention, May 21 -25, 2018, Opatija, Croatia, 2018 [COBISS: 31430951]
J. Slak, G. Kosec; Generic implementation of meshless local strong form method, ECT2018, 2018 [COBISS: 31829799]
J. Slak, G. Kosec; Refined RBF-FD solution of linear elasticity problem, 2018 3rd International Conference on Smart and Sustainable Technologies, SpliTech 2018, June 26-29, 2018, Split, Croatia, 2018 [COBISS: 31728167]
M. Depolli, R. Trobec, G. Kosec, K. Pereira, M. Abdel Wahab, S. Tomar, P. Bordas; Parallel performance of abaqus software for simulation of fretting fatigue, Proceedings of the Fifth International Conference on Parallel, Distributed, Grid and Cloud Computing for Engineering, PARENG 2017, 30-31 May 2017, Pécs, Hungary, 2017 [COBISS: 30529575]
J. Slak, G. Mejak, G. Kosec; Reševanje linearnih elastostatičnih problemov z brezmrežnimi metodami : magistrsko delo, 2017 [COBISS: 18122329] ::
G. Kosec; Stability analysis of a meshless method in irregular nodal distributions for flow problems, International journal of computational methods and experimental measurements, vol. 5, 2017 [DOI: 10.2495/CMEM-V5-N3-329-336][COBISS: 30069543]
G. Kosec, R. Trobec; Simulation of semiconductor devices with a local numerical approach, Engineering analysis with boundary elements, 2015 [DOI: 10.1016/j.enganabound.2014.07.013][COBISS: 27912487] ::
R. Trobec, G. Kosec; Parallel scientific computing : theory, algorithms, and applications of mesh based and meshless methods, 2015 [COBISS: 28468007]