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


Related projects

Multi-analysis of fretting fatigue using physical and virtual experiments
DTR in natural convection regime
Core research programme P2-0095

P-Lab team

Gregor Kosec
Jure Slak
Miha Rot
Mitja Jančič
Filip Strniša
Urban Duh
Blaž Rojc


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] ::
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] ::
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] ::
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] ::
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. Rot, M. Horvat, G. Kosec; Modalna dekompozicija naravne konvekcije v nenewtonskih tekočinah : magistrsko delo, 2020 [COBISS: 31313155] ::
J. Slak, G. Kosec; Adaptive RBF-FD method : PhD thesis, 2020 [COBISS: 31592707] ::
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]
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]
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] ::
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]
G. Kosec, J. Slak, M. Depolli, R. Trobec, K. Pereira, S. Tomar, T. Jacquemin, S. Bordas, M. 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] ::
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, 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]
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]
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] ::
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] ::
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]
J. Slak, G. Kosec; Generic implementation of meshless local strong form method, ECT2018, 2018 [COBISS: 31829799]
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]
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]
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]
M. Depolli, R. Trobec, G. Kosec, K. Pereira, M. 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] ::
R. Trobec, G. Kosec; Parallel scientific computing : theory, algorithms, and applications of mesh based and meshless methods, 2015 [COBISS: 28468007]
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] ::