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Mathematical Modeling and Numerical Simulations


Currently, 9 scientists work in the Department: two Professors  DSc, one Assoc. Professor DSc, two Assoc. Professors Dr,  two Assistant Professors. Dr, two Assistants. In addition two associate members (Assoc. Professors Dr) of the IMech work in the Department.






Prof. Daniel Danchev DSc -  Head


979 6447

Prof. Stefan Stefanov DSc


979 6463

Prof. Stanimir Iliev DSc


979 6488

Prof. Dr Nina Pesheva


979 6439

Assoc. Prof. Dr Kiril Shterev



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Assitant Prof. Dr Galin Valchev 105 979 6701

Assitant Prof. Dr Dobry Dankov


979 6488

Assitant Prof. Milan Rashevski


979 6456

Assist. Prof. Dr Pavel Iliev


979 6456

Assoc. Prof. Dr Nadezhda Bunzarova


979 6701

Assoc. Prof. Dr Sl. Slavchev (AM)*

326 979 6456

Assoc. Prof. Dr P. Gospodinov (AM)*


979 6463


AM -- Associate member


In the period 2017-2019 the staff of the Department  will work on the following science projects:


А) Exact and numerical results for the behavior of finite-dimensional inhomogeneous statistical-mechanical systems exhibiting phase transitions

Project DN 02/8 (ДН 02/8) 2018-2019 with the  National Science Fund of Bulgaria;

Project leader- Prof. DSc D. Danchev

The current project aims to provide exact results for basic statistical-mechanical models, as well as to obtain numerical results for them. We will study the modifications in the phase diagrams of such systems due to their finite size, the behavior of the order parameters profiles and the response functions. Special attention will be paid to the fluctuation induced interactions, including the Casimir effect, in model fluid systems undergoing phase transitions near the respective critical points of the infinite (bulk) or the finite systems. This topic is very important and currently actively researched since the above-mentioned forces bear a significant impact on the understanding of the behavior and manipulation of nano-devices. Currently, the Casimir effect and other similar phenomena are a research object in the quantum electrodynamics, the chromodynamics, the cosmology, the condensed matter physics, in some branches of biology, as well as in nano-technologies.

The interested reader is directed to the following important review papers in this field [1.1-1.5]. With respect to the current knowledge on the critical Casimir effect, whose properties are of main interest in the present project, to a certain degree the main results are summarized and discussed in the following review articles [1.6,1.7], some more specific aspects are discussed in [1.4,1.8,1.9]. As it is becoming clear, the study of the properties of Casimir effect in different fields, unavoidably needs and involves knowledge from mathematics as well as numerical methods, and computer systems.


[1.1] A. Rodriguez, P.-C. Hui, D. Woolf, S. Johnson, M. Lončar and F. Capasso, Classical and fluctuation-induced electromagnetic interactions in micron-scale systems: designer bonding, antibonding, and Casimir forces, Ann. Phys., 527(1-2), 45-80, 2015.

[1.2] G. Klimchitskaya and V. Mostepanenko, Casimir and van der Waals forces: Advances and problems, Proc. of Peter the Great St.Petersburg Polytechnic University, N1(517), 41-65, 2015.

[1.3] L. Woods, D. Dalvit, A. Tkatchenko, P. Rodriguez-Lopez, A. Rodriguez and R. Podgornik, A materials perspective on Casimir and van der Waals interactions, ArXiv e-prints, 2015.

[1.4] O. Vasilyev, Monte Carlo Simulation of Critical Casimir Forces. Order, Disorder and Criticality, vol. 4, ch. 2, 55-110, World Scientific, 2015.

[1.5] R. Zhao, Y. Luo and J. Pendry, Transformation optics applied to van der Waals interactions, Sci. Bull., 61(1), 59-67, 2016.

[1.6] M. Krech, Casimir Effect in Critical Systems, World Scientific, Singapore, 1994.

[1.7] J. Brankov, D. Dantchev and N. Tonchev, The Theory of Critical Phenomena in Finite-Size Systems – Scaling and Quantum Effects, World Scientific, Singapore, 2000.

[1.8] A. Gambassi and S. Dietrich, Critical Casimir forces steered by patterned substrates, Soft Matter, 7, 1247-1253, 2011.

[1.9] D. Dean, Non-equilibrium fluctuation-induced interactions, Phys. Scripta, 86(5), 058502, 2012.



B) Theoretical Investigation of nonequilibrium gas flows in micro / nano systems

Project DN 02/7 (2016-2019) with the National Science Fund of Bulgaria

Research area- Mathematical Sciences and Informatics;

Project leader: Prof. DSc Stefan Stefanov


The main driver for the development of micro and nano technologies is the basic research both experimental and theoretical, that generates the necessary new knowledge and ideas about the physical processes occurring in Micro-Electro-Mechanical Systems (MEMS). It is known that the knowledge of classical continuum mechanics, and more generally, continuum physics is based entirely on the assumption of continuity of matter, allowing the use of mathematical models describing the relevant physical processes with systems of partial differential equations and appropriate boundary conditions providing continuity. However, shortly speaking, such knowledge and models fails to be valid and accurate for processes in micro / nano-sized areas. In such systems, the processes can be highly non-equilibrium with presence of discontinuities caused by the discrete nature of molecular flow. To adequately describe the non-equilibrium processes and phenomena in MEMS it is necessary to use more or less new approaches, methods and knowledge of kinetic theory, statistical and quantum physics in order to investigate the relationship and transition between continuum and discrete environments, as well as emerging phenomena in intermediate states. This is a relatively new scientific field in which a large number of scientists from many countries are engaged. This type of research is adopted for important and topical at the institutional level and comes as a part of the scientific priorities of the European Commission. Accordingly, they are intended as various sub-areas in the Bulgarian national strategy for research. Routes to the efficient use of resources must be identified and implemented to provide more environmentally friendly and resource-oriented technologies in the near future. European roadmap process intensification identified several measures: miniaturization, improved heat transfer and recovery of lost heat.

The main objective of the proposed project is the Bulgarian team to participate with qualitative contributions in the research of phenomena and processes in the world of micro / nano scale in the field of microfluidics, in which researchers from the Institute have achieved significant scientific results. The team of scientists from the Institute of Mechanics at BAS offering this project has internationally recognized achievements in theoretical and numerical studies of nonequilibrium gas flows and phenomena in MEMS. This is evident from the list of publications in prestigious scientific journals, presented for each participant.


C) Mechano-mathematical modelling and numerical simulations of contemporary technological processes

Project leader: Assoc. Prof. DSc S. Iliev

(priority areas 1,2,3 and 4 of the National Science Strategy)


D) Theoretical Investigation of Rarefied Gas Dynamics of Monoatomic Gas, Gas Mixtures, and Chemically Reacting Gases and Fluid-Structure Interaction in micro/nano systems

Project KP-06-N32/6, with the National Science Fund of Bulgaria,

Rresearch area, of the project:

Mathematical Sciences and Informatics, Physical Sciences, Chemical Sciences, Technical Sciences

Project leader: Assic. Prof. Dr. Kiril Shterev.

During the last decade, the development of micro-electromechanical systems (MEMS) increased dramatically. The main driver for this development appears the fundamental research (experimental and theoretical) that generate necessary new knowledge and ideas about the physical processes occurring in Micro-Electro-Mechanical Systems.
To adequately describe the non-equilibrium processes and phenomena in MEMS is necessary to use new approaches, methods and knowledge of the kinetic theory, statistical and quantum physics to investigate the relationship and transition between continua and discrete environments, as well as emerging phenomena in intermediate states.
In the project proposal we plan to study the phenomena and processes in micro/nano gas flows which, in our opinion, have not been examined in substance or tested, but not sufficiently so far. These are rarefied gas flows of gas mixtures and chemically reacting gases in micro/nano-sized areas. Particular attention will be paid to the Fluid-Structure interaction in microfluidics. The team of this project for the first time presented a mathematical model and obtained the results of the fully coupled interaction of elastic beam and rarefied gas simulated by the method DSMC (Direct Monte Carlo Statistical Method). In this project it is planned to investigate the influence between the parameters of the rarefied gas flow and frequencies of vibration of small elastic beams.
The following key objectives are identified:
1. To explore and gain new knowledge of the rarefied gas flows of monoatomic gas, gas mixtures, and chemically reacting gases in microchannels with simple and complex geometry.
2. To develop mathematical models and examine fluid-structure interaction of rarefied gas and elastic beam where we will consider the impact of discrete fluctuations and micro impulses caused by the molecular structure of the gas on the characteristic frequencies of vibration of small elastic beams of various shapes (micro-consoles and plates - cantilevers in micro system).
Molecular (DSMC) and continuum models will be used for the modeling of the gas flows. The elastic element will be modelled as an elastic beam or an elastic rectangular plate. In both cases, geometrically nonlinear versions of beam or plate theory will be used (Euler-Bernoulli or Timoshenko in the case of beam and Kirchhoff or Reissner-Mindlin in the case of plate). The numerical implementation of project tasks requires large computational resources and will therefore be implemented and computed on clusters with CPU and GPU processors.
The results will be used in the design and optimization of micro-electro-mechanical systems as well as from our European partners




Public Archive:

A pressure based, iterative finite volume method is developed for calculation of compressible, viscous, heat conductive gas flows at all speeds. For more information click here.


Dynamic Meniscus Profile Method for Detertmination of the Dynamic Contact Angle in the Wilhelmy Plate Geometry is developed and availablel for use here.


Modified date:08-05-2024