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Computational Mathematics

Institute of Mathematics and Informatics

Bulgarian Academy of Sciences

The former Laboratory of Numerical Methods (presently Department of Computational Mathematics) was created in 1987 by some members of the Department of Mathematical Modeling. The aim was to further strengthen the direction of numerical methods for PDEs as the most important component in mathematical modeling for industrial applications. Prof. Raytcho Lazarov was appointed for its first Head. Further heads were Prof. Michail Kaschiev (1991-2004) and Assoc. Prof. Natalia Kolkovska.

- A number of applied projects were developed in collaboration with
- Institute of Metallurgy and Metal Sciences;
- Institute of Microelectronics;
- Technical University of Sofia;
- Joint Institute for Nuclear Research (Dubna, Russia);
- Institute of Mathematical Modeling of Russian Academy of Sciences;
- Texas A&M University;
- Darmstadt University of Technology (Germany);
- Institute of Science and Technology of University of Manchester;
- Engineering Department of Queen Mary College (University of London).

Department of Computational Mathematics

15PhD and a lot of MSc. students wrote their theses under the supervision of department members.

- The department was involved in the organization of 6International conferences on Numerical Methods and Applications in Sofia and Borovets.
- Members of the department participated in Scientific, Program or Organizing committees of more than 40 other conferences.
- In the period 1988-2003 about 15 scientists from the departmentgot regular positions or PhD/PostDoc fellowships at:
- Institute for Parallel Processing;
- Sofia University;
- Texas A&M University (USA);
- University of Texas in Austin (USA);
- University of California, Los Angeles (USA);
- Penn State University (USA);
- Technical University of Eindhoven (The Netherlands);
- University of Nijmegen (The Netherlands);
- Fraunhofer Institut Techno- und Wirtschaftsmathematik in Kaiserslautern (Germany);
- etc.

Department of Computational Mathematics

Research report for the period 2004-2008

- Scientific staff:
- Prof. DSc. Raytcho Lazarov – retired in 2008, currently at Texas A&M University;
- Prof. DSc. Mihail Kaschiev - deceased 2007;
- Assoc. Prof. Dr. Natalia Kolkovska;
- Assoc. Prof. Dr. Oleg Iliev – currently at FhG ITWM, Kaiserslautern;
- Dr. Ivan Bazhlekov;
- Dr. Milena Dimova;
- Dr. Ivan Georgiev – PostDoc at RICAM, Linz;
- Dr. Stanislava Stoilova;
- Dr. Daniela Vasileva;
- Dr. Ludmil Zikatanov – only in 2004, currently at Penn State University;
- Polya Dobreva – part-time in 2004-2005, currently at Institute of Mechanics.

Department of Computational Mathematics

Theoretical investigation and practical realization of numerical methods and algorithms for PDEs, systems of PDEs and integral equations- construction and analysis for

- finite element, finite difference and finite volume approximations;
- multilevel and domain decomposition methods;
- a posteriori error control and adaptive grid refinement.
- Mathematical modeling and numerical simulationof physical, fluid dynamic, chemical, electrostatic, thermodynamic, biomechanical problems, etc.
- formation and ionization of hydrogen like atoms and ions in magnetic fields;
- magnetosheath-magnetosphere 3D system;
- filtration processes, non-Newtonian and multiphase flows in plain and porous media;
- drop dynamics (breakup and coalescence) in complex non-Newtonian and viscoelastic multiphase flows in presence of surfactants;
- elasticity problems;
- glass crystallization processes;
- effects of electrostatic surface forces;
- formation of structures in nonlinear heat-transfer media.

Department of Computational Mathematics

- joint work and participation in scientific projects of Institute for Parallel Processing, BAS;
- joint work with Institute of Mechanics and Institute of Physical Chemistry, BAS;
- at national level
- participation in a scientific project of Technical University, Gabrovo;
- joint work with Faculty of Mathematics and Informatics, Faculty of Physics and Faculty of Chemistry, Sofia University;
- in Europe and world wide
- scientific projects and joint work with JINR, Dubna, Russia;
- participation in scientific projects and joint work with Czech (Institute of Geonics), Polish, Hungarian, Austrian (RICAM) Acad. Sci.;
- participation in scientific projects of EC FP5, EC FP6;
- joint work with FhG-ITWM, Kaiserslautern(Germany), CWI, Amsterdam(The Netherlands), Technical University of Eindhoven(The Netherlands),Texas A&M University(USA).

Department of Computational Mathematics

number of scientific papers published in journals abroad: 67 (listed in SCI expanded: 54);

- number of scientific papers published in Bulgarian journals: 3;
- number of scientific papers published in conference proceedings: 18;
- number of scientific reports (published by FhG-ITWM, RICAM, TAMU, JINR, etc.): 18;

- number of citations appeared in the period 2004-2008: more than 500, most cited: Prof. R. Lazarov: more than 300citations.

Department of Computational Mathematics

participation in teaching at Sofia University and South-West University of Blagoevgrad:

- lectures on Mathematics, Calculus, Numerical methods;
- seminars on Applied mathematics, Mathematical modeling, Numerical methods.
- post-graduate trainingat BAS: Theory of approximations.
- co-organization (with FMI, SU and IPP, BAS) of

Sixth International Conference on Numerical Methods and Applications,

August 20-24, 2006, Borovets, Bulgaria:

- 128 participants, 70 from abroad;
- 116 lecturers, 68 from abroad.
- organization of regular seminars on Computational mathematics.
- participation in editorial boards:
- Computational Methods in Applied Mathematics (Prof. R. Lazarov, Assoc.Prof. O. Iliev);
- East-West Journal on Numerical Mathematics (Prof. R. Lazarov);
- Numerical Methods for Partial Differential Equations (Prof. R. Lazarov);
- Mathematical Modelling and Analysis (Assoc.Prof. O. Iliev).

Department of Computational Mathematics

participation in councils, commissions and other expert bodies of external for BAS institutions:

- Council on Informatics and Mathematical modeling, National Supreme Commission for Attestation, 2004-2007 (Prof. M. Kaschiev);
- Council of the Faculty of Mathematics and Natural Sciences, South-West University, Blagoevgrad, 2002-2006 (Prof. M. Kaschiev);
- Expert Commission for Scientific Cooperation with JINR-Dubna, 2000-2007 (Prof. M. Kaschiev);
- Working Group WG 2.5, International Federation for Information Processing, 1988-present (Prof. R. Lazarov);
- International Society for Porous Media, 2008-present (Assoc.Prof. O. Iliev, president-elect 2009-).
- study/research visits:
- Dr. Ivan Bazhlekov, TUE, Eindhoven, The Netherlands, 01.01.2001-31.12.2004;
- Dr. Daniela Vasileva, CWI, Amsterdam, The Netherlands, 01.09.2003-31.05.2004;
- Prof. Michail Kaschiev, JINR, Dubna, Russia, 03.06.-27.06.04, 12.06.-01.07.2005;
- Dr. Stanislava Stoilova, JINR, Dubna, Russia, 16.06.-27.06.2004;
- Dr. Ivan Georgiev, Institute of Geonics, Ostrava, Czech Respublic, 25.06.-09.07.2005;
- Dr. Ivan Georgiev, RICAM, Linz, Austria, 03.10.-16.12.2005, 01.09.2008-31.08.2009.

Department of Computational Mathematics

Scientific awards/recognition

- Prof. R. Lazarov- Doctor Honoris Causa of Sofia University, 2006;
- Dr.I. Georgiev - Award of Bulgarian Academy of Sciences for Young Scientists, 2006;
- Prof. R. Lazarov- Medal of Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 2008;
- Prof. R. Lazarov - Pichoridis Distinguished Lectureship, University of Crete, Greece, June 2008;
- Prof. R. Lazarov - Erasmus Mundus Visiting Scholar Award, University of Kaiserslautern, July 2008 - June 2011.

Department of Computational Mathematics

A sample of scientific results

Department of Computational Mathematics

Prof. Mihail Kaschiev, Dr. Milena Dimova (with S. Vinitsky et al. (JINR, Dubna))

A new efficient numerical method for calculating the energy levels of low-lying exited states of a hydrogen atom in a strong magnetic field is developed. This method is based on the modern implementation of the Kantorovich approach to the parametric eigenvalue problems in spherical coordinates. The initial two-dimentional spectral problem for the Schrödinger equation is reduced to a one-dimentional spectral parametric problem for the angular variable and a finite set of ordinary second-order differential equations for the radial variable. The resulting systems are solved using high-order accuracy approximations of the finite element method.

The approach elaborated provides a useful tool for calculations of threshold phenomena in formation and ionization of (anti)hydrogen like atoms and ions in magnetic traps and channeling of ions in thin films.

Department of Computational Mathematics

The crystal growth in multi-component systems with crystal composition different from that of the ambient phaseis simulated. The mathematical problem is a special case of moving boundary problems. The boundary immobilization method is applied to solve numerically the diffusion equations in an unknown region. A variety of physical characteristics, as concentration profiles, the size of the growing crystal, are calculated for different physical parameters. Adequateinterpretation of the results is given.

Assoc.Prof. Natalia Kolkovska, Dr. Ivan Georgiev (with Prof. I. Avramov (Inst. Physical Chem.) and Prof. Chr. Russel (Fr.-Schiller Univ., Jena, Germany))

Time dependence of the size of the growing crystal.

Concentration profiles at different times

Department of Computational Mathematics

Numerical methods for solving second order elliptic problems with specific boundary condition, given by a sum of normal derivative and a second order elliptic operator in tangential variables are proposed and investigated. Optimal error estimates of the numerical methods in Sobolev spaces are proved. Similar theoretical results are established for elliptic problems with discontinuous coefficients and interface conditions of the same type.

Assoc.Prof. Natalia KolkovskaDepartment of Computational Mathematics

Assoc.Prof. Natalia Kolkovska, Dr. Daniela Vasileva (with Dr. R. Slavchov (Faculty of Chemistry, Sofia University))

An algorithm for numerical simulation of surface forces, acting on AFM (atomic force microscope) is developed. The mathematical model considers three phases (tip, water, dielectric) and two interface surfaces – tip-water and water-dielectric. On each interface the surface dielectric permittivities are modifying the conditions of the Gauss law. For this model a finite difference method in cylindrical coordinates on a non-uniform grid, aligned with both interfaces, is developed. The numerical experiments show that surface dielectric permittivities of tip-water and water-dielectric give a strong addition to the image force pulling the AFM tip toward the dielectric surface investigated.

Department of Computational Mathematics

Assoc.Prof. Oleg Iliev, Dr. Daniela Vasileva (with Dr. D. Stoyanov (FhG-ITWM) and Prof. W. Doerfler (Univ. Karlsruhe))

An adaptive refinement multigrid solver for numerical simulation of flow of non-Newtonian fluid in saturated porous media is developed. The mathematical model consists of the continuity equation and the generalized Darcy law. The numerical method is based on a finite volume discretization with mass conservation on the interfaces between the coarser and finer grids and second order accurate discretisation for the fluxes.

Results from numerical solution of various academic and practice-induced problems demonstrate that the adaptive local refinement approach allows to obtain the same accuracy as in the full grid case, but using significantly less memory and CPU time.

Department of Computational Mathematics

Assoc. Prof. Oleg Iliev, Dr. Daniela Vasileva

A local refinement algorithm for computer simulation of flow through oil filters is developed. The mathematical model is based on laminar incompressible Navier-Stokes equations for the flow in pure liquid zones and Brinkman extension to Darcy model for the flow in the porous zone. A finite volume method on cell-centered locally refined grids is used for the discretization and special attention is paid to the conservation of the mass on the interface between the coarse and the fine grid.

A variety of numerical experiments are performed and the results show that the solver could be successfully used for simulation of coupled flow in plain and porous media. The local refinement ensures a significant acceleration of the computations and saving of memory, which is very important in the case of 3D numerical simulations.

Department of Computational Mathematics

Dr. Ivan Bazhlekov (with Prof. H. Meijer and Dr. P. Anderson (TU Eindhoven))

A three-dimensional boundary integral method for deformable drops in viscous flows is developed. The method is based on a new nonsingular contour-integral representation. The contour integration overcomes the main difficulty with boundary-integral calculations: the singularities of the kernels. It also improves the accuracy of the calculations as well as the numerical stability.

Drop deformation and breakup in shear flow are shown in the figure. Topological transition at time t=53.6 is also shown. Simulations of large deformation and topological transition are possible due to the developed semi-automatic adaptive mesh refinement.

Department of Computational Mathematics

Dr. Ivan Bazhlekov (with Prof. A. Chesters; Prof. F. van de Vosse; Prof. H. Meijer and Dr. P. Anderson (TU Eindhoven))

Mathematical modeland correspondingnumerical methods for simulation of 2D and 3D drop coalescence in complex non-Newtonian and viscoelastic multiphase flowsare developed. In 2D case the problem is solved numerically by means of a finite difference method for the equations in the continuous phase and a boundary integral method or finite-element method in the drops. In 3D case the numerical method is based on the nonsingular boundary integral method. In this class of problems an important feature is the presence of a thin film of thickness 3-5 orders of magnitude smaller than the drop size. In order to improve the resolution in the film zone a higher-order interface approximation is introduced. Successfully are simulated drop-to-drop interaction (shown in the figure in the case of external compressional flow) as well as foam dynamics.

Department of Computational Mathematics

Numerical model for computer simulation of the effect of insoluble surfactants on drop dynamics is developed. The mathematical model consists of: Stokes equation in the fluid phases; stress-balance boundary condition on the interfaces; convection-diffusion equation on the evolving interface governs the distribution of the surfactant concentration, which in turn determines the interfacial tension. The numerical method is a combination of a three-dimensional boundary-integral method for the hydrodynamics and a finite-volume method to solve the coupled fluid dynamics and surfactant transport problem.

The model is applied successfully for 3D simulation of drop deformation and breakup (left figure), drop-to-drop interaction (right figure) and foam dynamics. The color bar represents the surfactant concentration.

Dr. Ivan Bazhlekov (with Prof. H. Meijer and Dr. P. Anderson (TU Eindhoven))Department of Computational Mathematics

drop deformation and breakup (movie)

Department of Computational Mathematics

drop-to-drop interaction (movie)

Department of Computational Mathematics

foam dynamics (8 drops in a larger drop, movie)

Department of Computational Mathematics

Self-similar solutions of a nonlinear heat-conduction equation with a volume source under blow-up conditions are considered. The self-similar problem is a BVP for a nonlinear elliptic equation with nonunique solution. An efficient numerical method for investigation of the eigenfunctions of the self-similar problem is developed. This method is based on the Continuous Analog of Newton Method and the Method of Finite Elements. The completely new types of egenfunctions depending on the values of the parameters of the medium and the choice of the initial approximations are obtained – two-dimensional eigenfunctions with “zero” regions and “spiral” egenfunctions.

Dr. Milena Dimova (with Prof. S. Dimova (Faculty of Math.&Inf., Sofia University))Department of Computational Mathematics

Dr. Ivan Georgiev (with Prof. S. Margenov (IPP-BAS))

An algorithm for the numerical solution of the Lamé equations of elasticity in the case of mesh anisotropy and coefficient jumps is developed. A preconditioned conjugate gradient (PCG) method is applied for iterative solution of the linear algebraic system obtained after non-conforming finite element discretization. Displacement decomposition of the stiffness matrix is used as a first step of the algorithm. At the second step, modified incomplete factorization MIC(0) is applied to a proper auxiliary M-matrix to get an approximate factorization of the obtained block-diagonal matrix.

Computer simulation of a pile foundation system in a multi-layer soil media: vertical strains; vertical stresses, vertical displacements

Department of Computational Mathematics

Dr. Ivan Georgiev (with Dr. J. Kraus (RICAM, AAS) and Prof. S. Margenov (IPP-BAS))

Algebraic multilevel iteration methods for three-dimensional elliptic problems discretized by a family of Rannacher Turek non-conforming finite elements are developed. The derived estimates of the constant in the strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality allow the efficient multilevel extension of the related two-level preconditioners. Representative numerical tests well illustrate the optimal complexity of the resulting iterative solver, also for the case of non-smooth coefficients.

128^3 voxels

256^3 voxels

μFEM simulation - microstructure analysis of bones

Department of Computational Mathematics

A multigrid adaptive mesh-refinement algorithm is developed for the solution of convection-diffusion problems. The method is based on discontinuous Galerkin (Baumann-Oden DG) discretization. The numerical experiments show that the algorithm may be successfully used for resolution of boundary and interior layers.

Further an adaptive semirefinement technique is developed and the comparison with the adaptive refinement algorithm shows that significantly less computer resources may be used for layers, almost parallel to the x or y axis.

Dr. Daniela Vasileva (with Prof. P.W.Hemker and A. Kuut (CWI, Amsterdam))Department of Computational Mathematics

Additional information about activities of the department's members (CVs, list of publications, etc.) may be found on the web-site of the Institute:

http://www.math.bas.bg/new/site/?call=USE~structure;&action=single&id=10&lang=en

Department of Computational Mathematics

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