Department of Computational Mathematics2 History The former Laboratory of Numerical Methods...

Department of Computational MathematicsDepartment of Computational Mathematics 22

HistoryHistory

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).


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

The department was involved in the organization of 6 International 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 department got 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.


Research report for the period 2004-2008Research report for the period 2004-2008

Scientific staff:Scientific staff:

1. Prof. DSc. Raytcho Lazarov – retired in 2008, currently at Texas A&M University;

2. Prof. DSc. Mihail Kaschiev - deceased 2007;

3. Assoc. Prof. Dr. Natalia Kolkovska;

4. Assoc. Prof. Dr. Oleg Iliev – currently at FhG ITWM, Kaiserslautern;

5. Dr. Ivan Bazhlekov;

6. Dr. Milena Dimova;

7. Dr. Ivan Georgiev – PostDoc at RICAM, Linz;

8. Dr. Stanislava Stoilova;

9. Dr. Daniela Vasileva;

10. Dr. Ludmil Zikatanov – only in 2004, currently at Penn State University;

11. Polya Dobreva – part-time in 2004-2005, currently at Institute of Mechanics.


Main fields of researchMain fields of research

Theoretical investigation and practical realization of numerical methods and algorithms Theoretical investigation and practical realization of numerical methods and algorithms

for PDEs, systems of PDEs and integral equationsfor 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 simulationMathematical modeling and numerical simulation of 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.


Formal andFormal and informal co-operation and relations informal co-operation and relations

within the Academy

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).


Most important dataMost important data

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 300 citations.


participation in teachingteaching 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 trainingpost-graduate training at BAS: Theory of approximations.

co-organizationco-organization (with FMI, SU and IPP, BAS) of

Sixth International Conference on Numerical Methods and ApplicationsSixth International Conference on Numerical Methods and Applications,

August 20-24, 2006, Borovets, Bulgaria:

128 participants, 70 from abroad;

116 lecturers, 68 from abroad.

organizationorganization of regular seminars on Computational mathematics.

participation in editorial boardseditorial 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).


participation in councils, commissions and other expert bodiescouncils, 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 visitsstudy/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.


Scientific awards/recognitionScientific 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.


A sample of scientific resultsA sample of scientific results


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

A new efficient numerical method for numerical method for calculating the energy levels of low-calculating the energy levels of low-lying exited states of a hydrogen atom lying exited states of a hydrogen atom in a strong magnetic fieldin 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.


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

The crystal growth in multi-component systems with crystal composition different from that of the crystal growth in multi-component systems with crystal composition different from that of the

ambient phaseambient phase is 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. Adequate interpretation of

the results is given.

Time dependence of the size of the growing crystal. Concentration profiles at different times


Assoc.Prof. Natalia Kolkovska

Numerical methods for solving second order elliptic problems with specific boundary conditionNumerical 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 variablessum 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 Kolkovska, Dr. Daniela Vasileva (with Dr. R. Slavchov (Faculty of Chemistry, Sofia University))

An algorithm for numerical simulation of surface forces, acting on AFMalgorithm 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.


An adaptive refinement multigrid solver for adaptive refinement multigrid solver for

numerical simulation of flow of non-numerical simulation of flow of non-

Newtonian fluid in saturated porous mediaNewtonian 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.

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


Assoc. Prof. Oleg Iliev, Dr. Daniela Vasileva

A local refinement algorithm for computer simulation of flow through oil filterslocal 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.



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

A three-dimensional boundary three-dimensional boundary integral method for deformable integral method for deformable drops in viscous flowsdrops 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.


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

Mathematical modelMathematical model and corresponding

numerical methods for simulation of 2D and 3D numerical methods for simulation of 2D and 3D

drop coalescence drop coalescence in complex non-Newtonian and in complex non-Newtonian and

viscoelastic multiphase flowsviscoelastic multiphase flows are 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.


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

Numerical model for computer simulation of the effect of insoluble surfactants on drop dynamicsNumerical 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.


drop deformation and breakup (movie)drop deformation and breakup (movie)


drop-to-drop interaction (movie)drop-to-drop interaction (movie)


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


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

Self-similar solutions of a nonlinear heat-conduction equation Self-similar solutions of a nonlinear heat-conduction equation with a volume source under blow-up conditionswith 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.


An algorithm for the numerical solution of the Lamé equations of elasticity in the case of mesh algorithm for the numerical solution of the Lamé equations of elasticity in the case of mesh

anisotropy and coefficient jumpsanisotropy 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

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


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

Algebraic multilevel iteration methods for three-dimensional elliptic problemsAlgebraic multilevel iteration methods for three-dimensional elliptic problems discretized by a

family of Rannacher Turek non-conforming finite elementsRannacher 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


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

A multimultigridgrid adaptive mesh-refinement algorithm adaptive mesh-refinement algorithm is developed for the solution of convection-convection-

diffusion problemsdiffusion 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.


Additional information about activities of the

department's members (CVs, list of publications, CVs, list of publications,

etc.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

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






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Department of Computational Mathematics2 History The former Laboratory of Numerical Methods...

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