Кафедра "Прикладна математика"

Постійне посилання колекціїhttps://repository.kpi.kharkov.ua/handle/KhPI-Press/4610

Офіційний сайт кафедри http://web.kpi.kharkov.ua/apm

Від 1981 року кафедра має назву "Прикладна математика", первісна назва – кафедра теоретичної й математичної фізики.

Кафедра теоретичної й математичної фізики була заснована в 1947 році. Організатором і першим завідувачем цієї кафедри був відомий вчений-математик, фахівець із конструктивної теорії функцій, член-кореспондент Української Академії наук Наум Ілліч Ахієзер. У 1970 році кафедра цілком чітко взяла курс на дослідження прикладних питань математики, і ще тоді припускалося перейменування кафедри в кафедру "Прикладна математика".

Кафедра входить до складу Навчально-наукового інституту комп'ютерного моделювання, прикладної фізики та математики Національного технічного університету "Харківський політехнічний інститут".

У складі науково-педагогічного колективу кафедри працюють: 1 доктор технічних наук, 1 доктор фізико-математичних наук, 5 кандидатів технічних наук, 4 кандидата фізико-математичних наук; 2 співробітника мають звання професора, 8 – доцента, 1 – старшого наукового співробітника.

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Algebraization in stability problem for stationary waves of the Klein-Gordon equation
(Харківський національний університет імені В. Н. Каразіна, 2019) Goloskubova, Nataliia; Mikhlin, Yuri V.
Nonlinear traveling waves of the Klein-Gordon equation with cubic nonlinearity are considered. These waves are described by the nonlinear ordinary differential equation of the second order having the energy integral. Linearized equation for variation obtained for such waves is transformed to the ordinary one using separation of variables. Then so-called algebraization by Ince is used. Namely, a new independent variable associated with the solution under consideration is introduced to the equation in variations. Integral of energy for the stationary waves is used in this transformation. An advantage of this approach is that an analysis of the stability problem does no need to use the specific form of the solution under consideration. As a result of the algebraization, the equation in variations with variable in time coefficients is transformed to equation with singular points. Indices of the singularities are found. Necessary conditions of the waves stability are obtained. Solutions of the variational equation, corresponding to boundaries of the stability/instability regions in the system parameter space, are constructed in power series by the new independent variable. Infinite recurrent systems of linear homogeneous algebraic equations to determine coefficients of the series can be written. Non-trivial solutions of these systems can be obtained if their determinants are equal to zero. These determinants are calculated up to the fifth order inclusively, then relations connecting the system parameters and corresponding to boundaries of the stability/ instability regions in the system parameter place are obtained. Namely, the relation between parameters of anharmonicity and energy of the waves are constructed. Analytical results are illustrated by numerical simulation by using the Runge-Kutta procedure for some chosen parameters of the system. A correspondence of the numerical and analytical results is observed.
Analysis of free vibration of porous power-law and sigmoid functionally graded sandwich plates by the R-functions method
(Shahid Chamran University of Ahvaz, 2023) Kurpa, Lidiya; Shmatko, Tetyana; Awrejcewicz, Jan; Timchenko, Galina; Morachkovska, Iryna
Investigation of free vibration of porous power and sigmoid-law sandwich functionally graded (FG) plates with different boundary conditions is presented in this paper. The FG sandwich plate includes three layers. The face layers are fabricated of functionally graded material (FGM) and middle layer (core) is isotropic (ceramic). Imperfect sigmoid FG sandwich plates with even and linear-uneven porosities and nonporous core layer are studied. Developed approach has been realized in the framework of a refined theory of the first-order shear deformation theory (FSDT) using variational methods and the R-functions theory. The analytical expressions are obtained for calculating the elastic characteristics with the assumption that the values of Poisson's ratio are the same for constituent FGM materials. For rectangular plates, the obtained results are compared with known results and a good agreement is obtained. Vibration analysis of a complex-shaped porous sandwich plate made of FGM has been performed. The effect of various parameters on the dynamic behavior of the plate, such as the type and values of porosity coefficients, power index, lay-up scheme, types of FGM, has been studied.
Analysis of Geometrically Nonlinear Vibrations of Functionally Graded Shallow Shells of a Complex Shape
(Marcílio Alves, 2017) Awrejcewicz, Jan; Kurpa, Lidiya; Shmatko, T.
Geometrically nonlinear vibrations of functionally graded shallow shells of complex planform are studied. The paper deals with a power-law distribution of the volume fraction of ceramics and metal through the thickness. The analysis is performed with the use of the R-functions theory and variational Ritz method. Moreover, the Bubnov-Galerkin and the Runge-Kutta methods are employed. A novel approach of discretization of the equation of motion with respect to time is proposed. According to the developed approach, the eigenfunctions of the linear vibration problem and some auxiliary functions are appropriately matched to fit unknown functions of the input nonlinear problem. Application of the R-functions theory on every step has allowed the extension of the proposed approach to study shallow shells with an arbitrary shape and different kinds of boundary conditions. Numerical realization of the proposed method is performed only for one-mode approximation with respect to time. Simultaneously, the developed method is validated by investigating test problems for shallow shells with rectangular and elliptical planforms, and then applied to new kinds of dynamic problems for shallow shells having complex planforms.
Analysis of structural performance of aluminium sandwich plates with foam-filled hexagonal cores
(2007) Sadowski, Tomasz; Burlayenko, V. N.
Applicatin of R-functions Theory to Nonlinear Vibration Problems of Laminated Shallow Shells with Cutouts
(NTU "KhPI", 2016) Kurpa, Lidiya; Timchenko, Galina; Osetrov, Andrey
In present work an effective method to research geometrically nonlinear free vibrations of elements of thinwalled constructions that can be modeled as laminated shallow shells with complex planform is applied. The proposed method is based on joint use of R-functions theory, variational methods and Bubnov-Galerkin procedure. It allows reducing an initial nonlinear system of motion equations of a shallow shell to the Cauchy problem. The mathematical formulation of the problem is performed in a framework of the refined first-order theory. The appropriate software is created within POLE-RL program system for polynomial results and using C+ + programs for splines. New problems of linear and nonlinear vibrations of laminated shallow shells with cutouts are solved. To confirm reliability of the obtained results their comparison with the ones obtained using spline-approximation and known in literature is provided. Effect o f boundary condition on cutout is studied.
Application of Homogenization FEM Analysis to Aluminum Honeycomb Core Filled with Polymer Foams
(2010) Burlayenko, Vyacheslav. N.; Sadowski, Tomasz
The effect of polyvinyl chloride (PVC) foam filler on elastic properties of a regular hexagonal aluminum honeycomb core is studied. The unit cell strain energy homogenization approach based on the finite element method (FEM) within ABAQUS code is applied for prediction of effective material constants of the foam-filled honeycomb core. The developed FE model is then used to observe a three-dimensional stress state over the hexagonal unit cell and, thereby, to assess the influence of the foam-filling on the distribution of the local interfacial stresses.
Application of R-Functions Theory to Study Parametric Vibrations and Dynamical Stability of Laminated Plates
(Точка, 2013) Kurpa, Lidiya; Mazur, Olga; Tsukanov, Igor
The problem of nonlinear parametric vibrations and stability analysis of the symmetric laminated plates is considered. The proposed method is based on multimode approximation of displacements and solving series auxiliary linear tasks. The main feature of the work is the application of the R-functions theory, which allows investigating parametric vibrations of plates with complex shape and different boundary conditions.
Application of the R-Functions Method for Nonlinear Bending of Orthotropic Shallow Shells on an Elastic Foundation
(NTU "KhPI", 2016) Kurpa, Lidiya; Lyubitska, Kateryna
Geometrically nonlinear behavior of orthotropic shallow shells subjected to the transverse load and resting on Winkler’s foundation is investigated. On base of the R-function theory and variational methods problem's solution for shells with complex plan form is proposed. The algorithm to finding upper and lower critical loads is developed. The stress-strain state of shallow shells with the complex planform is investigated including different boundary conditions, properties of material and elastic foundation.
Application of the R-Functions Theory to Problems of Nonlinear Dynamics of Laminated Composite Shallow Shells and Plates: Review
(NTU "KhPI", 2016) Kurpa, Lidiya
A review of studies performed using the R-functions theory to solve problems of nonlinear dynamics of plates and shallow shells is presented. The systematization of results and studies for the problems of free and parametric vibrations and for problems of static and dynamic stability is fulfilled. Expansion of the developed original method of discretization for nonlinear movement equations on new classes of nonlinear problems is shown. These problems include researches of vibrations of antisymmetric laminated cylindrical and spherical panels; laminated composite shallow shells with variable thicknesss of the layers; functionally graded (FG) shallow shells and others. The basic issues that arise when using RFM are described. The future prospects of using the theory of R-functions for solving problems of nonlinear dynamics of plates and shallow shells with complex form are formulated. First of all this is an algorithms development and creation of the associated software to apply multi-modes approximations; improvement of approximation tools for nonlinear problems; investigation of the cracked functionally graded shallow shells; FG panels under thermal environments; parametric vibrations, static and dynamical stability of the multilayered and FG plates and shells.
Application of the Variational-Structural Method to Investigate the Elasto-Plastic Bending of Thin Shells and Plates
(NTU "KhPI", 2016) Morachkovska, Irina; Timchenko, Galina; Lyubitska, Kateryna
The effective method basing on theory of R-functions and variational structural method is developedfor solving non-linear boundary problems. Elastic-plastic bending of thin shallow shells is considered. The problems are reduced to finding stationary points of suggested mixed variational functionals according to the initial linearization due to usage of subsequent loading and Newton-Kantorovich jointly with method of varying elastic parameters. The method is used for automatic calculations in «POLE» programming system for investigations of shell structural elements. The numerical justification of the method is given. New laws of nonlinear deformation of shallow shells and plates with complex shapes in plane are established.
Complex numbers and their application to representing curves and domains on the complex plane
(National Technical University "Kharkiv Polytechnic Institute", 2024) Dimitrova, S. D.; Girya, N. P.; Burlayenko, V. M.; Naboka, O. O.
The educational-methodological textbook focuses on an important topic in mathematical analysis – the calculus of complex functions with a single variable. This textbook extensively explores the fundamental theoretical concepts and offers solutions to standard problems. It incorporates exercises for study and a series of tasks for individual student work. Tailored for students and lecturers in higher technical educational institutions.
Creep damage anisotropy of thinwalled elements structures
(IPPT PAN, Poland, 2004) Morachkovsky, O. K.; Burlayenko, V. N.
The paper is devoted to the development and theoretical justification of an anisotropy creep damage material model. For the description of initial anisotropy and damage-induced anisotropy the second-order damage tensor has been used. The numerical method of anisotropy creep damage lifetime prediction in thin-walled elements of structures was elaborated on the basis of the proposed model and FE scheme. The results of anisotropy creep damage analysis in models of plates were discussed.
Dynamical instability of laminated plates with external cutout
(Elsevier Inc., 2016) Awrejcewicz, Jan; Kurpa, Lidiya; Mazur, Olga
A method to study dynamical instability and non-linear parametric vibrations of symmetrically laminated plates of complex shapes and having different cutouts is proposed. The first-order shear deformation theory (FSDT) and the classical plate theory (CPT) are used to formulate a mathematical statement of the given problem. The presence of cutoutses sentially complicates the solution of buckling problem, since the stress field is non-uniform. At first, a plane stress analysis is carried out using the variational Ritz method and the R-functions theory. The obtained results are applied to investigate buckling and parametric vibrations of laminated plates. The developed method uses the R-functions theory, and it may be directly employed to study laminated plates of arbitrary forms and different boundary conditions. Besides, the proposed method is numerical-analytical, what greatly facilitates a solution of similar-like non-linear problems. In order to show the advantage of the developed approach, instability zones and response curves for the layered cross- and angle-ply plates with external cutouts are constructed and discussed.
Dynamical stability and parametrical vibrations of the laminated plates with complex shape
(Marcílio Alves, 2013) Kurpa, Lidiya; Mazur, Olga; Tkachenko, Victoria
The problem of nonlinear vibrations and stability analysis for the symmetric laminated plates with complex shape, loaded by static or periodic load in-plane is considered. In general case research of stability and parametric vibrations is connected with many mathematical difficulties. For this reason we propose approach based on application of R-functions theory and varia-tional methods (RFM).The developed method takes into ac-count pre-buckle stress state of the plate. The proposed ap-proach is demonstrated on testing problems and applied to laminated plates with cutouts. The effects of geometrical pa-rameters, load, boundary conditions on stability regions and nonlinear vibrations are investigated.
Dynamics and fracture of impacted sandwich composites under time varying loads: Numerical modelling and simulations
(CongressLine Ltd., Hungary, 2017) Burlayenko, Vyacheslav. N.; Sadowski, Tomasz; Dimitrova, Svetlana
In this study, the dynamics and fracture of sandwich plates containing a pre-existing skin-to-core interfacial damage and subinterfacial core damage induced by an incident impact is examined. The dynamic response of sandwich plates with debonding that is allowed to be growing with a time, is simulated by using the finite element method within the ABAQUS code. The forced vibration analysis of impact-damaged sandwich plates is carried out accounting for contact and friction conditions within the debonded region in the simulations with ABAQUS. The damage mechanics approach implemented into ABAQUS via cohesive elements is used for modelling the debonding propagation under impulsive and harmonic loading. The influence of the skin-to-core debonding growth on the global nonlinear dynamics and strength of the sandwich plates is studied in detail.
Dynamics of sandwich plates weakened by single/multiple debonding
(University of Zielona Góra Press, 2009) Burlayenko, Vyacheslav. N.; Sadowski, Tomasz
The dynamic behavior of partially damaged at the skin/core interface sandwich plates with flexible honeycomb and polyvinyl chloride (PVC) foam cores are studied. The commercial finite element code ABAQUS is used to calculate natural frequencies and mode shapes of the sandwich plates with the debond zone. The effects of debonding size, debonding location and number of debonding on the modal parameters of damaged sandwich plates with various boundary conditions are investigated. The results of dynamic analysis illustrated that can be useful for analyzing practical problems related to the non-destructive damage detection of delaminated sandwich plates.
Elements of linear algebra and analytic geometry
(ФОП Панов А. М., 2020) Rudnyeva, G. V.
The book contains theoretical and practical material in linear algebra and analytic geometry in English. Theoretical part presents the proofs of the basic theorems and statements and the derivations of the formulas necessary to solve practical problems. Practical part of the book consists of practical tasks for each topic and variants of individual tasks. It is intended for students of technical universities studying higher mathematics in English, foreign students and teachers of higher mathematics.
FE modeling of delamination growth in interlaminar fracture specimens
(Politechnika Lubelska, 2008) Burlayenko, V. N.; Sadowski, Tomasz
Interlaminar fracture specimens like Double Cantilever Beam (DCB), End Notched Flexural (ENF), Single Leg Bending (SLB) etc. are widely used for studying the interlaminar toughness of composite laminates. The aim of this paper is to analysis delamination specimens within the framework of a meso-level damage modeling of composite laminates. In this case interlaminar interface is assumed as a damageable homogeneous layer between adjacent layers of the specimen bulk material. The degradation of the interlaminar connection can be taken into account by means either of an appropriate damage initiation criterion and damage evolution law or using fracture mechanics approach. Onset and growth of the delamination pre-existing crack in the fracture specimens are simulated by using both modeling possibility within commercial finite element code ABAQUSᵀᴹ. Comparisons between numerical predictions of used different finite element models as well as available experimental data have been performed.
FE modeling of dynamics of impact damaged sandwich plates with intermittent CONTACT in detached fragments
(Wydawnictwo-Drukarnia Liber Duo s.c., 2010) Burlayenko, V. N.; Sadowski, Tomasz
FE-analysis of dynamic creep-damage in thin-walled structures
(Springer-Verlag, 2002) Morachkovsky, Oleg; Breslavsky, Dmitry; Burlayenko, V. N.
The models for description of creep-damage behaviour in materials and thin shallow shells and plates deforming in conditions of joint action of static and fast cyclic load are given. The properties of the proposed material model were established by comparison of experimental and numerical data. The method for numerical simulation by in-house code of a dynamic creep and long-term strength of shallow shells and plates is created on the basis of the FEM. New laws of dynamic creep influence on stress-strain state, shaping and fracture of thin-walled elements of structures had been established by numerical calculations. With the purpose of verification of the created method of dynamic creep numerical simulation of rectangular plates made of an aluminium alloy were carried in order to verify the method of calculation.