Search results for: Differential geometry

Authors: Hadi Farhadian, Homayoon Katibeh

Abstract:

In this paper, groundwater seepage into Amirkabir tunnel has been estimated using analytical and numerical methods for 14 different sections of the tunnel. Site Groundwater Rating (SGR) method also has been performed for qualitative and quantitative classification of the tunnel sections. The obtained results of above mentioned methods were compared together. The study shows reasonable accordance with results of the all methods unless for two sections of tunnel. In these two sections there are some significant discrepancies between numerical and analytical results mainly originated from model geometry and high overburden. SGR and the analytical and numerical calculations, confirm high concentration of seepage inflow in fault zones. Maximum seepage flow into tunnel has been estimated 0.425 lit/sec/m using analytical method and 0.628 lit/sec/m using numerical method occured in crashed zone. Based on SGR method, six sections of 14 sections in Amirkabir tunnel axis are found to be in "No Risk" class that is supported by the analytical and numerical seepage value of less than 0.04 lit/sec/m.

Keywords: Water Seepage, Amirkabir Tunnel, Analytical Method, DEM, SGR.

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Authors: M. Bagheri, P. Mottaghizadeh

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Reliable information about tool temperature distribution is of central importance in metal cutting. In this study, tool-chip interface temperature was determined in cutting of ST37 steel workpiece by applying HSS as the cutting tool in dry turning. Two different approaches were implemented for temperature measuring: an embedded thermocouple (RTD) in to the cutting tool and infrared (IR) camera. Comparisons are made between experimental data and results of MSC.SuperForm and FLUENT software. An investigation of heat generation in cutting tool was performed by varying cutting parameters at the stable cutting tool geometry and results were saved in a computer; then the diagrams of tool temperature vs. various cutting parameters were obtained. The experimental results reveal that the main factors of the increasing cutting temperature are cutting speed (V ), feed rate ( S ) and depth of cut ( h ), respectively. It was also determined that simultaneously change in cutting speed and feed rate has the maximum effect on increasing cutting temperature.

Keywords: Cutting parameters, Finite element modeling, Temperature measurement, Tool-chip interface temperature.

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Authors: D.Katherasan, Madana Sashikant, S.Sandeep Bhat, P.Sathiya

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Bead-on-plate welds were carried out on AISI 316L (N) austenitic stainless steel (ASS) using flux cored arc welding (FCAW) process. The bead on plates weld was conducted as per L25 orthogonal array. In this paper, the weld bead geometry such as depth of penetration (DOP), bead width (BW) and weld reinforcement (R) of AISI 316L (N) ASS are investigated. Taguchi approach is used as statistical design of experiment (DOE) technique for optimizing the selected welding input parameters. Grey relational analysis and desirability approach are applied to optimize the input parameters considering multiple output variables simultaneously. Confirmation experiment has also been conducted to validate the optimized parameters.

Keywords: bead-on-plate welding, bead profiles, desirability approach, grey relational analysis

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Authors: Elham Zamiri

Abstract:

In this research, we have focused on numeral simulation of a fuel rod in order to examine distribution of heat temperature in components of fuel rod by Fluent software by providing steady state, single phase fluid flow, frequency heat flux in a fuel rod in nuclear reactor to numeral simulation. Results of examining different layers of a fuel rod consist of fuel layer, gap, pod, and fluid cooling flow, also examining thermal properties and fluids such as heat transition rate and pressure drop. The obtained results through analytical method and results of other sources have been compared and have appropriate correspondence. Results show that using heavy water as cooling fluid along with few layers of gas and pod have the ability of reducing the temperature from above 300 ^◦C to 70 ^◦C. This investigation is developable for any geometry and material used in the nuclear reactor.

Keywords: Nuclear fuel fission, numberal simulation, fuel rod, reactor, fluent software.

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Authors: Phang Chang, Phang Piau

Abstract:

Wavelet transform or wavelet analysis is a recently developed mathematical tool in applied mathematics. In numerical analysis, wavelets also serve as a Galerkin basis to solve partial differential equations. Haar transform or Haar wavelet transform has been used as a simplest and earliest example for orthonormal wavelet transform. Since its popularity in wavelet analysis, there are several definitions and various generalizations or algorithms for calculating Haar transform. Fast Haar transform, FHT, is one of the algorithms which can reduce the tedious calculation works in Haar transform. In this paper, we present a modified fast and exact algorithm for FHT, namely Modified Fast Haar Transform, MFHT. The algorithm or procedure proposed allows certain calculation in the process decomposition be ignored without affecting the results.

Keywords: Fast Haar Transform, Haar transform, Wavelet analysis.

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Authors: Huang Shengqin, Xiao Hong

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LES with mixed subgrid-scale model has been used to simulate aerodynamic performance of hypersonic configuration. The simulation was conducted to replicate conditions and geometry of a model which has been previously tested. LES Model has been successful in predict pressure coefficient with the max error 1.5% besides afterbody. But in the high Mach number condition, it is poor in predict ability and product 12.5% error. The calculation error are mainly conducted by the distribution swirling. The fact of poor ability in the high Mach number and afterbody region indicated that the mixed subgrid-scale model should be improved in large eddied especially in hypersonic separate region. In the condition of attach and sideslip flight, the calculation results have waves. LES are successful in the prediction the pressure wave in hypersonic flow.

Keywords: Hypersonic, LES, mixed Subgrid-scale model, experiment.

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Authors: M.Heenaye-Mamode Khan, R.K. Subramanian, N. A. Mamode Khan

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The quest of providing more secure identification system has led to a rise in developing biometric systems. Dorsal hand vein pattern is an emerging biometric which has attracted the attention of many researchers, of late. Different approaches have been used to extract the vein pattern and match them. In this work, Principle Component Analysis (PCA) which is a method that has been successfully applied on human faces and hand geometry is applied on the dorsal hand vein pattern. PCA has been used to obtain eigenveins which is a low dimensional representation of vein pattern features. Low cost CCD cameras were used to obtain the vein images. The extraction of the vein pattern was obtained by applying morphology. We have applied noise reduction filters to enhance the vein patterns. The system has been successfully tested on a database of 200 images using a threshold value of 0.9. The results obtained are encouraging.

Keywords: Biometric, Dorsal vein pattern, PCA.

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Authors: Randhir Singh Baghel, Govind Shay Sharma

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In this paper, we explore an ecosystem that contains a three-species food chain. The first and second species are in competition with one another for resources. However, the third species plays an important role in providing non-linear Crowley-Martin functional support for the first species. Additionally, the third species consumes the second species in a linear fashion, taking advantage of the available resources. This intricate balance ensures the survival of all three species in the ecosystem. A set of non-linear isolated first-order differential equations establish this model. We examine the system's stability at all potential equilibrium locations using the perturbed technique. Furthermore, by spending a lot of time observing the species in their natural habitat, the numerical illustrations at suitable parameter values for the model are shown.

Keywords: Competition, predator, response function, local stability, numerical simulations.

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Authors: S. A. Ahmadi, S. Sharif, R. Jamshidi

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In this article, the flow behavior around a NACA 0012 airfoil which is oscillating with different Reynolds numbers and in various amplitudes has been investigated numerically. Numerical simulations have been performed with ANSYS software. First, the 2- D geometry has been studied in different Reynolds numbers and angles of attack with various numerical methods in its static condition. This analysis was to choose the best turbulent model and comparing the grids to have the optimum one for dynamic simulations. Because the analysis was to study the blades of wind turbines, the Reynolds numbers were not arbitrary. They were in the range of 9.71e5 to 22.65e5. The angle of attack was in the range of -41.81° to 41.81°. By choosing the forward wind speed as the independent parameter, the others like Reynolds and the amplitude of the oscillation would be known automatically. The results show that the SST turbulent model is the best choice that leads the least numerical error with respect the experimental ones. Also, a dynamic stall phenomenon is more probable at lower wind speeds in which the lift force is less.

Keywords: Dynamic stall, Numerical simulation, Wind turbine, Turbulent Model

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Authors: M. A. Koroma, Z. Chuangyi, A. F., Kamara, A. M. H. Conteh

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In this work, we apply the Modified Laplace decomposition algorithm in finding a numerical solution of Blasius’ boundary layer equation for the flat plate in a uniform stream. The series solution is found by first applying the Laplace transform to the differential equation and then decomposing the nonlinear term by the use of Adomian polynomials. The resulting series, which is exactly the same as that obtained by Weyl 1942a, was expressed as a rational function by the use of diagonal padé approximant.

Keywords: Modified Laplace decomposition algorithm, Boundary layer equation, Padé approximant, Numerical solution.

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Authors: Nhon Van Do, Nam Hoai Le, Vien Chan Luong

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The Model for Knowledge Base of Computational Objects (KBCO model) has been successfully applied to represent the knowledge of human like Plane Geometry, Physical, Calculus. However, the original model cannot easyly apply in inorganic chemistry field because of the knowledge specific problems. So, the aim of this article is to introduce how we extend the Computional Object (Com-Object) in KBCO model, kinds of fact, problems model, and inference algorithms to develop a program for solving problems in inorganic chemistry. Our purpose is to develop the application that can help students in their study inorganic chemistry at schools. This application was built successful by using Maple, C# and WPF technology. It can solve automatically problems and give human readable solution agree with those writting by students and teachers.

Keywords: artificial intelligence, automated problem solving, knowledge base system, knowledge representation, reasoning strategy, education software/educational applications.

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Authors: Parastou Kharazmi, Folke Björk

Abstract:

Some of the main causes for degradation of polymeric materials are thermal aging, hydrolysis, oxidation or chemical degradation by acids, alkalis or water. The first part of this paper provides a brief summary of advances in technology, methods and specification of composite materials for relining as a rehabilitation technique for sewage systems. The second part summarizes an investigation on frequently used composite materials for relining in Sweden, the rubber filled epoxy composite and reinforced polyester composite when they were immersed in deionized water or in dry conditions, and elevated temperatures up to 80°C in the laboratory. The tests were conducted by visual inspection, microscopy, Dynamic Mechanical Analysis (DMA), Differential Scanning Calorimetry (DSC) as well as mechanical testing, three point bending and tensile testing.

Keywords: Composite, epoxy, polyester, relining, sewage.

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Authors: H. Golchoobian, S. Saedodin, M. H. Taheri, A. Sarafraz

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In this paper, natural convection in an attic is numerically investigated. The geometry of the problem is considered to be a triangular enclosure. ANSYS Fluent software is used for modeling and numerical solution. This study is for steady state. Four right-angled triangles with height to base ratios of 2, 1, 0.5 and 0.25 are considered. The behavior of various parameters related to its performance, including temperature distribution and velocity vectors are evaluated, and graphs for the Nusselt number have been drawn. Also, in this study, the effect of geometric shape of enclosure with different height-to-base ratios has been evaluated for three types of boundary conditions of winter, summer day and one another state. It can be concluded that as the bottom side temperature and ratio of base to height of the enclosure increases, the convective effects become more prominent and circulation happened.

Keywords: Enclosure, natural convection, numerical solution, Nusselt number, triangular.

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Authors: A. A. Shafie, M. F. Alias, M. H. Ali

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The main objective of this project is to build an autonomous microcontroller-based mobile robot for a local robot soccer competition. The black competition field is equipped with white lines to serve as the guidance path for competing robots. Two prototypes of soccer robot embedded with the Basic Stamp II microcontroller have been developed. Two servo motors are used as the drive train for the first prototype whereas the second prototype uses two DC motors as its drive train. To sense the lines, lightdependent resistors (LDRs) supply the analog inputs for the microcontroller. The performances of both prototypes are evaluated. The DC motor-driven robot has produced better trajectory control over the one using servo motors and has brought the team into the final round.

Keywords: Soccer robot, Obstacle detection, Differential drive, Line following.

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Authors: M. Jahangiri Lahkani, Gh. Ghassemi, N. Sohrabi, N. Rasooli

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Geometry optimizations of metal complexes of Salen(bis(Salicylidene)1,2-ethylenediamine) were carried out at HF and DFT methods employing Lanl2DZ basis set. In this work structural, energies, bond lengths and other physical properties between Mn2+,Cu2+ and Ni2+ ions coordinated by salen–type ligands are examined. All calculations were performed using Gaussian 98W program series. To investigate local aromaticities, NICS were calculated at all centers of rings. The higher the band gap indicating a higher global aromaticity. The possible binding energies have been evaluated. We have evaluated Frequencies and Zero-point energy with freq calculation. The NICS(Nucleous Independent Chemical Shift) Results show Ni(II) complexes are antiaromatic and aromaticites of Mn(II) complexes are larger than Cu(II) complexes. The energy Results show Cu(II) complexes are stability than Mn(II) and Ni(II) complexes.

Keywords: Frequency Calculation, Hartree-Fock (HF), Nucleous Independent Chemical Shift (NICS), Salen(bis(Salicylidene)1, 2-ethylenediamine).

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Authors: Amin Abedi, Naser Vosoughi, Mohammad Bagher Ghofrani

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Double heterogeneity of randomly located pebbles in the core and Coated Fuel Particles (CFPs) in the pebbles are specific features in pebble bed reactors and usually, because of difficulty to model with MCNP code capabilities, are neglected. In this study, characteristics of HTR-10, Tsinghua University research reactor, are used and not only double heterogeneous but also truncated CFPs and Pebbles are considered.Firstly, 8335 CFPs are distributed randomly in a pebble and then the core of reactor is filled with those pebbles and graphite pebbles as moderator such that 57:43 ratio of fuel and moderator pebbles is established.Finally, four different core configurations are modeled. They are Simple Cubic (SC) structure with truncated pebbles,SC structure without truncated pebble, and Simple Hexagonal(SH) structure without truncated pebbles and SH structure with truncated pebbles. Results like effective multiplication factor (Keff), critical height,etc. are compared with available data.

Keywords: Double Heterogeneity, HTR-10, MCNP, Pebble Bed Reactor, Stochastic Geometry.

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Authors: A. Aarabzadeh, R. Hizaji

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Reinforced Concrete (RC) deep beams are a special type of beams due to their geometry, boundary conditions, and behavior compared to ordinary shallow beams. For example, assumption of a linear strain-stress distribution in the cross section is not valid. Little study has been dedicated to fixed-end RC deep beams. Also, most experimental studies are carried out on simply supported deep beams. Regarding recent tendency for application of deep beams, possibility of using fixed-ended deep beams has been widely increased in structures. Therefore, it seems necessary to investigate the aforementioned structural element in more details. In addition to experimental investigation of a concrete deep beam under cyclic load, different failure mechanisms of fixed-ended deep beams under this type of loading have been evaluated in the present study. The results show that failure mechanisms of deep beams under cyclic loads are quite different from monotonic loads.

Keywords: Deep beam, cyclic load, reinforced concrete, fixed-ended.

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Authors: Marwan Abukhaled, Ibrahim Sadek

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This paper presents the vibrations suppression of a thermoelastic beam subject to sudden heat input by a distributed piezoelectric actuators. An optimization problem is formulated as the minimization of a quadratic functional in terms of displacement and velocity at a given time and with the least control effort. The solution method is based on a combination of modal expansion and variational approaches. The modal expansion approach is used to convert the optimal control of distributed parameter system into the optimal control of lumped parameter system. By utilizing the variational approach, an explicit optimal control law is derived and the determination of the corresponding displacement and velocity is reduced to solving a set of ordinary differential equations.

Keywords: Optimal control, Thermoelastic beam, variational approach, modal expansion approach

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Authors: Mala Nath, Nagamani Kompelli, Partha Roy, Snehasish Das

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Two new metal-based anticancer chemotherapeutic agents, [(Ph2Sn)2(HGuO)2(phen)Cl2] 1 and [(Ph3Sn)(HGuO)(phen)]- Cl.CH3OH.H2O 2, were designed, prepared and characterized by analytical and spectral (IR, ESI-Mass, 1H, 13C and 119Sn NMR) techniques. The proposed geometry of Sn(IV) in 1 and 2 is distorted octahedral and distorted trigonal-bipyramidal, respectively. Both 1 and 2 exhibit potential cytotoxicity in vitro against MCF-7, HepG-2 and DU-145 cell lines. The intrinsic binding constant (Kb) values of 1 (2.33 × 105 M-1) and 2 (2.46 × 105 M-1) evaluated from UV-Visible absorption studies suggest non-classical electrostatic mode of interaction via phosphate backbone of DNA double helix. The Stern- Volmer quenching constant (Ksv) of 1 (9.74 × 105 M-1) and 2 (2.9 × 106 M-1) determined by fluorescence studies suggests the groove binding and intercalation mode for 1 and 2, respectively. Effective cleavage of pBR322 DNA is induced by 1.Their interaction with DNA of cancer cells may account for potency.

Keywords: Anticancer agents, DNA binding studies, NMR spectroscopy, organotin.

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Authors: Marios Poulos, Sozon Papavlasopoulos, V. S. Belesiotis

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In the proposed method for Web page-ranking, a novel theoretic model is introduced and tested by examples of order relationships among IP addresses. Ranking is induced using a convexity feature, which is learned according to these examples using a self-organizing procedure. We consider the problem of selforganizing learning from IP data to be represented by a semi-random convex polygon procedure, in which the vertices correspond to IP addresses. Based on recent developments in our regularization theory for convex polygons and corresponding Euclidean distance based methods for classification, we develop an algorithmic framework for learning ranking functions based on a Computational Geometric Theory. We show that our algorithm is generic, and present experimental results explaining the potential of our approach. In addition, we explain the generality of our approach by showing its possible use as a visualization tool for data obtained from diverse domains, such as Public Administration and Education.

Keywords: Computational Geometry, Education, e-Governance, Semantic Web.

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Authors: Cătălin Mărculescu, Andrei Avram, Cătălin Pârvulescu, Marioara Avram, Cătălin Mihai Bălan

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The present microfluidic study is emphasizing the flow behavior within a Y shape micro-bifurcation in two similar flow configurations. We report here a numerical and experimental investigation on the velocity profiles evolution and secondary flows, manifested at different Reynolds numbers (Re) and for two different boundary conditions. The experiments are performed using special designed setup based on optical microscopic devices. With this setup, direct visualizations and quantitative measurements of the path-lines are obtained. A Micro-PIV measurement system is used to obtain velocity profiles distributions in a spatial evolution in the main flows domains. The experimental data is compared with numerical simulations performed with commercial computational code FLUENT in a 3D geometry with the same dimensions as the experimental one. The numerical flow patterns are found to be in good agreement with the experimental manifestations.

Keywords: Micro-PIV, numerical investigations, secondary flows, velocity profiles.

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Authors: M. Bobojanov, D. Rismukhamedov, F. Tuychiev, Kh. Shamsutdinov

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The article presents the results of experimental study of a two-speed asynchronous motor 4A80B6/4U3 with pole-changing winding on a fan drive VSUN 160x74-0.55-4 in static and dynamic modes. A prototype of a pole-changing Motor was made based on the results of the calculation and the performance and mechanical characteristics of the Motor were removed at the experimental stand, and useful capacities and other parameters from both poles were determined. In dynamic mode, the curves of changes of torque and current of the stator were removed by direct start, constant speed operation, by switching of speeds and stopping.

Keywords: Pole-changing winding, two speed asynchronous machine, basic scheme, winding factor, differential leakage factor.

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Authors: Ekaterina Artiukhina, Panagiotis Grammelis

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Torrefaction of biomass pellets is considered as a useful pretreatment technology in order to convert them into a high quality solid biofuel that is more suitable for pyrolysis, gasification, combustion, and co-firing applications. In the course of torrefaction, the temperature varies across the pellet, and therefore chemical reactions proceed unevenly within the pellet. However, the uniformity of the thermal distribution along the pellet is generally assumed. The torrefaction process of a single cylindrical pellet is modeled here, accounting for heat transfer coupled with chemical kinetics. The drying sub-model was also introduced. The nonstationary process of wood pellet decomposition is described by the system of non-linear partial differential equations over the temperature and mass. The model captures well the main features of the experimental data.

Keywords: Torrefaction, biomass pellets, model, heat and mass transfer.

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Authors: Zeinab Hooshyar, Alireza Mehdizadeh

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Endovascular aneurysm repair is a new and minimally invasive repair for patients with abdominal aortic aneurysm (AAA). This method has potential advantages that are incomparable with other repair methods. However, the enlargement of aneurysm in the absence of endoleak, which is known as endotension, may occur as one of post-operative compliances of this method. Typically, endotension is mainly as a result of pressure transmitted to aneurysm sac by endovascular installed graft. After installation of graft the aneurysm sac reduces significantly but remains non-zero. There are some factors which affect this pressure transmitted. In this study, the geometry features of installed vascular graft have been considered. It is inferred that graft neck angle and iliac bifurcation angle are two factors which can affect the drag force on graft and consequently the pressure transmitted to aneurysm.

Keywords: Endovascular graft, transmitted pressure, Drag force, Finite Element Modeling, neck angle, iliac bifurcation angle.

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Authors: E. Assareh, M.A. Behrang, M. Ghalambaz, A.R. Noghrehabadi, A. Ghanbarzadeh

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In this paper, a new approach is introduced to solve Blasius equation using parameter identification of a nonlinear function which is used as approximation function. Bees Algorithm (BA) is applied in order to find the adjustable parameters of approximation function regarding minimizing a fitness function including these parameters (i.e. adjustable parameters). These parameters are determined how the approximation function has to satisfy the boundary conditions. In order to demonstrate the presented method, the obtained results are compared with another numerical method. Present method can be easily extended to solve a wide range of problems.

Keywords: Bees Algorithm (BA); Approximate Solutions; Blasius Differential Equation.

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Authors: Olusheye Akinfenwa, Samuel Jator, Nianmin Yoa

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A block backward differentiation formula of uniform order eight is proposed for solving first order stiff initial value problems (IVPs). The conventional 8-step Backward Differentiation Formula (BDF) and additional methods are obtained from the same continuous scheme and assembled into a block matrix equation which is applied to provide the solutions of IVPs on non-overlapping intervals. The stability analysis of the method indicates that the method is L0-stable. Numerical results obtained using the proposed new block form show that it is attractive for solutions of stiff problems and compares favourably with existing ones.

Keywords: Stiff IVPs, System of ODEs, Backward differentiationformulas, Block methods, Stability.

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Authors: D. Mirauda, M. Greco, A. Volpe Plantamura

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The necessity of updating the numerical models inputs, because of geometrical and resistive variations in rivers subject to solid transport phenomena, requires detailed control and monitoring activities. The human employment and financial resources of these activities moves the research towards the development of expeditive methodologies, able to evaluate the outflows through the measurement of more easily acquirable sizes. Recent studies highlighted the dependence of the entropic parameter on the kinematical and geometrical flow conditions. They showed a meaningful variability according to the section shape, dimension and slope. Such dependences, even if not yet well defined, could reduce the difficulties during the field activities, and also the data elaboration time. On the basis of such evidences, the relationships between the entropic parameter and the geometrical and resistive sizes, obtained through a large and detailed laboratory experience on steady free surface flows in conditions of macro and intermediate homogeneous roughness, are analyzed and discussed.

Keywords: Froude number, entropic parameter, roughness, water discharge.

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Authors: Roman Novak, Matjaz Vencelj

Abstract:

The angular distribution of Compton scattering of two quanta originating in the annihilation of a positron with an electron is investigated as a quantum key distribution (QKD) mechanism in the gamma spectral range. The geometry of coincident Compton scattering is observed on the two sides as a way to obtain partially correlated readings on the quantum channel. We derive the noise probability density function of a conceptually equivalent prepare and measure quantum channel in order to evaluate the limits of the concept in terms of the device secrecy capacity and estimate it at roughly 1.9 bits per 1 000 annihilation events. The high error rate is well above the tolerable error rates of the common reconciliation protocols; therefore, the proposed key agreement protocol by public discussion requires key reconciliation using classical error-correcting codes. We constructed a prototype device based on the readily available monolithic detectors in the least complex setup.

Keywords: Compton scattering, gamma-ray polarization, quantumcryptography, quantum key distribution

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Authors: M. Fakharian, M. I. Khodakarami

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In this paper, a new trend for improvement in semianalytical method based on scale boundaries in order to solve the 2D elastodynamic problems is provided. In this regard, only the boundaries of the problem domain discretization are by specific subparametric elements. Mapping functions are uses as a class of higherorder Lagrange polynomials, special shape functions, Gauss-Lobatto- Legendre numerical integration, and the integral form of the weighted residual method, the matrix is diagonal coefficients in the equations of elastodynamic issues. Differences between study conducted and prior research in this paper is in geometry production procedure of the interpolation function and integration of the different is selected. Validity and accuracy of the present method are fully demonstrated through two benchmark problems which are successfully modeled using a few numbers of DOFs. The numerical results agree very well with the analytical solutions and the results from other numerical methods.

Keywords: 2D Elastodynamic Problems, Lagrange Polynomials, G-L-Lquadrature, Decoupled SBFEM.

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Authors: V. V. Zozulya

Abstract:

New theory for functionally graded (FG) shell based on expansion of the equations of elasticity for functionally graded materials (GFMs) into Legendre polynomials series has been developed. Stress and strain tensors, vectors of displacements, traction and body forces have been expanded into Legendre polynomials series in a thickness coordinate. In the same way functions that describe functionally graded relations has been also expanded. Thereby all equations of elasticity including Hook-s law have been transformed to corresponding equations for Fourier coefficients. Then system of differential equations in term of displacements and boundary conditions for Fourier coefficients has been obtained. Cases of the first and second approximations have been considered in more details. For obtained boundary-value problems solution finite element (FE) has been used of Numerical calculations have been done with Comsol Multiphysics and Matlab.

Keywords: Shell, FEM, FGM, legendre polynomial.

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