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Holomorphic functions are the key tool to construct representation formulae for the solutions for a manifold of plane problems, especially for the flow of a viscous fluid modelled by the Stokes system. Three-dimensional representation formulae can be constructed by tools of hypercomplex analysis, i.e. by working with monogenic functions playing the role of a three-dimensional analogue of holomorphic functions. However, several alternative constructions in hypercomplex setting are possible. In this paper, the three-dimensional representation of a general solution for the Stokes system, based on the functions of a reduced quaternionic variable, is presented. Moreover, an ill-posed Cauchy problem for the Stokes system, consisting in reconstruction of the velocity field in the interior from overdetermined boundary conditions given on a part of the boundary, is considered. It is shown, that if the domain is star-shaped, then the Cauchy problem can be reduced to the problem of the regular extension for a quaternionic function from the boundary conditions given on a part of its boundary.Holomorphic functions are the key tool to construct representation formulae for the solutions for a manifold of plane problems, especially for the flow of a viscous fluid modelled by the Stokes system. Three-dimensional representation formulae can be constructed by tools of hypercomplex analysis, i.e. by working with monogenic functions playing the role of a three-dimensional analogue of holomorphic functions. However, several alternative constructions in hypercomplex setting are possible. In this paper, the three-dimensional representation of a general solution for the Stokes system, based on the functions of a reduced quaternionic variable, is presented. Moreover, an ill-posed Cauchy problem for the Stokes system, consisting in reconstruction of the velocity field in the interior from overdetermined boundary conditions given on a part of the boundary, is considered. It is shown, that if the domain is star-shaped, then the Cauchy problem can be reduced to the problem of the regular extension for a quater...

This paper focuses on the acoustic backscattering from objects with non-convex shape in two-dimensional space. The scattering simulations are provided using the boundary integral formulation and Nystrom discretization in a single parameter approach. Several numerical examples covering three non-convex kite-shape cross sections and two wave numbers in acoustic domain are carried out to show the influence of shapes on acoustic scattering. Also, the material absorption properties are considered. The error on the accuracy of simulation results is considered in terms of wave numbers.This paper focuses on the acoustic backscattering from objects with non-convex shape in two-dimensional space. The scattering simulations are provided using the boundary integral formulation and Nystrom discretization in a single parameter approach. Several numerical examples covering three non-convex kite-shape cross sections and two wave numbers in acoustic domain are carried out to show the influence of shapes on acoustic scattering. Also, the material absorption properties are considered. The error on the accuracy of simulation results is considered in terms of wave numbers.

We numerically studied light propagation in the two-dimensional human neck model using the time-dependent radiative transfer equation (RTE) for an application of diffuse optical tomography to a thyroid cancer detection. Because complicated light propagation is induced by a heterogeneity of the human neck, highly forward-peaked scattering of light by the tissue volumes except the trachea, and the refractive-index mismatch at the trachea-tissue interface and external boundary, one needs a large number of discrete angular directions (ND) for the RTE calculations, meaning high computational loads. Toward a construction of an efficient numerical scheme of the RTE, in this paper, we focused on a numerical treatment of the highly forward-peaked scattering of light. Then, we investigated numerical errors of the RTE calculations using two kinds of renormalization approaches to the highly forward-directed phase function: the conventional Liu’s approach and developed approach by some of the authors. The investigations showed that the developed approach provided accurate results of the RTE calculations around the peak time regimes even when ND decreased. In the decay time regime, however, the developed approach provided less accurate results because in that regime the refractive-index mismatch at the trachea-tissue interface dominated in the light propagation over the highly forward-peaked scattering.We numerically studied light propagation in the two-dimensional human neck model using the time-dependent radiative transfer equation (RTE) for an application of diffuse optical tomography to a thyroid cancer detection. Because complicated light propagation is induced by a heterogeneity of the human neck, highly forward-peaked scattering of light by the tissue volumes except the trachea, and the refractive-index mismatch at the trachea-tissue interface and external boundary, one needs a large number of discrete angular directions (ND) for the RTE calculations, meaning high computational loads. Toward a construction of an efficient numerical scheme of the RTE, in this paper, we focused on a numerical treatment of the highly forward-peaked scattering of light. Then, we investigated numerical errors of the RTE calculations using two kinds of renormalization approaches to the highly forward-directed phase function: the conventional Liu’s approach and developed approach by some of the authors. The investigatio...

A coupled spectral and boundary element method based on Helmholtz equation for shallow water wave is presented in this paper. This combine method fulfils the advantage of both the methods, spectral discretization of the boundary using the Chebyshev’s polynomial and boundary integral equation which satisfy the Somerfield radiation boundary condition. The model validation has been performed on rectangular harbor for which the experimental, theoretical and analytical simulations already exist. The present model is implemented to estimate the wave amplification for various directional incoming waves propagating towards the realistic ports such as Paradip Port, Odisha, India and Pohang New harbor (PNH), Pohang, South Korea under the resonance conditions. The reliable simulation results suggest that the present numerical scheme is competent and can be implemented on any realistic harbor for estimation of wave behavior.A coupled spectral and boundary element method based on Helmholtz equation for shallow water wave is presented in this paper. This combine method fulfils the advantage of both the methods, spectral discretization of the boundary using the Chebyshev’s polynomial and boundary integral equation which satisfy the Somerfield radiation boundary condition. The model validation has been performed on rectangular harbor for which the experimental, theoretical and analytical simulations already exist. The present model is implemented to estimate the wave amplification for various directional incoming waves propagating towards the realistic ports such as Paradip Port, Odisha, India and Pohang New harbor (PNH), Pohang, South Korea under the resonance conditions. The reliable simulation results suggest that the present numerical scheme is competent and can be implemented on any realistic harbor for estimation of wave behavior.

In the paper a description of heat transfer in a one-dimensional two-layered metal film is considered. The fuzzy coupled lattice Boltzmann equations for electrons and phonons supplemented by appropriate boundary and initial conditions are applied to analyse the thermal process in a thin metal film. The model with fuzzy values of relaxation times and boundary-initial conditions for gold and titanium is proposed. The problem considered is solved by the fuzzy lattice Boltzmann method using α-cuts and the rules of directed interval arithmetic. The application of α-cuts allows one to avoid complicated arithmetical operations in the fuzzy numbers set. In the final part of the paper an example for a numerical solution is presented.In the paper a description of heat transfer in a one-dimensional two-layered metal film is considered. The fuzzy coupled lattice Boltzmann equations for electrons and phonons supplemented by appropriate boundary and initial conditions are applied to analyse the thermal process in a thin metal film. The model with fuzzy values of relaxation times and boundary-initial conditions for gold and titanium is proposed. The problem considered is solved by the fuzzy lattice Boltzmann method using α-cuts and the rules of directed interval arithmetic. The application of α-cuts allows one to avoid complicated arithmetical operations in the fuzzy numbers set. In the final part of the paper an example for a numerical solution is presented.

In the paper a kind of meshless discretization technique, called the Kansa method, is investigated in the context of problems with multiple boundary conditions. This numerical method uses interpolant composed of radial basis functions as well as collocation technique to discretize differential equations. To overcome the problem that appears for equations with multiple boundary conditions, where more equations should be associated with a boundary node than degrees of freedom that exist at this node, an extension of the method is proposed. The key idea lies in the modification of the interpolant with the use of Hermite formulation. The details of the approach are shown in the paper. Moreover, the special attention is paid to estimate respective value of the shape parameter included in radial functions to ensure stability of the solution process and high accuracy. To illustrate usefulness, accuracy and convergence of the method, it is employed to solve a test problem of the bending Kirchhoff plate.

In this paper we consider different methods for formulating boundary optimal control problems of either Dirichlet or Neumann type. On one hand, standard approaches typically have the drawback of searching for the optimal controls in proper sub-spaces of natural trace spaces. Therefore, we consider alternative formulations with lifting functions that have several advantages. First of all, boundary controls can be determined on natural trace spaces, without additional regularity as required by standard approaches. This also leads to numerical discretizations that have the same rate of convergence for the unknowns defined on the domain and for those on the boundary. A potential drawback of the method consists in the use of control functions defined over the problem domain instead of on its boundary, thus increasing the number of degrees of freedom of the problem. This drawback can be compensated by using lifting functions with restricted support, whose boundary contains the control boundary under interest. Numerical results solving the optimality systems in an all-at-once approach show that it is possible to use restricted functions which do not substantially change the minimum value of the target functional with respect to the case of lifting functions with non-restricted support.In this paper we consider different methods for formulating boundary optimal control problems of either Dirichlet or Neumann type. On one hand, standard approaches typically have the drawback of searching for the optimal controls in proper sub-spaces of natural trace spaces. Therefore, we consider alternative formulations with lifting functions that have several advantages. First of all, boundary controls can be determined on natural trace spaces, without additional regularity as required by standard approaches. This also leads to numerical discretizations that have the same rate of convergence for the unknowns defined on the domain and for those on the boundary. A potential drawback of the method consists in the use of control functions defined over the problem domain instead of on its boundary, thus increasing the number of degrees of freedom of the problem. This drawback can be compensated by using lifting functions with restricted support, whose boundary contains the control boundary under interest. N...

Spoon is a form of free-form surfaces. In this paper, a method for designing a spoon using partial differential equations (PDE) is discussed. Boundary curves are defined in term of Fourier series. By selecting the appropriate boundary curves, a design of spoon is generated using fourth order elliptic PDE. Then by solving the elliptic PDE, a smooth design of spoon can be generated. In addition, this paper also examined the impact of the choice of number of boundary curves in generating different spoon shapes.

To solve the problems in the DEM simulations of the screening process of a swing-bar sieve, in this paper we propose the real-virtual boundary method to build the geometrical model of the screen deck on a swing-bar sieve. The motion of the swing-bar sieve is modelled by the planer multi-body kinematics. A coupled model of the discrete element method (DEM) with multi-body kinematics (MBK) is presented to simulate the flowing and passing processes of soybean particles on the screen deck. By the comparison of the simulated results with the experimental results of the screening process of the LA-LK laboratory scale swing-bar sieve, the feasibility and validity of the real-virtual boundary method and the coupled DEM-MBK model we proposed in this paper can be verified. This work provides the basis for the optimization design of the swing-bar sieve with circular apertures and complex motion.

The investigations presented in this paper aim optimization sound protection properties of materials based on phononic crystals. For this purpose one need take into consideration phononic crystals passbands and wave transmission at the boundary of material components. This paper concerns low frequency acoustic waves propagation in one dimensional two component phononic crystals. The low frequency passband width dependencies on phononic crystals parameters are studied and the computations of acoustic wave transmission at the boundary of two phononic crystals or at the boundary of phononic crystal and uniform media are presented.

Several factors can affect the failure stress of a grain boundary, such as grain boundary structure, energy and excess volume, in addition to its interactions with dislocations. In this paper, we focus on the influence of grain boundary energy and excess volume at the boundary on the failure stress of a grain boundary in tantalum from molecular-dynamics simulations. Flyer plate simulations were carried out for a handful of boundary types with different energies and excess volumes. These boundaries were chosen as model systems to represent various boundaries observed in “real” materials. For a small, but representative, set of boundaries explored, no direct correlation was observed between the void nucleation stress of a boundary and either its energy and excess volume. This result suggests that average properties of grain boundaries alone are not sufficient indicators of the failure strength of a boundary.Several factors can affect the failure stress of a grain boundary, such as grain boundary structure, energy and excess volume, in addition to its interactions with dislocations. In this paper, we focus on the influence of grain boundary energy and excess volume at the boundary on the failure stress of a grain boundary in tantalum from molecular-dynamics simulations. Flyer plate simulations were carried out for a handful of boundary types with different energies and excess volumes. These boundaries were chosen as model systems to represent various boundaries observed in “real” materials. For a small, but representative, set of boundaries explored, no direct correlation was observed between the void nucleation stress of a boundary and either its energy and excess volume. This result suggests that average properties of grain boundaries alone are not sufficient indicators of the failure strength of a boundary.

In the paper the soft tissue freezing process is considered. The tissue sub-domain is subjected to the action of cylindrical cryoprobe. Thermal processes proceeding in the domain considered are described using the dual-phase lag equation (DPLE) supplemented by the appropriate boundary and initial conditions. DPLE results from the generalization of the Fourier law in which two lag times are introduced (relaxation and thermalization times). The aim of research is the identification of these parameters on the basis of measured cooling curves at the set of points selected from the tissue domain. To solve the problem the evolutionary algorithms are used. The paper contains the mathematical model of the tissue freezing process, the very short information concerning the numerical solution of the basic problem, the description of the inverse problem solution and the results of computations.In the paper the soft tissue freezing process is considered. The tissue sub-domain is subjected to the action of cylindrical cryoprobe. Thermal processes proceeding in the domain considered are described using the dual-phase lag equation (DPLE) supplemented by the appropriate boundary and initial conditions. DPLE results from the generalization of the Fourier law in which two lag times are introduced (relaxation and thermalization times). The aim of research is the identification of these parameters on the basis of measured cooling curves at the set of points selected from the tissue domain. To solve the problem the evolutionary algorithms are used. The paper contains the mathematical model of the tissue freezing process, the very short information concerning the numerical solution of the basic problem, the description of the inverse problem solution and the results of computations.

The paper presents data on the effect of a distributed roughness of the blunted nose of the cone on the position of the laminar-turbulent transition. The studies were carried out at Mach number 5.95. It is found that the position of the roughness plays an important role in the transition process, it is obtained that the roughness has the greatest effect on the transition when applied to an angle Θ ≈ 90 °. It was found that the presence of a roughness on the nose of the model, even at subcritical Reynolds numbers, affects the pulsation processes in the boundary layer.The paper presents data on the effect of a distributed roughness of the blunted nose of the cone on the position of the laminar-turbulent transition. The studies were carried out at Mach number 5.95. It is found that the position of the roughness plays an important role in the transition process, it is obtained that the roughness has the greatest effect on the transition when applied to an angle Θ ≈ 90 °. It was found that the presence of a roughness on the nose of the model, even at subcritical Reynolds numbers, affects the pulsation processes in the boundary layer.

Some automobile factories have segmented mixed-model production lines into shorter sub-lines according to part group, such as engine, trim, and powertrain. The effects of splitting a line into sub-lines have been reported from the standpoints of worker motivation, productivity improvement, and autonomy based on risk spreading. There has been no mention of the possibility of shortening the line length by altering the product sequence using sub-lines. The purpose of the present paper is to determine the conditions under which sub-lines reduce the line length and the degree to which the line length may be shortened. The line lengths for a non-split line and a line that has been split into sub-lines are compared using three methods for determining the working area, the standard closed boundary, the optimized open boundary, and real-life constant-length stations. The results are discussed by analyzing the upper and lower bounds of the line length. Based on these results, a procedure for deciding whether or not to split a production line is proposed.Some automobile factories have segmented mixed-model production lines into shorter sub-lines according to part group, such as engine, trim, and powertrain. The effects of splitting a line into sub-lines have been reported from the standpoints of worker motivation, productivity improvement, and autonomy based on risk spreading. There has been no mention of the possibility of shortening the line length by altering the product sequence using sub-lines. The purpose of the present paper is to determine the conditions under which sub-lines reduce the line length and the degree to which the line length may be shortened. The line lengths for a non-split line and a line that has been split into sub-lines are compared using three methods for determining the working area, the standard closed boundary, the optimized open boundary, and real-life constant-length stations. The results are discussed by analyzing the upper and lower bounds of the line length. Based on these results, a procedure for deciding whether or not...

The main purpose of this paper is examination of an application of modern parallel computing technique OpenMP to speed up the calculation in the numerical solution of parametric integral equations systems (PIES). The authors noticed, that solving more complex boundary problems by PIES sometimes requires large computing time. This paper presents the use of OpenMP and fast C++ linear algebra library Armadillo for boundary value problems modelled by 3D Laplace’s equation and solved using PIES. The testing example shows that the use of mentioned technologies significantly increases speed of calculations in PIES.

Boundary value problems, more precisely Dirichlet’s problem for a string equation, or for an equivalent system of first order equations have been first studied in the first half of last century ([1] – [9]). The interest to these problems has been big ever since, see e.g. [10, 11]. All these papers have looked into the boundary value problems in a finite domains in the plane. Strictly hyperbolic systems with more than two characteristics in infinite domains, have been studied in [12, 13]. The question of boundary value problems for a hyperbolic system of equations with more than two characteristics in finite domain on the plane, when a boundary conditions are prescribed at a whole boundary of the domain, evidently remained open. In this paper, we study this problem in a finite domain on the plane for a hyperbolic system of equations of the first order with constant coefficients and with three mutually distinct characteristics.

The paper proposes the use of rectangular Bezier surface patches for shape identification of the boundary for 3D potential problems modeled by the Laplace equation. The graphical representation of the boundary generated by Bezier patches is considered directly in mathematical formula of parametric integral equation systems (PIES) that allows to eliminate the need of dividing the boundary into elements. The numerical examples included in the paper demonstrate simplification in the boundary description by Bezier patches and applicability for boundary shape identification problems.

This paper presents the investigation of shear walls behavior in seismic response of tall buildings. In this study, shear wall system was tested for one building with total ten systems namely I, II, III, IV, V, VI, VII, VIII, IX, and X. Seismic loads were applied to the building models using response spectrum method. In the response spectrum data, the acceleration equals the ground acceleration 1.0 g on its direction and 0.3g in the opposite direction. The result indicated, the comparison of all model in term of displacement show model VI produced the lowest lateral displacement and base shear load of 5.8 mm and 2407 kN, respectively, while model II has the highest displacement of 11.57 mm. The study show, displacement of model VI was decreased up to 21% by adding boundary elements compared to model I which both of the model located at the same location. Thus, it can be concluded that, in term of displacement the optimum solution to resist seismic load by provide a shear wall with boundary elements near the core of the building as applied to Model VI.

This paper presents a topology optimization for compliance minimization of a two-dimensional linear elastic statically loaded structures. Solid isotropic material with penalization (SIMP) and level-set function (LSF) as a topology representation model were compared in this research. SIMP model discretizes a design domain as the relative material density raised to some power times of the material properties of solid material. Each grid value ranges from "0" for void to "1" if the material exists and pseudo materials will appear for a grid value between 0 and 1. This is not desirable because no physical material exists with the property described. While LSF model is defined by dividing the design domain into grids as the design variables with different values in each grid. Then, the prescribed threshold value will cut the LSF at a single level where the topology model will be presented by the level set that has a higher value than the threshold. This model can always provide a clear boundary and geometry information during the optimization process. Results show that SIMP model provides a slightly higher compliance than LSF model due to the presence of pseudo material. Thus, LSF model is better than SIMP model in performing minimum compliance optimization and its manufacturing capabilities.

In the paper we introduce Rv–generalized solution to the Stokes problem with singularity in a non-convex polygonal domain with one reentrant corner on its boundary. The weighted analogue of the LBB condition is proved. A new weighted finite element method is constructed. Results of numerical experiments have shown the efficiency of the method.

It has become more or less standard in scientific codes to utilize shared- and distributed-memory parallelism achieved by OpenMP and MPI and thus to use the computational power of all available cores. However, in recent years the theoretical peak performance of CPUs has also been rising due to the capabilities of vector processing units able to perform simultaneous computations on vectors of data. This concept, known as SIMD, becomes increasingly important, yet it is still quite neglected. This paper deals with the intra-node optimized code utilizing the SIMD registers solving the boundary integral equations, and with numerical experiments on two modern Intel’s processors, namely Xeon Phi 7250 and Xeon 8160.

Choice of interpolation stencils is critical in ensuring a robust reconstruction based sharp interface immersed boundary approach. This paper tries to assess the behavior of two pressure reconstruction stencils [Linear Pressure Interpolation (LPI) and Quadratic Pressure Interpolation (QPI)] on various parameters such as the order of accuracy, fluid to surface mesh ratio, under both stationary and moving conditions. For the proposed parametric study benchmark cases such as flow past rigid stationary sphere (at Re =100), flow past streamwise rotating sphere (at Re =100) are simulated. The studies point out that under all conditions Quadratic interpolation scheme performs better compared to the linear interpolation stencils.

The Szegӧ kernel plays an important role in conformal mapping and satisfies a boundary Kerzman-Stein integral equation. The series representation for the Szegӧ kernel for an annulus region is well known. In this paper, the Adomian decomposition method is applied to solve the boundary Kerzman-Stein integral equation. The method provides another series form for the Szegӧ kernel for the annulus region. The two series are shown to be equivalent. Some numerical examples are also presented to compare the speed of convergence of the two series.

This work concerns the issue of curve reconstruction of the handwriting image based on a Whale Optimization Algorithm (WOA). WOA is a novel nature-inspired meta-heuristic algorithm that mimics the social behavior of humpback whales and it is a new optimization technique for solving optimization problems. Optimization results based on mathematical problems and structural optimization problems demonstrate that WOA is very competitive compared to the state-of-the-art optimization methods. The objective of this paper is to find the optimal value for the control points of the cubic Bezier curve. It is important to construct the fitting curve that approximates a given set of data points optimally. There are four main steps required in reconstructing of the bitmap image, namely boundary extraction, corner point detection, parameterization and curve fitting. Since the aim of this study is to minimize the distance between the outline of the original image and the parametric curve, Sum Square of Error (SSE) has been used to calculate the error between the two curves. The best fitted parametric curve that represents all the given data points is then obtained.

The fundamental problem of a holomorphic extension of functions of one complex variable has a great importance in applications. This problem is well studied in the classical complex analysis, particularly, a lot of results related to the conditions for a holomorphic extension of a function into a domain from its boundary values are available. By the help of the Kolosov-Muskhelishvili formulae these results can be used in solution of ill-posed problems of the plane elasticity theory. Similar problems can be formulated for spatial problems of linear elasticity: identify displacement and stresses in a body from given values on a boundary. Thus, we have a three-dimensional ill-posed Cauchy problem for the elliptic Lame system. This problem can be reduced to the problem of a regular extension into a domain of the quaternion function defined on a part of its boundary. In this paper, we discuss problems of a regular extension of quaternion functions in the context of applications in the theory of elasticity.

In previous work, the generalised relation between the local values of temperature and the corresponding heat flux has been achieved by the use of a novel technique that involves generalised derivatives (in particular, derivatives of non-integer orders). Confluent hyper-geometric functions, known as Whittaker’s functions, appear in the course of the solution procedure, upon applying the Laplace transform to the original transport equation. The relation is written in the integral form and provides a relationship between the local values of the temperature and heat flux. In the case of a non-zero value of the boundary speed, the solution is in the form of an integral mapping, which should allow chaotic solutions under some set of parameters. This possibility is investigated in the present paper.

Exact solutions for three-dimensional nonlinear flows of a viscous incompressible fluid are examined. These solutions belong to Lin’s class of solutions, which are velocities linearly dependent on a part of coordinates. This allows these solutions to be used for the description of large-scale currents of the World Ocean. The obtained exact solution describes the flow of a vertical vortex fluid. A vertical twist in the fluid arises from the consideration of inertia forces and the nonuniform velocity distribution on the free boundary of the fluid layer. This solution allows one to describe counterflows of an incompressible fluid for flows in a thin layer. Thus, the obtained exact solution of the Navier-Stokes equations describes a new mechanism of momentum transfer in a fluid. The obtained solutions are analyzed in this paper. The existence of stagnation points for the flow of a vertical vortex fluid in an infinite layer with permeable boundaries is shown.

In this paper, domain discretization in dealing with domain integral in material nonlinearity plate bending analysis with boundary element method is presented. The domain discretization is conducted by using 9-nodes quadrilateral cells. The semi-discontinuous cell of 9-nodes quadrilateral cell is presented in order to avoid the coincident in between boundary element point and domain element point. The nonlinear term in the formula is analysed by considering that the material is assumed to undergo small strains. The von Mises criterion is used to evaluate the plastic zone and elastic perfectly plastic material behaviour is assumed. A total incremental method is implemented to solve the nonlinear system of equation. The numbers of boundary and domain cells are varied to study the convergence of the analysis. The number of boundary and domain cells has an influence on the accuracy of the results. The increased number of boundary and domain cells gives better results, however a relative coarser mesh can be implemented to have a good results. The current results have a good agreement with the previous results by other researchers.

In this paper, an approach to solving the problem of the determination of effective stress-strain, thermophysical and electrophysical properties of filled composite materials on the basis of the variational principles is developed. In the numerical implementation, it might be reduced to the use of the finite element method (FEM). An approach, possessing a certain generality, to finding effective physical-mechanical, thermophysical and electrophysical properties of filled polymer composites is proposed and described. Firstly, a general concept based on obtaining detailed parameters of the composite state under external impact taking into account the actual structure of the material is developed. Then, the averaging procedure aimed at obtaining effective properties of the composites is employed. Secondly, the realization of boundary problems is reduced, for all cases (physical-mechanical, thermophysical and electrophysical), to the formulation of the corresponding variable principles and their realization by the methods of computational mechanics, in particular, by the finite element method.

In the paper the problems of optimal control of distributed systems described the process of production of quartz optical fiber are considered. This process includes two main stages: the alloying of quartz tubes and the optical fiber drawing from a preform. The following problems of optimal control of distributed systems are formulated: two-dimensional problem with boundary control and one-dimensional problem with distributed observation which describe the process of quartz tubes alloying, and one-dimensional problem with boundary control and distributed observation for problem of quartz optical fiber drawing. For these problems the optimality systems are obtained and the algorithms for their realization are proposed.

Temperature dependent viscosity and Coriolis force were applied to the steady Benard-Marangoni convection where the lower boundary of a horizontal layer of the binary mixture is heated from below and cooled from above. The purpose of this paper is to study in detail the onset of convection with these effects. Few cases of boundary conditions are studied which are rigid-rigid, rigid-free and free-free representing the lower-upper boundaries. A detailed numerical calculation of the marginal stability curves was performed by using the Galerkin method and it is showed that temperature dependent viscosity and Soret number destabilize the binary fluid layer system and Taylor number act oppositely.

Non-circular cylindrical gear (NCCG) is a common form of non-circular gear. Different from the circular gear, the tooth profile equation of NCCG cannot be obtained. So it is necessary to use a numerical algorithm to calculate the tooth profile of NCCG. For this reason, this paper presents a simple and highly efficient numerical algorithm to obtain the tooth profile of NCCG. Firstly, the mathematical model of tooth profile envelope of NCCG is established based on the principle of gear shaping, and the tooth profile envelope of NCCG is obtained. Secondly, the polar radius and polar angle of shaper cutter tooth profile are chosen as the criterions, by which the points of NCCG tooth cogging can be screened out. Finally, the boundary of tooth cogging points is extracted by a distance criterion and correspondingly the tooth profile of NCCG is obtained.

The paper presents the micro-scale model of unconstrained solidification of the grain immersed in under-cooled liquid, based on the front tracking approach. For this length scale, the interface tracked through the domain is meant as the solid-liquid boundary. To prevent generation of huge meshes the energy transport equation is discretized on the adaptive control volume (c.v.) mesh. The coupling of dynamically changing mesh and moving front position is addressed. Preliminary results of simulation of a test case, the growth of single grain, are presented and discussed.

The paper presents first approach to the GPU-based parallelization of the coupled Immersed Boundary and Lattice Boltzmann Method. The proposed numerical simulator deals with fluid and heat flow in a domains with complex internal boundaries using Cartesian grid. The solution algorithm was parallelized with the aid of the CUDA architecture. Several heat and fluid flow problems, i.e., heated lid-driven flow and laminar natural convection in square domains without internal obstacles and isothermal flow past stationary cylinder were investigated. Satisfactory accelerations of the solution times were obtained for problems without internal boundaries. For test case with internal boundaries decrease in the parallel computing efficiency was observed as a results of numerical handling of the internal boundaries.

In this paper we consider a model case of stationary vibrations of a thin flat plate, one side of which is embedded, the opposite side is free, and the sides are freely leaned. In mathematical modeling there is a local boundary value problem for the inhomogeneous biharmonic equation in a rectangular domain. Boundary conditions are given on all boundary of the domain. We show that the considered problem is self-adjoint. Herewith the problem is ill-posed. We show that the stability of solution to the problem is disturbed. Sufficient conditions of existence of the problem solution are found.

A key input parameter to Detonation Shock Dynamics models is the angle that the propagating detonation wave makes with the charge edge. This is commonly referred to as the boundary angle, and is a property of the explosive/confiner material combination. Such angles can be determined: (i) experimentally from measured detonation wave-shapes, (ii) theoretically, or (iii) via hydrocode simulations using a reactive burn model. Of these approaches: (i) is difficult because of resolution, (ii) breaks down for certain configurations, while (iii) requires a well validated model.In this paper, the CREST reactive burn model, which has previously been successful in modelling a wide range of explosive phenomena, is used to simulate recent Detonation Confinement Sandwich Tests conducted at LANL using the insensitive high explosive PBX 9502. Simulated detonation wave-shapes in PBX 9502 for a number of different confiner materials and combinations closely match those recorded from the experiments. Boundary angles were su...

In this paper, material nonlinearity plate bending analysis with boundary element method is presented. The nonlinear term in the formula is analysed by considering that the material is assumed to undergo small strains. The von Mises criterion is used to evaluate the plastic zone and elastic perfectly plastic material behaviour is assumed. The domain integral due to material nonlinearity is evaluated using a cell discretization technique and a total incremental method is implemented to solve the nonlinear system of equation. The size of the increment, the number of boundary elements and the number of domain cells are varied to study the convergence of the analysis. The size of load increment shows big influence on the results. The smaller the size the better results can be obtained, however 200 steps to reach the final load is a reasonable size to get a good results. The number of boundary and domain cells has an influence on the accuracy of the results. The increased number of boundary as well as domain c...

In this paper, multi-scale modeling methodology has been applied to simulate the relatively thick composite panels subjected to high speed local impact loading. Instead of massive parallel processing, we propose to use surrogate modeling to bridge micro-scale and macro-scale. Multi-scale modeling of fracture phenomena of composite materials will consist of (1) micro-scale modeling of fiber-matrix structure using the unit-volume-element technique, which can incorporate the boundary effect, and the level set method for crack modeling, which can model the crack propagation independent of finite element mesh; (2) macro-scale simulation of composite panels under high strain-rate impact using material response calculated from micro-scale modeling; and (3) surrogate modeling to integrate the two scales. In order to validate the predictions, first we did the material level lab experiment such as tensile test. We also did the field test of bullet impact into composite panels made of 4 plies fiber. The impact veloc...

In this paper, we consider a model case of stationary vibrations of a thin flat plate, one side of which is embedded, the opposite side is free, and the sides are freely leaned. In mathematical modeling, there is a local boundary value problem for the biharmonic equation in a rectangular domain. Boundary conditions are given on all boundary of the domain. We show that the considered problem is self-adjoint. Herewith, the problem is ill-posed. We show that the stability of solution to the problem is disturbed. Necessary and sufficient conditions of existence of the problem solution are found. Spaces of the ill-posedness of the considered problem are constructed.

In the present paper, we investigate a nonlocal boundary problem for the Laplace equation in a half-disk, with opposite flows at the part of the boundary. The difference of this problem is the impossibility of direct applying of the Fourier method (separation of variables). Because the corresponding spectral problem for the ordinary differential equation has the system of eigenfunctions not forming a basis. A special system of functions based on these eigenfunctions is constructed. This system has already formed the basis. This new basis is used for solving the nonlocal boundary value problem. The existence and the uniqueness of the classical solution of the problem are proved.

There are four main processes involved during the reconstruction of bitmap images, which include boundary detection, corner points detection, parameterization and curve fitting. This paper presents the use of Artificial Bee Colony (ABC) algorithm during the process of parameterization and cubic Bezier curve was used for the reconstruction of generic shapes by using Least Square method. Since the purpose of this study is to minimize the distance between boundary of the original image and parametric curve, Sum Square Error (SSE) has been used to calculate the error given by these two curves. The results obtained by using ABC algorithm will be compared with the results obtained when using chord length method during the parameterization process. Note that, for the comparison purpose, the same images have been used and the procedure remains the same except for parameterization process. The finding of this study illustrates that the proposed method has successfully generated fitted cubic Bezier curve that resem...

The Ahlfors map and Szego kernel are both classically related to each other. Ahlfors map can be computed using Szego kernel without relying on the zeros of Ahlfors map. The Szego kernel is a solution of a Fredholm integral equation of the second kind with the Kerzman-Stein kernel. The exact zeros of the Ahlfors map are unknown except for the annulus region. This paper presents a numerical method for computing the zeros of the Ahlfors map of any bounded doubly connected region. The method depends on the values of Szego kernel, its derivative and the derivative of boundary correspondence function of Ahlfors map. Using combination of Nystrom method, GMRES method, fast multiple method and Newton’s method, the numerical examples presented here prove the effectiveness of the proposed method.

This paper deals with the mathematical model of a steady flow of a heat-conductive incompressible viscous fluid through a spatially periodic plane profile cascade. The corresponding boundary value problem is reduced to one spatial period. We prove the existence of a weak solution of a coupled problem, with various boundary conditions on the parts of the boundary. Particularly, the condition on the outflow is a variant of the so called “do nothing” boundary condition.

In this paper we consider a model case of stationary vibrations of a thin flat plate, one side of which is embedded, the opposite side is free, and the sides are freely leaned. In mathematical modeling there is a local boundary value problem for the biharmonic equation in a rectangular domain. Boundary conditions are given on all boundary of the domain. We show that the considered problem is self-adjoint. Herewith the problem is ill-posed. We show that the stability of solution to the problem is disturbed. Necessary and sufficient conditions of existence of the problem solution are found. Spaces of the ill-posedness of the considered problem are constructed.

Cryosurgery is a technique to eradicate abnormal biological tissues by freezing. The objective of cryosurgery is to maximize cryoinjury of tumor tissues while at the same time minimizing cryoinjury to the surrounding healthy tissues. The location and number of cryoprobes are important factors to obtain optimal cryosurgery in complex-shaped tumors. This paper presents multiprobe cryosurgery simulation with optimal cryoprobes location in target region. Bubble packing method is used to obtain optimal cryoprobes layout. We consider mathematical model of freezing by bioheat transfer equation in solid (frozen tissue), liquid (unfrozen tissue), and mushy region. This model is referred to as Stefan problem where the location of moving solid-mushy or mushy-liquid interface is not known and it is as part of the solution. We reformulate the bioheat equations into single enthalpy (energy) equation which can resolve the moving boundary between two phases. The first-order of Godunov method is adopted to obtain numerical solution of the phase change problem. This method is easily applied since we only solve one governing equation regardless the moving interface between two phases. For demonstration purposes, multiprobe cryosurgery for lung cancer case with complex geometry is simulated and interpreted. The numerical simulation for three to ten cryoprobes configuration shows that the nine cryoprobes layout has the smallest total defect in this case. The numerical results provide an important information for cryosurgeon before conducting effective cryosurgery protocol.

The paper is devoted to the solution of non-stationary boundary value problems (BVP) for an anisotropic elastic medium at antiplane deformation. The solutions in class of shock wave are considered for strong hyperbolic equation which describes the motion of the considered medium. The statement of two initial boundary problems are given. The conditions on the wave fronts are obtained. The energy conservation law and questions of the uniqueness of the solutions in the class of shock waves are considered. Dynamic analogues of Green and Gauss’s formulas constructed and their integral representations obtained.

Boundary inverse problems occupy an important place among the inverse problems of mathematical physics. They are connected with the problems of diagnosis, when additional measurements on one of the borders or inside the computational domain are necessary to restore the boundary regime in the other border, inaccessible to direct measurements. The boundary inverse problems belong to a class of conditionally correct problems, and therefore, their numerical solution requires the development of special computational algorithms.The paper deals with the solution of the boundary inverse problem for one-dimensional second-order parabolic equations, consisting in the restoration of boundary regime according to measurements inside the computational domain. For the numerical solution of the inverse problem it is proposed to use an analogue of a computational algorithm, proposed and developed to meet the challenges of identification of the right side of the parabolic equations in the works P.N.Vabishchevich and his st...

This contribution deals with unilateral contact problems with Tresca friction (given friction model) in hemitropic mi-cropolar elasticity. Based on a boundary integral approach such problems can be reduced to boundary variational inequalities. This suggests the use of boundary element methods for their numerical treatment. With higher order approximation this leads to a nonconforming approximation what can numerically be realized by means of Gauss-Lobatto quadrature.The contribution is based on the recent papers [7, 8] of the author and on joint work [3] with A. Gachechiladze, R. Gachechi-ladze, and D. Natroshvili.

The paper is devoted to the comparative analysis of features of some popular mechanical models of interphase boundaries in metal-ceramic composite materials. We have numerically studied the influence of the parameters of interface models on effective strength, strain-hardening coefficient and ultimate strain of a microscopic composite fragment including adjacent areas of the matrix, inclusions and the interface zone. We have shown that the most “powerful” and flexible mechanical model of an interphase boundary is the model of an “interface shell” with a gradient of mechanical properties.

In this paper, the dynamics of a three species food-web system is discussed. The food-web comprises of one predator and two logistically growing competing species. The predator species is taking food from one of the competitors with Holling type II functional response. Another competitor is the amensal species for the predator of first species. The system is shown to be positive and bounded. The stability of various axial points, boundary points and interior point has been investigated. The persistence of the system has been studied. Numerical simulation has been performed to show the occurrence of Hopf bifurcation and stable limit cycle about the interior point. The presence of second competitor and its interaction with predator gives more complex dynamics than the simple prey-predator system. The existence of transcritical bifurcation has been established about two axial points. The existence of periodic attractor having period-2 solution has been shown, when amensal coefficient is chosen as bifurcation...