Set Of Boundary Value Problems Research Articles

Computer homogenization of porous piezoceramics of different ferrohardness with random porous structure and inhomogeneous polarization field

The article is concerned with the homogenization problems, in which the effective moduli of porous piezoceramic composites are determined taking into account the inhomogeneity of the polarization field. The homogenization problems are solved by the finite element method in the framework of the theory of effective moduli and the Hill energy principle using the ANSYS package. To this end, in static problems of electroelasticity, the displacements and electric potential, which are linear in spatial variables, are specified on the boundary of a representative volume to provide constant stress and electric induction fields for a homogeneous reference medium. After solving a set of boundary value problems under different boundary conditions and determining the volume-averaged stress components and the electric induction vector, a complete set of effective moduli for the piezoelectric composite is calculated. A representative volume of the piezocomposite is created in the form of a regular finite element mesh consisting of cubic elements. Pores in the representative volume are assumed to be filled with a piezoelectric material with extremely small moduli. Finite elements with pore properties are selected according to a simple random algorithm. The inhomogeneous polarization field is found by solving an electrostatic problem, in which the polarization process in the representative volume is modeled based on a simplified linear formulation. The local coordinate systems for individual finite elements of the composite matrix are specified by the directions of the polarization vectors. In the following, when solving the problems of electroelasticity, these local coordinate systems associated with the elements of the piezoelectric matrix allow recalculating the material properties according to the formulas of transformation of the tensor components as the coordinate systems rotate. In addition, consideration is given to different models describing the change in the moduli of the material from an unpolarized state to a polarized one as a function of the polarization vector. Computational experiments were carried out for three types of piezoceramics: soft ferroelectric piezoceramics PZT-5H, piezoceramics PZT-4 of medium ferrohardness, and piezoceramics PZT-8 with higher degree of ferrohardness. The dependences of the effective moduli on porosity are compared for different laws of polarization inhomogeneity and different kinds of piezoceramic material of the composite matrix.

An Efficient and Accurate Approach to Electrical Boundary Layer Nanofluid Flow Simulation: A Use of Artificial Intelligence

Engineering and technological research groups are becoming interested in neural network techniques to improve productivity, business strategies, and societal development. In this paper, an explicit numerical scheme is given for both linear and nonlinear differential equations. The scheme is correct to second order. Additionally, the scheme’s consistency and stability are guaranteed. Backpropagation of Levenberg–Marquardt, the effect of including an induced magnetic field in a mathematical model for electrical boundary layer nanofluid flow on a flat plate, is quantitatively investigated using artificial neural networks. Later, the model is reduced into a set of boundary value problems, which are then resolved using the suggested scheme and a shooting strategy. The outcomes are also contrasted with earlier studies and the MATLAB solver bvp4c for validation purposes. In addition, neural networking is also employed for mapping input to outputs for velocity, temperature, and concentration profiles. These results prove that artificial neural networks can make precise forecasts and optimizations. Using a neural network to optimize the fluid flow in an electrical boundary layer while subjected to an induced magnetic field is a promising application of the suggested computational scheme. Fluid dynamics benefits greatly from combining numerical methods and artificial neural networks, which could lead to new developments in various fields. Results from this study may aid in optimizing fluid systems, leading to greater productivity and effectiveness in numerous technical fields.

Compactness of boundary value problems for impulsive integro-differential equation

In this paper, we establish sufficient conditions to show the compactness of solution set of boundary value problems for impulsive integro-differential equation using ?-Hilfer fractional operator in a appropriate Banach space. The method we use to show our result is based on fixed point theorems for Meir-Keeler condensing operators via measure of non-compactness, an example is presented to illustrate our method.

The modified finite element method for heat and mass transfer of unsteady reacting flow with mixed convection

This study reveals the extension of a mathematical model of heat and mass transfer of fluid flow over a sheet by incorporating the effect of non-linear mixed convection. The governing equations of flow phenomena are expressed as partial differential equations (PDEs). Similarity transformations are employed to get a dimensionless set of boundary value problems. Most of the existing relevant literature employed some solver to solve a set of differential equations, but this study implements the finite element method to tackle the boundary value problems. The employed finite element method is based on the Galerkin approach. For verifications of the obtained results, a set of linear and non-linear boundary value problems is also solved with Matlab solverbvp4c. The results are displayed in graphs by varying Grashof number, modified (solutal) Grashof number, non-linear convection parameters in heat and mass transfer, radiation parameter, Prandtl number, Schmidt number, and reaction rate parameter. Also, numerical values for the friction at the wall and local Nusselt and Sherwood numbers are given in tables. The problem in PDEs form is also solved with software that implements the finite element method to solve problems. The simulations are also provided, which is the outcome of the software. It is shown that the velocity profile escalates by growing values of thermal and solutal Grashof numbers. Problem-solving techniques from this study may be used in future research to address other unsolved heat transfer fluid physics issues.

The Numerical simulation of composite sandwich cylindrical shells reinforced with circular frames under local loadings

The paper considers a woven fabric (based on fiberglass) composite structure in the form of a foam core sandwich cylindrical shell reinforced with circular frames, under the action of local loads applied to the frames. A technique is described for obtaining a numerical solution (with the confirmed reliability) to the problem of the stress-strain state of this type of structure using two alternative computational models, one of which is based on the numerical integration method, and the other one is based on the finite element method. The first model is developed by adopting a scheme, in which the frames are considered as short cylindrical shells obeying the single normal hypothesis. The foam core sandwich sections connected to them are considered within the framework of the zig-zag theory of soft core sandwich shells based on the assumption of incompressibility of the core along thickness. In this case, the corresponding calculation problem is formulated in the form of systems of algebraic and differential (in partial derivatives) equations for each of the shell sections being considered, which are supplemented by kinematic and force conditions at their joints, as well as boundary conditions at the left and right ends of the structure. The problem solving for each harmonic number is reduced to solving a set of boundary value problems for systems of 8 and 12 linear ordinary differential equations of the first order, related by conditions at the specified joints and using the procedure for expanding the parameters of the stress-strain state and applied loads into Fourier series in the circumferential direction. The solution algorithm is constructed using the numerical integration procedure in the orthogonal sweep version combined with the displacement method procedure (to satisfy the conditions at the joints). The declared finite element model is built within the ABAQUS software package using S4 shell elements (for composite layers) and C3D20 volume elements (for frames and core). The formed model, when setting deliberately overestimated values of the corresponding elastic modules, can implement a situation close to fulfilling the set of hypotheses adopted when constructing the first model. Having fixed in such a situation the consistency of the calculation results on the basis of the alternative computational models constructed in this way and thereby confirming the reliability of the obtained numerical solution, we carry out the transition to the calculation using the real values of the mentioned modules and the analysis of their influence on the stress-strain state of the investigated sandwich shell. The presented example of the calculation of a composite sandwich cylindrical structure, one of the frames of which is under the action of two local axial loads, demonstrates the possibilities of the adopted modeling method.

Моделювання скінченних неоднорідностей дискретними особливостями

This work focuses on development of a mathematical apparatus that allows to perform an approximate description of inhomogeneities of finite sizes in a continuous bodies by arranging the sources given on sets of smaller dimensions. The structure and properties of source densities determine the adequacy of the model. The theory of differential forms and generalized functions underlies this study. The boundary value problems with nonsmooth coefficients are formulated. The solutions of such problems is sought in the form of weakly convergent series and as an alternative - an equivalent recurrent set of boundary value problems with jumps. A feature of this approach is the ability to consistently improve the adequacy of the description of inhomogeneity. This is important because it allows to qualitatively assess the impact of real characteristic properties on the accuracy of the model description. Reducing the dimensions of inhomogeneities allows the use of efficient methods such as the Green's function and boundary integral equations to obtain a semi-analytic solution for direct and inverse problems. The work is based on a number of partial problems that demonstrate the proposed approach in modeling of inhomogeneities. The problems of modeling of the set of finite defects in an oscillating elastic beam, the set of inhomogeneities of an arbitrary shape in an oscillating plate, fragile cracks in a two-dimensional elastic body under static loading are considered.

A studyof vibrations of a cylinder with a viscoelastic coating

The paper presents the problem of steady-state longitudinal-radial vibrations of an elastic hollow cylinder with a viscoelastic coating. The properties of the coating change along the radial coordinate only and are described by the variable Lame parameters and density. The ends of the cylinder are under sliding interface conditions, and the outer side surface is subject to periodic loads. Within the framework of the model of a standard viscoelastic body and following the principle of correspondence, the variable Lame parameters are replaced by the complex functions of the radial coordinate and vibration frequency. The solution was obtained using two approaches. Within the framework of the first approach, the solution is constructed using the method of separation of variables and is reduced to solving a set of boundary value problems for canonical systems of first-order differential equations with variable coefficients. The solution to each of these problems is obtained numerically using the shooting method. The second approach is based on the finite element method implemented in the FlexPDE package. We compared the obtained solutions for the given laws of variation of the Lame parameters and a fixed frequency by analyzing the graphs of the real and imaginary parts of radial displacement and stress components. The convergence of the solution found using the finite element method is shown as a function of the number of nodes for the values of the radial displacement functions measured at three points. The graphs of the amplitude-frequency characteristics measured on the outer surface are plotted for different values of the relaxation time. The effect of variable properties of the coating on the displacement function is estimated. The advantages of each approach are described, and the areas of their practical application are revealed.

Thermo-electro-mechanical phase-field modeling of paraelectric to ferroelectric transitions

Ferroelectric materials are widely used in engineering and science applications due to their large nonlinear thermo-electro-mechanical coupling. This paper develops and studies a phase-field model to describe paraelectric to ferroelectric phase transitions and the giant electrocaloric effect, a large adiabatic temperature change with the application of an electric field. Spatially inhomogeneous temperature distributions, latent heat due to phase changes, and heat conduction are included in the model. The model is derived from fundamental balances of force, charge, micro-force, and energy. The second law of thermodynamics further constrains the constitutive behaviors derived from a phenomenological free energy function for the material. The finite element method is applied to solve the governing equations for a selected set of boundary value problems. First, the motion of a paraelectric/ferroelectric phase boundary controlled by the application and removal of heat is simulated. Then, a model electrocaloric cooling device based on a multilayer ferroelectric capacitor is simulated through a full thermodynamic refrigeration cycle. The model geometry and boundary conditions are chosen to match realistic device configurations. The device is driven through a cycle with two adiabatic and two constant electric field legs, and compared with the analytically computed ideal bulk electrocaloric cooling cycle. Several inefficiencies arise in the device, including incomplete transformation, entropy loss due to phase boundary motion, and high energy zones with large stresses and closure domains at the electrode tip.

Degenerate boundary conditions on a geometric graph

The boundary conditions of the Sturm-Liouville problem defined on a star-shaped geometric graph of three edges are studied. It is shown that if the lengths of the edges are different, then the Sturm-Liouville problem does not have degenerate boundary conditions. If the lengths of the edges and the potentials are the same, then the characteristic determinant of the Sturm-Liouville problem can not be equal to a constant different from zero. But the set of Sturm-Liouville problems for which the characteristic determinant is identically equal to zero is an infinite (continuum). In this way, in contrast to the Sturm-Liouville problem defined on an interval, the set of boundary-value problems on a star-shaped graph whose spectrum completely fills the entire plane is much richer. In the particular case when the minor A124 for matrix of coefficients is nonzero, it does not consist of two problems, as in the case of the Sturm-Liouville problem given on an interval, but of 18 classes, each containing two to four arbitrary constants.

Coupled thermal stresses analysis in the composite elastic-plastic cylinder

The present study is devoted to the set of boundary value problems in the frameworks of coupled thermoelastoplasticity under axial symmetry conditions for a composite circular cylinder. Throughout the paper the conventional Prandtl–Reuss elastic–plastic model generalised on the thermal effects is used. The yield stress is assumed by linear function of the temperature. The plastic potential is chosen in the form of Tresca yield criterion and the associated plastic flow rule is derived. The adding process of a heated cylinder to another is simulated. The coupled thermal stresses are calculated during processes of cooling and material unloading. The elastic-plastic borders positions are calculated and plastic flow domains are localized. Numerical results are graphically analysed.

Symmetries, coverings, and decomposition of systems and trajectory generation

We derive relations between the notions of symmetry, covering, and decomposition of systems and trajectory generation. We show that any decomposition of a system determines a finite-dimensional covering of that system and is determined by it. We present conditions on vector fields under which any covering is obtained by factorization along the Lie algebra of such fields. On the basis of these relations, we study whether a point-to-point steering problem can be transformed into a set of boundary value problems of lower dimension.

A new fitted operator finite difference method to solve systems of evolutionary reaction-diffusion equations

In recent years, fitted operator finite difference methods (FOFDMs) have been developed for numerous types of singularly perturbed ordinary differential equations. The construction of most of these methods differed though the final outcome remained similar. The most crucial aspect was how the difference operator was designed to approximate the differential operator in question. Very often the approaches for constructing these operators had limited scope in the sense that it was difficult to extend them to solve even simple one-dimensional singularly perturbed partial differential equations. However, in some of our most recent work, we have successfully designed a class of FOFDMs and extended them to solve singularly perturbed time-dependent partial differential equations. In this paper, we design and analyze a robust FOFDM to solve a system of coupled singularly perturbed parabolic reaction-diffusion equations. We use the backward Euler method for the semi-discretization in time. An FOFDM is then developed to solve the resulting set of boundary value problems. The proposed method is analyzed for convergence. Our method is uniformly convergent with order one and two, respectively, in time and space, with respect to the perturbation parameters. Some numerical experiments supporting the theoretical investigations are also presented.

Wavelet Galerkin method for solving nonlinear singular boundary value problems arising in physiology

In this study, wavelet Galerkin method has been developed for solving nonlinear singular boundary value problems associated with physiology science. The quasilinearization technique is applied to reduce the given nonlinear problem to a sequence of linear problems. We modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using compactly supported cubic B-spline wavelets and Galerkin method. These wavelets are used as testing and weighting functions. The method is computationally attractive, and applications are demonstrated through illustrative examples.

Approaches to the mesokinetic theory of search for the optimal technological route for conversion of a raw material into the desired material

Information available on the synthesis of new materials demonstrates that the increased demand for materials can be satisfied only by passing from analogy-based and combinatorial search for material production technology to search based on the semiempirical theory of raw material conversion into the desired material. This theory is representable as a set of boundary-value problems on the kinetics of the elementary processes that lead to the formation of a material and are conducted in apparatuses varying in their design, with experimental data for substances of various natures generalized in these boundary-value problems. While accumulating experimental data, it is appropriate to carry out a four-step passage from virtual modeling of the operation of apparatuses to process optimization.

On solvability sets of boundary value problems for linear functional differential equations

On solvability sets of boundary value problems for linear functional differential equations

Thermomechanical Deformation in an Orthotropic Micropolar Thermoelastic Solid

The thermomechanical deformation in an orthotropic micropolar generalized thermoelastic half-space is investigated. Descarte’s method, along with the irreducible case of Cardon’s method, is used to obtain the roots of an eight-degree equation. Laplace and Fourier transform techniques are used to obtain the general solution for the set of boundary value problems. Particular types of boundary conditions have been taken to illustrate the utility of the approach. The transformed components of the stresses and temperature distribution have been obtained. A numerical inversion technique is employed to invert the integral transform, and the resulting quantities are presented graphically.

Spline solution of non-linear singular boundary value problems

A class of non-linear singular boundary value problems is solved by new methods based on non-polynomial cubic spline. We use the quasilinearization technique to reduce the given non-linear problem to a sequence of linear problems. We modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using a non-polynomial cubic spline approximation. Convergence of the methods is shown through standard convergence analysis. Numerical examples are given to illustrate the applicability and efficiency of our methods.

A method of convolution of Fourier expansions as applied to solving boundary value problems with intersecting interface lines

An efficient method of construction of solutions to a set of boundary value problems with additional interface conditions, more complicated boundary conditions, and so on on the basis of known solutions to classical boundary value problems is proposed. The method is based on the representation of solutions to classical and more complicated problems in the form of expansions into Fourier series with subsequent reduction of one series to the other. As a result, formulas directly expressing solutions to more complicated problems in terms of solutions to classical problems are obtained. On the basis of the well-known solution to the Dirichlet problem on a half plane, solutions to boundary value problems with interface conditions (including generalized conditions of the type of a crack and a screen) on intersecting straight lines for boundary conditions of the first and the third kind are obtained.

The numerical solution of non-linear singular boundary value problems arising in physiology

A class of non-linear singular ordinary differential equations, is solved by a new method based on non-polynomial cubic spline. We use the quesilinearization technique to reduce the given non-linear problem to a sequence of linear problems. We modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using non-polynomial cubic spline approximation. The resulting system of algebraic equations is solved by using a tri-diagonal solver. Computational results are provided to demonstrate the viability of the new method.

On lower and upper solutions without ordering on time scales

In order to enlarge the set of boundary value problems on time scales, for which we can use the lower and upper solutions technique to get existence of solutions, we extend this method to the case when the pair lacks ordering. We use the degree theory and a priori estimates to obtain the existence of solutions for the second-order Dirichlet boundary value problems. To illustrate a wider application of this result, we conclude with an example which shows that a combination of well and non-well ordered pairs can yield the existence of multiple solutions.