Single Unknown Function Research Articles

Simply Supported Polygon Plate Frequencies using Classical and Shear Deformation Theories

In this paper, the frequency equation relationship between classical plate theory and a third order single variable plate theory is derived. The third order single variable plate theory used herein is considerably simpler in the class of higher order plate theories. The theory uses a single variable to formulate the plates. The theory is governed by only one differential equation. As the formulation involves a single unknown function, plate analysis using this theory goes almost in the similar lines of classical plate theory. Hence, the complexity associated with plate analysis is considerably less compared to other higher order plate theories. This paper focuses on obtaining the frequency equation relationship between the third order single variable plate theory used herein and the classical plate theory. The relationships between the third order plate theory and classical plate theory frequency equations are derived. Using the derived equations, the frequencies have been calculated for the case of regular polygon plates with simply supported edge conditions. Frequencies calculated herein are compared with other shear deformation plate theory frequencies. The obtained plate frequency results are in good agreement and are accurate.

Parallel Mosaic-Skeleton Algorithm for the Numerical Solution of a Three-Dimensional Scalar Scattering Problem in Integral Form

A three-dimensional scalar stationary scattering problem is considered. It is formulated in the form of a weakly singular Fredholm boundary integral equation of the first kind with a single unknown function. The equation is approximated by a system of linear algebraic equations, which is then solved numerically by an iterative method. The mosaic-skeleton method is used at the stage of the approximate solution of this system in order to reduce the computational complexity of the approach.

ТОЧНОЕ РЕШЕНИЕ ЗАДАЧИ О НАГРУЖЕНИИ ДАВЛЕНИЯМИ ПОЛОГО ЦИЛИНДРА ИЗ НЕЛИНЕЙНО НАСЛЕДСТВЕННОГО МАТЕРИАЛА В СЛУЧАЕ СТЕПЕННЫХ ФУНКЦИЙ НЕЛИНЕЙНОСТИ

We study analytically the exact solution of the quasi-static problem for a thick-walled tube of physically non-linear viscoelastic material obeying the Rabotnov constitutive equation with two arbitrary material functions (a creep compliance and a function which governs physical non-linearity). We suppose that a material is homogeneous, isotropic and incompressible and that a tube is loaded with time-dependent internal and external pressures (varying slowly enough to neglect inertia terms in the equilibrium equations) and that a plain strain state is realized, i.e. zero axial displacements are given on the edge cross sections of the tube. We previously have obtained the closed form expressions for displacement, strain and stress fields via the single unknown function of time and integral operators involving this function, two arbitrary material functions of the constitutive relation, preset pressure values and radii of the tube and derive functional equation to determine this unknown resolving function. Assuming creep complience is arbitrary and choosing the material function governing non-linearity to be power function with a positive exponent, we construct exact solution of the resolving non-linear functional equation, calculate all the convolution integrals involved in the general representation for strain and stress fields and reduce it to simple algebraic formulas convenient for analysis and use. Strains evolution in time is characterized by creep compliance function and loading history. The stresses in this case depend on the current magnitudes of pressures only, they don't depend on creep compliance (i.e. viscoelastic properties of a material) and on loading history. The stress field coincides with classical solution for non-linear elastic material or elastoplastic material with power hardening (for non-decreasing pressure difference). We obtain criteria for increase, decrease or constancy of stresses with respect to radial coordinate in form of inequalities for the exponent value and for difference of pressures. Assuming creep compliance is arbitrary, we study analytically properties of strain and stress fields in a tube under internal pressure growing with constant rate and properties of corresponding stress-strain curves implying measurement of strains at a surface point of a tubular specimen.

Three-dimensional electric potential induced by a point singularity in a multilayered dielectric medium

A simple and effective method is proposed to derive the three-dimensional electric potential induced by a point singularity of any type in an N-phase dielectric medium composed of N−2 intermediate dielectric layers of equal thickness encased in two semi-infinite dielectric media. The point singularity can include a point charge or a point electric dipole. The original boundary value problem for the N-phase medium is reduced to the determination of a single unknown three-dimensional harmonic function through satisfaction of the continuity conditions across all of the N−1 perfect planar interfaces. The single harmonic function can be completely determined after analytically solving the resulting linear recurrence relations, which are independent of the type and the specific location of the singularity. When the singularity is a point charge, we obtain the self-energy of the point charge expressed in terms of a single function and the Coulomb force on the point charge expressed in terms of the negative derivative of this function.

HYDRODYNAMICS, DISTRIBUTION OF FLOWS AND THERMAL EFFICIENCY OF COIL HEAT EXCHANGERS IN UNITS OF HEAT-USING APPARATUS OF TUBE FURNACES

The theoretical foundations of construction, mathematical description and engineering calculation of heat exchangers of the serpentine type in blocks of heat-using equipment of tube furnaces and other types of reactors designed for carrying out endothermic reactions (in particular, reforming of natural gas with water vapor) are considered. It is shown that the thermal efficiency of heat exchangers of the coil type is significantly affected by the correct choice of parameters ensuring a uniform distribution of energy flows over the surface of heat-resistant heat exchange tubes. This technological problem is solved by compiling the heat balance and selecting the system of the corresponding equations, which allows to calculate the temperature contour of the coil heat exchanger, its hydrodynamic characteristics and the distribution of mass and heat flows through the heat exchange tubes. The use of the tensor form of the Boussinesq hypothesis is considered, with which the Reynolds equation describing a turbulent flow is transformed to a partial differential equation for a single unknown function and its averaged form is obtained. In relation to the problem under consideration, the correctness of the chosen approach was confirmed both theoretically and experimentally. It is shown that in the core of a turbulent flow with an intense suction or injection, the liquid behaves almost as ideal and the well-known Helmholtz – Friedmann theorem holds with the necessary accuracy. From the aforementioned averaged equation, expressions are obtained that are suitable for describing heat fluxes in channels with suction or injection. According to this theoretical model, thermal calculations of coil-type heat exchangers were carried out, a more accurate assessment of the temperature of the heated medium in each coil tube was made, and the temperature gradient of the external heat carrier over the cross section of the gas duct was found. For the first time in the practice of calculations when choosing the parameters of coils, a number of boundary conditions were taken into account, such as the condition of the coil layout, the necessary heat exchange surface, permissible restrictions on hydraulic resistance, etc.

A study of quantum field theories in AdS at finite coupling

We study the O(N) and Gross-Neveu models at large N on AdSd+1 background. Thanks to the isometries of AdS, the observables in these theories are constrained by the SO(d, 2) conformal group even in the presence of mass deformations, as was discussed by Callan and Wilczek [1], and provide an interesting two-parameter family of quantities which interpolate between the S-matrices in flat space and the correlators in CFT with a boundary. For the actual computation, we judiciously use the spectral representation to resum loop diagrams in the bulk. After the resummation, the AdS 4-particle scattering amplitude is given in terms of a single unknown function of the spectral parameter. We then “bootstrap” the unknown function by requiring the absence of double-trace operators in the boundary OPE. Our results are at leading nontrivial order in frac{1}{N} , and include the full dependence on the quartic coupling, the mass parameters, and the AdS radius. In the bosonic O(N) model we study both the massive phase and the symmetry-breaking phase, which exists even in AdS2 evading Coleman’s theorem, and identify the AdS analogue of a resonance in flat space. We then propose that symmetry breaking in AdS implies the existence of a conformal manifold in the boundary conformal theory. We also provide evidence for the existence of a critical point with bulk conformal symmetry, matching existing results and finding new ones for the conformal boundary conditions of the critical theories. For the Gross-Neveu model we find a bound state, which interpolates between the familiar bound state in flat space and the displacement operator at the critical point.

New Single-Source Surface Integral Equation for Magneto-Quasi-Static Characterization of Transmission Lines Situated in Multilayered Media

We recently proposed a novel single-source integral equation (SSIE) for accurate broadband resistance and inductance extraction and current flow modeling in 2-D conductors. The new surface integral equation is advantageous compared with the traditional volume electric field integral equation (V-EFIE) used for the inductance extraction, since the unknown function is defined on the surface of conductors as opposed to the volumetric unknown current density in V-EFIE. The new SSIE is also more suitable for the solution of inductance extraction problems than the traditional surface integral equation formulations, as it features only a single unknown surface function as opposed to having the unknown equivalent electric and magnetic surface current densities. The new equation also features only the electric field Green’s functions unlike the previously known SSIE formulations. The latter property makes the new SSIE equation particularly suitable to the inclusion of the multilayered substrate effect into the inductance extraction model. This paper describes the generalization of the new SSIE formulation to the case of transmission line models embedded into the multilayered lossy substrates. This paper also shows how the matrix sparsity in the method of moments discretization of the novel integral equation can be exploited to accelerate its numerical solution and reduce associated memory use. This sparsity arises due to the skin-effect-based attenuation of the fields in conductors’ cross sections leading to vanishing levels of the matrix elements corresponding to the distant interactions. Typical examples of inductance extraction in complex interconnects situated in lossy substrate are considered to validate the proposed techniques against traditional approaches.

A crack with surface effects in a piezoelectric material

We study the contribution of surface piezoelectricity to the anti-plane deformations of a hexagonal piezoelectric material weakened by a crack. The surface piezoelectricity is incorporated by using an extended version of the continuum-based surface/interface model of Gurtin and Murdoch. The original boundary value problem is finally reduced to a system of two coupled first-order Cauchy singular integro-differential equations by considering a distribution of line dislocations and electric-potential-dislocations on the crack. Through a diagonalization strategy, the coupled system can be transformed into two independent singular integro-differential equations, each of which contains only one single unknown function and can be numerically solved by the collocation method. Our solution demonstrates that the stresses, strains, electric displacements and electric displacements exhibit the logarithmic singularity at the crack tips. The obtained solution is further used to predict the size-dependent effective electroelastic properties of a piezoelectric solid containing multiple nanocracks with surface piezoelectricity within the framework of non-interaction approximation.

Blind X-Ray CT Image Reconstruction From Polychromatic Poisson Measurements

We develop a method for sparse image reconstruction from polychromatic computed tomography (CT) measurements under the blind scenario where the material of the inspected object and the incident-energy spectrum are unknown. We obtain a parsimonious measurement-model parameterization by changing the integral variable from photon energy to mass attenuation, which allows us to combine the variations brought by the unknown incident spectrum and mass attenuation into a single unknown mass-attenuation spectrum function; the resulting measurement equation has the Laplace integral form. The mass-attenuation spectrum is then expanded into first order B-spline basis functions. We derive a block coordinate-descent algorithm for constrained minimization of a penalized negative log-likelihood (NLL) cost function, where penalty terms ensure nonnegativity of the spline coefficients and nonnegativity and sparsity of the density map. The image sparsity is imposed using total-variation (TV) and $\ell_1$ norms, applied to the density-map image and its discrete wavelet transform (DWT) coefficients, respectively. This algorithm alternates between Nesterov's proximal-gradient (NPG) and limited-memory Broyden-Fletcher-Goldfarb-Shanno with box constraints (L-BFGS-B) steps for updating the image and mass-attenuation spectrum parameters. To accelerate convergence of the density-map NPG step, we apply a step-size selection scheme that accounts for varying local Lipschitz constant of the NLL. We consider lognormal and Poisson noise models and establish conditions for biconvexity of the corresponding NLLs. We also prove the Kurdyka-{\L}ojasiewicz property of the objective function, which is important for establishing local convergence of the algorithm. Numerical experiments with simulated and real X-ray CT data demonstrate the performance of the proposed scheme.

Application of Differential Quadrature method in free vibration analysis of nanobeams based on various nonlocal theories

Conducting experiments with nanoscale size is quite difficult to handle. In this regard, mathematical modeling and their solutions play a vital role in the field of nanotechnology. In the present work, the governing differential equations are converted into a single unknown function and then Differential Quadrature (DQ) method is employed to investigate free vibration of nanobeams based on different beam theories like Euler–Bernoulli, Timoshenko, Reddy and Levinson in conjunction with nonlocal elasticity theory for the first time. Application of DQ method in the governing differential equation converts the problem to a generalized eigenvalue problem and the solution of this gives the frequency parameters. It is observed that frequency parameters are overpredicted in Euler–Bernoulli beam theory than other types of beam theories. It is also concluded that the effect of nonlocal parameter is more in case of higher vibration modes. Present results are compared with other available results and are found to be in good agreement.

Calculation of the oscillations of shells of revolution with a liquid using the finite element method

A new version of the finite element method for calculation of axisymmetric and nonaxisymmetric oscillations of shells of revolution partially filled with an ideal incompressible liquid is presented. Narrow annular strips of the shell together with thin layers of the liquid contained in them are considered to be finite elements. Displacements of a liquid layer are approximated by a single unknown function, which represents an axial displacement in an assumed form, using an exact analytical solution of the continuity equation with a boundary condition of continuity at the shell surface. The matrices of the stiffness and inertia of the shell and added liquid mass matrices have been obtained.

Collapse of a self-similar cylindrical scalar field with non-minimal coupling II: strong cosmic censorship

We investigate self-similar scalar field solutions to the Einstein equations in whole cylinder symmetry. Imposing self-similarity on the spacetime gives rise to a set of single variable functions describing the metric. Furthermore, it is shown that the scalar field is dependent on a single unknown function of the same variable and that the scalar field potential has exponential form. The Einstein equations then take the form of a set of ODEs. Self-similarity also gives rise to a singularity at the scaling origin. We extend the work of Condron and Nolan (2014 Class. Quantum Grav. 31 015015), which determined the global structure of all solutions with a regular axis in the causal past of the singularity. We identified a class of solutions that evolves through the past null cone of the singularity. We give the global structure of these solutions and show that the singularity is censored in all cases.

Collapse of a self-similar cylindrical scalar field with non-minimal coupling

We investigate self-similar scalar field solutions to the Einstein equations in whole cylinder symmetry. Imposing self-similarity on the spacetime gives rise to a set of single variable functions describing the metric. Furthermore, it is shown that the scalar field is dependent on a single unknown function of the same variable and that the scalar field potential has exponential form. The Einstein equations then take the form of a set of ODEs. Self-similarity also gives rise to a singularity at the scaling origin. We discuss the number of degrees of freedom at an arbitrary point and prove existence and uniqueness of a 2-parameter family of solutions with a regular axis. We discuss the evolution of these solutions away from the axis toward the past null cone of the singularity, determining the maximal interval of existence in each case.

Subanalytic solutions of linear difference equations and multidimensional hypergeometric sequences

We consider linear difference equations with polynomial coefficients over C and their solutions in the form of sequences indexed by the integers (sequential solutions). We investigate the C -linear space of subanalytic solutions, i.e., those sequential solutions that are the restrictions to Z of some analytic solutions of the original equation. It is shown that this space coincides with the space of the restrictions to Z of entire solutions and that the dimension of this space is equal to the order of the original equation. We also consider d -dimensional ( d ≥ 1 ) hypergeometric sequences, i.e., sequential and subanalytic solutions of consistent systems of first-order difference equations for a single unknown function. We show that the dimension of the space of subanalytic solutions is always at most 1 , and that this dimension may be equal to 0 for some systems (although the dimension of the space of all sequential solutions is always positive). Subanalytic solutions have applications in computer algebra. We show that some implementations of certain well-known summation algorithms in existing computer algebra systems work correctly when the input sequence is a subanalytic solution of an equation or a system, but can give incorrect results for some sequential solutions.

Contact interaction of two elastic half spaces with circular groove

We present the method of functions of intercontact gaps extended to axially symmetric contact problems for two half spaces one of which contains a circular surface groove. This approach is based on the construction of integral representations for the displacements and stresses in each body of the contact couple via the functions defined in an unknown (but bounded) region of the contact interface. In the analyzed case of frictionless contact of the half spaces, the integral representations contain a single unknown function, namely, the height of the gap. As examples, the analytic solutions are obtained for some shapes of the initial groove. Thus, for some shapes of the groove, the normal contact stresses exhibit peaks at the point corresponding to the edge of the initial defect. The indicated feature of the distribution of stresses is discussed.

Fracture dynamics analysis of mode III interface moving crack on functionally graded coating

The interface moving crack between the functionally graded coating and infinite substrate structure with free boundary is investigated in this paper. By application of the interface bonding conditions of the two media, all the quantities have been represented by means of a single unknown function. With the help of the exponent model of the shear modulus and density, the dual integral equation of moving crack problem is obtained by Fourier transform. The displacement is expanded into series form using Jacob Polynomial, and then the semi-analytic solution of dynamic stress intensity factor is derived by Schmidt method. Dynamic stress intensity factor is influenced by those parameters such as crack velocity, graded parameter and coating height.

복합재료 구조 요소의 탄성문제에 대한 해

본 연구는 일반적인 적층 복합재료의 구조요소에서 탄성영역에 대한 해석적 해에 대한 방법을 나타낸 것이다. 혼합된 경계조건 하에서 2차원 평면응력탄성문제는 변위포텐셜함수라 불리는 단일미지함수로 표현된 1/4 부분미분방정식의 해로 축소시켰으며, 응력과 변위의 모든 성분은 어떠한 경계조건에도 적합한 방법을 만드는 동일한 변위포텐셜항으로 표현하였다. 이 방법은 각도를 가진 적층판과 90도 적층판으로 각각 구성된 구조요소의 두 가지 특별문제에 대해서 해석적인 해를 얻는데 적용된다. 본 연구에서 나타낸 몇 가지 수치적인 결과는 두 가지로 적층된 유리섬유복합재료에 관한 것이다. 연구결과는 지지된 하중의 임계영역에서 모든 경계조건이 정확히 만족되어 크게 신뢰할 만한 결과를 나타내었다. 이는 혼합된 어떠한 경계조건하에서도 복합재료의 구조요소에서 탄성영역에 대한 정확한 해석적 해를 얻는 데 적용시킬 수 있을 뿐 아니라 단순한 문제를 해결하는 데도 신뢰할 만한 결과를 얻을 수 있음을 입증한 것이다. The present study describes a method for analytical solution to elastic field in structural elements of general symmetric laminated composite materials. The two dimensional plane stress elasticity problems under mixed boundary conditions are reduced to the solution of a single fourth order partial differential equation, expressed in terms of a single unknown function, called displacement potential function. In addition, all the components of stress and displacement are expressed in terms of the same displacement potential function, which makes the method suitable for any boundary conditions. The method is applied to obtain analytical solutions to two particular problems of structural elements consisting of an angle-ply laminate and a cross-ply laminate, respectively. Some numerical results are presented for both the problems with reference to the glass/epoxy composite. The results are highly accurate and reliable as all the boundary conditions including those in the critical regions of supports and loads are satisfied exactly. This verifies the method as a simple and reliable one as well as capable to obtain exact analytical solution to elastic field in structural elements of composite materials under mixed and any other boundary conditions.

Formulation of reduced surface integral equations for the electromagnetic wave scattering from three‐dimensional layered dielectric bodies

A reduction procedure is developed for an arbitrarily shaped layered dielectric body using for each interface a single unknown function to which the classical surface electric and magnetic currents are related by some surface operators. These operators and single functions are determined recursively from one interface to the next. This allows us to derive the field everywhere from the solution of a surface integral equation in only one vector function relative to only the interface between the layered body and the source region. Since the reduction operators are independent of the structure of the outside region and of the given field source, and also invariant under translation and rotation, the analysis of the three‐dimensional electromagnetic wave scattering and propagation for systems of multilayered or/and multiply nested dielectric bodies based on reduced single integral equations is substantially more efficient than that based on existing coupled integral equation formulations using electric and magnetic currents on all the interfaces, especially for configurations with identical such bodies arbitrarily located and oriented with respect to each other.

Dimensions of solution spaces of H-systems

An H - system is a system of first-order linear homogeneous recurrence equations for a single unknown function T , with coefficients which are polynomials with complex coefficients. We consider solutions of H -systems which are of the form T : dom ( T ) → C where either dom ( T ) = Z d , or dom ( T ) = Z d ∖ S and S is the set of integer singularities of the system. It is shown that any natural number is the dimension of the solution space of some consistent H -system, and that in the case d ≥ 2 there are H -systems whose solution space is infinite dimensional. The relationship between dimensions of solution spaces in the two cases dom ( T ) = Z d and dom ( T ) = Z d ∖ S is investigated. We prove that every consistent H -system H has a non-zero solution T with dom ( T ) = Z d . Finally we give an appropriate corollary to the Ore–Sato theorem on possible forms of solutions of H -systems in this setting.

Mode III Yoffe Dynamic Crack Problem on the Interface between Dissimilar Materials

A moving crack in a laminated structure with free boundary subjected to anti-plane shear loading is investigated in this paper. Using the bonding conditions of the interface between different media, all the quantities in our question have been represented with a single unknown function, and the problem is transformed into a dual integrated equation with the method of Fourier transform. The equation is solved using Schmidt method. Finally the numerical results show the relationships among the dynamic stress intensity factor and crack velocity, the height of different laminated material, shear moduli of different laminated material.