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The scheme for solving of a mixed problem with general boundary conditions is proposed for a heat equation \[a(x)\frac{\partial T}{\partial \tau}= \frac{\partial}{\partial x} \left(\lambda(x)\frac{\partial T}{\partial x}\right)\] with coefficient $a(x)$ that is thegeneralized derivative of a function of bounded variation, $\lambda(x)>0$, $\lambda^{-1}(x)$ is a bounded and measurable function. The boundary conditions have the form $$\left\{ \begin{array}{l}p_{11}T(0,\tau)+p_{12}T^{[1]}_x (0,\tau)+ q_{11}T(l,\tau)+q_{12}T^{[1]}_x (l,\tau)= \psi_1(\tau),\\p_{21}T(0,\tau)+p_{22}T^{[1]}_x (0,\tau)+ q_{21}T(l,\tau)+q_{22}T^{[1]}_x (l,\tau)= \psi_2(\tau),\end{array}\right.$$ where by $T^{[1]}_x (x,\tau)$ we denote the quasiderivative $\lambda(x)\frac{\partial T}{\partial x}$. A solution of this problem seek by thereduction method in the form of sum of two functions $T(x,\tau)=u(x,\tau)+v(x,\tau)$. This method allows to reduce solving of proposed problem to solving oftwo problems: a quasistationary boundary problem with initialand boundary conditions for the search of the function $u(x,\tau)$ and a mixed problem with zero boundaryconditions for some inhomogeneous equation with an unknown function $v(x,\tau)$. The first of these problems is solved through the introduction of the quasiderivative. Fourier method andexpansions in eigenfunctions of some boundary value problem forthe second-order quasidifferential equation $(\lambda(x)X'(x))'+ \omega a(x)X(x)=0$ are used for solving of the second problem. The function $v(x,\tau)$ is represented as a series in eigenfunctions of this boundary value problem. The results can be used in the investigation process of heat transfer in a multilayer plate.
44A unified modeling approach for rotating flexible shaft-disk systems with general boundary and coupling conditions
A unified modeling approach for rotating flexible shaft-disk systems with general boundary and coupling conditions
42Thermoelasticity of multilayered plates with imperfect interfaces
Thermoelasticity of multilayered plates with imperfect interfaces
71Nonlinear dynamics of straight fluid-conveying pipes with general boundary conditions and additional springs and masses
Nonlinear dynamics of straight fluid-conveying pipes with general boundary conditions and additional springs and masses
Identification of patterns of non-stationary laminar flow of a viscous fluid at the inlet section of plane-parallel pressure flow
The inlet region of a plane-parallel pressure flow is the subject of this study. The patterns of variations in the hydrodynamic inlet region under unsteady plane-parallel pressure flow of a viscous fluid are examined in this research. Based on the boundary layer equation and flow characteristics, the boundary conditions of the problem were determined and a boundary value problem was formulated. The problem's boundary conditions were established and a boundary value problem was developed based on the boundary layer equation and flow characteristics. In order to find patterns of velocity change over time and over the length of the inlet region under general boundary conditions, a method for integrating the boundary value conditions was created. Solutions for scenarios with a constant and parabolic velocity distribution in the inlet region were derived from the general solutions. Regularities of pressure and velocity change were found along the entire hydrodynamic inlet region. Using computer analysis, graphs of velocity changes over time at various points along the entire length of the inlet region are constructed. The patterns of velocity distribution along the entire length of the inlet region depending on time can be seen using graphs. This allows one to estimate the length of the hydrodynamic inlet region and calculate the fluid flow velocity at any point in this region. The findings enable revealing the essence of the processes running in an hydropneumatic automation system's transition sections. Based on the revealed regularities of the hydrodynamic parameters of viscous incompressible liquid during unsteady flows, it is possible to correctly design of the automatic systems' channels of regulating units ensuring their smooth and accurate operation
Read more41Generalized boundary conditions for the Aharonov–Bohm effect combined with a homogeneous magnetic field
The most general admissible boundary conditions are derived for an idealized Aharonov–Bohm flux intersecting the plane at the origin on the background of a homogeneous magnetic field. A standard technique based on self-adjoint extensions yields a four-parameter family of boundary conditions; the other two parameters of the model are the Aharonov–Bohm flux and the homogeneous magnetic field. The generalized boundary conditions may be regarded as a combination of the Aharonov–Bohm effect with a point interaction. Spectral properties of the derived Hamtonians are studied in detail.
Read more31Vibrations of Composite Laminated Circular Panels and Shells of Revolution with General Elastic Boundary Conditions via Fourier-Ritz Method
A Fourier-Ritz method for predicting the free vibration of composite laminated circular panels and shells of revolution subjected to various combinations of classical and non-classical boundary conditions is presented in this paper. A modified Fourier series approach in conjunction with a Ritz technique is employed to derive the formulation based on the first-order shear deformation theory. The general boundary condition can be achieved by the boundary spring technique in which three types of liner and two types of rotation springs along the edges of the composite laminated circular panels and shells of revolution are set to imitate the boundary force. Besides, the complete shells of revolution can be achieved by using the coupling spring technique to imitate the kinematic compatibility and physical compatibility conditions of composite laminated circular panels at the common meridian with θ = 0 and 2π. The comparisons established in a sufficiently conclusive manner show that the present formulation is capable of yielding highly accurate solutions with little computational effort. The influence of boundary and coupling restraint parameters, circumference angles, stiffness ratios, numbers of layer and fiber orientations on the vibration behavior of the composite laminated circular panels and shells of revolution are also discussed.
Read more11A Dichotomy of PDE Sturmian Theory
Parallel Sturmian comparison theorems are proved for hyperbolic and elliptic linear differential equations of second order, with special emphasis on cylindrical domains with the time axis parallel to the axis of the cylinder. These theorems have the following form: If a boundary problem I for a partial differential equation in a bounded domain has a nontrivial solution, then every solution of a second boundary problem II of the same type has a zero in the domain if the coefficient functions and boundary functions of II majorize those of I. The results extend and unify earlier results given for either hyperbolic equations or elliptic equations. In particular, the majorization hypotheses in Theorem 3, in the form of inequalities between the data functions in the differential equations and in the boundary conditions, indicate parallel sets of sufficient conditions for the Sturmian conclusion in the elliptic and hyperbolic cases. Counterexamples are given to show that two known theorems in the elliptic case do not extend to the hyperbolic case. Extensions to singular boundaries and mixed problems are included. The dichotomy referred to in the title pertains to the different but parallel Sturmian theories for elliptic and hyperbolic boundary problems: Results of similar type are true for both types of problems and yet other results true in the elliptic case are false in the hyperbolic case.
Read more11Efficient spectral-Galerkin methods for polar and cylindrical geometries
Efficient spectral-Galerkin methods for polar and cylindrical geometries
9Imposing slip conditions on curved boundaries for 3D incompressible flows with a very high-order accurate finite volume scheme on polygonal meshes
The conventional no-slip boundary condition does not always hold in several fluid flow applications and must be replaced with appropriate slip conditions according to the wall and fluid properties. However, not only slip boundary conditions are still a subject of discussion among fluid dynamicists, but also their numerical treatment is far from being well-developed, particularly in the context of very high-order accurate methods. The complexity of these conditions significantly increases when the boundary is not aligned with the chosen coordinate system and, even more challenging, when the fluid slips along a curved boundary. The present work proposes a simple and efficient numerical treatment of general slip boundary conditions on arbitrary curved boundaries for three-dimensional fluid flow problems governed by the incompressible Navier–Stokes equations. In that regard, two critical challenges arise: (i) achieving very high-order of convergence with arbitrary curved boundaries for the classical no-slip boundary conditions and (ii) extending the developed numerical techniques to impose general slip boundary conditions. The conventional treatment of curved boundaries relies on generating curved meshes to eliminate the geometrical mismatch between the physical and computational boundaries and achieve high-order of convergence. However, such an approach requires sophisticated meshing algorithms, cumbersome quadrature rules on curved elements, and complex non-linear transformations. In contrast, the reconstruction for off-site data (ROD) method handles arbitrary curved boundaries approximated with linear piecewise elements while employing polynomial reconstructions with specific linear constraints to fulfil the prescribed boundary conditions. For that purpose, the general slip boundary conditions are reformulated on a local orthonormal basis to allow a straightforward application of the ROD method with scalar boundary conditions. The Navier–Stokes equations are discretised with a staggered finite volume method, and the numerical fluxes are computed solely on the polygonal mesh elements. Several benchmark test cases of fluid flow problems in non-trivial three-dimensional curved domains are addressed and confirm that the proposed method effectively achieves up to the eighth-order of convergence.
Read more36Three-Dimensional Green’s Functions in an Anisotropic Half-Space With General Boundary Conditions
This paper derives, for the first time, the complete set of three-dimensional Green’s functions (displacements, stresses, and derivatives of displacements and stresses with respect to the source point), or the generalized Mindlin solutions, in an anisotropic half-space z>0 with general boundary conditions on the flat surface z=0. Applying the Mindlin’s superposition method, the half-space Green’s function is obtained as a sum of the generalized Kelvin solution (Green’s function in an anisotropic infinite space) and a Mindlin’s complementary solution. While the generalized Kelvin solution is in an explicit form, the Mindlin’s complementary part is expressed in terms of a simple line-integral over [0,π]. By introducing a new matrix K, which is a suitable combination of the eigenmatrices A and B, Green’s functions corresponding to different boundary conditions are concisely expressed in a unified form, including the existing traction-free and rigid boundaries as special cases. The corresponding generalized Boussinesq solutions are investigated in details. In particular, it is proved that under the general boundary conditions studied in this paper, the generalized Boussinesq solution is still well-defined. A physical explanation for this solution is also offered in terms of the equivalent concept of the Green’s functions due to a point force and an infinitesimal dislocation loop. Finally, a new numerical example for the Green’s functions in an orthotropic half-space with different boundary conditions is presented to illustrate the effect of different boundary conditions, as well as material anisotropy, on the half-space Green’s functions.
Read moreStatic and Vibration Analysis of Two-phase Peridynamic Rods or Strings under General Boundary Conditions — Exact, FDM and FEM Approaches
Static and dynamic analysis of two-phase peridynamic one-dimensional systems is thoroughly investigated in this paper for various boundary conditions by exact and numerical approaches. Such a nonlocal study concerns the axial or torsional responses of two-phase peridynamic rods or the behavior of a two-phase peridynamic string. For a normalized exponential kernel, the integro-differential equation of the two-phase peridynamic problem can be converted as a linear higher-order differential equation with constant coefficients. Exact static solutions of fixed–fixed and fixed–free two-phase peridynamic axial rod (TPPAR) under uniform distributed loading and concentrated force are analytically derived. It is shown that the pure peridynamic rod problem can be ill-posed due to a conflict between the natural boundary condition and the peridynamic constitutive boundary condition. The two-phase peridynamic problem may be justified from physical arguments, and is also seen as a regularized peridynamic problem. The exact eigen-frequencies of the two-phase peridynamic rods are also analytically obtained for various boundary conditions, including fixed–free, fixed–fixed or free–free boundary conditions. These theoretical approaches are corroborated by numerical investigations based on Finite Difference Method (FDM), Finite Element Method (FEM), exact FEM and variational iterative methods. Two FDM schemes are presented for the statics and the vibration of two-phase peridynamic rods, with convergence orders of 1 and 2, respectively. Moreover, three different FE procedures are proposed as: (1) FEM for direct solving the integro-differential equations of two-phase peridynamic one-dimensional system, (2) FEM for discretizing the pure differential equations, and (3) Exact FEM to obtain both the exact stiffness matrix and exact dynamic stiffness matrix of two-phase peridynamic one-dimensional system, based on Eisenberger’s methodology. The order of convergence for the first and the second proposed FEM is equal to 2 and 4, according to the type of selected finite elements. Moreover, the third proposed FEM based on the exact shape function of the peridynamic problem gives the exact solution from one single FEM element. In addition, the variational iterative method of Stodola–Vianello is successfully applied to the two-phase peridynamic eigenvalue problems.
Read more14Solution to Vibrations of Double-Beam Systems under General Boundary Conditions
The problem of vibrations of double-beam systems is of technological interest and has been extensively studied in the literature. However, a general solution is yet to be developed for the general boundary conditions. In this paper, a general solution of the free and forced vibrations of double-beam systems under classical, nonclassical, and mixed boundary conditions is derived. The obtained mode-shape solutions are exact and explicit. The characteristic frequency equation of a double-beam system is of order-4 and order-8 for the classical boundary conditions and the nonclassical boundary conditions, respectively. The solution of forced vibrations is developed with the classical modal expansion technique. In addition to the theory, applications of the proposed method are illustrated by numerical examples. The obtained frequencies for special problems are compared with the literature, and the agreement between the proposed method and the literature is remarkably good. This shows that the proposed shape function method may be applied reliably to determine the transverse vibration mode shapes and natural frequencies of the double-beam system under the general boundary conditions. The solutions of free and forced vibrations of the double-beam systems presented in the paper could easily be programmed into a computer for vibration analyses.
Read more7Second Order Differential Equations with General Boundary Conditions
This article is concerned with second order linear differential systems involving a parameter together with integral boundary conditions. The objective is to establish the existence of eigenvalues for the boundary problems and to determine the oscillatory behavior of the associated solutions. The methods employed in the paper are derived primarily from the work of W. M. Whyburn, and the results presented here represent an extension of some of his work on similar boundary problems.
Read more27Vibration analyses of general thin and moderately thick laminated composite curved beams with variable curvatures and general boundary conditions
An exact semianalytical method for vibration analysis of general thin and moderately thick laminated composite curved beams with variable curvatures and general boundary conditions is presented. In the framework of the first-order shear deformation theory, the method combines the variational principle and multilevel partition technique. As one of the innovation points, the general boundary conditions are enforced by using the virtual boundary spring technology. Each of the fundamental beam unknowns is then invariantly expanded as Jacobi polynomials. The convergence study and numerical verifications of the laminated composite curved beams with various boundary conditions are carried out.
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17Kernel density estimation with linked boundary conditions
Kernel density estimation on a finite interval poses an outstanding challenge because of the well‐recognized bias at the boundaries of the interval. Motivated by an application in cancer research, we consider a boundary constraint linking the values of the unknown target density function at the boundaries. We provide a kernel density estimator (KDE) that successfully incorporates this linked boundary condition, leading to a non‐self‐adjoint diffusion process and expansions in nonseparable generalized eigenfunctions. The solution is rigorously analyzed through an integral representation given by the unified transform (or Fokas method). The new KDE possesses many desirable properties, such as consistency, asymptotically negligible bias at the boundaries, and an increased rate of approximation, as measured by the AMISE. We apply our method to the motivating example in biology and provide numerical experiments with synthetic data, including comparisons with state‐of‐the‐art KDEs (which currently cannot handle linked boundary constraints). Results suggest that the new method is fast and accurate. Furthermore, we demonstrate how to build statistical estimators of the boundary conditions satisfied by the target function without a priori knowledge. Our analysis can also be extended to more general boundary conditions that may be encountered in applications.
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