Applications of GIFTS III to structural engineering problems

The model problem of a plane angle for a second-order elliptic system subject to Dirichlet, mixed, and Neumann boundary conditions is analyzed. For each boundary condition, the existence of solutions of the form $r^λv$ is reduced to spectral analysis of a particular matrix. Focusing on Dirichlet and mixed boundary conditions, optimal bounds on $|\Re λ|$ are derived, employing tools from numerical range analysis and accretive operator theory. The developed framework is novel and recovers known bounds for Dirichlet boundary conditions. The results for mixed boundary conditions are new and represent the central contribution of this work. Immediate applications of these findings are new regularity results for linear second-order elliptic systems subject to mixed boundary conditions.

Based on consolidation equations proposed for unsaturated soil, an analytical solution for 1D consolidation of an unsaturated single-layer soil with nonhomogeneous mixed boundary condition is developed. The mixed boundary condition can be used for special applications, such as tests occur in laboratory. The analytical solution is obtained by assuming all material parameters remain constant during consolidation. In the derivation of the analytical solution, the nonhomogeneous boundary condition is first transformed into a homogeneous boundary condition. Then, the eigenfunction and eigenvalue are derived according to the consolidation equations and the new boundary condition. Finally, using the method of undetermined coefficients and the orthogonal relation of the eigenfunction, the analytical solution for the new boundary condition is obtained. The present method is applicable to various types of boundary conditions. Several numerical examples are provided to investigate the consolidation behavior of an unsaturated single-layer soil with mixed boundary condition.

Boundary control of small solutions to fluid–structure interactions arising in coupling of elasticity with Navier–Stokes equation under mixed boundary conditions

Boundary value problems (BVP) arise in many mathematical model of frontier science. In short, a BVP is a second order ordinary differential equation with boundary conditions prescribed at two distinct points. Usually, we prescribe three types of boundary conditions: Dirichlet, Neumann and Mixed. Among them mixed boundary conditions is little difficult to approach. One of the method to solve the BVP with mixed boundary conditions is the shooting method. In this method, we first convert the BVP into an initial value problems (IVP) and then solve it. We then check whether the solution of IVP is a solution of the BVP. There are standard ways of solving IVPs. Recently, some alternate methods are also used to solve them. For example, several authors use Haar wavelet. Frames are considered as an extension to wavelet. In this paper, we construct operational matrices from a frame and then apply them to solve the IVP which is obtained from the given BVP by shooting method. This method is simple and easy to use because we construct the operational matrix once, store it and then apply it any times we want. Moreover, we also show that the given operational matrix method converges quadratically.

When quantum field theory is formulated in terms of transition amplitudes between initial and final configurations, it is important to impose mathematically consistent boundary conditions on the fields. The choice of boundary conditions is also important when dealing with elliptic complexes and index theorems on bounded manifolds. Although Dirichlet and Neumann boundary conditions are adequate for many purposes, they are not always appropriate. However, a number of the remaining cases can be handled by imposing Dirichlet conditions on certain field components and Neumann conditions on the others. Quantum gravity and supergravity, spinor field theory, and various elliptic complexes all lend themselves to mixed boundary conditions of this sort. The mathematical significance of mixed boundary conditions is investigated, and a simple rule is obtained for determining which field components should obey which boundary conditions. It will be shown that gauge transformations on a bounded manifold give rise to intrinsic gauge transformations within the boundary. This result is used to define gauge-invariant and BRS-invariant quantum amplitudes for boundary configurations. The most general form for mixed boundary conditions on the Dirac operator and the de Rham complex will also be derived.

According to finite-size scaling theory, the correlation length in an Ising strip with infinite length and a width of L lattice constants has the form ${\ensuremath{\xi}}_{L}$(T)=Lf(c(T-${T}_{c}$)L) in the scaling regime L\ensuremath{\gg}1,\ensuremath{\Vert}T-${T}_{c}$\ensuremath{\Vert}/${T}_{c}$\ensuremath{\ll}1. Here f(x) is a universal scaling function, which does, however, depend on the boundary conditions at the edges of the strip. Utilizing the correspondence between the two-dimensional Ising model and the one-dimensional quantum Ising model in a transverse field, we calculate the explicit form of the function f(x) for spin-spin and energy-energy correlations in strips with periodic, free, fixed, and mixed free-fixed boundary conditions.

In this paper we show that the h-p spectral element method developed in [3,8,9] applies to elliptic problems in curvilinear polygons with mixed Neumann and Dirichlet boundary conditions provided that the Babuska-Brezzi inf-sup conditions are satisfied. We establish basic stability estimates for a non-conforming h-p spectral element method which allows for simultaneous mesh refinement and variable polynomial degree. The spectral element functions are non-conforming if the boundary conditions are Dirichlet. For problems with mixed boundary conditions they are continuous only at the vertices of the elements. We obtain a stability estimate when the spectral element functions vanish at the vertices of the elements, which is needed for parallelizing the numerical scheme. Finally, we indicate how the mesh refinement strategy and choice of polynomial degree depends on the regularity of the coefficients of the differential operator, smoothness of the sides of the polygon and the regularity of the data to obtain the maximum accuracy achievable.

Smoothed particle hydrodynamics (SPH) is a meshfree method and has been widely used in the field of fluid dynamics. However, there is still a lack of robust approach to implement the far-field boundary condition in the simulation of free stream flow past the rigid body. To solve this problem, on the basis of Lastiwka et al.’s work, an improved non-reflecting boundary condition, called mixed characteristic boundary condition is proposed in this paper. The mixed characteristic boundary condition is tested by two-dimensional incompressible flows with a perturbation at the initial moment. The numerical results show that the mixed characteristic boundary condition can effectively suppress the reflection of waves on far-field boundaries. Finally, flows past a hydrofoil are simulated within the framework of the kernel gradient free SPH in two kinds of non-reflecting boundary condition. The results indicate that the present boundary condition is more robust and suitable for SPH than the one adopted by Lastiwka et al.

Nonexistence of Positive Solutions for Singular Hyperbolic Differential Inequalities

A fourier method for initial value problems with mixed boundary conditions

This paper presents two complementary classes of analytical benchmark problems for the one-dimensional wave equation governing longitudinal vibration of a prismatic rod with mixed (clamped–free) boundary conditions. The first benchmark class consists of classical initial-value problems and includes both compatible and incompatible initial data at the space–time corners, highlighting their influence on convergence, regularity, and termwise differentiation of displacement, velocity, and axial force series representations. The second benchmark class prescribes the displacement at two time instants (initial and final time), leading to a fundamentally different modal structure and revealing spectral conditioning effects governed by the ratio L/(cte). The derived closed-form solutions provide reference configurations for verification of transient numerical solvers, particularly in scenarios where classical smooth compatibility assumptions are not satisfied.

Paper is devoted to determine the solution of a non-stationary heat equation in an axial symmetry cylindrical coordinates subject to a nonhomogeneous mixed discontinuous boundary conditions of the first and of the second kind inside the disk of an finite surface cylinder. The solution of the given mixed boundary value problem is obtained with the aid of a classical methods and based on the application of a new type of a dual series equations (DSE) with a Bessel function of the first kind of order zero as a kernel. The solution of obtained DSE which is discussed in this paper is introduced to a Fredholm integral equation of the first kind.

Consider the problem of finding a complex function $f = u + iv$, which is holomorphic in a given domain, with the real part u taking prescribed values on one part of the boundary, and the imaginary part v taking prescribed values on the remainder of the boundary. (This is essentially equivalent to solving Laplace’s equation subject to mixed Dirichlet and Neumann boundary conditions.) By virtue of the Cauchy integral formula, the unknown boundary values of u and v satisfy a $2 \times 2$ system of singular integral equations, which can be solved in a certain class of weighted $L_p $ spaces, by applying a result of I. Gohberg and N. Krupnik. The values of both u and v are then known over the whole of the boundary, and so f can be computed using the Cauchy integral formula.

Radiative entropy generation in a gray absorbing, emitting, and scattering planar medium at radiative equilibrium

The vibration response of isotropic and orthotropic plates with mixed boundary conditions is numerically modeled using a solution that is based on the differential quadrature method (DQM). The DQM is applied to each region and with the imposition of appropriate boundary conditions; the problem is transformed into a standard eigenvalue problem. The δ technique is used to treat the various boundary conditions. The results also demonstrate the efficiency of the method in treating this class of engineering problem.