Satisfy Boundary Conditions Research Articles

The Application Of The Laplace Transform In The Heat Equation

The one dimensional of homogeneous heat equation is two ordinary partial differential equations which general form , so needs solution. The one dimensional of homogeneous heat equation the method is ever been solved by Fourier series and Fourier Integrals. The other method used to in solving one dimensional of homogeneous heat equation is Laplace Transform, which is used to transform into ordinary differential equation is one dimension for initial conditions . Inverse general solution that satisfies boundary conditions is found, for every t and initial conditions that have been transform is . From the result, it an obtain temperature on the poin x in the solid at time t.

Separable solutions to nonlinear anisotropic diffusion equation in elliptic coordinates.

Two- and three-dimensional exact solutions of the nonlinear diffusion equation are proved to exist in elliptic coordinates subject to an arbitrary piecewise constant azimuthal anisotropy. Degrees of freedom traditionally used to satisfy boundary conditions are instead employed to ensure continuity and conservation of mass across contiguity surfaces between subdomains of distinct diffusivities. Not all degrees of freedom are exhausted thereby, and conditions are given for the inclusion of higher harmonics. Degrees of freedom associated with one isotropic subdomain are always available to satisfy boundary conditions. The second harmonic is pivotal in the solution construction as well as the identification of partial symmetries in the domain partition. The anisotropy gives rise to an unconventional mixed type critical point that combines saddle and node-like characteristics. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Torsional instability of functionally graded porous toroidal shell segments including elastic edge constraints and elevated temperature

This article aims to analyze the nonlinear instability of toroidal shell segment (TSS) made of functionally graded porous material (FGPM), exposed to elevated temperature and subjected to uniform torsion. The volume fraction of two material constituents is varied through the shell thickness according to a power-law function, while porous distribution into FGPM is modeled in forms of cosine functions. Governing equations in terms of deflection and stress function are established within the framework of classical shell theory incorporating geometrical nonlinearity. Two edges of the TSS are assumed to be simply supported and elastically restrained. Multi-term analytical solutions are assumed to satisfy boundary conditions and Galerkin method is applied to derive closed-form expressions of nonlinear load–deflection relations and buckling loads. Parametric studies are carried out to analyze the separate and combined influences of material and geometry properties, porous coefficient and distributions, degree of edge constraint and elevated temperature on the buckling resistance and postbuckling strength of torsionally loaded FGPM TSSs.

Behaviours of disco-seismological c-mode oscillations around the inner edge of discs

ABSTRACT Accretion flows in the innermost region of relativistic discs are transonic. At the transonic radius, the differential equation describing wave motions with constant frequency becomes singular. This implies that for an oscillation of constant frequency to be realized in the innermost region of discs, the oscillation needs to satisfy some regularity conditions (boundary conditions) at the sonic radius. In this paper, under the assumption that the unperturbed disc structure around the sonic radius is rather smooth in the radial direction, behaviours of the fourth-order ordinary differential equation describing non-axisymmetric c-mode oscillations are examined around the sonic radius. Among four linearly independent wave solutions, one is always singular at the sonic radius and is outside of our interest. In remaining three wave modes satisfying boundary conditions, the advection term in equation of motion is a main contributor in determining the structure of the oscillations. Due to this, the waves are tightly wound or change their amplitude sharply in a narrow region around the sonic radius. This characteristic is especially notable in one of three wave modes. Possibility of the oscillations being swallowed into the central source with no reflection at the sonic radius is briefly discussed.

Asymptotic behavior of an intrinsic rank-based estimator of the Pickands dependence function constructed from B-splines

A bivariate extreme-value copula is characterized by its Pickands dependence function, i.e., a convex function defined on the unit interval satisfying boundary conditions. This paper investigates the large-sample behavior of a nonparametric estimator of this function due to Cormier et al. (Extremes 17:633–659, 2014). These authors showed how to construct this estimator through constrained quadratic median B-spline smoothing of pairs of pseudo-observations derived from a random sample. Their estimator is shown here to exist whatever the order m ge 3 of the B-spline basis, and its consistency is established under minimal conditions. The large-sample distribution of this estimator is also determined under the additional assumption that the underlying Pickands dependence function is a B-spline of given order with a known set of knots.

A face-based immersed boundary method for compressible flows using a uniform interpolation stencil

In this study, an immersed boundary method developed for compressible viscous flows (Ramakrishnan, R., Girdhar, A., & Ghosh, S. (2016). Immersed Boundary Methods for Compressible Laminar Flows) is modified to improve their stability and robustness. The embedded object is represented as a set of line segments in two dimensions with their outward unit normal vectors specified. A forcing method that leverages the finite volume approach is used, wherein the solution at the cell interfaces that lie near the boundaries of the embedded solid is reconstructed to implicitly satisfy boundary conditions at the immersed surface. The proposed immersed boundary method is validated for transonic inviscid flow past a bump in a channel, supersonic flow past a circular cylinder, transonic viscous flow past a NACA0012 airfoil, and supersonic viscous flow past a circular cylinder. The results are compared with simulations from the literature using contours of flow properties, surface pressure, or Mach number plots and show good agreement.

Free vibrations of circular and sectorial plates by using the shape functions with trigonometric and hyperbolic terms

In this paper, free vibration of different plates in the polar coordinate is studied by using the Rayleigh–Ritz method and calculating strain and kinetic energies. The investigated plates include annular sectorial and circular, sectorial and circular with fixed center, circular sectorial, and solid circular. In this work, trigonometric and hyperbolic terms are used to express the shape function of each plate so that the shape function approximately satisfies boundary conditions and also regularity conditions for the solid plates. The mentioned plates can have any different types of the boundary conditions and the presented method is versatile and has no limitations. For the solid plates, the shape function is calculated in such a way that it can be used to calculate strain energy of the plate. Some types of the plates are examined as the case studies and the results are also calculated by the ABAQUS CAE for validation of the proposed method.

Periodic traveling waves for a diffusive SIR epidemic model with general nonlinear incidence and external supplies

In this paper, we propose a non-autonomous reaction–diffusion SIR infectious disease model with nonlinear incidence, taking fully into account the effects of periodic environmental factors as well as population dynamics on disease transmission in space, and investigate the existence of periodic traveling wave solutions satisfying boundary conditions. Specifically, we first define the basic reproductive number R0 and critical wave speed c∗, which will directly determine the existence of periodic traveling waves. Then, by considering a truncation problem and using fixed-point theorem, some estimation and limit techniques, the sufficient conditions on the existence of periodic traveling waves satisfying some boundary conditions are deduced for every wave speed c>c∗ when R0>1, and the nonexistence of periodic traveling waves is also obtained for any c>0 when R0<1. Finally, some numerical examples are given to verify the theoretical results.

Ab initio method of calculating invariant measure for turbulent flow

We present a method of calculating the invariant measure (IM) of hydrodynamical systems describing incompressible viscous fluid flow in bounded 3D volume V. Using a basis { w I } of zero-divergence vector functions on V satisfying boundary conditions we transform the Navier–Stokes equations (NSE) for the velocity field v ( x , t) = ∑ I v I (t) w I ( x ) into a simple system of ordinary differential equations for v I (t). The fact that fluid consists of a finite number of molecules in thermal motion implies that the number N of variables v I is finite too and that the system of equations for v I (t) contains a white-noise term. We prove that all solutions of this system are global, the IM exists and is unique. Its density ψ(v) defined on the phase space satisfies an elliptic partial differential equation (PDE). Expanding into Hermite functions He μ (v) φ ε (v) (where φ ε is a Gaussian function on , He μ are Hermite polynomials orthonormal with the weight φ ε and are IM mean values of He μ ) we transform this PDE into an infinite system of linear algebraic equations for and suggest some methods for solving it numerically. From known first- and second-order we could easily calculate mean turbulent velocity and Reynolds stress at any point x . Cylindrical pipe flow is used to illustrate the method everywhere in the paper. All conclusions are derived from the mathematical model by rigorous mathematics, with no further assumptions. All approximations can be arbitrarily improved.

Temperature Distribution Comparison in Smooth and Stepped Functions

In this article the distribution of temperature in both smooth and stepped functions is compared. The study uses a model of one dimensional N-cross-sections domain of a vase shaped medium, where approximation of the solution in the first case uses smooth functions, and in the second one, stepped functions. The temperature distribution is described by heat equation. Analysis of temperature distribution in both cases is based on finding eigenvalues and their corresponding eigen-functions which satisfy boundary conditions at given endpoints. Mathcad software was applied to determine the eigenvalues and their corresponding eigen-functions, together with the temperature distribution in the media of concern. The temperature distribution in both cases was found to be basically the same. The problem solution for each case is presented and an example of a one-dimensional vase shaped domain of length 4 units for each case is also given.

A localized spatiotemporal particle collocation method for long-time transient homogeneous diffusion analysis

This short communication proposes a novel localized spatiotemporal particle collocation method (LSPCM) to analyze long-time transient homogeneous diffusion problems. A time-dependent general solution of diffusion equation without singularity or near singularity at origin is developed. The LSPCM bypasses Laplace transformation or finite difference method to treat the time derivative of the diffusion equation, and can directly approximate the transient homogeneous diffusion problems. Each discrete node in computational domain is only supported by its nearby nodes covered by the local spatiotemporal subdomain. The physical variable of each discrete node is approximated by the linear combination of the physical variable of its nearby nodes and the time-dependent general solutions. Through satisfying boundary conditions and continuity conditions, the LSPCM eventually results in a large-scale sparse linear system which is easy to store and solve. Numerical experiments show that the LSPCM can accurately simulate long-time transient homogeneous diffusion problems by using a very long-time interval.

Novel approach to apply existing constitutive soil models to the modelling of interfaces

AbstractA novel scheme to apply constitutive (continuum) soil models to the mechanical modelling of soil‐structure interfaces is presented. Opposite to existing approaches, all normal stress components of the interface depend directly on the normal stress components of the adjacent continuum. This allows to satisfy boundary conditions not only for the continuum but also for the interface. Interface models based on Hypoplasticity with intergranular strain extension and on Sanisand (version of 2004) are formulated. They are implemented in the framework of a mortar contact discretisation technique in a finite element code, which is available for download. A numerical differentiation scheme is used to secure consistent derivatives of contact forces with respect to displacement. The performance of the proposed interface models is evaluated by comparison with data from (cyclic) simple interface shear tests and an existing (hypoplastic) interface model proposed by Stutz et al. (2016). Subsequently, a large‐scale cyclic interface shear test with complex geometry is simulated using the different interface models. It is concluded that only the novel interface formulation allows to appropriately consider the boundary conditions of the large‐scale interface shear test in the interface.

Complex Variable Solution for Stress and Displacement of Layered Soil with Finite Thickness

The static problem of a layered isotropic elastic body is a very useful research subject in relation to the analysis and design of foundation works. Due to the complexity of the problem, there is no analytical solution to the problem so far. This study provides an efficient analytical approach to accurately calculate the displacement and stress fields of the soil. The constraints of bedrock on soil, different soil layer thickness and the shear stress of the foundation on soil were all taken into account in the analysis. In this study, each layer is regarded as an isotropic elastomer with infinite width, and the layers are in complete contact. By using conformal mapping, each layer is mapped to a unit circle, and the two complex potential functions are expanded into Taylor series with unknown coefficients. These unknown coefficients are obtained by satisfying boundary conditions and continuity conditions. The boundary and continuity conditions were verified in this paper. As a validation step, we compared the analytical results for the settlement with the results of the ANSYS numerical simulations and found good agreement. Parametric analyses were also carried out to investigate the influence of different distribution forms of base pressure on surface settlement, and the effects of layered properties on the surface settlement and stress field.

The Scattering of SH Wave in a Nano Hole Embedded the Infinite Inhomogeneous Medium

Scattering of the shear waves by a nano-sized cylindrical hole embedded the inhomogeneous is investigated in this study. The Helmholtz equation with a variable coefficient is transformed the standard Helmholtz equation by the complex function method and the conformal mapping method. By wave function expanding method, the analytical expressions of the displacement field and stress field in the inhomogeneous medium are obtained. Considering the surface effect and using the generalized Young-Laplace equation, we obtain the boundary conditions at nano arbitrary-shaped hole, then the field equations satisfying boundary conditions are attributed to solving a set of infinite algebraic equations. Numerical results show that when the radius of the cylindrical cavity shrinks to nanometers, surface energy becomes a dominant factor that affects the dynamic stress concentration factor (DSCF) around the cylindrical cavity. The influence the density variation of the inhomogeneity on the DSCF is discussed at the same time.

Fast Cooperative Trajectory Generation of Unmanned Aerial Vehicles Using a Bezier Curve-Based Shaping Method

This study uses a Bezier curve-based shaping (BCBS) approach based on the dynamic model to quickly generate 3D cooperative trajectories for unmanned aerial vehicles (UAVs). It can facilitate the solution of the flight trajectory problem that occurs after the reallocation of mission points. Trajectory coordinates are expanded based on the Bezier curves that naturally satisfy boundary conditions (BCs); trajectories with the minimum flight time and those satisfying the dynamic constraints and collision-free conditions can be obtained by adjusting the Bezier coefficients. A three-UAV flight simulation is carried out to verify the effectiveness of the proposed method. Further, the performance of the proposed method is compared with that of the Gauss pseudospectral method (GPM). The simulation results of the BCBS method are used as the initial values for GPM optimization. It is demonstrated that the BCBS method requires a lower computation time than the direct solver, which is only 0.07% of the latter, and obtains similar optimization results (3.34% difference). This is considerably important for the rapid generation of optimized trajectories with the limited computing power of onboard computers. Furthermore, this method is expected to achieve online collaborative trajectory generation for multiple UAVs in view of its high computational efficiency.

FREE VIBRATION ANALYSIS OF SINGLY CURVED CLAMPED SHELLS USING THE ISOGEOMETRIC FINITE STRIP METHOD

&amp;lt;jats:p&amp;gt;A hybrid method for the spatial discretization of two-dimensional domains is recently derived and applied to the problem of free vibrations of simply-supported singly curved shells. This new method follows from a tensor product of NURBS functions and a carefully selected series that satisfies boundary conditions a priori. The formulation unifies spatial discretization schemes of the semi-analytical Finite strip method and the Isogeometric analysis. In this paper, the method is improved by implementing the capability to deal with clamped-clamped boundary conditions. The numerical analysis shows that the method has favorable accuracy per DOF, in comparison with the standard finite elements.&amp;lt;/jats:p&amp;gt;

Application of exponential functions in weighted residuals method in structural mechanics. Part 3: infinite cylindrical shell under concentrated forces

Solution for cylindrical shell under concentrated force is a fundamental problem which allow to consider many other cases of loading and geometries. Existing solutions were based on simplified assumptions, and the ranges of accuracy of them still remains unknown. The common idea is the expansion of them into Fourier series with respect to circumferential coordinate. This reduces the problem to 8th order even differential equation as to axial coordinate. Yet the finding of relevant 8 eigenfunctions and exact relation of 8 constant of integrations with boundary conditions are still beyond the possibilities of analytical treatment. In this paper we apply the decaying exponential functions in Galerkin-like version of weighted residual method to above-mentioned 8th order equation. So, we construct the sets of basic functions each to satisfy boundary conditions as well as axial and circumferential equilibrium equations. The latter gives interdependencies between the coefficients of circumferential and axial displacements with the radial ones. As to radial equilibrium, it is satisfied only approximately by minimizations of residuals. In similar way we developed technique for application of Navier like version of WRM. The results and peculiarities of WRM application are discussed in details for cos2j concentrated loading, which methodologically is the most complicated case, because it embraces the longest distance over the cylinder. The solution for it clearly exhibits two types of behaviors – long-wave and short-wave ones, the analytical technique of treatment of them was developed by first author elsewhere, and here was successfully compared. This example demonstrates the superior accuracy of two semi analytical WRM methods. It was shown that Navier method while being simpler in realization still requires much more (at least by two orders) terms than exponential functions.

Hydrodynamics of active particles in viscosity gradients

The introduction of a particle into a preexisting nonuniform viscosity field will generally modify the viscosity field in order to satisfy boundary conditions on the particle surface. Here we discuss the effect of this disturbance viscosity on the dynamics of active particles, as well as characterize the relative importance of the local changes in stress versus the nonlocal flow changes due to nonuniform viscosity.

Canonical polynomials in the problem about the bend of circular plate of variable thickness

The problem about the bend of circular plate of variable thickness under specifically selected laws of rigidity change is reduced to the ordinary differential equation with variable coefficients of polynomial kind. The construction of the approximate solution of equation that satisfies boundary conditions is realized by means of canonical polynomials and $\tau$-method of Lantzosh.

TM Standing Wave Solution of Maxwell Equations in a Self-Defocusing Kerr Media and its Application to a Thin-Film Waveguide

We applied the dielectric function method to solve analytically L-NL-L structure problems with negative Kerr nonlinearity. A damped wave in linear and a periodic standing wave in non-linear media had to be matched at boundaries. We gave a formulation of boundary conditions that did not explicitly include a film thickness. The boundary-value of a dielectric function can be expressed through the constant of non-trivial integral of motion. Using it, one generates a family of matched solutions satisfying boundary conditions. Then arbitrary film thickness can be checked against this family of solutions in search of matches. As a result, all fitted solutions are determined straightforwardly.