Sixth-order Partial Differential Equation Research Articles

Blow-Up Phenomena for a Sixth-Order Partial Differential Equation with a General Nonlinearity

Blow-Up Phenomena for a Sixth-Order Partial Differential Equation with a General Nonlinearity

A Linear Theory of Local Thermal Circulations

The dynamics of local thermal circulations (LTCs) are examined by constructing a linear theory based on Boussinesq equations in the planetary boundary layer (PBL). Linear theory arranges LTCs into a sixth-order partial differential equation of temperature, which can be solved by using the Fourier transform and inverse Fourier transform. The analytic solution suggests that the horizontal distribution of the temperature anomaly is basically determined by surface heating, while the vertical distribution of the temperature anomaly is a combination of exponential decay and an Ekman spiral. For shallow PBL cases where the Ekman elevation is much smaller than the vertical scale of motion, the higher-order partial differential terms that represent the Ekman spiral structure can be ignored so that the equations reduce to a second-order partial differential equation. Compared with the numerical results, this so-called low-order approximation does not induce dramatic errors in the temperature distribution. However, to avoid distortion in the forced atmospheric circulation, the eddy viscosity in the motion equations should be replaced with the Rayleigh form, which is the common practice in LTCs. For deep PBL cases where the Ekman elevation is comparable to the vertical scale of motion, both the exponential decay and Ekman spiral structure play roles in the forced atmospheric circulation. The most significant influence is that there exist three additional compensating forced circulation cells that surround the direct forced circulation cell.

Random motions in [formula omitted] with orthogonal directions

This paper is devoted to the detailed analysis of three-dimensional motions in R3 with orthogonal directions switching at Poisson times and moving with constant speed c>0. The study of the random position at an arbitrary time t>0 on the surface of the support, forming an octahedron Sct, is completely carried out on the edges and faces (Fct). In particular, the motion on the faces Fct is analyzed by means of a transformation which reduces it to a three-directions planar random motion. This permits us to obtain an integral representation on Fct in terms of integral of products of first order Bessel functions. The investigation of the distribution of the position p=p(t,x,y,z) inside Sct implied the derivation of a sixth-order partial differential equation governing p (expressed in terms of the products of three D’Alembert operators). A number of results, also in explicit form, concern the time spent on each direction and the position reached by each coordinate as the motion develops. The analysis is carried out when the incoming direction is orthogonal to the ongoing one and also when all directions can be uniformly chosen at each Poisson event. If the switches are governed by a homogeneous Poisson process many explicit results are obtained.

On the dynamics and stability of size-dependent symmetric FGM plates with electro-elastic coupling using meshless local Petrov-Galerkin method

In present study, the equations of motion of sandwich symmetric functionally graded size-dependent plates integrated with piezoelectric layers are derived using Hamilton’s variational principle (with modified Lagrangian) based on a higher-order nonlocal strain gradient theory in conjunction with Reddy’s third-order shear deformation plate kinematics. For the first time, a set of sixth-order partial differential equations are derived and solved by meshless local Petrov-Galerkin method to investigate the bifurcation buckling and free vibration of plates under electromechanical in-plane forces with considered nanostructure and various boundary conditions. The shifted and scaled moving least-squares approach is utilized for approximation of unknown variables, and the Gauss weight function is used as test function for deriving local weak form of governing equations. The Gauss-Legendre quadrature is employed for numerical evaluation of weak forms. A power-law distribution and a half cosine variation are applied to model the variation of materials properties and electric potentials, respectively. The obtained results are verified with well-known solutions in the literature. The effect of nonlocal parameter, material length scale parameter, power-law index, predefined electric field, axial compressive and tensile forces, volume ratio of FG core and piezoelectric layers, and stiffness of Pasternak foundation on the critical buckling loads and natural frequencies are presented and discussed comprehensively. The results obtained can be used in the analysis, design and control of composite nanostructures that are widely used in MEMS and NEMS devices.

Applicability and efficacy of Galerkin-based approximation for solving the buckling and dynamics of nanobeams with higher-order boundary conditions

In this research effort, the general nonlocal elasticity theory is applied to an axially-loaded Euler-Bernoulli nanobeam with varying boundary conditions. In applying the general nonlocal elasticity theory, a modified version of Eringen's nonlocal elasticity theory, a sixth-order partial differential equation is obtained for the transverse governing equation of motion. Obtaining the sixth-order equation created the need for additional non-classical beam boundary conditions. These boundary conditions are determined using the weighted residual approach and are physically interpreted as higher-order force and moment terms. This study investigates the buckling and dynamic responses of nanobeams using a Galerkin-based approximation with different trial mode shapes, as it relates to the exact solution. As such, comparisons are made between using three distinct types guess functions in the approximation, namely, the exact unloaded classical mode shapes, exact unloaded Eringen nonlocal mode shapes, and exact unloaded general nonlocal mode shapes. Unloaded mode shapes refer to the mode shapes in which the applied axial load is neglected or set to zero. Additional studies are conducted focusing on direct comparison between Eringen nonlocal and general nonlocal beams for both equal and unequal nonlocal parameters. It is shown that selection of an appropriate guess function that satisfies all boundary conditions is necessary for the approximate solution. Additionally, the response of Eringen and general nonlocal nanobeams with equal nonlocal parameters and hinged-hinged boundary conditions is identical using both the exact and approximate solutions. Finally, it is demonstrated that the general nonlocal dynamic response follows a different trajectory than the Eringen nonlocal dynamic response as a function of the applied axial load. The findings of this investigation may be used by other researchers to simplify analysis procedures and to better understand the fundamental relationships existing in the size-dependent governing equations and boundary conditions for nanobeam-based systems.

Numerical solutions of a gradient-elastic Kirchhoff plate model on convex and concave geometries using isogeometric analysis

Abstract In this work, we study numerical solutions of a gradient-elastic Kirchhoff plate model on convex and concave geometries. For a convex plate, we first show the well-posedness of the model. Then, we split the sixth-order partial differential equation (PDE) into a system of three second-order PDEs. The solution of the resulting system coincides with that of the original PDE. This is verified with convergence studies performed by solving the sixth-order PDE directly (direct method) using isogeometric analysis (IGA) and the system of second-order PDEs (split method) using both IGA and C0 finite elements. Next, we study a concave pie-shaped plate, which has one re-entrant point. The well-posedness of the model on the concave domain is proved. Numerical solutions obtained using the split method differ significantly from that of the direct method. The split method may even lead to nonphysical solutions. We conclude that for gradient-elastic Kirchhoff plates with concave corners, it is necessary to use the direct method with IGA.

3D Modelling with C2 Continuous PDE Surface Patches

In this paper, we present a new modelling method to create 3D models. First, characteristic cross section curves are generated and approximated by generalized elliptic curves. Then, a vector-valued sixth-order partial differential equation is proposed, and its closed form solution is derived to create PDE surface patches from cross section curves where two adjacent PDE-surface patches are automatically stitched together. With the approach presented in this paper, C2 continuity between adjacent surface patches is well-maintained. Since surface creation of the model is transformed into the generation of cross sectional curves and few undetermined constants are required to describe cross sectional curves accurately, the proposed approach can save manual operations, reduce information storage, and generate 3D models quickly.

Generating bicubic B-spline surfaces by a sixth order PDE

As the solutions of partial differential equations (PDEs), PDE surfaces provide an effective way for physical-based surface design in surface modeling. The bicubic B-spline surface is a useful tool for surface modeling in computer aided geometric design (CAGD). In this paper, we present a method for generating bicubic B-spline surfaces with the uniform knots and the quasi-uniform knots from the sixth order PDEs. From the given boundary condition, based on the cubic B-spline basis representation and the PDE mask, the resulting bicubic B-spline surface can be generated uniquely. The boundary condition is more flexible and can be applied for curvature-continuous surface design, surface blending and hole filling. Some representative examples show the effectiveness of the presented method.

Study on vibration characteristics of sandwich beam with BCC lattice core

In this study, the vibration characteristics of a sandwich lattice beam have been studied using an analytical approach and finite element(FE) analyses. In the analytical approach, a sixth-order partial differential equation for expressing the motion in terms of the transverse displacement has been solved, which is assumed that the sandwich core is deformed mainly by the shear load. Here, a new theoretical approach for predicting the equivalent shear modulus GC∗ of the lattice core is proposed. The analytical prediction of the natural frequency, accounting for transverse shear, agrees well with the FE results. Also, the vibration response of the first-order natural frequency f1 exhibits a upwardly convex curve as the strut diameter increases, suggesting that there is a maximum value of the frequency f1 for a given facesheet thickness tf. Moreover, it is observed that the maximum frequency of the sandwich lattice beam is always greater than that for a continuum beam, which ensures the effectiveness of the lattice core in enhancing the vibration response. Therefore, vibration suppression in aircraft structures can be achieved by introducing lattice sandwich panels.

Parity-Time Symmetry and Exceptional Points for Flexural-Gravity Waves in Buoyant Thin-Plates

We derive and apply a transfer matrix method (M-matrix) coupling liquid surface waves and flexural-gravity waves in buoyant thin elastic plates. We analyze the scattering matrix (S-matrix) formalism for such waves propagating within a Fabry-Perot like system, which are solutions of a sixth order partial differential equation (PDE) supplied with adequate boundary conditions. We develop a parity-time (PT)-symmetry theory and its applications to thin elastic floating plates. The sixth order PDE governing the propagation of these waves leads to six by six M and S matrices, and results in specific physical properties of the PT-symmetric elastic plate systems. We show the effect of geometry and gain/loss on the asymmetric propagation of flexural-gravity waves, as well as a Fano-like line-shape of the reflection signature. Importantly, we show the possibility of obtaining coherent perfect absorber-laser (CPAL) using simple thin structures.

Highly dispersive optical solitons of equation with various polynomial nonlinearity law

Abstract An nonlinear sixth-order partial differential equation for description of propagation pulse with various polynomial nonlinearities is considered. Traveling wave solutions are used to obtain the system of equations for real and imaginary parts. Solitary waves are shown to be for some conditions of the parameters of the equation. Optical solitons for nonlinear differential equation with various polynomial nonlinearity law are found and demonstrated.

A stable and efficient meshfree Galerkin method with consistent integration schemes for strain gradient thin beams and plates

Abstract The strain gradient (SG) theory, incorporating with thin beam and plate models, can effectively describe size effects of micro- and nano-structures. However, since these models are determined by a sixth-order partial differential equation that requires the C2 continuity of deflection in a Galerkin weak form, it is difficult to make stable and efficient numerical analysis. In this paper, a meshfree Galerkin method is presented for SG thin beams and plates. To satisfy the continuity and convergence requirement, moving least square or reproducing kernel shape functions are employed with cubic approximation bases. To pass the patch test, integration constraints are derived and consistent integration schemes are proposed with nodal smoothed derivatives instead of standard ones on evaluating points. Numerical results show that consistent integration is superior to the standard Gauss integration in convergence, accuracy and efficiency.

Natural Frequency of Rotating Single-Walled Carbon Nanotubes with Considering Gyroscopic Effect

This paper investigates the bending vibration of rotating single-walled carbon nanotubes (SWCNTs) based on nonlocal theory. To this end, the rotating SWCNTs system modeled as a beam with a circular cross section and the Euler-Bernoulli beam theory (EBT) is applied with added effects such as rotary inertia, gyroscopic effect and rotor mass unbalance. Using nonlocal theory, two coupled sixth order partial differential equations that govern the vibration of rotating SWCNTs are derived. To obtain the natural frequency and dynamic response of the nanorotor system, the equation of motion for the rotating SWCNTs are solved. It is found that there are two frequencies in the frequency spectrum. The positive rootintroduced as forward whirling mode, while the negative root represents backward whirling mode. The detailed mathematical derivations are presented while the emphasis is placed on investigating the effect of the several parameters such as, tube radius, angular velocity and small scale parameter on the vibration behavior of rotating nanotubes. It is explicitly shown that the vibration of a spinning nanotube is significantly influenced by these effects. To validate the accuracy and efficiency of this work, the results obtained herein are compared with the existing theoretical and experimental results and good agreement is observed. To the knowledge of authors, the vibration of rotating SWCNTs considering gyroscopic effect has not investigated analytically yet and then the results generated herein can be served as a benchmark for future works.

Scattering theory and cancellation of gravity-flexural waves of floating plates

We combine theories of scattering for linearized water waves and flexural waves in thin plates to characterize and achieve control of water wave scattering using floating plates. This requires manipulating a sixth-order partial differential equation with appropriate boundary conditions of the velocity potential. Making use of multipole expansions, we reduce the scattering problem to a linear algebraic system. The response of a floating plate in the quasistatic limit simplifies, considering a distinct behavior for water and flexural waves. Unlike similar studies in electromagnetics and acoustics, scattering of gravity-flexural waves is dominated by the zeroth-order multipole term and this results in non-vanishing scattering cross-section also in the zero-frequency limit. Potential applications lie in floating structures manipulating ocean waves.

C2 Continuous Blending of Time-Dependent Parametric Surfaces

Surface blending is widely applied in mechanical engineering. Creating a smooth transition surface of C2 continuity between time-dependent parametric surfaces that change their positions and shapes with time is an important and unsolved topic in surface blending. In order to address this issue, this paper develops a new approach to unify both time-dependent and time-independent surface blending with C2 continuity. It proposes a new surface blending mathematical model consisting of a vector-valued sixth-order partial differential equation and blending boundary constraints and investigates a simple and efficient approximate analytical solution of the mathematical model. A number of examples are presented to demonstrate the effectiveness and applications. The proposed approach has the advantages of (1) unifying time-independent and time-dependent surface blending, (2) always maintaining C2 continuity at trimlines when parametric surfaces change their positions and shapes with time, (3) providing effective shape control handles to achieve the expected shapes of blending surfaces but still exactly satisfy the given blending boundary constraints, and (4) quickly generating C2 continuous blending surfaces from the approximate analytical solution with easiness, good accuracy, and high efficiency.

A finite element solution to the generalized Onsager model of fluid flow in a countercurrent gas centrifuge

ABSTRACT Based on the Onsager Equation, a linearized sixth-order partial differential equation describing the flow in the volume of the rotor of a gas centrifuge is solved using a finite element algorithm employed by the CurvSOL code. The results are compared to those from Pancake, an existing code that uses eigenfunction expansion to solve the Onsager equation. Excellent agreement is demonstrated between the solutions of the wall temperature gradient and scoop drive mechanisms for both the Rome and Iguaçu centrifuge designs. Results suggest rotor speed and radius play key roles in the influence of wall curvature on the flow solution.

A Comparison Study of HO-CFD and DSC-RSK with Small Computational Bandwidths for Solving Some Classes of Boundary-Value and Eigenvalue Problems

This paper conducts a comparison analysis of high order central finite difference (HO-CFD) method and discrete singular convolution-regularized Shannon kernel (DSC-RSK) scheme with small computational bandwidths for solving some classes of boundary-value and eigenvalue problems. Second-, fourth- and sixth-order partial differential equations are taken into account. New strategies to generate parameters [Formula: see text] in DSC-RSK are proposed to ensure minimum errors for each case, and the influence of parameters [Formula: see text] with more decimal places is analyzed. Apart from the existing matched interface and boundary (MIB) scheme, a new double-parameter MIB scheme is also proposed. The influence of small computational bandwidths is discussed in detail. Numerical results by using HO-CFD and DSC-RSK are presented and compared to illustrate the performance of both methods in small bandwidth limit. Some remarkable conclusions have been drawn at the end of this study.

A finite-difference approach for solving pure quasi-P-wave equations in transversely isotropic and orthorhombic media

Starting from the dispersion relation and setting S-wave velocity along symmetry axes to zero, pseudoacoustic-wave equations have been proposed to describe the kinematics of acoustic wavefields in transversely isotropic (TI) and orthorhombic media. To date, the numerical stability of the pseudoacoustic-wave equations has been improved by developing coupled systems of wave equations; however, most simulations still suffer from S-wave artifacts that are the fundamental solutions of the fourth- and sixth-order partial differential equations. Pure quasi-P-wave equations accurately describe the traveltimes of P-waves in TI and orthorhombic media and are free of S-wave artifacts. However, it is difficult to directly solve the pure quasi-P-wave equations using conventional finite-difference schemes due to the presence of pseudo-differential operators. We approximated these pseudo-differential operators by algebraic expressions, whose coefficients can be determined by minimizing differences between the true and approximated values of the pseudo-differential operators in the wavenumber domain. The derived new coupled systems involve modified acoustic-wave equations and a Poisson’s equation that can be solved by conventional finite-difference stencils and fast Poisson’s solver. Several 2D and 3D numerical examples demonstrate that the simulations based on the new systems are free of S-wave artifacts and have correct kinematics of quasi-P-waves in TI and orthorhombic media.

Dynamic behaviour of an infinite saturated transversely isotropic porous media under fluid-phase excitation

One of the required fundamental solutions of a poroelastodynamic problem is the solution of the saturated porous medium under a fluid flux or pressure. In the framework of coupled formulations of the displacement and pore-fluid pressure proposed by Biot, the fundamental solutions for a transversely isotropic fluid-saturated porous full-space are analytically presented when the domain of interest is affected by a time-harmonic fluid either ring flux or ring pressure. To this end, a solely scalar potential function is used to uncouple the Biot's coupled partial differential equations, where a sixth order partial differential equation achieved, from which the potential function is determined with the aid of Hankel integral transform and Fourier expansion series. The analytical fundamental displacements, stresses and pore-fluid pressure are presented in the form of one-dimensional semi-infinite integrals, which are due to inverse Hankel integral transforms. The integrals are also degenerated for the isotropic material in mechanical point of view. Because of the complexity of the integrands, the integrals are evaluated numerically. To this end, an adaptive numerical quadrature is adopted and coded in MATHEMATICA software. Some numerical results have been provided to illustrate the displacements, stresses and pore-fluid pressure.

A mixed finite element method for a sixth-order elliptic problem

We consider a saddle point formulation for a sixth order partial differential equation and its finite element approximation, for two sets of boundary conditions. We follow the Ciarlet-Raviart formulation for the biharmonic problem to formulate our saddle point problem and the finite element method. The new formulation allows us to use the $H^1$-conforming Lagrange finite element spaces to approximate the solution. We prove a priori error estimates for our approach. Numerical results are presented for linear and quadratic finite element methods.