ON CRITICAL EXPONENTS FOR SEMILINEAR HEAT EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

This paper deals with the blow up properties of solutions to semilinear heat equation u t- Δ u=u p in R N +×(0,T) with the nonlinear boundary condition -ο u ο x 1 = u q for x 1=0,t∈(0,T) .It has been proved that if max( p,q) ≤1,every nonnegative solution is global.When min (p,q) >1 by letting α=1p-1 and β=12(q-1) it follows that if max (α,β)≥N2 ,all nontrivial nonnegative solutions are nonglobal,whereas if max (α,β)<N2 ,there exist both global and nonglobal solutions.Moreover,the exact blow up rates are established.

Multi-soliton solutions of coupled Lakshmanan-Porsezian-Daniel equations with variable coefficients under nonzero boundary conditions

This paper aims to investigate the multi-soliton solutions of the coupled Lakshmanan–Porsezian–Daniel equations with variable coefficients under nonzero boundary conditions.These equations are utilized to model the phenomenon of nonlinear waves propagating simultaneously in non-uniform optical fibers.By analyzing the Lax pair and the Riemann–Hilbert problem,we aim to provide a comprehensive understanding of the dynamics and interactions of solitons of this system.Furthermore,we study the impacts of group velocity dispersion or the fourth-order dispersion on soliton behaviors.Through appropriate parameter selections,we observe various nonlinear phenomena,including the disappearance of solitons after interaction and their transformation into breather-like solitons,as well as the propagation of breathers with variable periodicity and interactions between solitons with variable periodicities.

Riemann–Hilbert problem for the defocusing Lakshmanan–Porsezian–Daniel equation with fully asymmetric nonzero boundary conditions

The Riemann–Hilbert approach is demonstrated to investigate the defocusing Lakshmanan–Porsezian–Daniel equation under fully asymmetric nonzero boundary conditions.In contrast to the symmetry case,this paper focuses on the branch points related to the scattering problem rather than using the Riemann surfaces.For the direct problem,we analyze the Jost solution of lax pairs and some properties of scattering matrix,including two kinds of symmetries.The inverse problem at branch points can be presented,corresponding to the associated Riemann–Hilbert.Moreover,we investigate the time evolution problem and estimate the value of solving the solutions by Jost function.For the inverse problem,we construct it as a Riemann–Hilbert problem and formulate the reconstruction formula for the defocusing Lakshmanan–Porsezian–Daniel equation.The solutions of the Riemann–Hilbert problem can be constructed by estimating the solutions.Finally,we work out the solutions under fully asymmetric nonzero boundary conditions precisely via utilizing the Sokhotski–Plemelj formula and the square of the negative column transformation with the assistance of Riemann surfaces.These results are valuable for understanding physical phenomena and developing further applications of optical problems.

Effect of boundary conditions on shakedown analysis of heterogeneous materials

The determination of the ultimate load-bearing capacity of structures made of elastoplastic heterogeneous materials under varying loads is of great importance for engineering analysis and design. Therefore, it is necessary to accurately predict the shakedown domains of these materials. The static shakedown theorem, also known as Melan's theorem, is a fundamental method used to predict the shakedown domains of structures and materials. Within this method, a key aspect lies in the construction and application of an appropriate self-equilibrium stress field(SSF). In the structural shakedown analysis, the SSF is typically constructed by governing equations that satisfy no external force(NEF) boundary conditions. However, we discover that directly applying these governing equations is not suitable for the shakedown analysis of heterogeneous materials. Researchers must consider the requirements imposed by the Hill-Mandel condition for boundary conditions and the physical significance of representative volume elements(RVEs). This paper addresses this issue and demonstrates that the sizes of SSFs vary under different boundary conditions, such as uniform displacement boundary conditions(DBCs), uniform traction boundary conditions(TBCs), and periodic boundary conditions(PBCs). As a result, significant discrepancies arise in the predicted shakedown domain sizes of heterogeneous materials. Built on the demonstrated relationship between SSFs under different boundary conditions, this study explores the conservative relationships among different shakedown domains, and provides proof of the relationship between the elastic limit(EL) factors and the shakedown loading factors under the loading domain of two load vertices. By utilizing numerical examples, we highlight the conservatism present in certain results reported in the existing literature. Among the investigated boundary conditions, the obtained shakedown domain is the most conservative under TBCs.Conversely, utilizing PBCs to construct an SSF for the shakedown analysis leads to less conservative lower bounds, indicating that PBCs should be employed as the preferred boundary conditions for the shakedown analysis of heterogeneous materials.

Existence of Entropy Solution for Degenerate Parabolic-Hyperbolic Problem Involving p(x)-Laplacian with Neumann Boundary Condition

We consider a strongly non-linear degenerate parabolic-hyperbolic problem with p(x)-Laplacian diffusion flux function. We propose an entropy formulation and prove the existence of an entropy solution.

Least Square Finite Element Model for Analysis of Multilayered Composite Plates under Arbitrary Boundary Conditions

Laminated composites are widely used in many engineering industries such as aircraft, spacecraft, boat hulls, racing car bodies, and storage tanks. We analyze the 3D deformations of a multilayered, linear elastic, anisotropic rectangular plate subjected to arbitrary boundary conditions on one edge and simply supported on other edge. The rectangular laminate consists of anisotropic and homogeneous laminae of arbitrary thicknesses. This study presents the elastic analysis of laminated composite plates subjected to sinusoidal mechanical loading under arbitrary boundary conditions. Least square finite element solutions for displacements and stresses are investigated using a mathematical model, called a state-space model, which allows us to simultaneously solve for these field variables in the composite structure’s domain and ensure that continuity conditions are satisfied at layer interfaces. The governing equations are derived from this model using a numerical technique called the least-squares finite element method (LSFEM). These LSFEMs seek to minimize the squares of the governing equations and the associated side conditions residuals over the computational domain. The model is comprised of layerwise variables such as displacements, out-of-plane stresses, and in- plane strains, treated as independent variables. Numerical results are presented to demonstrate the response of the laminated composite plates under various arbitrary boundary conditions using LSFEM and compared with the 3D elasticity solution available in the literature.

Boundary Conditions for Momentum and Vorticity at an Interface between Two Fluids

Boundary conditions for momentum and vorticity have been precisely derived, paying attention to the physical meaning of each mathematical expression of terms rigorously obtained from the basic equations: Navier-Stokes equation and the equation of vorticity transport. It has been shown first that a contribution of fluid molecules crossing over a conceptual surface moving with fluid velocity due to their fluctuating motion is essentially important to understanding transport phenomena of momentum and vorticity. A notion of surface layers, which are thin layers at both sides of an interface, has been introduced next to elucidate the transporting mechanism of momentum and vorticity from one phase to the other at an interface through which no fluid molecules are crossing over. A fact that a size of δV, in which reliable values of density, momentum, and velocity of fluid are respectively defined as a volume-averaged mass of fluid molecules, a volume-averaged momentum of fluid molecules and a mass-averaged velocity of fluid molecules, is not infinitesimal but finite has been one of the key factors leading to the boundary conditions for vorticity at an interface between two fluids. The most distinguished characteristics of the boundary conditions derived here are the zero-value conditions for a normal component of momentum flux and tangential components of vorticity flux, at an interface.

Absorbing Boundary Conditions for Simulating Water Waves Near Solid Bodies

The objective of this paper is to present a new method for designing absorbing or non-reflective boundary conditions (ABC) or (NRBC), illustrated by the case study of the modelling of a solid body in water, specifically the capillary gravity waves generated by its motion at the surface. The study analyses the flow of an inviscid, barotropic, and compressible fluid around the stationary solid body. The dynamic behaviour of the fluid is analysed using a two-dimensional coupled Neumann-Kelvin model extended with capillarity and inertia terms. For computational purposes, it is necessary to truncate the unbounded spatial domain with artificial boundaries and then introduce appropriate absorbing boundary conditions. The propagation of short wavelength waves in a convective fluid medium with significant differences in properties between the interior and the surface of the fluid presents a number of difficulties in the design of these conditions. The results are illustrated numerically and commented upon.

Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics

Material identification is critical for understanding the relationship between mechanical properties and the associated mechanical functions.However,material identification is a challenging task,especially when the characteristic of the material is highly nonlinear in nature,as is common in biological tissue.In this work,we identify unknown material properties in continuum solid mechanics via physics-informed neural networks(PINNs).To improve the accuracy and efficiency of PINNs,we develop efficient strategies to nonuniformly sample observational data.We also investigate different approaches to enforce Dirichlet-type boundary conditions(BCs)as soft or hard constraints.Finally,we apply the proposed methods to a diverse set of time-dependent and time-independent solid mechanic examples that span linear elastic and hyperelastic material space.The estimated material parameters achieve relative errors of less than 1%.As such,this work is relevant to diverse applications,including optimizing structural integrity and developing novel materials.

Dynamic characterization of tailing dam using fully coupled dynamic analysis with different boundary conditions-a case study

The seismic response analysis of a tailing dam is studied using a fully coupled effective stress approach in conjunction with an advanced multi yield surface plastic constitutive model for tailing material.Strain controlled static and cyclic triaxial tests were carried out to obtain the constitutive model for the tailing material.The tailing materials were collected from the Rampura Agucha tailing dam(Rajasthan State,India).A 2D nonlinear finite element(FE)model was then developed using different boundary conditions from the tailing embankment constructed using the downstream and upstream method of rising using OpenSees software.In first case,the model boundary was fixed in both the X and Y directions,and in the second case,viscous dashpots were introduced for both side and horizontal boundaries.The model was validated with experimental results on tailing material.Analyses were carried out considering five different earthquake motions,which were applied at the base.Comparisons of the different boundary conditions in terms of displacement flow vectors,pore pressure and stress-strain curves during shaking are presented.From the analysis,it was observed that the viscous boundary condition replicates the actual field conditions more accurately than the fixed boundary condition.In addition,it was found that the tailing embankment constructed by the downstream and upstream method of rising is not susceptible to liquefaction and lateral spreading for earthquake motions,even for a magnitude>5.5.

UNIQUENESS OF INVERSE TRANSMISSION SCATTERING WITH A CONDUCTIVE BOUNDARY CONDITION BY PHASELESS FAR FIELD PATTERN

In this paper,we establish the unique determination result for inverse acoustic scattering of a penetrable obstacle with a general conductive boundary condition by using phaseless far field data at a fixed frequency.It is well-known that the modulus of the far field pattern is invariant under translations of the scattering obstacle if only one plane wave is used as the incident field,so it is impossible to reconstruct the location of the underlying scatterers.Based on some new research results on the impenetrable obstacle and inhomogeneous isotropic medium,we consider different types of superpositions of incident waves to break the translation invariance property.

Effect of spin on the instability of THz plasma waves in field-effect transistors under non-ideal boundary conditions

Terahertz(THz) radiation can be generated due to the instability of THz plasma waves in field-effect transistors(FETs). In this work, we discuss the instability of THz plasma waves in the channel of FETs with spin and quantum effects under non-ideal boundary conditions. We obtain a linear dispersion relation by using the hydrodynamic equation, Maxwell equation and spin equation. The influence of source capacitance, drain capacitance, spin effects, quantum effects and channel width on the instability of THz plasma waves under the non-ideal boundary conditions is investigated in great detail. The results of numerical simulation show that the THz plasma wave is unstable when the drain capacitance is smaller than the source capacitance;the oscillation frequency with asymmetric boundary conditions is smaller than that under non-ideal boundary conditions;the instability gain of THz plasma waves becomes lower under non-ideal boundary conditions. This finding provides a new idea for finding efficient THz radiation sources and opens up a new mechanism for the development of THz technology.

Artificial boundary condition for Klein-Gordon equation by constructing mechanics structure

An innovative local artificial boundary condition is proposed to numerically solve the Cauchy problem of the Klein-Gordon equation in an unbounded domain.Initially,the equation is considered as the axial wave prop-agation in a bar supported on a spring foundation.The numerical model is then truncated by replacing the half-infinitely long bar with an equivalent mechanical structure.The effective frequency-dependent stiffness of the half-infinitely long bar is expressed as the sum of rational terms using Pade approximation.For each term,a corresponding substructure composed of dampers and masses is constructed.Finally,the equivalent mechan-ical structure is obtained by parallelly connecting these substructures.The proposed approach can be easily implemented within a standard finite element framework by incorporating additional mass points and damper elements.Numerical examples show that with just a few extra degrees of freedom,the proposed approach effec-tively suppresses artificial reflections at the truncation boundary and exhibits first-order convergence.

Free Vibration Analysis of Rectangular Plate with Cutouts under Elastic Boundary Conditions in Independent Coordinate Coupling Method

Based on Kirchhoff plate theory and the Rayleigh-Ritz method,the model for free vibration of rectangular plate with rectangular cutouts under arbitrary elastic boundary conditions is established by using the improved Fourier series in combination with the independent coordinate coupling method(ICCM).The effect of the cutout is taken into account by subtracting the energies of the cutouts from the total energies of the whole plate.The vibration displacement function of the hole domain is based on the coordinate system of the hole domain in this method.From the continuity condition of the vibration displacement function at the cutout,the transition matrix between the two coordinate systems is constructed,and the mass and stiffness matrices are completely obtained.As a result,the calculation is simplified and the computational efficiency of the solution is improved.In this paper,numerical examples and modal experiments are presented to validate the effectiveness of the modeling methods,and parameters related to influencing factors of the rectangular plate are analyzed to study the vibration characteristics.

One-and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

We address the issue of point value reconstructions from cell averages in the context of third-order finite volume schemes,focusing in particular on the cells close to the boundaries of the domain.In fact,most techniques in the literature rely on the creation of ghost cells outside the boundary and on some form of extrapolation from the inside that,taking into account the boundary conditions,fills the ghost cells with appropriate values,so that a standard reconstruction can be applied also in the boundary cells.In Naumann et al.(Appl.Math.Comput.325:252–270.https://doi.org/10.1016/j.amc.2017.12.041,2018),motivated by the difficulty of choosing appropriate boundary conditions at the internal nodes of a network,a different technique was explored that avoids the use of ghost cells,but instead employs for the boundary cells a different stencil,biased towards the interior of the domain.In this paper,extending that approach,which does not make use of ghost cells,we propose a more accurate reconstruction for the one-dimensional case and a two-dimensional one for Cartesian grids.In several numerical tests,we compare the novel reconstruction with the standard approach using ghost cells.

Entropy Formulation for Triply Nonlinear Degenerate Elliptic-Parabolic-Hyperbolic Equation with Zero-Flux Boundary Condition

In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.

Fast High Order Algorithm for Three-Dimensional Helmholtz Equation Involving Impedance Boundary Condition with Large Wave Numbers

Acoustic fields with impedance boundary conditions have high engineering applications, such as noise control and evaluation of sound insulation materials, and can be approximated by three-dimensional Helmholtz boundary value problems. Finite difference method is widely applied to solving these problems due to its ease of use. However, when the wave number is large, the pollution effects are still a major difficulty in obtaining accurate numerical solutions. We develop a fast algorithm for solving three-dimensional Helmholtz boundary problems with large wave numbers. The boundary of computational domain is discrete based on high-order compact difference scheme. Using the properties of the tensor product and the discrete Fourier sine transform method, the original problem is solved by splitting it into independent small tridiagonal subsystems. Numerical examples with impedance boundary conditions are used to verify the feasibility and accuracy of the proposed algorithm. Results demonstrate that the algorithm has a fourth- order convergence in and -norms, and costs less CPU calculation time and random access memory.

Effect of boundary slip on electroosmotic flow in a curved rectangular microchannel

The aim of this study is to numerically investigate the impact of boundary slip on electroosmotic flow(EOF) in curved rectangular microchannels. Navier slip boundary conditions were employed at the curved microchannel walls. The electric potential distribution was governed by the Poisson–Boltzmann equation, whereas the velocity distribution was determined by the Navier–Stokes equation. The finite-difference method was employed to solve these two equations. The detailed discussion focuses on the impact of the curvature ratio, electrokinetic width, aspect ratio and slip length on the velocity. The results indicate that the present problem is strongly dependent on these parameters. The results demonstrate that by varying the dimensionless slip length from 0.001 to 0.01 while maintaining a curvature ratio of 0.5 there is a twofold increase in the maximum velocity. Moreover, this increase becomes more pronounced at higher curvature ratios. In addition, the velocity difference between the inner and outer radial regions increases with increasing slip length. Therefore, the incorporation of the slip boundary condition results in an augmented velocity and a more non-uniform velocity distribution. The findings presented here offer valuable insights into the design and optimization of EOF performance in curved hydrophobic microchannels featuring rectangular cross-sections.

Application of the double absorbing boundary condition in seismic modeling

We apply the newly proposed double absorbing boundary condition（DABC）（Hagstrom et al., 2014） to solve the boundary reflection problem in seismic finite-difference（FD） modeling. In the DABC scheme, the local high-order absorbing boundary condition is used on two parallel artificial boundaries, and thus double absorption is achieved. Using the general 2D acoustic wave propagation equations as an example, we use the DABC in seismic FD modeling, and discuss the derivation and implementation steps in detail. Compared with the perfectly matched layer（PML）, the complexity decreases, and the stability and fl exibility improve. A homogeneous model and the SEG salt model are selected for numerical experiments. The results show that absorption using the DABC is considerably improved relative to the Clayton–Engquist boundary condition and nearly the same as that in the PML.

COMPACT FOUR-COMPONENT 2-D FDFD METHOD WITH EQUIVALENT SURFACE IMPEDANCE BOUNDARY CONDITION FOR MULTILAYER METAL-COATED WAVEGUIDE

A compact four-component two-dimensional （2-D） finite-difference frequency domain （FDFD） method with the equivalent surface impedance boundary condition is used to analyze the dispersion characteristics of multilayer metal-coated waveguides. According to the equivalent surface impedance boundary condition,the relationship between transverse field components on the boundary can be easily depicted. Once the eigen equation is solved,the propagation constant can be obtained as the eigen value for a given frequency. Results of the proposed method agaree well with those of high frequency structure simulator（HFSS）.