GENERAL DECAY FOR A VISCOELASTIC EQUATION OF VARIABLE COEFFICIENTS WITH A TIME-VARYING DELAY IN THE BOUNDARY FEEDBACK AND ACOUSTIC BOUNDARY CONDITIONS

A variable coefficient viscoelastic equation with a time-varying delay in the boundary feedback and acoustic boundary conditions and nonlinear source term is considered.Under suitable assumptions, general decay results of the energy are established via suitable Lyapunov functionals and some properties of the convex functions. Our result is obtained without imposing any restrictive growth assumption on the damping term and the elements of the matrix A and the kernel function g.

The effects of variable specific heat on the stability of hypersonic boundary layer on a flat plate

The gas temperature within hypersonic boundary layer flow is so high that the specific heat of gas is no longer a constant but relates to temperature. How variable specific heat influences on boundary layer flow stability is worth researching. The effect of the variable specific heat on the stability of hypersonic boundary layer flows is studied and compared with the case of constant specific heat based on the linear stability theory. It is found that the variable specific heat indeed has some effects on the neutral curves of both the first-mode and the second-mode waves and on the maximum rate of growth also. Therefore, the relationship between specific heat and temperature should be considered in the study of the stability of the boundary layer.

Variable Curvature Modeling Method of Soft Continuum Robots with Constraints

The inherent compliance of continuum robots holds great promise in the fields of soft manipulation and safe human–robot interaction.This compliance reduces the risk of damage to the manipulated object and its surroundings.However,continuum robots possess theoretically infinite degrees of freedom,and this high flexibility usually leads to complex deformations when subjected to external forces and positional constraints.Describing these complex deformations is the main challenge in modeling continuum robots.In this study,we investigated a novel variable curvature modeling method for continuum robots,considering external forces and positional constraints.The robot configuration curve is described using the developed mechanical model,and then the robot is fitted to the curve.A ten-section continuum robot prototype with a length of 1 m was developed in order to validate the model.The feasibility and accuracy of the model were verified by the ability of the robot to reach target points and track complex trajectories with a load.This work was able to serve as a new perspective for the design analysis and motion control of continuum robots.

Complex variable solution for boundary value problem with X-shaped cavity in plane elasticity and its application*

A new type of displacement pile, the X-section cast-in-place concrete （XCC） pile, has recently been developed in China. Extensive field tests and laboratory experi- ments are undertaken to evaluate its performance and quantify the non-uniform deforma- tion effect （NUDE） of the X-shaped cross section during installation. This paper develops a simplified theoretical model that attempts to capture the NUDE. Based on the theory of complex variable plane elasticity, closed-form solutions of the stress and displacement for the X-shaped cavity boundary value problem are given. Subsequently, the analytical solution is used to evaluate the NUDE, the concrete filling index （CFI）, and the perimeter reduction coefficient of the XCC pile cross section. The computed results are compared with field test results, showing reasonable agreement. The present simplified theoretical model reveals the deformation mechanism of the X-shaped cavity and facilitates applica- tion of the newly developed XCC pile technique in geotechnical engineering.

Boundary Layer Flow past a Stretching Cylinder and Heat Transfer with Variable Thermal Conductivity

The boundary layer flow of viscous incompressible fluid over a stretching cylinder has been considered to study flow field and temperature field. Due to non-linearity, a numerical approach called Keller-box technique has been used to compute the values of velocity function f and temperature field at different points of dynamic region. The expressions for skin friction and Nusselt number have also been obtained. The dependence of velocity profile and temperature profile on the dimensionless parameter of practical interest has been analyzed in detail by graphs. The dependence of Skin friction and Nusselt number has been seen through tables.

A numerical method based on boundary integral equations and radial basis functions for plane anisotropic thermoelastostatic equations with general variable coefficients

A boundary integral method with radial basis function approximation is proposed for numerically solving an important class of boundary value problems governed by a system of thermoelastostatic equations with variable coe?cients. The equations describe the thermoelastic behaviors of nonhomogeneous anisotropic materials with properties that vary smoothly from point to point in space. No restriction is imposed on the spatial variations of the thermoelastic coe?cients as long as all the requirements of the laws of physics are satis?ed. To check the validity and accuracy of the proposed numerical method, some speci?c test problems with known solutions are solved.

Visco-Elastic Boundary Layer Flow past a Stretching Plate and Heat Transfer with Variable Thermal Conductivity

In the present paper, the boundary layer flow of Walters Liquid B Model over a stretching plate has been considered to solve heat flow problem with variable conductivity. First, using similarity transformation, the velocity components of velocity have been obtained. Then, the heat flow problem has been considered in two ways: 1) prescribed surface temperature (PST), and 2) prescribed stretching plate heat flux (PHF) in case of variable conductivity. Due to variable conductivity, temperature profile has its two part- one mean tempera-ture and other temperature profile induced due to variable conductivity. The related results have been dis-cussed with the help of graphs.

Prediction of hypersonic boundary layer transition on sharp wedge flow considering variable specific heat

When the air temperature reaches 600 K or higher, vibration is excited. The specific heat is not a constant but a function of temperature. Under this condition, the transition position of hypersonic sharp wedge boundary layer is predicted by using the improved eN method considering variable specific heat. The transition positions with different Mach numbers of oncoming flow, half wedge angles, and wall conditions are computed condition, the nearer to the Mach number The results show that for the same oncoming flow condition and wall transition positions of hypersonic sharp wedge boundary layer move much leading edge than those of the flat plate. The greater the oncoming flow the closer the transition position to the leading edge.

An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of theboundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming （SQP） algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method.

A SPECIAL METHOD OF FOURIER SERIES WHICH IS EQUAL TO THE METHOD OF SEPARATION OF VARIABLES ON BOUNDARY VALUE PROBLEM

By the separation of singularity, a special Fourier series solution of the boundary value problem for plane is obtained, which can satisfy all boundary conditions and converges rapidly. II is proved that the solution is equal to the result of separation of variables. As a result, the non-linear characteristic equations resulting from the method of separation of variables are transformed into polynomial equations that can provide a foundation for approximate computation and asymptotic analysis.

An Unsteady Two-Dimensional Complex Variable Boundary Element Method

The Complex Variable Boundary Element Method (CVBEM) procedure is extended to modeling applications of the two-dimensional linear diffusion partial differential equation (PDE) on a rectangular domain. The methodology in this work is suitable for modeling diffusion problems with Dirichlet boundary conditions and an initial condition that is equal on the boundary to the boundary conditions. The underpinning of the modeling approach is to decompose the global initial-boundary value problem into a steady-state component and a transient component. The steady-state component is governed by the Laplace PDE and is modeled using the Complex Variable Boundary Element Method. The transient component is governed by the linear diffusion PDE and is modeled by a linear combination of basis functions that are the products of a two-dimensional Fourier sine series and an exponential function. The global approximation function is the sum of the approximate solutions from the two components. The boundary conditions of the steady-state problem are specified to match the boundary conditions from the global problem so that the CVBEM approximation function satisfies the global boundary conditions. Consequently, the boundary conditions of the transient problem are specified to be continuously zero. The initial condition of the transient component is specified as the difference between the initial condition of the global initial-boundary value problem and the CVBEM approximation of the steady-state solution. Therefore, when the approximate solutions from the two components are summed, the resulting global approximation function approximately satisfies the global initial condition. In this work, it will be demonstrated that the coupled global approximation function satisfies the governing diffusion PDE. Lastly, a procedure for developing streamlines at arbitrary model time is discussed.

A Conceptual Numerical Model of the Wave Equation Using the Complex Variable Boundary Element Method

In this work, a conceptual numerical solution of the two-dimensional wave partial differential equation (PDE) is developed by coupling the Complex Variable Boundary Element Method (CVBEM) and a generalized Fourier series. The technique described in this work is suitable for modeling initial-boundary value problems governed by the wave equation on a rectangular domain with Dirichlet boundary conditions and an initial condition that is equal on the boundary to the boundary conditions. The new numerical scheme is based on the standard approach of decomposing the global initial-boundary value problem into a steady-state component and a time-dependent component. The steady-state component is governed by the Laplace PDE and is modeled with the CVBEM. The time-dependent component is governed by the wave PDE and is modeled using a generalized Fourier series. The approximate global solution is the sum of the CVBEM and generalized Fourier series approximations. The boundary conditions of the steady-state component are specified as the boundary conditions from the global BVP. The boundary conditions of the time-dependent component are specified to be identically zero. The initial condition of the time-dependent component is calculated as the difference between the global initial condition and the CVBEM approximation of the steady-state solution. Additionally, the generalized Fourier series approximation of the time-dependent component is fitted so as to approximately satisfy the derivative of the initial condition. It is shown that the strong formulation of the wave PDE is satisfied by the superposed approximate solutions of the time-dependent and steady-state components.

Dirichlet boundary value problem with variable growth

In this paper, we study higher order elliptic partial differential equations with variable growth, and obtain the existence of solutions in the setting of Wm,p（x） spaces by means of an abstract result for variational inequalities obtained by Gossez and Mustonen. Our result generalizes the corresponding one of Kováik and Rákosník.

An extended variable-grid global ocean circulation model and its preliminary results of the equatorial Pacific circulation

To investigate the interaction between the tropical Pacific and China seas a variable-grid global ocean circulation model with fine grid covering the area from 20°S to 50°N and from 99° to 150°E is developed. Numerical computation of the annually cyclic circulation fields is performed. The results of the annual mean zonal currents and deep to abyssal western boundary currents in the equatorial Pacific Ocean are reported. The North Equatorial Current,the North Equatorial Countercurrent, the South Equatorial Current and the Equatorial Undercurrent are fairly well simulated. The model well reproduces the northward flowing abyssal western boundary current.From the model results a lower deep western boundary current east of the Bismarck-Solomon-New Hebrides Island chain at depths around 2 000 m has been found. The model results also show that the currents in the equatorial Pacific Ocean have multi-layer structures both in zonal currents and western boundary currents, indicating that the global ocean overturning thermohaline circulation appears of multi-layer pattern.

A SCALAR AUXILIARY VARIABLE (SAV) FINITE ELEMENT NUMERICAL SCHEME FOR THE CAHN-HILLIARD-HELE-SHAW SYSTEM WITH DYNAMIC BOUNDARY CONDITIONS

In this paper,we consider the Cahn-Hilliard-Hele-Shaw(CHHS)system with the dynamic boundary conditions,in which both the bulk and surface energy parts play important roles.The scalar auxiliary variable approach is introduced for the physical system;the mass conservation and energy dissipation is proved for the CHHS system.Subsequently,a fully discrete SAV finite element scheme is proposed,with the mass conservation and energy dissipation laws established at a theoretical level.In addition,the convergence analysis and error estimate is provided for the proposed SAV numerical scheme.

Coupling of high order multiplication perturbation method and reduction method for variable coefcient singular perturbation problems

Based on the precise integration method （PIM）, a coupling technique of the high order multiplication perturbation method （HOMPM） and the reduction method is proposed to solve variable coefficient singularly perturbed two-point boundary value prob lems （TPBVPs） with one boundary layer. First, the inhomogeneous ordinary differential equations （ODEs） are transformed into the homogeneous ODEs by variable coefficient dimensional expansion. Then, the whole interval is divided evenly, and the transfer ma trix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efficient.

Coordinating optimization-based sliding mode variable structure control for electro-hydraulic servo system

A sliding mode variable structure control （SMVSC） based on a coordinating optimization algorithm has been developed. Steady state error and control switching frequency are used to constitute the system performance indexes in the coordinating optimization, while the tuning rate of boundary layer width （BLW） is employed as the optimization parameter. Based on the mathematical relationship between the BLW and steady-state error, an optimized BLW tuning rate is added to the nonlinear control term of SMVSC. Simulation experiment results applied to the positioning control of an electro-hydraulic servo system show the comprehensive superiority in dynamical and static state performance by using the proposed controller is better than that by using SMVSC without optimized BLW tuning rate. This succeeds in coordinately considering both chattering reduction and high-precision control realization in SMVSC.

Iterative Method on Well Bore Boundary in Numerical Modeling of Variably Saturated Flow

An iterative method was developed for incorporating the well bore boundary into the finite difference model of water flow in variably saturated porous media. Six cases were presented involving groundwater pumping or injection to demonstrate the advantages of the iterative method over the traditional method. For the iterative method, the total flux gradually approached the well discharge and the flux profile was non-uniform. And the iterative method took into account the variation of well bore water table. Compared to the traditional method, the iterative method can simulate the variably saturated flow caused by pumping or injection more realistically.

Design of Variable Structure Control by Using the L_2 Gain Method

The robust stabilization problem for a family of nonlinear plants with mismatch uncertainties is addressed, and a solution is presented based on variable structure control theory and H∞ control theory. A kind of boundary layer is built near the ideal switch surface which can eliminate chattering in the switch surface. The proposed control method with L2 gain can guarantee exponential stability of a system state with no parameter uncertainty and exter- nal disturbance, while it can guarantee state ultimate boundness if parameter uncertainty and external disturbance exist. In the proposed design method, stability of the closed-loop system is analyzed by adopting the Lyapurtov func- tion approach. Finally the numerical simulation results show that the proposed smooth variable structure controller has good pelformance without chattering in the switch surface.

Topology Optimization of Sound-Absorbing Materials for Two-Dimensional Acoustic Problems Using Isogeometric Boundary Element Method

In this work,an acoustic topology optimizationmethod for structural surface design covered by porous materials is proposed.The analysis of acoustic problems is performed using the isogeometric boundary elementmethod.Taking the element density of porousmaterials as the design variable,the volume of porousmaterials as the constraint,and the minimum sound pressure or maximum scattered sound power as the design goal,the topology optimization is carried out by solid isotropic material with penalization(SIMP)method.To get a limpid 0–1 distribution,a smoothing Heaviside-like function is proposed.To obtain the gradient value of the objective function,a sensitivity analysis method based on the adjoint variable method(AVM)is proposed.To find the optimal solution,the optimization problems are solved by the method of moving asymptotes(MMA)based on gradient information.Numerical examples verify the effectiveness of the proposed topology optimization method in the optimization process of two-dimensional acoustic problems.Furthermore,the optimal distribution of sound-absorbingmaterials is highly frequency-dependent and usually needs to be performed within a frequency band.