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There is today a growing need to accurately model the angular scattering response of metasurfaces for optical analog processing applications. However, the current metasurface modeling techniques are not well suited for such a task since they are limited to small angular spectrum transformations, as shall be demonstrated. The goal of this work is to overcome this limitation by improving the modeling accuracy of these techniques and, specifically, to provide a better description of the angular response of metasurfaces. This is achieved by extending the current methods, which are restricted to dipolar responses and weak spatially dispersive effects, so as to include quadrupolar responses and higher-order spatially dispersive components. The accuracy of the newly derived multipolar model is demonstrated by predicting the angular scattering of a dielectric metasurface placed in a vacuum. This results in a modeling accuracy that is substantially better than the standard dipolar model.

The flow of a compressible fluid with slip through a cylinder with an asymmetric local constriction has been considered both numerically, as well as analytically. For the numerical work, a particle-based method whose dynamics is governed by the multiparticle collision (MPC) rule has been used together with a generalized boundary condition that allows for slip at the wall. Since it is well known that an MPC system corresponds to an ideal gas and behaves like a compressible, viscous flow on average, an approximate analytical solution has been derived from the compressible Navier–Stokes equations of motion coupled to an ideal gas equation of state using the Karman–Pohlhausen method. The constriction is assumed to have a polynomial form, and the location of maximum constriction is varied throughout the constricted portion of the cylinder. Results for centerline densities and centerline velocities have been compared for various Reynolds numbers, Mach numbers, wall slip values and flow geometries.

The boundary feedback control of networks of freeway traffic is considered in this article by means of the partial differential equation (PDE)-based techniques. The control and measurements are all located at the boundaries of each link. We have established the boundary control model for the system, a linear hyperbolic system of balance laws, which includes not only the traffic flow dynamics of the network described by combining the linearized Aw-Rascle-Zhang (ARZ) traffic flow model of each link but also the integrated on-ramping metering and variable speed limit control modeled as the boundary condition. As the traffic demand of the network is fluctuated, the boundary input-to-state stability (ISS) controller is designed to suppress the disturbance and regulate the traffic flow into the boundedness regions of the desired states. Based on a novel Lyapunov function, some sufficient conditions in terms of the matrix inequalities are derived for the ISS boundary stabilization in the L²-norm. The numerical simulation is given to illustrate the effectiveness of the developed boundary feedback control. Moreover, an interesting traffic experiment is carried out by using the traffic simulation software, AIMSUN, and some specific traffic data are collected and analyzed to verify the feasibility and effectiveness of the boundary control.

This paper presents a mathematical modelling of a novel multiphysics (electro-mechanic coupling) problem of the shear wave propagation in laminated structures (piezoelectric - hydrogel - elastic substrate) using the wave mode method while finding the general solution for each medium. A dynamic mechanical equilibrium equation (for transverse deflection) and Maxwell equation (for electric potential) are solved in a coupled manner over each domain resulting a general closed-form solution. A specific analytical solution is then obtained enforcing the boundary conditions at the top of piezoelectric layer and the interface continuity requirements between each layer and the elastic half-space. A novel approach, to uncouple the coupled equations, is presented resulting in a final systematic solution. The effect of x coordinate (thickness) on the field variables is carefully examined for the electrically open and short cases. The jump in the stress and electric displacement components (causing delamination due to shearing mode of fracture) is present across all the interfaces due to the different bulk material constants. The key contribution of the current work is demonstrating the influence of fixed-charge concentration inside the hydrogel layer on the shear wave propagation. The present study thus provides a new concept for adjusting and controlling the elastic wave propagation in the composite structures, and provides a proper theoretical understanding of wave propagation in the damage tolerance-based design of piezotronics devices.

In order to simulate the elastic wave propagation in unbounded solids, this paper introduces a non-local method to construct absorbing boundary conditions in the bond-based peridynamic theory. To construct non-local absorbing boundary conditions in the form of absorbing layers with increasing damping, we first re-derived the particle motion equilibrium equation of the bond-based peridynamic theory, based on the principle of virtual work and Lagrange's equation on the premise of considering material damping. We also give the expression for the inter-particle damping force density which satisfies the balance laws of linear momentum and angular momentum. This paper also provides a process and algorithm for realizing the non-local absorbing boundary conditions in the bond-based peridynamic theory, and adopts the Fortran language for computer program compilation. Finally, we simulate the elastic wave propagation in semi-unbounded and unbounded solids as numerical examples. The results show that such non-local absorbing boundary conditions can stably absorb and attenuate incident elastic waves. Since it is derived in the time domain, this method does not require frequency-domain operations, such as Fourier and Laplace transformations, nor does it require wave field splitting.

This paper presents a numerical investigation of the laminar flow and pressure drop characteristics of planar Tee joints, a canonical flow of interest for the thermal-hydraulic design of oil-immersed power transformer windings. After formulating the problem in nondimensional form, the steady, constant property flow in planar 90 ∘ Tee joints is computed numerically by integrating the Navier–Stokes equations with fully developed upstream and downstream boundary conditions. The analysis assumes a straight-through configuration in which the straight duct holds flow in the same direction before and after the junction, whereas the flow in the side branch can either divide from the incoming flow or combine with it. The analysis starts with the description of the flow patterns that emerge in the dividing and combining cases for all mass split ratios, 0 ≤ m 1 ≤ 1 , and a wide range of straight duct to side branch width ratios, 1 ≤ α ≤ 3 , and Reynolds numbers of the common branch, 0 < Re3 ≤ 200, values that are representative of the cooling oil flow in oil-immersed transformer winding. Flow maps for planar Tee joints are then presented, showing the existence of different regions in the (Re3, m 1 )-plane that exhibit different number and location of recirculation zones. Pressure distributions and secondary loss coefficients are then computed and analyzed, providing a numerical database that is used to develop new local pressure loss correlations for planar Tee joints in an accompanying paper.

Assume the initial data of compressible Euler equations has finite energy and total mass. We can construct a sequence of solutions of one-dimensional compressible Navier-Stokes equations (density-dependent viscosity) with stress-free boundary conditions, so that, up to a subsequence, the sequence of solutions of compressible Navier-Stokes equations converges to a finite-energy weak solution of compressible Euler equations. Hence the inviscid limit of the compressible Navier-Stokes is justified. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., α = 1 , γ = 2 ).

The strain gradient elasticity theory is adopted to capture the small-scale effect of micro/nanostructures to derive the governing equation by using Hamilton's principle. A mesh-free method on the basis of moving Kriging interpolation, of which shape function possesses C2 continuum, is developed to investigate the vibration of the strain gradient plate. Owing to the inherent Kronecker delta function property of the moving Kriging interpolation, essential boundary conditions can be directly implemented. One superiority of the moving Kriging interpolation over the traditional moving least square approximation is that it can ensure the continuity and stability at the third order derivatives, which is confirmed by comparison with analytic solutions. Parametric studies in terms of small-scale parameters, boundary conditions and side length on the frequencies and mode shapes are carried out. Natural frequencies evaluated by strain gradient elasticity theory are found to be lower than those of classical plate model. Some lower natural frequencies corresponding to ultrahigh-order mode shapes appear. This demonstrates that the mesh-free moving Kriging interpolation method has high precision and good stability in the construction of the C2 continuum shape functions.

This work presents a method for conforming fluid flow to tensor fields in particle-in-cell simulations. Our focus is on the use of tensor fields as a medium to deflect fluids for computer graphics applications. The anisotropic fluid simulation here presented adapts the Fluid-Implicit-Particle (FLIP) method, by including the tensor field information in the advection and projection steps. Numerical solutions are proposed for adapting the anisotropic advection to the particle-in-cell model and for solving the anisotropic Helmholtz projection for Marker-and-Cell (MAC) grids. Suitable boundary conditions are also provided to ensure proper anisotropic fluid behavior. To evidence the capabilities of this approach, we present simulations with different tensor fields composed of isotropic, linear, and planar tensors. Experimental results show that our method is capable of deflecting or withholding flow in specific directions, according to the characteristics of the underlying tensor field, producing visual outcomes typically encountered in anisotropic media, such as permeable-like effects.

In this paper, a meshless discrete scheme based on the generalized finite difference method (GFDM) is proposed to solve the biharmonic interface problem. This scheme turns the interface problem to be two non-interface subproblems coupled with the interface conditions. The interface conditions act as the boundary conditions of the subproblems because the interface plays the role of their boundary. Therefore, the proposed method has the advantage of dealing with problems with complex geometrical interfaces. Due to the GFDM approximates the derivatives of the unknown variables by using a linear combination of the values for nearby nodes, the proposed method also has the advantage in dealing with the interface condition with the jump of derivatives. Four numerical examples are provided to verify the accuracy and the stability of the GFDM for biharmonic interface problems. The numerical results show that the H1 errors can reach almost the same convergence rate of the L∞ and L2 errors. The convergence rates are almost and even higher than fourth-order when the fourth-order Taylor Series expansion is adopted in the GFDM.

From the formal expansion of the solutions of Euler-Voigt equations in R + 2 with no-slip boundary conditions, the boundary layer equations of Euler-Voigt equations to Euler equations are obtained. In case of the analytic data, one obtains the local existence and uniqueness of the solutions for the boundary layer equations by abstract Cauchy-Kovalevskaya theorem.

The detailed study on convection heat transfer and flow structure characteristic of air-jet studied with impinging vertically on a hemisphere plate. The inner surface of the cylindrical-shaped plate is subject to constant heat flux boundary conditions. Hemispherical plate subjected to comparative computations with Reynolds number (Re) ranging from 3000, 6000, and 9000 for the round nozzle impinging jet.

In this paper, we present new Lyapunov-type inequalities for Hilfer-Katugampola fractional differential equations. We first give some unique properties of the Hilfer-Katugampola fractional derivative, and then by using these new properties we convert the multi-point boundary value problems of Hilfer-Katugampola fractional differential equations into the equivalent integral equations with corresponding Green's functions, respectively. Finally, we make use of the Banach's contraction principle to derive the desired results, and give a series of corollaries to show that the current results extend and enrich the previous results in the literature.

The quasi-steady translational motion of an incompressible couple stress fluid confined between two eccentric spherical surfaces is investigated. It is assumed that the motion is generated by allowing the internal sphere to translate with a constant velocity. The Stokesian assumption of low Reynolds numbers is considered so that the nonlinear terms of the equation of motion are neglected. The superposition principle is employed to construct the general solution of the governing equations in the presence of spherical surfaces using two spherical systems of coordinates relative to the centers of the two spherical surfaces. The no-slip boundary conditions on the spherical boundaries are applied. In addition, the conditions of vanishing couple stresses on the bounding surfaces are imposed. The boundary collocation technique is utilized numerically to satisfy the boundary conditions on the spherical surfaces. The drag force experienced by the fluid flow on the spherical particle is obtained and represented numerically through table and graphs. The numerical results show that the values of the drag force increases monotonically with the increase in the radii ratio. It increases also with the increase of the couple stress viscosity parameter. The obtained results are consistence with the results available in the literature for viscous fluids.

Cryoablation is the most commonly used type of cryosurgery. Today, this treatment is insufficiently automated and therefore less accurate than alternative technologies that significantly limits and gradually reduces the scope of its application. However, from low temperature engineering point of view cryosurgery has the untapped potential of the improving its accuracy. In this paper three major methodological problems of cryosurgical equipment and procedure improving are considered: the impact on target isotherm location (1) of the soft biotissue moisture content solidification latent heat variation, (2) of the cryoprobe constructions and operating at maximum cooling power including the fluid cryogen features and the accuracy of boundary condition replacing a cryoprobe and (3) target isotherm fixation approach by modifying the cryoprobe operating mode. The numerical results showed that the target isotherm location depends insignificantly on different moisture content of soft biotissue, then it is not a limiting factor for cryoexposure prediction. Also numerical results showed that the theoretically minimum temperature of cryoprobe surface significantly overestimates their movement, this leads to a significant difference between the experimental and simulation results. The typical shape and movement of cryonecrosis isotherm at refined maximum cryoprobe cooling power for two typical constructions of cryoprobes and two cryogens are presented. The novel algorithm of modifying the cryoprobe operation mode for controlled precision cryosurgery is proposed and validated. All this complements the previously obtained data and shows the prospects for performing precision cryosurgery. The total average uncertainty of target isotherm location during the studied preliminary predicting stage of temperature field movement (virtual cryoablation) is estimated no more than ±0.41 mm. This information is expected to be useful for improving the quality of cryosurgery planning algorithms (e.g. for tumor treatment).

The steady low-Reynolds-number rotation of a spherical soft particle (a hard core coated with a permeable porous layer) in a viscous fluid within a nonconcentric spherical cavity about their common diameter is semi-analytically studied. To solve the Stokes and Brinkman equations for the fluid velocity, a solution is constituted by the general solutions in two spherical coordinate systems originated from the particle and cavity centers and the boundary conditions are satisfied by a collocation technique. Numerical results of the hydrodynamic torque on the soft sphere are obtained as a function of the core-to-particle radius ratio, particle-to-cavity radius ratio, relative center-to-center distance of the particle and cavity, and ratio of the particle radius to the permeation length in the porous layer over the entire ranges. The effect of the cavity on the torque of a rotating soft particle is weaker than that of a corresponding hard particle (or soft one with lower permeability or thinner thickness of its porous layer). While the normalized torque of a soft sphere in general is an increasing function of the particle-to-cavity radius ratio, a weak minimum of it (surprisingly, less than the value of an unconfined particle) may occur for a particle with a small to mediate core-to-particle radius ratio and a high permeability inside a nonconcentric cavity at a moderate value of the particle-to-cavity radius ratio. Also, this torque in general is an increasing function of the eccentricity of the particle location, but it may decrease slightly with an increase in the eccentricity for a particle with a small to mediate core-to-particle radius ratio and a high permeability.

In this paper we study the solvability of different boundary value problems for the two dimensional steady incompressible Euler equation and for the magneto-hydrostatic equation. Two main methods are currently available to study those problems, namely the Grad-Shafranov method and the vorticity transport method. We describe for which boundary value problems these methods can be applied. The obtained solutions have non-vanishing vorticity.

In this article we present the discrete-time isoholonomic problem of the planar Purcell's swimmer and solve it using the discrete-time Pontryagin maximum principle. The 3-link Purcell's swimmer is a locomotion system moving in a low Reynolds number environment. The kinematics of the system evolves on a principal fiber bundle. A structure-preserving discrete-time kinematic model of the system is obtained in terms of the local form of a discrete connection. An adapted version of the Discrete Maximum Principle on matrix Lie groups is then employed to come up with the necessary optimality conditions for an optimal transfer from a given initial state while minimizing the mechanical energy expended in the presence of constraints on the controls. These necessary conditions appear as a two-point boundary value problem and are solved using a numerical technique. Results from numerical experiments are presented to illustrate the algorithm and compared with the existing results for a similar case in the literature.

We develop the idea that a natural link between Boltzmann schemes and finite volumes exists naturally: the conserved mass and momentum during the collision phase of the Boltzmann scheme induces general expressions for mass and momentum fluxes. We treat a unidimensional case and focus our development in two dimensions on possible flux boundary conditions. Several test cases show that a high level of accuracy can be achieved with this scheme.

The chapter reviews fundamental theories, formulas, solution methods, and a set (toolbox) of MATLAB functions for static analysis of Euler–Bernoulli beams. The system requirements for the MATLAB toolbox—a PC with Win 98SE/NT/2000 and XP or Mac with OS 9.x and up, and the software MATLAB (version 5.x and up) installed on the computer. A fundamental problem in the static analysis of Euler-Bernoulli beams is stated as the given external loads and boundary disturbances that determines the beam response (displacement, rotation, bending moment, and shear force), which is governed by a given equation and boundary conditions. There are many solution methods available for the static analysis of Euler-Bernoulli beams. The chapter presents three analytical solution methods for the beam problem—method of singularity functions, the boundary value approach, and distributed transfer function method (DTFM). The MATLAB functions are developed based on the distributed transfer function method, which is an exact analytical modeling and solution method for many one-dimensional distributed systems including bars, shafts, strings, and beams.

The aim of the work. The problem of numerical simulation of longitudinal, transverse and surface waves on the free surface of an elastic half-plane is considered. Methods. To solve the non-stationary dynamic problem of elasticity theory with initial and boundary conditions, the finite element method in displacements was used. Using the finite element method in displacements, a linear problem with initial and boundary conditions was led to a linear Cauchy problem. A quasiregular approach to solving a system of second-order linear ordinary differential equations in displacements with initial conditions and to approximating the area under study is proposed. The method is based on the schemes: point, line and plane. The study area is divided by spatial variables into triangular and rectangular finite elements of the first order. According to the time variable, the study area is divided into linear end elements with two nodal points. The Fortran-90 algorithmic language was used in the development of the software package. Results. Some information is given about numerical modeling of elastic stress waves in an elastic half-plane with a concentrated wave action in the form of a Delta function. The estimated area under study has 12 008 001 nodal points. A system of equations consisting of 48 032 004 unknowns is solved. The change of elastic contour stress on the free surface of the half-plane at different points is shown. The amplitude of Rayleigh surface waves is significantly greater than the amplitudes of longitudinal, transverse, and other waves with a concentrated vertical action in the form of a triangular pulse on the surface of an elastic half-plane. After surface Rayleigh waves, a dynamic process is observed in the form of standing waves.

In this paper, we present the numerical solution of the Stefan problem to calculate the temperature of the tungsten sample heated by the laser pulse. Mathematical modeling is carried out to analyze field experiments, where an instantaneous heating of the plate to 9000 K is observed due to the effect of a heat flow on its surface and subsequent cooling. The problem is characterized by nonlinear coefficients and boundary conditions. An important role is played by the evaporation of the metal from the heated surface. Basing on Samarskiis approach, we choose to implement the method of continuous counting considering the heat conductivity equation in a uniform form in the entire domain using the Dirac delta function. The numerical method has the second order of approximation with respect to space, the interval of smoothing of the coefficients is 5 K. As a result, we obtain the temperature distributions on the surface and in the cross section of the sample during cooling.

This paper presents nonlinear vibration behavior of shear deformable bidirectional porous beams with non-uniform porosity distribution. A new porosity distributions are proposed to maximize natural frequencies of the porous beam. It is assumed that closed-cell voids are distributed along the thickness and length of the beam. Hamilton principle is used to derive nonlinear governing equations using Reddy beam theory considering von Karman geometrical nonlinearity. Generalized differential quadrature method, harmonic balance approach along with the Picard iterative approach are used to discretize and solve the nonlinear dynamic equations. The variation of nonlinear frequency versus the vibration amplitude is highlighted considering various influential factors such as geometrical parameters, porosity distributions, beam theories and boundary conditions. It is shown that the newly defined porosity distribution can increase the frequencies in comparison to the conventional porosity distribution. Comparing beams of the same weight shows that porosities near the center of the beam further increases frequencies in comparison to the other porosity distributions. Moreover, bidirectional porosity increases the frequencies in comparison to the unidirectional porosity. Frequencies of the bidirectional porous Reddy beam depend not only on the bending stiffness, as in the case of unidirectional porous beam, but also on the first derivative of the bending stiffness.

This paper strives to derive the analytical solutions for the dynamic analysis of stepped multi-span rotor system, where the rotating shaft's internal damping effect is also considered. Specifically, the steady-state whirl analysis and free vibration analysis is given herein. For the steady-state whirl analysis, firstly, the stepped shaft is spilt to several uniform segments and their governing equations are established by Hamilton principle. Subsequently, by applying variable separation method and Laplace transform method to each segment's governing equation, their steady-state responses in terms of determinate interpolation functions multiplied by unknown boundary constants are derived. Then based on the transfer matrix method, each segment's boundary constants are determined with consideration of the compatibility conditions of each two adjacent segments and the end boundary conditions. For the free vibration analysis, one only need to remove the forced term in the steady-state form of transfer matrix and the rotor's characteristic equation will be obtained. Through the theoretical derivation, it is found that this analytical approach can be generalized to any isotropic elastic boundary conditions. To validate the proposed method, two case studies are proposed, where the finite element method is used as benchmark method. For the first one, the influence of internal damping on a uniform rotor's damped whirl characteristics and its stability is analyzed. The damped whirl characteristics analysis reveals that viscous internal damping result in the destabilization of forward modes as long as the spinning speed becomes higher than the critical speed. This phenomenon is also demonstrated by the time responses analysis. For the second one, the dynamic responses of a two-span three steps rotor-bearing system with isotropic viscoelastic boundary conditions are determined. All of simulated results show a great agreement between analytical results and FEM results.

In this paper, we present an indirect computational procedure based on the truncated second kind Chebyshev wavelets for finding the solutions of Caputo fractional time-invariant linear optimal control systems which the functional cost consists of the finite-time quadratic cost function. Utilizing the operational matrices of second Chebyshev wavelets (SCWs) of the Riemann–Liouville fractional integral, the corresponding linear two-point boundary value problem (TPBVP), obtained from the fractional Euler–Lagrange equations, is reduced to a coupled Sylvester-type matrix equation. An equivalent linear matrix form using the Kronecker product is constructed. The upper bound of the error of the SCWs approximation and the convergence of the proposed method are investigated. Low computational complexity and flexible accuracy are two important superiorities of this approach. Numerical experiments provide satisfactory results compared to the exiting techniques.

The purpose of the study - receiving of exact analytical solutions of statics problems of anisotropic plates, resting on an elastic subgrade and subjected to an action of cyclic symmetrical loads. The method of compensating loads is used for solving of the formulated problems. The basic and the compensating solutions are determined. The new approach, connected with the use of Nielsens equation for receiving of the solutions, is applied. For the first time by means of the method of compensating loads the exact analytical solutions of the cycle symmetric flexure of anisotropic circular plates, resting on the elastic subgrade, are received. Various boundary conditions and the loads, distributed along circumferences and over ring surfaces, are considered. The problem of anisotropic infinite plate with the circular opening, resting on the elastic subgrade, is also examined. All the solutions are obtained in the closed form and expressed in terms of Bessel functions.

In this paper, we prove that there exists a unique weak solution to the Dirichlet boundary value problem for second order elliptic operators whose coefficients are signed measures. Our approach is probabilistic, which also gives a representation of the solution. The heat kernel estimates and the theory of additive functionals play a crucial role in our approach.

International audience; Many-particle time domain methods are rising alternatives to particle-in-cell (PIC) or frequency methods to simulate the wave–beam interactions in traveling wave tubes (TWTs). We focus on two of those: our Hamiltonian discrete model DIMOHA is compared analytically against the pseudospectral method RUBEUS. Although based on two completely different approaches—the Gel’fand transform for DIMOHA and the telegraphist circuit for RUBEUS—we surprisingly find out that they share perfectly parallel sets of equations and variables. However, we conclude that DIMOHA is more flexible than RUBEUS in terms of pitch tapering and absorbing boundary conditions. It also shows excellent stability for steady-state simulation, allowing us to explain some discrepancies of RUBEUS with experimental results. These come from a standing wave pattern which is detectable in the vicinity of the sever. Finally, DIMOHA is tested for the first time with ultra-short pulses and exhibits excellent agreement with RUBEUS.

A finite difference scheme along with a fourth-order approximation is examined in this article for finding the solution of time-fractional diffusion equation with fourth-order derivative in space subject to homogeneous and non-homogeneous boundary conditions. Caputo fractional derivative is used to describe the time derivative. The time-fractional diffusion equation of order $$0< \gamma < 1$$ is transformed into Volterra integral equation which is then approximated by linear interpolation. A novel numerical scheme based on fractional trapezoid formula for time discretization followed by Stephenson’s scheme to discretize the fourth order space derivative is developed for the linear diffusion equation. Afterward, convergence and stability are investigated thoroughly showing that the proposed scheme is unconditionally stable and hold convergence accuracy of order $$O(\tau ^{2}+h^{4})$$ . The numerical examples are presented in accordance with the theoretical results showing the efficiency and accuracy of the presented numerical technique.

In this paper, we study second-order and fourth-order elliptic problems which include not only a Poisson equation in the bulk but also an inhomogeneous Laplace--Beltrami equation on the boundary of the domain. The bulk and the surface PDE are coupled by a boundary condition that is either of Dirichlet or Robin type. We point out that both the Dirichlet and the Robin type boundary condition can be handled simultaneously through our formalism without having to change the framework. Moreover, we investigate the eigenvalue problems associated with these second-order and fourth-order elliptic systems. We further discuss the relation between these elliptic problems and certain parabolic problems, especially the Allen--Cahn equation and the Cahn--Hilliard equation with dynamic boundary conditions.

We discuss boundary value problems for the characteristic stationary von Neumann equation (stationary sigma equation) and the stationary Wigner equation in a single spatial dimension. The two equations are related by a Fourier transform in the non-spatial coordinate. In general, a solution to the characteristic equation does not produce a corresponding Wigner solution as the Fourier transform will not exist. Solution of the stationary Wigner equation on a shifted k-grid gives unphysical results. Results showing a negative differential resistance in IV-curves of resonant tunneling diodes using Frensley’s method are a numerical artefact from using upwinding on a coarse grid. We introduce the integro-differential sigma equation which avoids distributional parts at $$k=0$$ k = 0 in the Wigner transform. The Wigner equation for $$k=0$$ k = 0 represents an algebraic constraint needed to avoid poles in the solution at $$k=0$$ k = 0 . We impose the inverse Fourier transform of the integrability constraint in the integro-differential sigma equation. After a cutoff, we find that this gives fully homogeneous boundary conditions in the non-spatial coordinate which is overdetermined. Employing an absorbing potential layer double homogeneous boundary conditions are naturally fulfilled. Simulation results for resonant tunneling diodes from solving the constrained sigma equation in the least squares sense with an absorbing potential reproduce results from the quantum transmitting boundary with high accuracy. We discuss the zero bias case where also good agreement is found. In conclusion, we argue that properly formulated open boundary conditions have to be imposed on non-spatial boundaries in the sigma equation both in the stationary and the transient case. When solving the Wigner equation, an absorbing potential layer has to be employed.

In this paper, we deal with a specific type of quasilinear boundary value problem with Dirichlet boundary conditions and with p -Laplacian. We show two ways of proving the existence of nontrivial weak solutions. The first one uses the mountain pass theorem, the other one is based on our new minimax theorem. This method is novel even for p = 2 . In the paper, we also present a numerical algorithm based on the introduced approach. The suggested algorithm is illustrated on numerical examples and compared with a current approach to demonstrate its efficiency.

In this paper, a new scheme is proposed to solve the Stokes interface problem. The scheme turns the Stokes interface problem into two coupled Stokes non-interface subproblems and adds a mixed boundary condition to overcome the numerical pressure oscillation. Since the interface becomes the boundary of the subproblems, the scheme has the advantage to deal with the interface problem with complex geometry. Furthermore, a generalized finite difference method (GFDM) is adopted to solve the coupled Stokes non-interface subproblems. The GFDM is developed from the Taylor series expansions and moving-least squares approximation. Due to the flexibility of the GFDM, it is convenient to handle the complex boundary conditions that appeared in the proposed scheme. The numerical examples verify the accuracy and stability of the GFDM to solve the Stokes interface problem with the mixed boundary conditions. Moreover, for some given numerical examples, the proposed scheme is more accurate than the classical formula of the pressure Poisson equation, especially in terms of pressure.

The paper deals with the numerical realization of the 3D Stokes flow subject to threshold slip boundary conditions. The weak velocity–pressure formulation leads to an inequality type problem that is approximated by a mixed finite element method. The resulting algebraic system is non-smooth. Besides the pressure, three additional Lagrange multipliers are introduced: the discrete normal stress releasing the impermeability condition and two discrete shear stresses regularizing the non-smooth slip term. Eliminating the discrete velocity component we obtain the minimization problem for the smooth functional, expressed in terms of the pressure, the normal, and the shear stresses. This problem is solved either by a path following variant of the interior point method or by the semi-smooth Newton method. Numerical scalability is illustrated by computational experiments.

We consider an overdetermined fourth order boundary value problem in which the boundary value of the Laplacian of the solution is prescribed, in addition to the homogeneous Dirichlet boundary condition. It is known that, in the case where the prescribed boundary value is a constant, this overdetermined problem has a solution if and only if the domain under consideration is a ball. In this paper, we study the shape of a domain admitting a solution to the overdetermined problem when the prescribed boundary value is slightly perturbed from a constant. We derive an integral identity for the fourth order Dirichlet problem and a nonlinear weighted trace inequality, and the combination of them results in a quantitative stability estimate which measures the deviation of a domain from a ball in terms of the perturbation of the boundary value.

The paper considers the features of the formation of an acoustic field by a spherical source with complicated properties in a regular plane-parallel waveguide, which is of practical importance in marine instrumentation and oceanographic research. The calculation algorithm is based on the use of the Helmholtz equation and the Fourier method for each partial region and the conjugation conditions on their boundaries. The presented calculation allows one to get rid of the idealized boundary conditions on the source surface, with the subsequent determination of the excitation coefficients of the waveguide modes within the framework of the Sturm-Liouville problem. In this case, the attraction of the boundary conditions on the surface and the bottom of the sea, as well as the Sommerfeld conditions, makes it possible to obtain the real distribution of the field in the vertical sections of the waveguide. The obtained frequency dependences of the pressure and vibrational velocity components show their amplitude-phase differences, which reach 90 degrees, which partially explains the appearance of singular points in the intensity field in a regular waveguide. It has been determined that multiple reflections of sound waves from the boundaries of the working space and the space of the waveguide cause oscillations of the pressure components with a change in the amplitude level up to 6dB. It was found that with an increase in the size of the source, a kind of resonance is formed in the working space, the frequency of which depends on the depth of the sea and corresponds to the region kr=x=5.8. It was found that when the acoustic field is formed in the working space, the frequency response of the impedance components is represented as a multiresonant dependence formed on the basis of the frequency characteristics of the lower modes and their combinations. Experimental studies have shown that the results of calculations of the mode composition of the acoustic field of the emitter, obtained in the conditions of the pool, correspond to the spatial characteristics of the mode components of the acoustic field with an error of up to 3dB

The motion of a body can be expressed relative to the present configuration of the body, known as the relative motion description, besides the classical Lagrangian and the Eulerian descriptions. When the time increment from the present state is small enough, the nonlinear constitutive equations can be linearized relative to the present state so that the resulting system of boundary value problems becomes linear. This formulation is based on the well-known ``small-on-large'' idea, and can be implemented for solving problems with large deformation in successive incremental manner. In fact, the proposed method is a process of repeated applications of the well-known “small deformation superposed on finite deformation” in the literature. This article presents these ideas applied to thermoelastic materials with a brief comment on the exploitation of entropy principle in general. Some applications of such a formulation in numerical simulations are briefly reviewed and a numerical result is shown.

Dielectric elastomers (DEs) deform and change shape when an electric field is applied across them. They are flexible, resilient, lightweight, and durable and as such are suitable for use as soft actuators. In this paper a physics-based and control-oriented model is developed for a DE tubular actuator using a physics-lumped parameter modeling approach. The model derives from the nonlinear partial differential equations (PDE) which govern the nonlinear elasticity of the DE actuator and the ordinary differential equation (ODE) that governs the electrical dynamics of the DE actuator. With the boundary conditions for the tubular actuator, the nonlinear PDEs are numerically solved and a quasi-static nonlinear model is obtained and validated by experiments. The full nonlinear model is then linearized around an operating point with an analytically derived Hessian matrix. The analytically linearized model is validated by experiments. Proportional–Integral–Derivative (PID) and $$H_{\infty }$$ control are developed and implemented to perform position reference tracking of the DEA and the controllers’ performances are evaluated according to control energy and tracking error.

We analyze a fully discrete pseudo-spectral scheme for the numerical solution of the initial-periodic boundary value problem for a nonlinear family of Boussinesq systems. The equations are discretized in space by the Fourier collocation method and in time by an explicit fourth order Runge–Kutta scheme. For the resulting fully discrete scheme, we prove error bounds of spectral accuracy in space and of fourth-order accuracy in time.

In the current article, we have performed computational analysis on convection heat transfer of a hybrid nanofluid in occurrences where porous media and the effect of magnetic force are involved. In order to assess the time-fractional derivatives, Caputo’s notion is utilized while the Darcy–Forchheimer model is applied due to the involvement of the porous medium. Moreover, the boundary conditions are assumed to be nonuniform through the equilibrium between the surface tension and shear stress over a semi-infinite permeable flat surface. Keeping in view the complexity of the fractional derivative model and nonuniform boundary conditions, the problem in question is a complicated one. Accordingly, the coupled momentum and energy equation is linearized and the finite difference scheme is then applied and implemented in MATLAB Code R2020b. Furthermore, we have also offered a comprehensive analysis regarding error and convergence of the proposed numerical method. The newly introduced numerical technique to determine the numerical solutions and some unique and interesting deductions are established. From the computational results, one can conclude that the fluid motion in both hybrid and single nanofluids slows down due to magnetic field, porosity, and inertia coefficient as the magnetic and electric fields are synchronized due to the formation of the Lorentz force and viscous interference. We believe that our proposed numerical technique regarding employment of the fractional model for heat transfer application to the hybrid nanofluid over a semi-infinite nonuniform permeable surface along with variable heat flux is not found in the literature so far. Furthermore, the obtained results will be a valuable addition to fractional calculus from an engineering point of view.

This research aims to study the performance of local maximum-entropy approximation (LME)-based element-free Galerkin meshfree (EFG) method and its integration in the heat conduction application. EFG methods have undergone significant development over the past two decades and have come to the forefront to solve partial differential equations. Being non-polynomial functions, LME is smooth and appears to be a viable substitute for the approximation in EFG methods. It possesses weak Kronecker delta property that allows the implementation of essential boundary conditions like FEM. In the present work, stabilized conforming nodal integration (SCNI) and its modified version with additional stability called modified SCNI (MSCNI) is used to perform the integration of LME-based EFG and tested against different discretization node sets. Poisson heat conduction equation with a different set of boundary conditions is chosen to study these integration schemes and compared with several Gaussian integration point schemes. It is found that the 3 or 4 point Gauss integration scheme is optimal for unstructured discretization and MSCNI is optimal for distorted discretization. SCNI and MSCNI are observed to be converging faster than the other methods, irrespective of the grid type.

Purpose: To investigate the effect of viscous dissipation and axial conduction on heat transfer for laminar forced convection flow through a circular pipe was studied. Constant wall temperature boundary condition was imposed on the circular pipe and the flow is assumed to be hydro dynamically developed and thermally developing. Methodology: Numerical solutions were obtained to observe the variations of non-dimensional bulk mean temperature, Nusselt number and wall heat from (or) to the fluid for Brinkman numbers   1.0, 0.5, 0.2,0.0 and Peclet numbers 10,50,100,300 and 500. Results: It was found that the heat generated by viscous dissipation was higher for higher Brinkman number and it delays the thermal entrance development of the fluid. Unique Contribution to Theory and Practice: Effect of axial conduction becomes negligible when the value of Peclet number crosses hundred

We study the Stokes eigenvalue problem under Navier boundary conditions in $$C^{1,1}$$ C 1 , 1 -domains $$\Omega \subset \mathbb {R}^3$$ Ω ⊂ R 3 . Differently from the Dirichlet boundary conditions, zero may be the least eigenvalue. We fully characterize the domains where this happens and we show that the ball is the unique domain where the zero eigenvalue is not simple, it has multiplicity three. We apply these results to show the validity/failure of a suitable Poincaré-type inequality. The proofs are obtained by combining analytic and geometric arguments.

A kind of Riemann problem of (λ,k) bi-analytic functions are studied by using the theory of boundary value problems for analytic functions. The solutions of Riemann problem of (λ,k) bi-analytic fun...

Herein, a miscellaneous contact problem of the theory of elasticity in the upper half-plane is considered. The boundary is a real semi-axis separated into four parts, on each of which the boundary conditions are set for the real or imaginary part of two desired analytical functions. Using new unknown functions, the problem is reduced to an inhomogeneous Riemann boundary value problem with a piecewise constant 2 × 2 matrix and four singular points. A differential equation of the Fuchs class with four singular points is constructed, the residue matrices of which are found by the logarithm method of the product of matrices. The single solution of the problem is represented in terms of Cauchy-type integrals when the solvability condition is met.

Context. The question of constructing a method of two-sided approximations for finding a positive solution of the Dirichlet problem for a semilinear elliptic equation based on the use of the Green’s functions method is considered. The object of research is the first boundary value problem (the Dirichlet problem) for a second-order semilinear elliptic equation. Objective. The purpose of the research is to develop a method of two-sided approximations for solving the Dirichlet problem for second-order semilinear elliptic equations based on the use of the Green’s functions method and to study its work in solving test problems. Method. Using the Green’s functions method, the initial first boundary value problem for a semilinear elliptic equation is replaced by the equivalent Hammerstein integral equation. The integral equation is represented in the form of a nonlinear operator equation with a heterotone operator and is considered in the space of continuous functions, which is semi-ordered using the cone of nonnegative functions. As a solution (generalized) of the boundary value problem, it was taken the solution of the equivalent integral equation. For a heterotone operator, a strongly invariant cone segment is found, the ends of which are the initial approximations for two iteration sequences. The first of these iterative sequences is monotonically increasing and approximates the desired solution to the boundary value problem from below, and the second is monotonically decreasing and approximates it from above. Conditions for the existence of a unique positive solution of the considered Dirichlet problem and two-sided convergence of successive approximations to it are given. General guidelines for constructing a strongly invariant cone segment are also given. The method developed has a simple computational implementation and a posteriori error estimate that is convenient for use in practice. Results. The method developed was programmed and studied when solving test problems. The results of the computational experiment are illustrated with graphical and tabular informations. Conclusions. The experiments carried out have confirmed the efficiency and effectiveness of the developed method and make it possible to recommend it for practical use in solving problems of mathematical modeling of nonlinear processes. Prospects for further research may consist the development of two-sided methods for solving problems for systems of partial differential equations, partial differential equations of higher orders and nonstationary multidimensional problems, using semi-discrete methods (for example, the Rothe’s method of lines).

The first confacial pentaoctahedron comprised of transition metal ions namely ZnII FeIIIA FeIIIB FeIIIA ZnII has been synthesized by using a dinucleating nonadentate ligand. The face-sharing bridging mode enforces short ZnII ⋅⋅⋅FeIIIA and FeIIIA ⋅⋅⋅FeIIIB distances of 2.83 and 2.72 A, respectively. Ab-initio CASSCF/NEVPT2 calculations provide significant negative zero-field splittings for FeIIIA and FeIIIB with |DA |>|DB | with the main component along the C3 axis. Hence, a spin-Hamiltonian comprised of anisotropic exchange, zero-field, and Zeeman term was employed. This allowed by following the boundary conditions from the theoretical results the simulation in a theory-guided parameter determination with Jxy =+0.37, Jz =-0.32, DA =-1.21, EA =-0.24, DB =-0.35, and EB =-0.01 cm-1 supported by simulations of high-field magnetic Mossbauer spectra recorded at 2 K. The weak but ferromagnetic FeIIIA FeIIIB interaction arises from the small bridging angle of 84.8° being at the switch from anti- to ferromagnetic for the face-sharing bridging mode.

In this paper, we investigate the following system where $$\Omega \subset {\mathbb {R}}^2$$ is a smoothly bounded planer domain. It is proved that under no-flux boundary conditions for both n and c as well as under Dirichlet boundary conditions for u, an associated initial-boundary value problem is globally solvable in a generalized sense for any initial data lying in appropriately regular space, and that the derived globally defined generalized solutions have the boundedness of certain temporal average, which especially avoids the occurrence of collapse into persistent Dirac-type measures at any finite time. Furthermore, in particular, for some solutions emanating from the initial data with small entire initial mass of n, there exists some waiting time from which on the solutions become smooth and asymptotically stabilize to some constant equilibrium.

The problem of obtaining the form of a spatial supersonic nozzle cross section that intersects the circular critical cross section of the nozzle and an output circuit inscribed in the given dimensions, which has the greatest thrust among all possible admissible forms. is considered. The Lagrange functional, in which all gas-dynamics equations and boundary condition are taken into account with the help of variable Lagrange multipliers, is compiled. The first variation of the functional is written out. The equations and constraints turning the first variation into zero form an adjoint problem for the Lagrange multipliers and an optimality condition. A numerical algorithm for the consistent solution of gas-dynamics equations and the adjoint problem is developed. Numerical examples are given.