Laplace Equation In Domains Research Articles

Strain gradient and Green–Naghdi-based thermoelastic wave propagation with energy dissipation in a Love–Bishop nanorod resonator under thermal shock loading

The thermoelastic wave propagation analysis is developed in a Love–Bishop nanorod resonator employing the strain gradient elasticity and Green–Naghdi theories with energy dissipation for the first time. The assumed nanorod resonator is subjected to thermal shock loading. The variational principle is applied to derive the coupled system of partial differential equations for temperature and displacement fields in a Love–Bishop nanorod. An analytical solution is proposed to solve the governing equations in Laplace domain. To obtain the temporal variation of fields’ variables, a proper Laplace inversion technique is employed in the problem. The size effects in the nano-sized Love–Bishop rods are taken into account with five small-scale parameters including three higher-order materials length parameters, the micro-length inertia and thermal parameters. Two types of thermal shock loadings such as laser pulse-induced suddenly increasing temperature are utilized in this work, and the laser shock-induced thermoelastic wave propagations in both temperature and displacement fields are illustrated. The effects of the higher-order materials length parameters and the micro-length inertia and thermal parameters on the propagation of thermal and elastic waves are studied in detail.

A method for dynamic response of a multilayered pavement loaded by FWD

AbstractThe dynamic response of a viscoelastic multilayered pavement under falling weight deflectometer (FWD) loading is an attractive topic for the pavement engineers. Starting from the governing equations, the Laplace transform is first applied to suppress the time variable. The ordinary differential equations (ODE) of the governing equations in Laplace domain are derived by the cylindrical vector functions systems. On the basis of the facts that the top and bottom boundary conditions dual appear, a novel analytical method called dual variable and dual boundary (DVDB) for the ODE is proposed combined with dual variable and position method (DVP). The final solution in time domain is obtained by the inverse Laplace transform that is accelerated by Epsilon algorithm. The solutions of the thin layer issues and imperfect interface conditions are also investigated. Some numerical results from DVDB for the dynamic response of multilayered pavement loaded by FWD discrete pulse series are compared and discussed. It can be concluded that the DVDB combined with Laplace transform can efficiently solve the dynamic response of multilayered pavement in time domain. Furthermore, it can also be suited for some cases, such as material anisotropy properties, thin layers in the structure and interface conditions.

Vibration analysis of pipes conveying fluid resting on a fractional Kelvin-Voigt viscoelastic foundation with general boundary conditions

In this paper, the stability of pipes conveying fluid with viscoelastic fractional foundation is investigated. The pipe is fixed at the beginning while the pipe end is constrained with two lateral and rotational springs. The fluid flow effect is modeled as a lateral distributed force, containing the fluid inertia, Coriolis and centrifugal forces. The pipe is modeled using the Euler-Bernoulli beam theory and a fractional Kelvin-Voigt model is employed to describe the viscoelastic foundation. The equation of motion is derived using the extended Hamilton’s principle. Presenting the derived equation in Laplace domain and applying the Galerkin method, a set of algebraic equations is extracted. Calculating the determinant of the coefficients of the extracted algebraic equations results in the stability margin of the pipe. Some run is done and effects of some physical parameters such as stiffness and damping of fractional viscoelastic foundation, fractional order parameter and the end lateral and rotational stiffness of end springs on the stability boundary of the pipe are considered and some conclusions are drawn.

Modeling of Capacitance in A Supercapacitor

A general supercapacitor model is developed by incorporating the effective surface area in the presence of pores. An analytical solution has been derived from the generic equation in Laplace domain based on the porous-electrode theory. This model demonstrates the effects of solution to matrix conductivity ratio, separator to electrode resistance ratio and discharge current density on the electrochemical impedance, capacitance and energy density of supercapacitor. The electrochemical impedance, capacitance and energy density of supercapacitor are calculated in this work. The maximum capacitance of 12.71 F/cm2was computed in low frequency range in this device. The proposed model can be applied to simulate the characteristics of polymer-based supercapacitor in near future.

Modeling of Capacitance in A Supercapacitor

A general supercapacitor model is developed by incorporating the effective surface area in the presence of pores. An analytical solution has been derived from the generic equation in Laplace domain based on the porous-electrode theory. This model demonstrates the effects of solution to matrix conductivity ratio, separator to electrode resistance ratio and discharge current density on the electrochemical impedance, capacitance and energy density of supercapacitor. The electrochemical impedance, capacitance and energy density of supercapacitor are calculated in this work. The maximum capacitance of 12.71 F/cm2 was computed in low frequency range in this device. The proposed model can be applied to simulate the characteristics of polymer-based supercapacitor in near future.

Strain gradient based dynamic response analysis of heterogeneous cylindrical microshells with porosities under a moving load

Forced vibration of a porous functionally graded (FG) cylindrical microshell due to a moving point load with constant velocity is studied for the first time. Through the thickness of microshell, there are even-type or uneven-type porosities. Therefore, material properties of the microshell become porosity-dependent and are described via modified power-law function. For micro-scale shells, small size effects due to non-uniform strain field can be considered via strain gradient theory (SGT). At first, the governing equations of the microshell are converted to new equations in Laplace domain. Then, time response of the microshell will be obtained implementing inverse Laplace transform technique. It will be demonstrated that forced vibration characteristics of a FG microshell rely on the velocity of moving load, strain gradients, porosity percentage, material/porosity distribution and its geometry.

A semi-analytical collocation Trefftz scheme for solving multi-term time fractional diffusion-wave equations

This study presents a semi-analytical boundary-only collocation technique for solving multi-term time-fractional diffusion-wave equations. In the present collocation technique, the Laplace transformation is first implemented to convert time-fractional diffusion-wave equation to a series of time-independent nonhomogeneous equations in Laplace domain. Then the composite multiple reciprocity method (CMRM) is applied to construct a high-order homogeneous equation, which has the same solution with one of time-independent nonhomogeneous equations in Laplace domain. The collocation Trefftz scheme with high-order T-complete functions is used to obtain the semi-analytical solution of high-order homogeneous equation with boundary-only collocation in Laplace domain. Finally the numerical Laplace inversion scheme is introduced to invert the Laplace domain solutions back into the time-dependent solutions of time-fractional diffusion-wave equations. The proposed method is easy-to-implement and flexible for irregular domain problems. It evades costly convolution integral calculation in time fractional derivation approximation, and avoids the effect of time step on numerical accuracy and stability. Error analysis and numerical investigations show that the proposed collocation method is highly accurate, computationally efficient, and numerically stable for multi-term time fractional diffusion-wave equations.

Semi-analytical transient pressure solution for power-law polymer injection in a vertical well

Enhanced Oil Recovery (EOR) is a process that extends reservoir life through field operations that aim to increase reservoir displacement efficiency. Injection of polymeric solutions, which is a chemical EOR process, improves sweep efficiency by decreasing the mobility of the injected fluids. One challenge in polymer flooding is to model the non-Newtonian rheological behavior of the injected solutions due to the complex interdependence between fluid flow velocity, fluid viscosity and shear rate. Proper reservoir management requires a thorough understanding of non-Newtonian fluid flow behavior through porous media, thus the need for new and improved mathematical models that describe the transient pressure behavior in wells. In this work we consider the flow of non-Newtonian pseudoplastic fluids in homogeneous isotropic porous media. To describe the rheological polymer behavior, we used the well-known power-law model, which leads to a nonlinear problem. In this paper, a novel semi-analytical solution is obtained by combining spatial domain discretization with pseudotime, so the flow problem is rewritten as a system of linear equations in Laplace domain. We also presented a variation of the proposed solution by assuming a pseudostationary behavior of the non-Newtonian velocity. Both approaches are compared to previously published approximated analytical solutions. One interesting aspect of our work is that it can be extended to more realistic polymer rheological models.

Transient natural convection flow between vertical concentric cylinders heated/cooled asymmetrically

This paper investigates the impact of asymmetric heating or cooling of cylinder surfaces on transient natural convection flow in between vertical concentric cylinders. The outer surface of inner cylinder is assumed to be heated with temperature greater than the cooled inner surface of the outer cylinder. Closed form expressions are obtained by using the well-known Laplace transform technique to solve the governing partial differential equations in Laplace domain, whereas the Riemann-sum approximation is used to invert to time domain. Results show that the role of buoyancy force distribution parameter is to increase temperature, velocity, skin-friction and volume flow rate for both air and water. Further, reverse flow formation can be controlled by using suitable buoyancy force distribution parameter.

The boundary element method applied to 3D magneto-electro-elastic dynamic problems

Due to the coupling properties, the magneto-electro-elastic materials possess a wide number of applications. They exhibit general anisotropic behaviour. Three-dimensional transient analyses of magneto-electro-elastic solids can hardly be found in the literature. 3D direct boundary element formulation based on the weakly-singular boundary integral equations in Laplace domain is presented in this work for solving dynamic linear magneto-electro-elastic problems. Integral expressions of the three-dimensional fundamental solutions are employed. Spatial discretization is based on a collocation method with mixed boundary elements. Convolution quadrature method is used as a numerical inverse Laplace transform scheme to obtain time domain solutions. Numerical examples are provided to illustrate the capability of the proposed approach to treat highly dynamic problems.

A numerical method for the solution of exterior Neumann problems for the Laplace equation in domains with corners

In this paper we propose a new boundary integral method for the numerical solution of Neumann problems for the Laplace equation, posed in exterior planar domains with piecewise smooth boundaries. Using the single layer representation of the potential, the differential problem is reformulated as a classical boundary integral equation. The use of a smoothing transformation and the introduction of a modified Gauss–Legendre quadrature formula for the approximation of the singular integrals, which turns out to be convergent, leads us to apply a Nyström type method for the numerical solution of the integral equation. We solve some test problems and present the numerical results in order to show the efficiency of the proposed procedure.

Transient magnetohydrodynamic free convective flow in vertical micro-concentric annuli

This article analyses the transient magnetohydrodynamic free convective flow in vertical micro-concentric annuli in the presence of velocity slip and temperature jump at the outer surface of the inner cylinder and the inner surface of the outer cylinder. The Laplace transform technique has been used to find the solutions for the velocity and temperature fields by solving the governing partial differential equations in Laplace domain. However, the Riemann-sum approximation method is used to invert the Laplace domain to the time domain. The solution derived is validated by assenting comparison with exact solutions derived for the steady state which has been derived separately. An excellent agreement was found for transient and steady state at large value of time. The solution obtained for the velocity has been used to compute the skin friction, while the temperature has been used to compute the Nusselt number. The effect of various flow parameters entering into the problem such as time, Prandtl number, curvature radius ratio, Hartmann number, rarefaction parameter, and fluid–wall interaction parameter are discussed with the aid of line graphs.

Transient analysis of axially functionally graded Timoshenko beams with variable cross-section

Abstract The present study goals to analyze transient analysis of axially functionally graded (AFG) Timoshenko beams with variable cross-section. Both the materials and geometrical properties of the AFG tapered beam varying along the longitudinal direction. In the analysis, the influences of rotary inertia and shear deformation are taken into account. Complementary functions method (CFM) is applied to solve the differential equations in Laplace domain. By using the modified Durbin's algorithm, the results are transformed to the time domain. For the verification of the recommended method, numerous problems which were solved in the literature are handled with the developed method and benchmarked with the findings in the literature.

Exact solution of Schrodinger equation for two state problem with time dependent coupling

The present work focuses on the exact solution of the time dependent Schrodinger equation involving two potentials coupled by a time dependent Dirac Delta function potential. The problem involving the partial differential equations in two variables can be reduced to a single integral equation in Laplace domain and by knowing the wave function at the origin we can derive the wave function everywhere. Solutions for the different time variation of the strength of the Dirac Delta function potentials has been derived.

Exact solution of time-dependent Schrodinger equation for two state problem in Laplace domain

The present work focuses on the exact solution of the time dependent Schrodinger equation involving two potentials coupled by a Dirac Delta potential. The problem involving the partial differential equations in two variables can be reduced to a single integral equation in Laplace domain and by knowing the wave function at the origin we can derive the wave function everywhere. Solutions for the bound state and partly unbound state along with propagator are derived for the first potential.

The transient analysis of a conducting crack in magneto-electro-elastic half-space under anti-plane mechanical and in-plane electric and magnetic impacts

This paper investigated the fracture behavior of a magneto-electro-elastic material subjected to transient electrical, magnetic and mechanical loads. The “smart” medium contains a straight-line crack, which is parallel to its poling direction and free boundary surface. The Fourier and Laplace transform techniques are used to reduce the problem to the solution of one Fredholm integral equation in Laplace domain and second equation in real domain. The Laplace inversion yields the result in the time domain. The equation in real domain is solved exactly. The semipermeable crack-face magneto-electric boundary conditions are utilized. Field intensity factors of stress, electric displacement, magnetic induction, crack displacement, electric and magnetic potentials and the energy release rate are determined. The electric displacement and magnetic induction of crack interior are discussed. Strong coupling between stress and electric and magnetic field near crack tips has been found. Numerical results are presented, and some conclusions are drawn.

A fractional differential constitutive model for dynamic stress intensity factors of an anti-plane crack in viscoelastic materials

Fractional differential constitutive relationships are introduced to depict the history of dynamic stress intensity factors (DSIFs) for a semi-infinite crack in infinite viscoelastic material subjected to anti-plane shear impact load. The basic equations which govern the anti-plane deformation behavior are converted to a fractionalwave-like equation. By utilizing Laplace and Fourier integral transforms, the fractional wave-like equation is cast into an ordinary differential equation (ODE). The unknown function in the solution of ODE is obtained by applying Fourier transform directly to the boundary conditions of fractional wave-like equation in Laplace domain instead of solving dual integral equations. Analytical solutions of DSIFs in Laplace domain are derived by Wiener-Hopf technique and the numerical solutions of DSIFs in time domain are obtained by Talbot algorithm. The effects of four parameters α, β, b1, b2 of the fractional differential constitutive model on DSIFs are discussed. The numerical results show that the present fractional differential constitutive model can well describe the behavior of DSIFs of anti-plane fracture in viscoelastic materials, and the model is also compatible with solutions of DSIFs of anti-plane fracture in elastic materials.

Time domain boundary element formulation for partially saturated poroelasticity

Abstract Based on the Mixture theory and the principles of continuum mechanics, a dynamic three-phase model for partially saturated poroelasticity is established as well as the corresponding governing equations in Laplace domain. The three-dimensional fundamental solutions are deduced following Hormander's method. Based on the weighted residual method, the boundary integral equations are established. The boundary element formulation in time domain for partially saturated media is obtained after regularization by partial integration, spatial discretization, and the time discretization with the Convolution Quadrature Method. The proposed formulation is validated with the semi-analytical one-dimensional solution of a column. Studies with respect to the spatial and temporal discretization show its sensitivity on a fine enough mesh. A half-space example allows to study the wave fronts. Finally, the proposed formulation is used to compute the vibration isolation of an open trench.

ON THE EXPONENTIAL CONVERGENCE OF THE METHOD OF FUNDAMENTAL SOLUTIONS

It is well known that the method of fundamental solutions (MFS) is a numerical method of exponential convergence. In other words, the logarithmic error is proportional to the node number of spatial discretization. In this study, the exponential convergence of the MFS is demonstrated by solving the Laplace equation in domains of rectangles, ellipses, amoeba-like shapes, and rectangular cuboids. In the solution procedure, the sources of the MFS are located as far as possible and the instability resulted from the ill-conditioning of system matrix is avoided by using the multiple precision floating-point reliable (MPFR) library. The results converge faster for the cases of smoother boundary conditions and larger area/perimeter ratios. For problems with discontinuous boundary data, the exponential convergence is also accomplished using the enriched method of fundamental solutions (EMFS), which is constructed by the fundamental solutions and the local singular solutions. The computation is scalable in the sense that the required time increases only algebraically.

Fundamental solutions for a partially saturated poroelastic continuum

AbstractThe Boundary Element Method is quite suitable for solving dynamic semi‐infinite or infinite linear problems. In order to establish the boundary integral equations, one crucial condition is the knowledge of corresponding fundamental solutions. For a partially saturated poroelastic continuum, the governing equations in Laplace domain are formulated based on the theory of mixtures, and the related fundamental solutions are derived by using Hörmanders method. The singular behavior of the fundamental solutions are investigated by a series expansion with respect to the variable r. Finally, some exemplary fundamental solutions are calculated to visualize the principal behavior as well, and comparisons with the related results of saturated poroelasticity are given. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)