Solving Laplace Equation Research Articles

Solution of Laplace Equation by Modified Differential Transform Method

In this paper, we applied the modified two-dimensional differential transform method to solve Laplace equation. Laplace equation is one of Elliptic partial differential equations. These kinds of differential equations have specific applications models of physics and engineering. We consider four models with two Dirichlet and two Neumann boundary conditions. The simplicity of this method compared to other iteration methods is shown here. It is worth mentioning that here only a few number of iterations are required to reach the closed form solutions as series expansions of some known functions.

Radioactive source localization employing resistive electrode array (REA) detector.

Objective.In this feasibility study, we explore an application of a Resistive Electrode Array (REA) for localization of a radioactive point source. The inverse problem posed by multichannel REA detection is studied from mathematical perspective and involves the questions of the minimal configuration of the conductive leads that can achieve this goal. The basic configuration consists of a circularly shaped REA with four opposite electrical lead-pairs at its perimeter.Approach.A robust mathematical reconstruction method for a 3D radioactive source relative to the REA is presented. The characteristic empirical Green's function for the detector response of the REA is determined by numerically solving Laplace equations with appropriate boundary conditions. Based on this model, Monte Carlo simulations of the inverse problem with Gaussian noise are performed and the overall accuracy of the localization is investigated.Main results.The results show a 3D error distribution of localization which is uniform in the (x,y)-plane of the REA and strongly correlated in the orthogonalz-axis. The overall accuracy decreases with higher distance of the source to the detector which is intuitive due to approximate flux dependence following the inverse square law. Further, a saturation in accuracy regarding the number of electrical leads and a linear dependence of the reconstruction error on the measurement noise level are observed.Significance.A broad range of REA detector configurations and their characteristics are investigated by this study for radioactive source localization allowing diverse practical applications with detector diameters ranging from millimeters to meters.

A Computational Conformal Geometry Approach to Calculate the Large Deformation of Plates/Shells With Arbitrary Shapes

Abstract High-accuracy numerical methods to solve the nonlinear Föppl–von Kármán (FvK) equations usually work well only in simple domains such as rectangular regions. Computational conformal geometry (CCG) provides a systematic method to transform complicated surfaces into simple domains, preserving the orthogonal frames such that the corresponding FvK equations can be solved by more effective numerical methods. Based on CCG, we proposed a general method for solving large deformation and nonlinear vibration of plate/shell structures with arbitrary shapes. The method can map any complex surface conformal to a rectangular region, and then FvK equations are solved in the rectangular region to study nonlinear vibration problems of any arbitrary shape plates/shells. The conform map is calculated by solving Laplace equations on a fine Delauney triangular mesh on the surface, which is numerically robust, and the map is harmonic and subsequently C∞ smooth, such that all the evaluations and spatial derivatives required by high accuracy methods at the regular nodes can be accurately and efficiently calculated. A variational function that is equivalent to the FvK equations is provided, such that the FvK equations can be solved by multiple numerical methods. The degree-of-freedom in solving the FvK equations is usually much less than that in the finite element methods described by displacements. The effectiveness of the proposed approach is verified by several benchmark examples, and the current method is suitable for calculating the large deflections and nonlinear dynamical responses of plates/shallow shells with arbitrary shapes.

How far does electrical stimulation activate white matter tracts? A computational modeling study

ObjectiveThe aim of this study was to model how the different parameters of electrical stimulation (intensity, pulse shape, probe geometry) influence the extent of white matter activation. MethodsThe electrical potentials generated by the stimulating electrodes were determined by solving Laplace equation. The temporal evolution of membrane potentials at each nodes of Ranvier of an axon was then computed by solving the coupled system of differential equations describing membrane dynamics and cable propagation. ResultsRegions of unilateral propagation were observed for monophasic pulses delivered with a bipolar probe aligned along the tract. For biphasic pulses, the largest activation areas and depths were found with a high inter-electrode-distance (IED) bipolar probe, oriented orthogonally to the tract. The smallest activation areas and depths were found for bipolar stimulations with the probe aligned parallel to the tract and low IED. For isotropic white matter regions, the activation area and depth were three times larger than for anisotropic white matter tracts. ConclusionsBipolar probes with biphasic pulses offer the greatest versatility: an orthogonal orientation acts as two monopolars (increased sensitivity when searching for a tract), whereas a parallel orientation corresponds to a single monopolar (increased specificity). Activation is more superficial when stimulating highly anisotropic tracts. SignificanceThis knowledge is essential for interpreting the behavorial effects of stimulation and the recordings of axono-cortical evoked potentials.

Identification and Investigation of Extreme Events Using an Arbitrary Lagrangian–Eulerian Approach With a Laplace Equation Solver and Coupling to a Navier–Stokes Solver

AbstractIncreased deployment of offshore wind turbines is seen as an important pathway to increase green renewable energy production. Improved and rapid identification of extreme events and evaluation of hydrodynamic loads due to such events is essential to reduce the cost of energy production. Numerical modeling to pre-screen sea states and to identify the crucial events to prioritize model tests will make a major contribution to reduce design times and costs for such structures. In this effort, a highly efficient and nonlinear numerical model based on the Laplace equations is used to generate undisturbed wave kinematics. Such a simulation is used to identify extreme wave events in a sea state realization, and further, the wave loading due to such events are evaluated using Morison formula. Events screened in this manner can then be transferred to a high-resolution model such as a Navier–Stokes equation-based solver to investigate the hydrodynamics in detail. The implementation and application of such an approach in the open-source hydrodynamic model REEF3D is presented in this work.

The kernel regularized learning algorithm for solving Laplace equation with Dirichlet boundary

In this paper, we give an investigation on the problem of solving Laplace equation with the kernel regularized regression. We provide a Sobolev type space corresponding to the Dirichlet boundary value problem on a compact domain, and with which define a reproducing kernel space (RKS), which is used as the hypothesis space for constructing kernel regularized learning algorithm. We give theory analysis for the convergence of the learning algorithm, bound an upper bound for the error. The discussions show that the learning rate is controlled by a [Formula: see text]-functional corresponding to the RKS. As an application we give the learning rate in case that the domain is the unit ball. The simulations show that the algorithm has better fitting performance. The investigations show that the problem of solving an elliptic boundary problem can be attributed to constructing an orthonormal basis with respect to a bilinear form corresponding to the boundary value condition.

Dynamic Simulation of Electron Emission Characteristics Based on Different Numerical Models of a Nanoscale Micro-Protrusion

It is well known that electron emission is one of the key factors to cause electrical breakdown in vacuum gaps. In previous studies, it tended to use simplified numerical models with uniform electric field on emission region to investigate the electron emission characteristics at the cathode micro-protrusion and the influence of the materials of the micro-protrusions. The objective of this article is to introduce an improved numerical model of a nanoscale micro-protrusion with nonuniform electrical field on the emission region, which could be used for micro-protrusions with various irregular geometries. The improved numerical model is used in this article to study the dynamic characteristics of the electron emission of Cu and Cr micro-protrusions under different macroscopic electric fields. First, the electric field distribution is obtained by solving Laplace equation based on finite difference method (FDM). Then, the emission current and Nottingham effect are obtained by calculating the general thermal-field (GTF) equations based on the electric field distribution. Finally, the dynamic characteristics of the electron emission current and temperature distribution are iteratively calculated considering the Joule heating and Nottingham effect. The calculation results of the improved numerical model are compared with those of the previous simplified numerical model. The results show that for Cu and Cr micro-protrusions, the Joule heating, and Nottingham effect would play different roles in the emission process. In general, it is more likely for the Cr micro-protrusions to evaporate than the Cu micro-protrusions. The results of this article may provide some useful information to further understand the vacuum breakdown initiated by electron emission.

Comparative Study of Analytic and Numerical Solutions of Steady-state Temperature Distribution on Semi-circular Plate Using Laplace Equation

In this paper numerical methods have been used to solve two dimensional steady state heat flow problem in polar coordinates with Dirichlet boundary conditions inside a semi-circular plate and the work focuses on the numerical methods for solving Laplace equation; finite difference schemes and Gauss elimination method. The numerical solution is compared with exact solution of the same problem. Finally, we analyze the absolute error in different number of iterations to check the accuracy of schemes.

Three-dimensional direct current invisibility cloak produced with bulk materials

By virtue of eliminating the characteristic signals of physical objects, invisibility cloaks can also prevent the objects from being detected by instruments using electrical current. However, most existing designs of direct current cloaks cannot be directly applied in the natural environment. In this paper, we present a three-dimensional direct current cloak by solving Laplace equation in electric conduction. The cloak is composed of a bilayer structure with homogeneous and isotropic bulk materials. The designed structure, having withstood both conventional electrical potential measurements and tests by a commercially-available detector, is highly effective in suppressing current characteristic signals in close simulation of natural environmental conditions. The cloak is expected to facilitate the study of characteristic signal control in medical and geological research in the field of current detection, such as tumor diagnosis and mineral exploration.

A novel efficient numerical solution of Laplace equation with mixed boundary conditions

This paper describes a new type of method to solve Laplace equation subject to mixed boundary conditions in two-dimensional domains for electrostatics. The electric potential ϕ is represented by harmonic Steklov eigenfunctions, obtained from a certain Steklov eigenproblem. Error estimates for this method are provided in terms of given boundary data of solutions. The key idea of the method is that Steklov eigenfuncitons could construct the orthonormal basis of the space of solutions. Some results of computational simulations on polygonal domains are presented to support the effectiveness of the new method.

Finite element approach for numerical integration over family of eight node linear quadrilateral element for solving Laplace equation

This paper describes the finite element solution of the boundary value problem or Laplace equation in curved and square region, the boundary region is divided into arbitrary triangles, and every triangle is divided in to three eight node quadrilateral elements, to generate a mesh over rectangular and curved region, performance of the new approach are demonstrated with numerical examples.

A velocity decomposition method combining potential and viscous flow for the estimation of submarine's hydrodynamic coefficients

Estimating hydrodynamic coefficients is important in the design of submarines. To enhance the efficiency of the computational fluid dynamics (CFD) method for evaluating the hydrodynamic performance of submarines, we have developed a velocity decomposition solver for the simulation of a moving body based on the open-source CFD platform OpenFOAM. The total velocity is decomposed into potential flow velocity and non-potential flow velocity. The potential flow velocity is further decomposed into unit components, which can be computed by the proposed method with less interpolation cost. Further, the original potential flow velocity can be recovered by linear summation of these unit velocity components. This avoids solving Laplace equation at each time step. The non-potential flow velocity is solved by using a complementary Navier–Stokes equation in a smaller domain as compared to that used in the conventional CFD method. In addition, a buffer source term is introduced for the transition of non-potential flow velocity to potential flow velocity. Oblique towing test and planar motion mechanism test are numerically simulated using the proposed method, which reveal good consistency between the experimental data and simulation results. This validates the efficiency and reliability of the proposed method in estimating the hydrodynamic coefficients of submarines.

Investigation of sloshing effects on lateral stability of tank vehicles during turning maneuver

An analytical model is developed in this article to study the effects of liquid sloshing on dynamics of articulated vehicle carrying liquid in turning maneuver. To investigate the interaction between the fluid and the vehicle, a four-axle tractor-semitrailer tanker with 32 degrees of freedom has been used. The method of separation of variables is used to solve Laplace equation together with the wall and free surface boundary conditions for fluid sloshing. Based on the results, the effects of fluid density, fill volume of the tanker, vehicle speed, and steer angle on lateral stability of the vehicle have been studied. Simulation results showed that decrease in the fill volume and fluid density leads to decrease in the normalized dynamic normal load factor of the rear axle of the semitrailer. According to the results, the frequency of the normalized roll angle of the semitrailer is independent of vehicle velocity and steering angle.

A ROBUST NUMERICAL METHODOLOGY FOR STEADY-STATE CAVITATING FLOWS

Modeling of cavitation is critical for the design of centrifugal pumps, marine propellers, and fuel injectors. Cavitating flows often exhibit transient characteristics due to the sudden growth and collapse of vapor cavities. A numerically robust steady-state approach can provide quick analysis with reasonable accuracy saving significant simulation time as compared to transient simulations. As we demonstrate in this paper, the effective steady-state modeling of cavitating flow is strongly governed by several key factors, such as mesh resolution, pressure velocity coupling, pseudo-transient method, and initialization techniques along with the linearization of mass transfer terms and the inclusion of turbulence diffusion treatment. The pseudo-transient method improves numerical robustness of steady-state simulations by improving diagonal dominance and providing a local under-relaxation based on a pseudo-time-step size. The coupled approach of solving continuity and momentum equations, the linearization of mass transfer source terms, and the modeling of turbulent diffusion play an important role in the solution stability and faster convergence. A hybrid initialization technique provides start-up stability by solving Laplace equations for pressure and velocity. The Zwart-Gerber-Belamri cavitation model, which originates from the simplification of Rayleigh-Plesset equation, is chosen for simulating cavitating flow within a homogeneous mixture multiphase framework. A highly resolved mesh has been used in the cavitating region to obtain accurate results. In addition, an overset mesh has also been used to demonstrate the ease of mesh generation and parametrization in complex geometries. The aforementioned methods are utilized in two different simulations: (i) flow in a cavitating orifice and (ii) flow in a fuel injector. A realistic evolution of the cavitation cloud in the first case and the inception of the classic "hydraulic flip" in the second clearly demonstrate the robust nature of the numerical method and its ability to accurately capture complex multiphase flow and mass transfer.

Numerical simulation of flow with evaporation in triangular grooves

We study the flow of an evaporating liquid film in triangular open microchannel. Above the surface of the liquid flows a gas. The sear-stress on the gas-liquid surface drags the liquid and removes the vapor. It models either flow in a channel or a flow in a single groove. The novelty of this work is that we simulate gas-driven flow with evaporation. We use Boundary element method to solve Laplace equation with mixed boundary conditions. This method gives the normal derivative of vapor concentration at the surface of the liquid. The resulting evaporation flux is growing sharply near the contact line. Integral evaporation coefficient slightly depends on the depth of the liquid. The upper gas flow drives the liquid film, resisting a velocity drop and preventing the dry spot formation.

Rebar Shape Time-Evolution During a Reinforced Concrete Corrosion Test: An Electrochemical Model

An electrochemical model is presented to calculate the rebar shape time-evolution in reinforced mortar specimens during forced corrosion tests. This provides a more realistic description than the usually used geometric models. The current distribution along the rebar perimeter is calculated by using Finite Element Method (FEM) to solve Laplace equation. Then, Faraday’s law is used to relate current distribution to rebar volume increase due to corrosion products creation. The shape of the rebar section is obtained as a function of corrosion time.

Noncontact Human Body Voltage Measurement Using Microsoft Kinect and Field Mill for ESD Applications

A technique for measuring the voltage on a person by combining an overhead mounted electrostatic field sensor with Microsoft Kinect image recognition software and hardware is presented in this paper. The Kinect's built-in skeleton detection method was used to determine the posture, height, and location of a person relative to the position of the field sensor. The voltage is estimated using field strength and geometry information. Two different algorithms are proposed for calculating the human body voltage using field strength data captured by the field sensor and human body coordinates recognized by the Kinect. First, using calibration data, the human body voltage was obtained by charging a person to a known voltage and walking through the field of vision, then a correction was performed to convert the field sensor data into the human body voltage. The second algorithm is based on solving Laplace equation. The human body is modeled by using a superposition of spheroids, and the E-field is solved analytically. Results of the algorithms were compared, with the proposed methodology providing less than a ±15% error margin for the human body voltage over a detection area of 2.0 m × 2.5 m.

The method of two-point angular basis function for solving Laplace equation

In this paper, an approach to improve the method of angular basis function (MABF) proposed by Young et al. (2015) is proposed. Instead of using lnr in the method of fundamental solution (MFS), the MABF adopts θ to construct the solution. However, since the nature of θ introduces multiple values along the branch cut such that to avoid the branch cut passing through the domain is an important issue (Li et al., 2018). Noticing this difficulty, Alves et al. (2018) first proposed a remedy which used a pair of two points to restrict the discontinuity appearing only along the line segment between two points, and they named this approach as cracklets. In this article, the two-point angular basis function (cracklets) is investigated deeply. We explain why for a multiply connected domain with a logarithm singular solution the cracklets will encounter failure. To resolve this difficulty, one can adopt the proposed method (cracklets) with the MFS or one can use domain decomposition method to separate the domain into several singly connected domains. Seven numerical examples are provided to show the validity of this method, and examples for dealing with the multiply connected domain are focused to support our claims.

Theoretical analysis of extreme wave oscillation in Paradip Port using a 3-D boundary element method

A mathematical model is developed to estimate the amplification of multidirectional random waves in the Paradip port, Odisha, India under the resonance conditions. A 3-D boundary element method with the consideration of variable bathymetry is utilized for solving Laplace equation in an irregular domain including the partial reflection boundary condition. The current numerical scheme is validated through the comparison of simulation results with previous well-defined studies along with measurement data for the rectangular harbor. Six key recorder stations are chosen as port location of moored ship to analyze the wave response in the Paradip port for multidirectional random waves. Further, wave spectrum is also estimated based on Fast Fourier Transformation (FFT) with respect to wave period at six recorder stations. Actual bathymetry of Paradip port is utilized to estimate the wave height ratio at each record station for the incident waves with different directions. Further, the resonance frequencies are determined for each directional incident wave and safe locations in Paradip port is also identified based on the simulation results. Thus, the proposed numerical scheme can be utilized to foster the prediction of wave profile in the realistic ports or harbors with complex geometries for the safe navigation.

Hydro-elastic vibration analysis of functionally graded rectangular plate in contact with stationary fluid

This study investigates the free vibration of a moderately thick rectangular plate, which is composed of functionally graded materials and floating on incompressible fluid. Material properties are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fraction of the constituent. The governing equations of the plate are analytically derived based on the first-order shear deformation theory with consideration of rotational inertial effects and transverse shear stresses. Applied pressure on the free surface of the plate is obtained by the velocity potential function together with Bernoulli’s equation. The equation governing on the oscillatory behaviour of the fluid is obtained by solving Laplace equation with satisfying the boundary conditions. The natural frequencies and shape modes of the rectangular plate are determined by decoupling and solving the motion equations system. Then, analyses of the numerical results of free vibrations and the effects of the different parameters such as thickness to length of the plate, boundary conditions, fluid density, index of volume fraction and the height of the fluid on the frequencies are investigated. Finally, the results of this research in limit case is compared and validated with the results of other researchers and finite element model.