Three-dimensional Laplace Equation Research Articles

The 3-D problem of temperature and thermal flux distribution around defects with temperature-dependent material properties

The analytical solution of 3-D heat conduction problem, including the temperature and thermal flux fields, is one of the important problems that have not been completely solved in solid mechanics. Considering the temperature dependence of material parameters makes the problem more difficult. In this paper, we first reduce the 3-D temperature-dependent heat conduction problem to the solution of 3-D Laplace equation by introducing the intermediate function. Then, the generalized ternary function is proposed, and the general solution of 3-D Laplace equation is given. Finally, the analytical solutions of three specific problems are obtained and the corresponding temperature-thermal flux fields are discussed. The results show that the thermal flux field of 3-D temperature dependent problem is the same as the classical constant thermal conductivity approach result, while the temperature field is different from the classical result. Thermal flux at a planar defect boundary has r?1/2 singularity, and its intensity is proportional to the fourth root of defect width. On the other hand, when blocked by a planar defect, the thermal flux distribution will re-adjusted so that it overflows at the same rate from all parts of the planar defect boundary.

An analytical method for the solution of earthquake-induced hydrodynamic forces and wave forces on multiple vertical cylinders with arbitrary smooth sections

Analytical methods are developed to investigate the interaction of water with multiple cylinders with arbitrary smooth cross-sections subjected to horizontal ground motion and incident linear waves. Based on the boundary conditions of the seafloor and the water surface, the three-dimensional Laplace equation was transformed into the two-dimensional (2D) Helmholtz equation. Based on the condition that the coefficient function of the cylinder surface can be expanded into a Fourier series, radiation theory and diffraction theory are extended to solve earthquake-induced hydrodynamic forces and wave forces on multiple cylinders. The proposed analytical solution consists of the high-order scattered wave pressures due to diffracted waves. The present method was verified using two double cylinder arrays with elliptical and round-ended cross-sections. Finally, this method was extended to multiple scattering analysis for the interaction of water under earthquake and wave loads.

Calculation of singular integrals on elements of three-dimensional problems by triple-reciprocity boundary element method

Accurate numerical evaluation of integrals arising in the boundary element method is fundamental to achieving useful results via this solution technique. In this paper, a new technique is considered to evaluate the weak singular integrals that arise in the solution of three-dimensional Laplace's equation. This new application of the triple-reciprocity boundary element method is proposed for the calculation of singular integrals. A formulation of the boundary element method is utilized, and a method for the direct numerical integration of the two-dimensional surface using a two-dimensional interpolation method is proposed. In numerical integral calculation, the numerical integration of arbitrary shape is possible, and integration in the case of two-dimensional integration is approximately changed into a one-dimensional integration by using the Green's second identity. In the introduced line integral, there is no singularity. To evaluate the efficiency of this method, several numerical examples are given.

The Method of Fundamental Solutions for the 3D Laplace Inverse Geometric Problem on an Annular Domain

In this paper, we are interested in an inverse geometric problem for the three-dimensional Laplace equation to recover an inner boundary of an annular domain. This work is based on the method of fundamental solutions (MFS) by imposing the boundary Cauchy data in a least-square sense and minimisation of the objective function. This approach can also be considered with noisy boundary Cauchy data. The simplicity and efficiency of this method is illustrated in several numerical examples.

Analytical Solution of the Three-Dimensional Laplace Equation in Terms of Linear Combinations of Hypergeometric Functions

We present some solutions of the three-dimensional Laplace equation in terms of linear combinations of generalized hyperogeometric functions in prolate elliptic geometry, which simulates the current tokamak shapes. Such solutions are valid for particular parameter values. The derived solutions are compared with the solutions obtained in the standard toroidal geometry.

Apply the Sturm-Liouville problem with Green’s function to linear system

In this paper we will study the Sturm-Liouville problem, we find the eigenvalues are the solution of the Sturm Liouville problem and the eigenfunctions are corresponding solutions. Thus, we study Construction of the Green’s Function to solving the first order differential n- dimensional linear system, and application for Fourier series. And we show that the Green’s Function Solution to the two- and three-dimensional Laplace and Poisson Equations.

On the Cauchy Problem for the Three-Dimensional Laplace Equation

In the present work, using the Carleman function, a harmonic function and its derivatives are restored by the Cauchy data on a part of the boundary of the region. It is shown that the effective construction of the Carleman function is equivalent to the construction of a regularized solution to the Cauchy problem. We assume that a solution to the problem exists and is continuously differentiable in a closed domain with precisely given Cauchy data. For this case, we establish an explicit formula for the continuation of the solution and its derivative, as well as the regularization formula for the case when under the indicated conditions instead of the initial Cauchy data their continuous approximations are given with a given error in the uniform metric. The stability estimates for the solution to the Cauchy problem in the classical sense are obtained.

Задача Коши для трехмерного уравнения Лапласа

Задача Коши для трехмерного уравнения Лапласа

A scalar prototype problem of deep abrasive drilling with an infinite free boundary: an asymptotic modeling study

A vanishing at infinity solution of the three-dimensional Laplace equation is sought in an a priori unknown domain with overdetermined boundary conditions, the right-hand sides of which depend on the unit normal to the free boundary. A perturbation analysis of the nonstandard free boundary problem that models deep abrasive drilling has been performed under the assumption that the free boundary is close to the surface of a given semi-infinite cylinder, the longitudinal position of which depends on the boundary data, and the leading-order asymptotic solution has been developed.

A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace Equation

In this article, an inverse problem with regards to the Laplace equation with non-homogeneous Neumann boundary conditions in a three-dimensional case is investigated. To deal with this problem, a regularization method (mollification method) with the bivariate de la Vallée Poussin kernel is proposed. Stable estimates are obtained under a priori bound assumptions and an appropriate choice of the regularization parameter. The error estimates indicate that the solution of the approximation continuously depends on the noisy data. Two experiments are presented, in order to validate the proposed method in terms of accuracy, convergence, stability, and efficiency.

Hypercomplex method for solving linear PDEs

Algebraic-analytic approach to constructing solutions for given partial differential equations were investigated in many papers. In particular, in papers [1-14]. It involves solving two problems. Problem (P 1) is to describe all the sets of vectors \( e_1, e_2, \ldots, e_d \), which satisfy the characteristic equation (or specify the procedure by which they can be found). And the problem (P 2) is to describe all the components of monogenic (i.e., continuous and differentiable in sense Gateaux) functions. In particular, for the equation (4) we must describe the components of the function \( \Phi(\zeta) = \exp \zeta \). Note that in the papers [15, 16] a constructive description of all analytic functions with values is obtained in an arbitrary finite-dimensional commutative associative algebra over the field \(\mathbb{C}\). The Theorem 5.1 of the paper [17] states that it is enough to limit the study of monogenic functions in algebras with the basis of \( \{1, \eta_1, \eta_2, \ldots, \eta_{n-1} \} \), where \( \eta_1, \eta_2, \ldots, \eta_{n-1} \) are nilpotents. In addition, in [18] it is showed that in each algebra with a basis of the form \( \{1, \eta_1, \eta_2, \ldots, \eta_{n-1} \} \) the equation (3) has solutions. That is, the problems (P 1) and (P 2) are completely solved on the classes of commutative associative algebras with the basis \( \{1, \eta_1, \eta_2, \ldots, \eta_{n-1}\} \). It is worth noting that in a finite-dimensional algebra a decomposition of monogenic functions has a finite number of components, and therefore, it generates a finite number of solutions of a given partial differential equations. In this paper, we propose a procedure for constructing an infinite number of families of solutions of given linear differential equations with partial derivatives with constant coefficients. We use monogenic functions that are defined on some sequences of commutative associative algebras over the field of complex numbers. To achieve this goal, we first study the solutions of the so-called characteristic equation on a given sequence of algebras. Further, we investigate monogenic functions on the sequence of algebras and study their relation with solutions of partial deferential equations. The proposed method is used to construct solutions of some equations of mathematical physics. In particular, for the three-dimensional Laplace equation and the wave equation, for the equation of transverse oscillations of the elastic rod and the conjugate equation, a generalized biharmonic equation and the two-dimensional Helmholtz equation. We note that this method yields all analytic solutions of the two-dimensional Laplace equation and the two-dimensional biharmonic equation (Goursat formula).

Thermo-hydrodynamic performance of tilted non-circular micropolar lubricated journal bearings

PurposeThe purpose of this study is to examine the thermo-hydrodynamic performance of tilted non-circular journal bearings lubricated with a micropolar fluid. The investigated bearing types are two- and three-lobe journal bearings with finite length.Design/methodology/approachFor this purpose, modified Reynolds, energy and three-dimensional Laplace equations are solved numerically by using generalized differential quadrature method. The effects of micropolarity characteristics of lubricants, such as characteristic length and coupling number, as well as tilt angle as a design parameter, on the performance of non-circular two- and three-lobe journal bearings are studied.FindingsThe results show that the tilt angle can affect the temperature and pressure profiles causing variation in the performance of non-circular bearings. Increasing coupling number and decreasing characteristic length cause the load-carrying capacity to decrease because of the increase in maximum oil temperature of the fluid film of lubricant and decrease in the minimum oil base viscosity. So, it is possible to select suitable values of tilt angle for achieving optimum performance of these bearings.Originality/valueThe non-circular bearings suggest several design parameters such as tilt angle for designers. By considering thermal effects for micropolar lubricant, the requirements of a specific application can be fulfilled.

A modified dual-level algorithm for large-scale three-dimensional Laplace and Helmholtz equation

A modified dual-level algorithm is proposed in the article. By the help of the dual level structure, the fully-populated interpolation matrix on the fine level is transformed to a local supported sparse matrix to solve the highly ill-conditioning and excessive storage requirement resulting from fully-populated interpolation matrix. The kernel-independent fast multipole method is adopted to expediting the solving process of the linear equations on the coarse level. Numerical experiments up to 2-million fine-level nodes have successfully been achieved. It is noted that the proposed algorithm merely needs to place 2–3 coarse-level nodes in each wavelength per direction to obtain the reasonable solution, which almost down to the minimum requirement allowed by the Shannon’s sampling theorem. In the real human head model example, it is observed that the proposed algorithm can simulate well computationally very challenging exterior high-frequency harmonic acoustic wave propagation up to 20,000 Hz.

Integral theorems for monogenic functions in an infinite-dimensional space with a commutative multiplication

We consider monogenic functions taking values in a topological vector space being an expansion of a certain infinite-dimensional commutative Banach algebra associated with the three-dimensional Laplace equation. We establish also integral theorems for monogenic functions taking values in the mentioned algebra and the mentioned topological vector space.

Numerical Procedure for Modeling of Light Emitting Diode with Mesh-Like Electrode

A computational procedure is presented for numerical modeling of the light emitting diode (LED) with top p-electrode designed as a mesh with the strips of rectangular cross section. Isotropic light emission in the LED’s active region and light reflection from the bottom electrode are considered. Three-dimensional Laplace equation for electric potential is solved by finite element method. The numerical model incorporates mapped infinite element to account for potential decay far away from the LED structure and finite element model developed for boundary condition at semiconductor-air interface in the mesh opening. Simulation results demonstrate the effect of the mesh’s geometrical parameters on the total output power.

Directional Method of Fundamental Solutions for Three-dimensional Laplace Equation

We propose a simple extension of the two-dimensional method of fundamental solutions (MFS) to a two-dimensional like MFS for the numerical solution of the three-dimensional Laplace equation in an arbitrary interior domain. In the directional MFS (DMFS) the directors are planar orientations, which can take the geometric anisotropy of the problem domain into account, and more importantly the order of the logarithmic singularity with $\ln R$ of the new fundamental solution is reduced than that of the conventional three-dimensional fundamental solution with singularity $1/r$. Some numerical examples are used to validate the performance of the DMFS.

Trapped modes in a wave guide with a circular cylinder

ABSTRACTTrapped mode in three-dimensional waveguide with parallel vertical walls a distance 2d apart, when there is a circular cylinder of radius placed symmetrically between them, is investigated. The potential is the solution of the three-dimensional Laplace equation in the fluid region between the cylinder and the parallel walls and the normal derivatives of potential are zero on both the walls and the cylinder. It is shown that for a cylinder of sufficiently small radius there exists a trapped mode, having a frequency close to the cutoff frequency, which is antisymmetric about the centerline of the guide and symmetric about a line through the center of the cylinder perpendicular to the centerline. The method used in this paper is to utilize the Fourier transform of complex form to the formula derived in the Appendix which can easily transform the solution in polar coordinates into the form useful for the rectangular coordinates. Thus, it is possible to express the unknown function in the solution in rectangular coordinates in terms of the coefficients of the solution in polar coordinates.

Unified Construction of Green’s functions for Poisson’s equation in inhomogeneous media with diffuse interfaces

Green’s functions for Poisson’s equation in inhomogeneous media with material interfaces have many practical applications. In the present work, we focus on Green’s functions for Poisson’s equation in inhomogeneous media with diffuse material interfaces where a gradual and continuous transition in the material constant is assumed in a small region around the interfaces between different materials. We present a unified general framework for calculating Green’s functions for Poisson’s equation in such inhomogeneous media and the framework can apply to all eleven orthogonal coordinate systems in which the three-dimensional Laplace equation is separable. Within this framework, the idea on how to design the so-called quasi-harmonic diffuse interface is discussed, formulations for building Green’s function for Poisson’s equation in an inhomogeneous medium with such a diffuse interface is elaborated, and a robust numerical method for calculating Green’s functions for Poisson’s equation in inhomogeneous media with general diffuse interfaces is developed. Several practically relevant separable coordinate systems are briefly surveyed at the level of definition and basic facts relevant for implementing the unified framework in these coordinate systems.

Enhancement of Digital PID Controller Performance for Blood Glucose Level of Diabetic Patients using Disparate Tuning Techniques

Objectives: To design digital PID controller by using CHR-I and CHR-II tuning techniques, as it helps in finding out the tuning parameters of controllers for a specific system. Transformation of analog to digital PID controller using various transformation techniques like first order hold method, impulse-invariant mapping, Tustin approximation and zero-pole mapping equivalents and also the mathematical modeling of blood glucose level, such that a system injects the exact amount of insulin into the body of diabetic patient to maintain his/her glucose level to the normal range. Method/Statistical Analysis: The differential equation of the blood glucose level is formulated and then it is converted to three-dimensional Laplace equation using forward Laplace transform. Using the Laplace transform the differential equation of the blood glucose is converted into a s-domain equation. Then, using the s-domain equation as the equation of the system and the Tuning techniques, CHR-I and CHR-II, the tuning parameters (Kp, Ki and Kd) are acquired. Then, it is converted into digital, i.e. in z-domain, by applying disparate transformation techniques. Findings: On analyzing the acquired equation, it is depicted that on tuning the controller with CHR-I tuning technique the system exhibits zero overshoot which is most reliable and efficient for diabetic patient. Also, a considerable settling time of 6.3362 seconds is also achieved. Application/Improvement: Therefore, a system that can inject the exact amount of insulin into the patient's blood and bring the blood glucose level to the normal range, by automatically calculating the amount of insulin required, from the available status of blood glucose level, is being achieved.

An extension of monogenic functions and spatial potentials

We obtain explicitly principal extensions of analytic functions of the complex variable into an infinite-dimensional commutative Banach algebra associated with the three-dimensional Laplace equation. We consider an extension of differentiable in the sense of Gâteaux functions with values in a topological vector space being an expansion of the mentioned algebra and its relations to spatial potentials.