Laplace Equation Research Articles

Re-constructing the river bed using the streamline-generation method

River bed reconstruction plays an essential part in supporting the hydrodynamic simulation and understanding the morphological processes of a river. The streamlines can be solved using Laplace equations. The equation is first numerically solved in a computational environment and then adapted to the whole considered physical field to solve the resulting streamlines in a physical domain. One of the goals of this research is to determine the bottom line of the riverbed, and doing this through a field survey is very expensive, the methods presented in this research help a lot in reducing costs. By determining the concave line, the path of the flood in the river bed is determined, so one of the practical achievements and efficiencies of this research is flood trending at a low cost. In the present study, the bottom line is determined for the meandering Qinhe River, a distributary of the Yellow River, China, and Gaz River, located in Khuzestan Province, Iran. The method is based on the following steps:•Reconstruction of 2D river based on the Laplace equation.•Use the Finite Element Method to solve streamlines in the physical domain.•Use the Finite Difference Method to solve streamlines in the physical domain.

Dynamics of the interaction of a pair of thin evaporating droplets on compliant substrates

The dynamics of the interaction of a system of two thin volatile liquid droplets resting on a soft viscoelastic solid substrate are investigated theoretically. The developed model fully considers the effect of evaporative cooling and the generated Marangoni stresses due to the induced thermal gradients, while also accounting for the effect of the gas phase composition and the diffusion of vapour in the atmosphere of the droplets. Using the framework of lubrication theory, we derive evolution equations for both the droplet profile and the displacement of the elastic solid, which are solved in combination with Laplace's equation for the vapour concentration in the gas phase. A disjoining-pressure/precursor-film approach is used to describe contact-line motion. The evolution equations are solved numerically, using the finite-element method, and we present a thorough parametric analysis to investigate the physical properties and mechanisms that affect the dynamics of droplet interactions. The results show that the droplets interact through both the soft substrate and the gas phase. In the absence of thermocapillary phenomena, the combined effect of non-uniform evaporation due to the increased vapour concentration between the two droplets and elastocapillary phenomena determines whether the drop–drop interaction is attractive or repulsive. The Marangoni stresses suppress droplet attraction at the early stages of the drying process and lead to longer droplet lifetimes. For substrates with intermediate stiffness, the emergence of spontaneous symmetry breaking at late stages of evaporation is found. The rich dynamics of this complex system is explored by constructing a detailed map of the dynamic regimes.

Regularity for solutions of elliptic $$p(x)-$$Laplacian type equations with lower order terms and hardy potential

Regularity for solutions of elliptic $$p(x)-$$Laplacian type equations with lower order terms and hardy potential

SOLUTION OF A MIXED BOUNDARY VALUE PROBLEM FOR THE LAPLACE EQUATION BY THE METHOD OF DIVIDING AN OBJECT IN A RECTANGULAR PARALLELEPIPED

The aim of the study is to solve the boundary value problem for the Laplace equation in a rectangular parallelepiped by the method of separation of variables and to estimate the obtained Fourier variable separation constants.
 
 Materials and methods. Methods of mathematical physics were used to solve the boundary value problem for the Laplace equation. The initial problem was divided into three standard ones, in which the inhomogeneous boundary conditions were given only on two parallel sides, for the rest of the problem they being assumed to be equal to 0.
 
 Results. The boundary value problem for the Laplace equation in a rectangular parallelepiped has been broken down into three problems. Partial solutions to these problems under given boundary conditions have been obtained. The Fourier variable separation constants are estimated.
 
 Findings. The solution to the Laplace’s equation for a parallelepiped is the sum of the solutions to three partial problems. The boundary functions of a parallelepiped are odd periodic over two variable functions whose periods are equal to the lengths of the corresponding sides of the parallelepiped. The Fourier constants of partial solutions to the problem are the coefficients of the expansion of the boundary periodic functions of two variables into a trigonometric Fourier series. In two-dimensional series of the solution to the Laplace’s equation for odd-numbered harmonics and for a set of simultaneously even and odd harmonics, the Fourier constants differ only in signs.

OUTER CYLINDRICAL SPACES OF A HOLLOW CYLINDER OF FINITE LENGTH

Introduction. The solution of the Laplace equation by the method of separation of Fourier variables for cylindrical domains makes it possible to study electromechanical devices in 3D format.
 
 The purpose of the study is to obtain analytical expressions for magnetic potentials and inductions of external three-dimensional spaces of electromechanical devices in a cylindrical coordinate system.
 
 Materials and methods. The unknown constants in the expressions for the regions under consideration are found from the boundary conditions of the magnetic field at the common boundaries of the electromechanical device with its outer space: the scalar magnetic potentials and magnetic inductions are the same, the current sheets experience a jump. When obtaining analytical expressions, the methods of mathematical physics were used.
 
 Research results. To find the Fourier constants, the outer three-dimensional spaces of classical electromechanical devices (generators, motors) are considered, which in a cylindrical coordinate system are represented by both hollow and solid cylinders joining the active regions of the devices with their cylindrical or end surfaces. General expressions for magnetic potentials and inductions of external spaces are obtained based on the solution of the Laplace equation as the Sturm–Liouville problem by the Fourier method of separation of variables. The magnetic fields of the outer spaces of the original hollow cylinder are analyzed: an outer cylinder with an infinitely large radius; internal solid cylinder; end cylinders of finite length. These data make it possible to increase the required number of equations to find the required number of Fourier constants in the analytical calculation of electromechanical devices by the method of separation of variables.
 
 Findings. Analytical expressions for the boundary values of magnetic potentials and inductions of the external space adjacent to the active regions of electromechanical devices with air gaps of both radial and axial form are obtained.

The boundary value problem for the two-dimensional Laplace equation with transmission conditions given on open Lipschitz curve

We consider boundary value problem for Laplace equation in bounded two-dimensional Lipschitz domain with transmission boundary condition given upon open curve. This conditions includes itself the jump of solution of boundary value problem and the meaning of boundary value of its normal derivative. We prove the equivalence of considered boundary value problem and obtained variational problem. As a result we prove existence and uniqueness of solution of the posed problems in appropriate functional spaces. Based on the integral representation of the solution the considered boundary value problem is reduced to the system of boundary integral equation which has unique solution.

One-Dimensional Subspaces of the SL(n,mathbb {R}) Chiral Equations

In this work we find solutions of the (n+2)-dimensional Einstein Field Equations (EFE) with n commuting Killing vectors in vacuum. In the presence of n Killing vectors, the EFE can be separated into blocks of equations. The main part can be summarized in the chiral equation (alpha g_{, bar{z}} g^{-1})_{, z} + (alpha g_{, z} g^{-1})_{, bar{z}} = 0 with gin SL(n,mathbb {R}). The other block reduces to the differential equation , (ln f alpha ^{1-1/n})_{, z} = 1/2 , alpha ,{textrm{tr}}( g_{, z} g^{-1})^2 and its complex conjugate. We use the ansatz g = g(xi ) , where xi satisfies a generalized Laplace equation, so the chiral equation reduces to a matrix equation that can be solved using algebraic methods, turning the problem of obtaining exact solutions for these complicated differential equations into an algebraic problem. The different EFE solutions can be chosen with desired physical properties in a simple way.

Existence and multiplicity solutions for a singular elliptic p(x)-Laplacian equation

This paper deals with the existence and multiplicity of nontrivial weak solutions for the following equationinvolving variable exponents:\begin{align*}\begin{cases}-\vartriangle_{p(x)}u+\dfrac{\vert u\vert^{r-2}u}{|x|^{r}}=\lambda h(x,u),&in ~\Omega,\\u=0,&on~\partial\Omega,\end{cases}\end{align*}where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$ with smooth enough boundary which is subject to Dirichlet boundary condition.Using a variational method and Krasnoselskii's genus theory, we would show the existence andmultiplicity of the solutions. Next, we study closedness of set of eigenfunctions, such that $p(x)\equiv p$.

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.

Study of the precipitation trend of asphaltenes and waxes in crude oil using computational chemistry and statistical thermodynamics methods

Asphaltenes aggregation and waxes in petroleum were simulated considering five molecular structures containing functional groups: asphaltenes, resins, and non-polar hydrocarbons in a heavy oil crude. The properties of molecules were calculated using the density functional theory. The van der Waals constants associated with the pairwise interactions and the Hamaker constants for the dispersed phase (formed by asphaltenes and waxes) and the dispersion medium (created by non-polar hydrocarbons) were estimated. Sparse pass size as a function of its composition is calculated using Qyle groups and the Young–Laplace equation. We also assessed the potential colloidal pair corresponding to the interactions between the asphaltene and resin aggregates for different dispersion media, the attractive energies between the asphaltene molecules (−60 10–20 J ≈ ) are more than twice as high as the attractive energies concerning the rest of the molecules present ( × -30–10] 10–20 ≈ × J). It is predicted that attraction energy between the aggregates decreases with the increase of the presence of resin in the dispersed phase, This is explained by the fact that the increase in the concentration n of resin can lead to a decrease in the relative size d of the aggregates by up to 90 % (for n = 0.1, d = 0.1 and for n = 0.9, d = 0.1). This decrease in aggregate size is manifested in a theoretically calculated effective viscosity reduction of up to 50 %.

Non-Stationary Helical Flows for Incompressible Couple Stress Fluid

We explored here the case of three-dimensional non-stationary flows of helical type for the incompressible couple stress fluid with given Bernoulli-function in the whole space (the Cauchy problem). In our presentation, the case of non-stationary helical flows with constant coefficient of proportionality α between velocity and the curl field of flow is investigated. In the given analysis for this given type of couple stress fluid flows, an absolutely novel class of exact solutions in theoretical hydrodynamics is illuminated. Conditions for the existence of the exact solution for the aforementioned type of flows were obtained, for which non-stationary helical flow with invariant Bernoulli-function satisfying to the Laplace equation was considered. The spatial and time-dependent parts of the pressure field of the fluid flow should be determined via Bernoulli-function if components of the velocity of the flow are already obtained. Analytical and numerical findings are outlined, including outstanding graphical presentations of various types of constructed solutions, in order to elucidate dynamic snapshots that show the timely development of the topological behavior of said solutions.

Analytical Solution for Cylindrical Shell of Permeable Material in Fractional Dimensional Space

We have investigated the Laplacian equation in fractional dimensional space (FDS) that is widely used in physics to describe many complex phenomena. Using this concept, we have applied it on a cylindrical shell of permeable material to find the analytical solution of electric potential in FDS. The derivation of this problem is performed by applying Gegenbauer polynomials. The general solution has been obtained in a closed form in the FDS and can be applied to the cylindrical shell for different materials inside the cylinder core and outside the shell. By setting the fractional parameter α = 3, the derived solution is retrieved for the integer order.

Computing thickness of irregularly-shaped thin walls using a locally semi-implicit scheme with extrapolation to solve the Laplace equation: Application to the right ventricle

Cardiac Magnetic Resonance (CMR) Imaging is currently considered the gold standard imaging modality in cardiology. However, it is accompanied by a tradeoff between spatial resolution and acquisition time. Providing accurate measures of thin walls relative to the image resolution may prove challenging. One such anatomical structure is the cardiac right ventricle. Methods for measuring thickness of wall-like anatomical structures often rely on the Laplace equation to provide point-to-point correspondences between both boundaries. This work presents limex, a novel method to solve the Laplace equation using ghost nodes and providing extrapolated values, which is tested on three different datasets: a mathematical phantom, a set of biventricular segmentations from CMR images of ten pigs and the database used at the RV Segmentation Challenge held at MICCAI’12. Thickness measurements using the proposed methodology are more accurate than state-of-the-art methods, especially with the coarsest image resolutions, yielding mean L1 norms of the error between 43.28% and 86.52% lower than the second-best methods on the different test datasets. It is also computationally affordable. Limex has outperformed other state-of-the-art methods in classifying RV myocardial segments by their thickness.

Gradient estimates for singular $p$-Laplace type equations with measure data

We are concerned with interior and global gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, whose prototype is given by the p -Laplace equation -\Delta_{p} u=\mu with p\in (1,2) . The cases when p\in (2-\frac{1}{n},2) and p\in (\frac{3n-2}{2n-1},2-\frac{1}{n}] were studied in Duzaar and Mingione [J. Funct. Anal. 259 (2010), 379–418] and Nguyen and Phuc [J. Funct. Anal. 278 (2020), art. 108391] respectively. In this paper, we improve the results of Nguyen and Phuc and address the open case when p\in(1,\frac{3n-2}{2n-1}] . Interior and global modulus of continuity estimates of the gradients of solutions are also established.

A variational quantum algorithm-based numerical method for solving potential and Stokes flows

This paper presents a numerical method based on the variational quantum algorithm to solve potential and Stokes flow problems. In this method, the governing equations for potential and Stokes flows can be respectively written in the form of Laplace's equation and Stokes equations using velocity potential, stream function and vorticity formulations. Then the finite difference method and the generalized differential quadrature (GDQ) method are applied to discretize the governing equations. For the prescribed boundary conditions, the corresponding linear systems of equations can be obtained. These linear systems are solved by using the variational quantum linear solver (VQLS), which resolves the potential and Stokes flow problems equivalently. To the best of authors' knowledge, this is the first study that incorporates the GDQ method which is inherently a high-order discretization method with the VQLS algorithm. Since the GDQ method can utilize much fewer grid points than the finite difference method to approximate derivatives with a higher order of accuracy, the size of the input matrix for the VQLS algorithm can be smaller. In this way, the computational cost may be saved. The performance of the present method is comprehensively assessed by two representative examples, namely, the potential flow around a circular cylinder and Stokes flow in a lid-driven cavity. Numerical results validate the applicability and accuracy of the present VQLS-based method. Furthermore, its time complexity is evaluated by the heuristic scaling, which demonstrates that the present method scales efficiently in the number of qubits n and the precision ε. This work brings quantum computing to the field of computational fluid dynamics. By virtue of quantum advantage over classical methods, promising advances in solving large-scale fluid mechanics problems of engineering interest may be prompted.

Recursive analytical quadrature for the close evaluation of Laplace and Helmholtz layer potentials over flat boundary elements in [formula omitted

A method for the analytical evaluation of singular and nearly singular layer potentials arising in the collocation boundary element method for the Laplace and Helmholtz equation is developed for flat boundary elements with polynomial shape functions. The method is based on dimension-reduction via the divergence theorem and a Recursive scheme for evaluating the resulting line Integrals for Polynomial Elements (RIPE). It is used to evaluate single layer, double layer, adjoint double layer, and hypersingular potentials, for both the Laplace and the Helmholtz kernels. It naturally supports nearly singular, singular, and hypersingular integrals under a single framework. The developed recursive algorithm allows accurate evaluation of layer potentials associated with O(p2) density functions used in a pth order boundary element in O(p3) time for the Laplace case.

On the Method of Transformations: Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations

Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg–de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave–solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons.

Frequency-dependent Q simulation and viscoacoustic reverse time migration based on the fractional Zener model

Seismic attenuation is a basic physical property of the earth, which significantly affects the characteristics of seismic wavefields. Accurately simulating wave propagation in the earth is essential to image subsurface structures. Some prevailing methods (e.g., the standard linear solid and fractional Laplacian equation) to describe seismic wave propagation in attenuating media are mainly based on the constant- Q model (CQM), which is valid at room temperature and pressure. However, laboratory measurements suggest that the quality factor Q is a function of frequencies in some regions. To simulate the frequency-dependent Q effect, we derive a viscoacoustic wave equation from the stress-strain relationship of the fractional Zener model (FZM) with variable fractional orders. During the implementation, we separate the real and imaginary parts of the modulus and introduce a low-rank decomposition method to solve the FZM equation. Because the amplitude dissipation and phase dispersion are decoupled, we establish a compensated reverse time migration ( Q-RTM) algorithm to mitigate adverse effects caused by seismic attenuation and improve the quality of seismic migration in frequency-dependent attenuating media. A two-layer model and the BP gas chimney model are used to perform Q-RTM tests. A low-pass filter with a Tukey window function is applied to suppress numerical instability during the compensation. Numerical results demonstrate that our FZM Q-RTM approach can produce high-resolution images with corrected reflector positions and amplitudes. Because the CQM equation ignores the frequency dependence of Q, it may lead to overcompensation in Q-RTM.

Analytical Determination of Irrotational Flow Profiles in Open-Channel Transitions

Transitional free surface flow profiles with a critical point occur with weak vorticity and viscosity effects and thus can be modeled with an irrotational flow approach. Important examples in hydraulic engineering include flow over low obstacles, transition structures in canals, and flow over high spillways. While solving Laplace’s equation is relatively simple, the determination of the unknown free surface and energy head of the flow is challenging. Both hydraulic quantities need to be iterated before solving the Laplace equation. Former models iterated the energy head on a trial-and-error basis, assuming that the linked free surface profile is smooth, i.e., free of waves. The iteration of the free surface for a given head is frequently accomplished using the Newton–Rapshon method, which is difficult for the challenging case of spillway flow, giving no solution in some cases. An alternative method of computing irrotational flow profiles in transitional flows involving a critical point is proposed in this work. The model contains three elements: mapping of Laplace’s equation to directly track the streamlines, determination of the critical point and unknown energy head using a critical flow condition for irrotational flows, and determination of the water surface position using an exact analytical solution. The proposed model is favorably compared with experimental data from different sources and CFD results, indicating a reasonable agreement.

Analytical Thermal Model of Slot for Electric Propulsion Motor in Electric Aircraft

Accurate prediction of temperature is critical for evaluating electric propulsion motor performance. This paper presents an analytical thermal model of slot for the prediction of temperature. According to the heat transfer in the slot, the shortcomings of the traditional winding model are pointed out. This paper adopts a more advanced layered model. The number of layers is calculated analytically. Based on Laplace's equation and Poisson's equation, the thermal resistance and temperature rise of each layer are analytically calculated. The layered windings and insulation are equivalent to a heat-generating entity whose thermal resistance and temperature rise are equal to those of the layered slot. The equivalent thermal conductivity of heat-generating entity can be obtained. The slot is equivalent to a heat-generating entity through the analytical method, which reduces the computational complexity of the simulation. The experiments are employed to validate the accuracy of the proposed model.