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Solution of differential equations resulting from separation of Laplace equation in various coordinate systems
The mathematics involved in numerically solving for the plane boundary value of the Laplace equation by the grid method is developed. The approximate solution of a boundary value problem for the domain of the Laplace equation by the grid method consists of finding u at the grid corner which satisfies the equation at the internal corners (u=Du) and certain boundary value conditions at the boundary corners.
General potential function in triaxially ellipsoidal coordinates for Laplace equation, examining near earth satellites and Vinti potential
The main purpose of this study is the modification of the parametric integral equations systems (PIES) method, to include the NURBS curves into its mathematical formulas. We want to create an opportunity for defining the boundary shape using NURBS curves without the classical discretization process. We present an inclusion of the NURBS curves into PIES, and its application in process of modeling and solving the boundary value problems, on Laplace’s equation example. The correctness and accuracy of such modeling is confirmed by the examples. Obtained solutions are compared with these obtained using linear segments and B-spline curves. Additionally, we verify the correctness of modeling and modifying the shape using analytical solutions. The NURBS curves significantly expand the possibilities and increase the accuracy of the boundary shape modeling which is an important aspect of the numerical solving of boundary value problems.
Radial heat equation and Laplace transforms, discussing representations alternative to two solutions
Numerical analysis of Laplace equation with nonlinear boundary conditions
The series solution to Laplace's equation in a helical coordinate system is derived and refined using symmetry and chirality arguments. These functions and their more commonplace counterparts are used to model solenoidal magnetic fields via linear, multidimensional curve-fitting. A judicious choice of functional forms, a small number of free parameters and sparse input data can lead to highly accurate, fine-grained modeling of solenoidal magnetic fields, including helical features arising from the winding of the solenoid, with overall field accuracy at better than one part per million.
Finite difference solution to mixed boundary value problem for Laplace equation
Wave equation in quarter space, showing particular solution of Laplace equation with wave velocity increase
In this paper, we introduce a diffuse interface model to simulate multi-component two-phase flow with partial miscibility based on a realistic equation of state (e.g. Peng–Robinson equation of state). Because of partial miscibility, thermodynamic relations are used to model not only interfacial properties but also bulk properties, including density, composition, pressure, and realistic viscosity. As far as we know, this effort is the first time to use diffuse interface modeling based on equation of state for modeling of multi-component two-phase flow with partial miscibility. In numerical simulation, the key issue is to resolve the high contrast of scales from the microscopic interface composition to macroscale bulk fluid motion since the interface has a nanoscale thickness only. To efficiently solve this challenging problem, we develop a multi-scale simulation method. At the microscopic scale, we deduce a reduced interfacial equation under reasonable assumptions, and then we propose a formulation of capillary pressure, which is consistent with macroscale flow equations. Moreover, we show that Young–Laplace equation is an approximation of this capillarity formulation, and this formulation is also consistent with the concept of Tolman length, which is a correction of Young–Laplace equation. At the macroscopical scale, the interfaces are treated as discontinuous surfaces separating two phases of fluids. Our approach differs from conventional sharp-interface two-phase flow model in that we use the capillary pressure directly instead of a combination of surface tension and Young–Laplace equation because capillarity can be calculated from our proposed capillarity formulation. A compatible condition is also derived for the pressure in flow equations. Furthermore, based on the proposed capillarity formulation, we design an efficient numerical method for directly computing the capillary pressure between two fluids composed of multiple components. Finally, numerical tests are carried out to verify the effectiveness of the proposed multi-scale method.
Integral representations involving Laplace transforms developed for use in solutions of radial heat equation
Identification in partial differential equations by Laplace equation
Various finite difference techniques used to solve Laplace's equation are compared. Curvilinear coordinate systems are used on two dimensional regions with irregular boundaries, specifically, regions around circles and airfoils. Truncation errors are analyzed for three different finite difference methods. The false boundary method and two point and three point extrapolation schemes, used when having the Neumann boundary condition are considered and the effects of spacing and nonorthogonality in the coordinate systems are studied.
A variety of problems in material science and fluid dynamics require the solution of Laplace's equation in multiply connected domains. Integral equation methods are natural candidates for such problems, since they discretize the boundary alone, require no special effort for free boundaries, and achieve superalgebraic convergence rates on sufficiently smooth domains in two space dimensions, regardless of shape. Current integral equation methods for the Dirichlet problem, however, require the solution of M independent problems of dimension N, where M is the number of boundary components and N is the total number of points in the discretization. In this paper, we present a new boundary integral equation approach, valid for both interior and exterior problems, which requires the solution of a single linear system of dimension N + M. We solve this system by making use of an iterative method (GMRES) combined with the last multipole method for the rapid calculation of the necessary matrix vector products. For a two-dimensional system with 200 components and 100 points on each boundary, we gain a speedup of a factor of 100 from the new analytic formulation and a factor of 50 from the fast multipole method. The resulting scheme brings large scale calculations in extremely complex domains within practical reach.
Axisymmetric equilibrium shapes of conducting drops and bubbles, (1) pendant or sessile on one face of a circular parallel-plate capacitor or (2) free and surface-charged, are found by solving simultaneously the free boundary problem consisting of the augmented Young-Laplace equation for surface shape and the Laplace equation for electrostatic field, given the surface potential. The problem is nonlinear and the method is a finite element algorithm employing Newton iteration, a modified frontal solver, and triangular as well as quadrilateral tessellations of the domain exterior to the drop in order to facilitate refined analysis of sharply curved drop tips seen in experiments. The stability limit predicted by this computer-aided theoretical analysis agrees well with experiments.
Axisymmetric equilibrium shapes and stability of isolated charged drops are found by solving simultaneously the Young-Laplace equation for surface shape and the Laplace equation for the electric field. Families of two-, three-, and four-lobed shapes that branch from the trunk family of spheres are treated systematically by means of the Galerkin/finite element method and a tessellation that deforms with the free surface. The results show that at the limit found by Rayleigh in 1882 the spherical family exchanges stability with a family of two-lobed shapes, a transcritically bifurcating family, one arm of which proves to consist of stable shapes. The results are reinforced by those of approximating the stable drop shapes as oblate spheroids. Thus oblate drops carrying charge in excess of the Rayleigh limit ought to be seen in experiments, though none have yet been reported.
A highly conducting charged drop that is surrounded by a fluid insulator of another density can be levitated by suitably applying a uniform electric field. Axisymmetric equilibrium shapes and stability of the levitated drop are found by solving simultaneously the augmented Young-Laplace equation for surface shape and the Laplace equation for the elecric field, together with constraints of fixed drop volume, charge, and center of mass. The means are a method of subdomains, finite element basis functions, and Galerkin's method of weighted residuals, all facilitated by a large-scale computer. Shape families of fixed charge are treated systematically by first-order continuation. Previous analyses by Abbas et al. in 1967 and Abbas and Latham in 1969, in which the shapes of levitated drops are approximated as spheroids, are corrected. The new analysis shows that drops charged to less than the Rayleigh limit lose shape stability at turning points, with respect to external field strength, and that the instability seen in experiments of Doyle et al. in 1964 and others is not a bifurcation to a family of two-lobed shapes, but rather is a related imperfect bifurcation.
The inversion of the Laplace-Beltrami operator and the computation of the Hodge decomposition of a tangential vector field on smooth surfaces arise as computational tasks in many areas of science, from computer graphics to machine learning to computational physics. In this work, we present a high-order accurate pseudo-spectral approach, applicable to closed surfaces of genus one in three-dimensional space, with a view toward applications in plasma physics and fluid dynamics.