Calculation of vector potentials from scalar potentials for 3D finite difference solutions

1990 ◽  
Vol 26 (2) ◽  
pp. 686-689 ◽  
Author(s):  
W. Muller ◽  
G. Szymanski
Keyword(s):  
Finite Difference ◽  
Vector Potentials ◽  
Scalar Potentials

scholarly journals CONTINUITY PROPERTIES OF SCHRÖDINGER SEMIGROUPS WITH MAGNETIC FIELDS

2000 ◽  
Vol 12 (02) ◽  
pp. 181-225 ◽  
Author(s):  
KURT BRODERIX ◽  
DIRK HUNDERTMARK ◽  
HAJO LESCHKE
Keyword(s):  
Brownian Bridge ◽  
Integral Kernel ◽  
Dirichlet Boundary Conditions ◽  
Dimensional Euclidean Space ◽  
Kato Class ◽  
Vector Potentials ◽  
Continuity Properties ◽  
Scalar Potentials ◽  
General Configuration ◽  
Zero Vector

The objects of the present study are one-parameter semigroups generated by Schrödinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Kato-like conditions. The configuration space is supposed to be an arbitrary open subset of multi-dimensional Euclidean space; in case that it is a proper subset, the Schrödinger operator is rendered symmetric by imposing Dirichlet boundary conditions. We discuss the continuity of the image functions of the semigroup and show local-norm-continuity of the semigroup in the potentials. Finally, we prove that the semigroup has a continuous integral kernel given by a Brownian-bridge expectation. Altogether, the article is meant to extend some of the results in B. Simon's landmark paper [Bull. Amer. Math. Soc.7 (1982) 447] to non-zero vector potentials and more general configuration spaces.

The Potentials

2019 ◽  
pp. 43-60
Author(s):  
Richard Freeman ◽  
James King ◽  
Gregory Lafyatis
Keyword(s):  
Wave Equations ◽  
Electric Quadrupole ◽  
Quadrupole Moments ◽  
Retarded Time ◽  
Vector Potentials ◽  
Electric And Magnetic Fields ◽  
Lorenz Gauge ◽  
The One ◽  
Scalar Potentials ◽  
Approximate Expressions

The concepts of scalar and vector potentials are introduced and the electric and magnetic fields are shown to be derived from specific forms of these potentials. The choice of these forms is restricted by gauge considerations, and the Lorenz gauge is introduced as the one most applicable for radiation. Using this, the wave equations prescribing the potentials in terms of the source conditions are presented. The modifications of vector and scalar potentials to account for speed of light and causality lead to the concept of “retarded time.” The potentials can be expressed in terms of moments of the source along with concepts of “near,” “intermediate,” and “far” zones to facilitate derivation of approximate expressions for the potentials evaluated at appropriate distances from the source. Finally, expressions for the vector potential in terms of the electric and magnetic dipole, and electric quadrupole moments of the source in the approximation zones are presented.

scholarly journals Acceleration of Augmented EFIE Using Multilevel Complex Source Beam Method

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Lianning Song ◽  
Yongpin Chen ◽  
Ming Jiang ◽  
Jun Hu ◽  
Zaiping Nie
Keyword(s):  
Computational Cost ◽  
Low Frequency ◽  
Electric Field Integral Equation ◽  
Vector Potentials ◽  
Multilevel Algorithm ◽  
Complex Source ◽  
The Matrix ◽  
Aggregation And Disaggregation ◽  
Scalar Potentials ◽  
Fast Multipole Algorithm

The computation of the augmented electric field integral equation (A-EFIE) is accelerated by using the multilevel complex source beam (MLCSB) method. As an effective solution of the low-frequency problem, A-EFIE includes both current and charge as unknowns to avoid the imbalance between the vector potentials and the scalar potentials in the conventional EFIE. However, dense impedance submatrices are involved in the A-EFIE system, and the computational cost becomes extremely high for problems with a large number of unknowns. As an exact solution to Maxwell’s equations, the complex source beam (CSB) method can be well tailored for A-EFIE to accelerate the matrix-vector products in an iterative solver. Different from the commonly used multilevel fast multipole algorithm (MLFMA), the CSB method is free from the problem of low-frequency breakdown. In our implementation, the expansion operators of CSB are first derived for the vector potentials and the scalar potentials. Consequently, the aggregation and disaggregation operators are introduced to form a multilevel algorithm to reduce the computational complexity. The accuracy and efficiency of the proposed method are discussed in detail through a variety of numerical examples. It is observed that the numerical error of the MLCSB-AEFIE keeps constant for a broad frequency range, indicating the good stability and scalability of the proposed method.

scholarly journals WEAK VECTOR AND SCALAR POTENTIALS: APPLICATIONS TO POINCARÉ'S THEOREM AND KORN'S INEQUALITY IN SOBOLEV SPACES WITH NEGATIVE EXPONENTS

2010 ◽  
Vol 08 (01) ◽  
pp. 1-17 ◽  
Author(s):  
CHÉRIF AMROUCHE ◽  
PHILIPPE G. CIARLET ◽  
PATRICK CIARLET
Keyword(s):  
Sobolev Spaces ◽  
Three Dimensional ◽  
Vector Potentials ◽  
Korn’S Inequality ◽  
Scalar Potentials ◽  
Simply Connected ◽  
Dimensional Domain ◽  
Weak Vector

In this paper, we present several results concerning vector potentials and scalar potentials with data in Sobolev spaces with negative exponents, in a not necessarily simply-connected, three-dimensional domain. We then apply these results to Poincaré's theorem and to Korn's inequality.

scholarly journals Lp-THEORY FOR VECTOR POTENTIALS AND SOBOLEV'S INEQUALITIES FOR VECTOR FIELDS: APPLICATION TO THE STOKES EQUATIONS WITH PRESSURE BOUNDARY CONDITIONS

2012 ◽  
Vol 23 (01) ◽  
pp. 37-92 ◽  
Author(s):  
CHÉRIF AMROUCHE ◽  
NOUR EL HOUDA SELOULA
Keyword(s):  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Vector Fields ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Vector Potentials ◽  
Free Function ◽  
Scalar Potentials ◽  
Lp Theory ◽  
Pressure Boundary Conditions

In a three-dimensional bounded possibly multiply connected domain, we give gradient and higher-order estimates of vector fields via div and curl in Lp-theory. Then, we prove the existence and uniqueness of vector potentials, associated with a divergence-free function and satisfying some boundary conditions. We also present some results concerning scalar potentials and weak vector potentials. Furthermore, we consider the stationary Stokes equations with non-standard boundary conditions of the form u × n = g × n and π = π0 on the boundary Γ. We prove the existence and uniqueness of weak, strong and very weak solutions. Our proofs are based on obtaining Inf–Sup conditions that play a fundamental role. We give a variant of the Stokes system with these boundary conditions, in the case where the compatibility condition is not verified. Finally, we give two Helmholtz decompositions that consist of two kinds of boundary conditions such as u⋅n and u × n on Γ.

Finite-difference simulation of borehole EM measurements in 3D anisotropic media using coupled scalar-vector potentials

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. G225-G233 ◽  
Author(s):  
Junsheng Hou ◽  
Robert K. Mallan ◽  
Carlos Torres-Verdín
Keyword(s):  
Electrical Conductivity ◽  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Vector Potential ◽  
Anisotropic Media ◽  
Anisotropic Rock ◽  
Partial Differential ◽  
Vector Potentials ◽  
Potential Formulation

This paper describes the implementation and successful validation of a new staggered-grid, finite-difference algorithm for the numerical simulation of frequency-domain electromagnetic borehole measurements. The algorithm is based on a coupled scalar-vector potential formulation for arbitrary 3D inhomogeneous electrically anisotropic media. We approximate the second-order partial differential equations for the coupled scalar-vector potentials with central finite differences on both Yee’s staggered and standard grids. The discretization of the partial differential equations and the enforcement of the appropriate boundary conditions yields a complex linear system of equations that we solve iteratively using the biconjugate gradient method with preconditioning. The accuracy and efficiency of the algorithm is assessed with examples of multicomponent-borehole electromagnetic-induction measurements acquired in homogeneous, 1D anisotropic, 2D isotropic, and 3D anisotropic rock formations. The simulation examples consider vertical and deviated wells with and without borehole and mud-filtrate invasion regions. Simulation results obtained with the scalar-vector coupled potential formulation favorably compare in accuracy with results obtained with 1D, 2D, and 3D benchmarking codes in the dc to megahertz frequency range for large contrasts of electrical conductivity. Our numerical exercises indicate that the coupled scalar-vector potential equations provide a general and consistent algorithmic formulation to simulate borehole electromagnetic measurements from dc to megahertz in the presence of large conductivity contrasts, dipping wells, electrically anisotropic media, and geometrically complex models of electrical conductivity.

Bound states of spinless particles with Coulomb interaction in the momentum representation

1989 ◽  
Vol 67 (10) ◽  
pp. 992-995 ◽  
Author(s):  
F. Domínguez-adame
Keyword(s):  
Wave Function ◽  
Large Particle ◽  
Bound States ◽  
Bound State ◽  
Momentum Representation ◽  
Oscillatory Behaviour ◽  
Vector Potentials ◽  
Particle Momentum ◽  
Klein Gordon ◽  
Scalar Potentials

Bound-state solutions of the Klein–Gordon Coulomb equation for vector and scalar potentials are investigated in the momentum representation. The collapse of the particle to the center for strong vector potentials is found. The corresponding wave function shows an anomalous oscillatory behaviour for large particle momentum. Particle collapse for strong scalar potentials does not exist.

SOLUTION OF PRIDE'S EQUATIONS THROUGH POTENTIALS

2006 ◽  
Vol 17 (06) ◽  
pp. 877-908 ◽  
Author(s):  
BORIS D. PLYUSHCHENKOV ◽  
VICTOR I. TURCHANINOV
Keyword(s):  
Computer Code ◽  
Linear Combinations ◽  
Acoustic Disturbance ◽  
Electrokinetic Effect ◽  
Vector Potentials ◽  
Displacement Vectors ◽  
Different Types ◽  
Spatial Coordinates ◽  
Em Field ◽  
Scalar Potentials

In 1994, S. Pride offered a system of equations describing interdependent propagation of acoustic and electromagnetic (EM) fields in porous media saturated by a fluid electrolyte (electrokinetic effect). We proved that the displacement vectors of frame and of pore fluid can be represented as a linear combinations of gradients of two scalar potentials and rotors of two vector potentials provided the coefficients of Pride's equations are independent of spatial coordinates. Each of these potentials satisfies Helmholtz equation with appropriate complex velocity. This representation underlies the computer code created by us for exploring borehole acoustics logging problems in axial-symmetric radially layered medium. Noteworthy, layers can be of any of the following types: uniform porous medium saturated by fluid electrolyte, uniform nonporous elastic medium, compressible nonviscous fluid or compressible viscous fluid. Results of numerical experiments are presented and discussed. The most important of them is that the back influence on acoustic disturbance of EM field excited by this disturbance is negligibly small in usually used frequency band and for realistic formation parameters. The mechanism of generation and propagation of different types of EM waves during acoustic logging is discussed as well.

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