Local molecular field theory for the treatment of electrostatics

We examine in detail the theoretical underpinnings of previous successful applications of local molecular field (LMF) theory to charged systems. LMF theory generally accounts for the averaged effects of long-ranged components of the intermolecular interactions by using an effective or restructured external field. The derivation starts from the exact Yvon-Born-Green hierarchy and shows that the approximation can be very accurate when the interactions averaged over are slowly varying at characteristic nearest-neighbor distances. Application of LMF theory to Coulomb interactions alone allows for great simplifications of the governing equations. LMF theory then reduces to a single equation for a restructured electrostatic potential that satisfies Poisson's equation defined with a smoothed charge density. Because of this charge smoothing by a Gaussian of width sigma, this equation may be solved more simply than the detailed simulation geometry might suggest. Proper choice of the smoothing length sigma plays a major role in ensuring the accuracy of this approximation. We examine the results of a basic confinement of water between corrugated wall and justify the simple LMF equation used in a previous publication. We further generalize these results to confinements that include fixed charges in order to demonstrate the broader impact of charge smoothing by sigma. The slowly-varying part of the restructured electrostatic potential will be more symmetric than the local details of confinements.Comment: To be published in J Phys-Cond Matt; small misprint corrected in Eq. (12) in V

Evaluation of three approaches for simulating 3D time-domain electromagnetic data

We implemented and compared the implicit Euler time-stepping approach, the inverse Fourier Transform-based approach, and the Rational Arnoldi method for simulating 3D transient electromagnetic data. We utilize the finite-element method with unstructured tetrahedral meshes for the spatial discretization supporting irregular survey geometries and anisotropic material parameters. Both, switch-on and switch-off current waveforms, can be used in combination with direct current solutions of Poisson problems as initial conditions. Moreover, we address important topics such as the incorporation of source currents and opportunities to simulate impulse as well as step response magnetic field data with all approaches for supporting a great variety of applications. Three examples ranging from simple to complex real-world geometries and validations against external codes provide insight into the numerical accuracy, computational performance, and unique characteristics of the three applied methods. We further present an application of logarithmic Fourier transforms to convert transient data into the frequency domain. We made all approaches available in the open-source Python toolbox custEM, which previously supported only frequency-domain electromagnetic data. The object-oriented software implementation is suited for further elaboration on distinct modeling topics and the presented examples can serve for benchmarking other codes

Random walk on fixed spheres for Laplace and Lamé equations

The Random Walk on Fixed Spheres (RWFS) introduced in our previous paper is presented in details for Laplace and Lam'e equations governing static elasticity problems. The approach is based on the Poisson type integral formulae written for each disc of a domain consisting of a family of overlapping discs. The original differential boundary value problem is equivalently reformulated in the form of a system of integral equations defined on the intersection surfaces (arches, in 2D, and caps, if generalized to 3D spheres). To solve the obtained system of integral equations, a Random Walk procedure is constructed where the random walks are living on the intersecting surfaces. Since the spheres are fixed, it is convenient to construct also discrete random walk methods for solving the system of linear equations approximating the system of integral equations. We develop here two classes of special Monte Carlo iterative methods for solving these systems of linear algebraic equations which are constructed as a kind of randomized versions of the Chebyshev iteration method and Successive Over Relaxation (SOR) method. It is found that in this class of randomized SOR methods, the Gauss-Seidel method has a minimal variance. In our prevoius paper we have concluded that in the case of classical potential theory, the Random Walk on Fixed Spheres considerably improves the convergence rate of the standard Random Walk on Spheres method. More interesting, we succeeded there to extend the algorithm to the system of Lam'e equations which cannot be solved by the conventional Random Walk on Spheres method. We present here a series of numerical experiments for 2D domains consisting of 5, 10, and 17 discs, and analyze the dependence of the variance on the number of discs and elastic constants. Further generalizations to Neumann and Dirichlet-Neumann boundary conditions are also possible

Images and plane waves : efficient field computation in electromagnetics and acoustics

In some simple or canonical problems, analytical solutions offer the most efficient way to compute the electromagnetic or acoustic fields. For arbitrary geometries, efficient numerical methods are needed. This thesis contains new or improved solutions of classical electrostatic problems and some further developments of a variant of the fast multipole method (FMM). In the first part, the electrostatic problems of pairs of both orthogonally intersecting and non-intersecting conducting spheres are solved using Kelvin's image theory. A new efficient method for evaluating the polarizability of two non-intersecting spheres is presented. Novel analytical solutions, and also computationally efficient approximative solutions, are obtained by applying Kelvin's inversion to the electrostatic image solution of a conducting wedge. Integral equation methods are popular for both electrodynamic and acoustic scattering problems. However, to be able to use very large number of unknowns, fast iterative methods, such as the fast multipole method, must be used. In the second part of this thesis, a new broadband variant of the multilevel fast multipole algorithm (MLFMA) is described and used for both acoustic and electromagnetic scattering problems. In particular, the implementation overcomes the low-frequency breakdown of the MLFMA using a combination of the spectral representation of the Green's function and Rokhlin's translation formula.reviewe