A coordinate transformation approach for efficient repeated solution of Helmholtz equation pertaining to obstacle scattering by shape deformations

Monte Carlo simulations of Helmholtz scattering from randomly positioned array of scatterers by utilizing coordinate transformations in finite element method

“…This section outlines the theoretical background for determining the material parameters of a transformation medium by using the principle of form invariance of Maxwell's equations under a general coordinate transformation [40][41][42][43][44][45][46][47][48]. An arbitrary coordinate transformation, r →r = T(r), transforms each point P in the original space Ω to another pointP in the transformed spaceΩ, where r = (x, y, z) andr = (x,ỹ,z) are the position vectors of the points P andP in the original and transformed coordinate systems, respectively.…”

“…We applied this idea to multiscale problems and called such transformation materials ''software metamaterials'' in [40]. In addition, we used this technique for modeling curved geometries that do not conform to a Cartesian grid especially in finite difference methods [41] and for modeling stochastic electromagnetic problems having significant uncertainty, such as rough surface scattering problems [42][43][44][45][46]. In the current paper, our aim is to employ the transformation invariance of Maxwell's equations for the solution of scattering from randomly positioned obstacles to realize multiple realizations in Monte Carlo simulations (corresponding to different positions of the obstacles) in a fast and efficient manner by using a single and simpler mesh and by locating transformation medium within the computational domain.…”

Full-wave algorithm to model effects of bedding slopes on the response of subsurface electromagnetic geophysical sensors near unconformities

“…We remark that exploitation of T.O. techniques to facilitate numerical field computation based on brute-force algorithms-rather than the eigenfunction-based spectral approach considered here-was recently proposed in [59,60]. We moreover emphasize that the 2-D Fourier integral is capable of modeling propagation and scattering behavior of these interface-flattening media, which possess azimuthal non-symmetric material tensors; the two stated 1-D integral transforms, which are inherently restricted to modeling azimuthal-symmetric media, by contrast lack this numerical modeling capability.…”

Scattering by stochastic boundaries: hybrid low- and high-order quantification algorithms

“…We are not aware of any previous numerical results that combine such large scale C316 stochastic sampling with efficient high-order stochastic boundary scattering simulations for this important class of model particles. Previous approaches involve low-order scattering simulations (based on perturbation methods, the method of moments, and the finite element method) applied in combination with mc, qmc, and high order integration methods (including gpc) for scatterers whose randomness consists of random deformations of a fixed shape [2,11,14,17].…”