The application of the boundary element method in BEM++ to small extreme Chebyshev ice particles and the remote detection of the ice crystal number concentration of small atmospheric ice particles

Calderón preconditioning of PMCHWT boundary integral equations for scattering by multiple absorbing dielectric particles

“…The use of boundary integral equations and their discretization using boundary element methods (BEMs) is well-established in the electrical engineering community but has only recently started getting serious attention in the atmospheric physics community [34,35,36]. BEM is a "numerically exact" method which can provide highly accurate simulations of scattering by complex ice crystal shapes.…”

“…Early applications of BEM to the simulation of light scattering by simple ice crystals include [37] and [38]. More comprehensive studies of complex crystal shapes (including hexagonal columns with conventional and stepped cavities, bullet rosettes and Chebyshev ice particles) have been given recently by Groth et al [34] and Baran and Groth [35], using the open-source BEM software library Bempp [39], available at www.bempp.com. In [34,35] it was shown (by comparison with a T-matrix method) that, using a discretization with at least 10 boundary elements per wavelength, Bempp can compute far-field quantities with 1% relative error, with reciprocity also satisfied to a similar accuracy.…”

“…More comprehensive studies of complex crystal shapes (including hexagonal columns with conventional and stepped cavities, bullet rosettes and Chebyshev ice particles) have been given recently by Groth et al [34] and Baran and Groth [35], using the open-source BEM software library Bempp [39], available at www.bempp.com. In [34,35] it was shown (by comparison with a T-matrix method) that, using a discretization with at least 10 boundary elements per wavelength, Bempp can compute far-field quantities with 1% relative error, with reciprocity also satisfied to a similar accuracy. The results in [34,35] were obtained using a standard desktop machine and were limited to size parameters up to 15, but Bempp supports parallelization and hence with HPC architectures much larger problems can be tackled.…”

“…In [34,35] it was shown (by comparison with a T-matrix method) that, using a discretization with at least 10 boundary elements per wavelength, Bempp can compute far-field quantities with 1% relative error, with reciprocity also satisfied to a similar accuracy. The results in [34,35] were obtained using a standard desktop machine and were limited to size parameters up to 15, but Bempp supports parallelization and hence with HPC architectures much larger problems can be tackled. We end this brief literature review of BEM in atmospheric physics applications by mentioning recent work by Yu et al [36], in which boundary integral equations were applied to scattering by multiple dielectric particles illuminated by unpolarized high-order Bessel vortex beams, and also by Groth et al [40], where a "hybrid numerical-asymptotic" BEM was presented for 2D high frequency scalar dielectric scattering problems which achieves fixed accuracy with frequency-independent computational cost.…”

“…One therefore requires a preconditioning strategy. In [34,35], algebraic preconditioning was applied, based on H-matrices. But this incurs a large memory overhead.…”