Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D

We present an effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular Laplace boundary integral operators and layer potentials in two and three spatial dimensions. The method relies on the use of Green's third identity and local Taylor-like interpolations of density functions in terms of harmonic polynomials. The proposed technique effectively regularizes the singularities present in boundary integral operators and layer potentials, and recasts the latter in terms of integrands that are bounded or even more regular, depending on the order of the density interpolation. The resulting boundary integrals can then be easily, accurately, and inexpensively evaluated by means of standard quadrature rules. A variety of numerical examples demonstrate the effectiveness of the technique when used in conjunction with the classical trapezoidal rule (to integrate over smooth curves) in two-dimensions, and with a Chebyshev-type quadrature rule (to integrate over surfaces given as unions of non-overlapping quadrilateral patches) in three-dimensions

The boundary integral method for magnetic billiards

We introduce a boundary integral method for two-dimensional quantum billiards subjected to a constant magnetic field. It allows to calculate spectra and wave functions, in particular at strong fields and semiclassical values of the magnetic length. The method is presented for interior and exterior problems with general boundary conditions. We explain why the magnetic analogues of the field-free single and double layer equations exhibit an infinity of spurious solutions and how these can be eliminated at the expense of dealing with (hyper-)singular operators. The high efficiency of the method is demonstrated by numerical calculations in the extreme semiclassical regime.Comment: 28 pages, 12 figure

A fast and well-conditioned spectral method for singular integral equations

We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in ${\cal O}(m^2n)$ operations using an adaptive QR factorization, where $m$ is the bandwidth and $n$ is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to ${\cal O}(m n)$ operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far- and near-field solution. The Julia software package SingularIntegralEquations.jl implements our method with a convenient, user-friendly interface

Minimal energy problems for strongly singular Riesz kernels

We study minimal energy problems for strongly singular Riesz kernels on a manifold. Based on the spatial energy of harmonic double layer potentials, we are motivated to formulate the natural regularization of such problems by switching to Hadamard's partie finie integral operator which defines a strongly elliptic pseudodifferential operator on the manifold. The measures with finite energy are shown to be elements from the corresponding Sobolev space, and the associated minimal energy problem admits a unique solution. We relate our continuous approach also to the discrete one, which has been worked out earlier by D.P. Hardin and E.B. Saff.Comment: 31 pages, 2 figure

Extended Boundary Element Method approach for Direct and Accurate Evaluation of Stress Intensity Factors

This thesis introduces an alternative method to evaluate Stress Intensity Factors (SIFs) in computational fracture mechanics directly, using the Extended Dual Boundary Element Method (XBEM) for 2D problems. A novel auxiliary equation introduced which enforces displacement continuity at the crack tip to yield a square system. Additionally, the enrichment method has been extended to 3D, so that the J-integral with XBEM and a direct technique are used to evaluate SIFs. This includes a complete description of the formulation of enrichment functions, a substitution of the enriched form of displacement into boundary integral equations, treatment of singular integrals, assembly of system matrices and the introduction of auxiliary equations to solve the system directly. The enrichment approach utilizes the Williams expansions to enrich crack surface elements for accurate evaluation of stress intensity factors. Similar to other enrichment methods, the new approach can yield accurate results on coarse discretisations, and the enrichment increases the 2D problem size by only two degrees of freedom per crack tip. In the case of 3D, the number of the new degrees of freedom depends on the desired number of crack front points where SIFs need to be evaluated. The auxiliary equations required to yield a square system are derived by enforcing continuity of displacement at the crack front. The enrichment approach provides the values of singular coefficients KI, KII and KIII directly in the solution vector; without any need for postprocessing such as the J-integral. Numerical examples are used to compare the accuracy of these directly computed SIFs to the J-integral processing of both conventional and XBEM approximations