arXiv subjects

Michael J. Carley

Publications and source records attributed to Michael J. Carley.

4 recordsLinked to original sources

arXiv2026

Translation of transient acoustic fields

A method is presented for the translation of acoustic field data from a source to a target region. Field data are represented as spherical harmonic expansions on spheres surrounding the source and target regions respectively and expansions are translated using a ``point and shoot'' method using the Kirchhoff--Helmholtz integral to carry out an axial translation from one sphere to the other. The principal motivation for the method is its use in a time-domain Fast Multipole Method, and test cases reflective of this application are presented. The method converges to six digits for appropriate values of parameters and for the values of $N$ considered here computational effort scales approximately as $N^{2}$ where $N$ is the order of spherical harmonic expansion for the field data. The method is causal and thus avoids artifacts generated in methods which are not based on intrinsically causal formulations.

physics.comp-ph↗

arXiv2025

Numerical evaluation of the Kirchhoff-Helmholtz integral outside a sphere

A method is presented for the fast evaluation of the transient acoustic field generated outside a spherical surface using surface data on the sphere. The method employs Lebedev quadratures, which are optimal integration on the sphere, and Lagrange interpolation and differentiation in an advanced time algorithm for the evaluation of the transient field. Numerical testing demonstrates that the approach gives near machine-precision accuracy and a speed-up in evaluation time which depends on the order of quadrature rule employed but breaks even with direct evaluation at a number of field points about 1.15 times the number of surface quadrature nodes, making the method an efficient means of evaluating the field generated by a large number of sources.

math.NA↗

arXiv2023

A Fast Multipole Method for axisymmetric domains

The Fast Multipole Method (FMM) for the Poisson equation is extended to the case of non-axisymmetric problems in an axisymmetric domain, described by cylindrical coordinates. The method is based on a Fourier decomposition of the source into a modal expansion and the evaluation of the corresponding modes of the field using a two-dimensional tree decomposition in the radial and axial coordinate. The field coefficients are evaluated using a modal Green's function which can be evaluated using well-known recursions for the Legendre function of the second kind, and whose derivatives can be found recursively using the Laplace equation in cylindrical coordinates. The principal difference between the cylindrical and Cartesian problems is the lack of translation invariance in the evaluation of local interactions, leading to an increase in computational effort for the axisymmetric domain. Results are presented for solution accuracy and convergence and for computation time compared to direct evaluation. The method is found to converge well, with ten digit accuracy being achieved for the test cases presented. Computation time is controlled by the balance between initialization and the evaluation of local interactions between source and field points, and is about two orders of magnitude less than that required for direct evaluation, depending on expansion order.

math.NA↗

arXiv2022

Spherical polar coordinate transformation for integration of singular functions on tetrahedra

A method is presented for the evaluation of integrals on tetrahedra where the integrand has an integrable singularity at one vertex. The approach uses a transformation to spherical polar coordinates which explicitly eliminates the singularity and facilitates the evaluation of integration limits. The method is also implemented in an adaptive form which gives convergence to a required tolerance. Results from the method are compared to the output from an exact analytical method for one tetrahedron and show high accuracy. In particular, when the adaptive algorithm is used, highly accurate results are found for poorly conditioned tetrahedra which normally present difficulties for numerical quadrature techniques. The approach is also demonstrated for evaluation of the Biot-Savart integral on an unstructured mesh in combination with a fixed node quadrature rule and demonstrates good convergence and accuracy.

math.NA↗