Quadrature by Expansion: A New Method for the Evaluation of Layer Potentials

Integral equation methods for the solution of partial differential equations, when coupled with suitable fast algorithms, yield geometrically flexible, asymptotically optimal and well-conditioned schemes in either interior or exterior domains. The practical application of these methods, however, requires the accurate evaluation of boundary integrals with singular, weakly singular or nearly singular kernels. Historically, these issues have been handled either by low-order product integration rules (computed semi-analytically), by singularity subtraction/cancellation, by kernel regularization and asymptotic analysis, or by the construction of special purpose "generalized Gaussian quadrature" rules. In this paper, we present a systematic, high-order approach that works for any singularity (including hypersingular kernels), based only on the assumption that the field induced by the integral operator is locally smooth when restricted to either the interior or the exterior. Discontinuities in the field across the boundary are permitted. The scheme, denoted QBX (quadrature by expansion), is easy to implement and compatible with fast hierarchical algorithms such as the fast multipole method. We include accuracy tests for a variety of integral operators in two dimensions on smooth and corner domains

Precipitation versus partitioning kinetics during the quenching of low-carbon martensitic steels

Abstract Low-carbon, low-alloy steels undergo auto-tempering and carbon partitioning to austenite during quenching to martensite. The microstructures of two such steels quenched at two cooling rates have been evaluated using electron microscopy to characterise lath and carbide precipitate morphologies, and the results have been compared with theoretical predictions based on the Thermo-Calc modules DICTRA and TC-Prisma. The modelling tools predicted the carbon depletion rates due to diffusion from the bcc martensite laths into austenite and the precipitation of cementite in the ferrite matrix. The predictions showed a satisfactory agreement with the metallographic results, indicating that the Thermo-Calc based software can aid in the design of new low-carbon martensitic steels