A locally-integrated meshless (LIM) method applied to advection-diffusion problems

Abstract

Paper presented to the 10th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Florida, 14-16 July 2014.

A new numerical solution scheme coined as the LIM (Locally-Integrated Meshless) method is formulated in this paper. As a Meshless formulation the LIM Method relies uniquely on a scattered non-ordered data point distribution within the domain of interest and does not require connectivity or polygonalization. In the LIM method formulation the field variable is approximated within localized overlapping regions containing a predetermined number of data points, as a linear combination of predefined expansion functions. These expansion functions are chosen in this study to be the wellknown Hardy Multiquadrics Radial-basis functions (RBF) with center at the data points within each localized region. A weighted-residual integration is applied on each region to minimize the difference between the approximated field and the exact one. In order to circumvent the integration of an unavailable exact field, the residual integral is decomposed into a collocation integral (weights are set to Dirac delta functions) and a fully-weighted integral, resulting in a formulation where the coefficients of the expansion can be expressed directly in terms of the values of the field variable at the data points. This approach allows mitigating the issues that arise when performing direct collocation where the resulting derivative fields are sensitive and inaccurate due to the ‘anchoring’ of the expansion to the data points. In contrast, the LIM yields derivative fields that are smooth and accurate as they are rendered from a ‘non-anchored’ expansion while providing the ability to express the expansion coefficients, and thus the derivative fields, directly in terms of the scattered data within each local region. This combination of features is critical for the implementation of the LIM method as a robust, stable, and accurate solution scheme of governing differential equations.