A LOCALLY-INTEGRATED MESHLESS (LIM) METHOD APPLIED TO ADVECTION- DIFFUSION PROBLEMS

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. INTRODUCTION In recent history, the area of physics-based engineering simulation has seen rapid increases in both computer workstation performance as well as common engineering model complexity, both driven largely in part by advances in memory density and availability of clusters and multi-core processors. While the increase in computation time due to model complexity has been largely offset by the increased performance of modern workstations, the increase in model setup time due to model complexity has continued to rise. As such, the major time requirement for solving an engineering model has transitioned from computation time to problem setup time. This is due to the fact that developing the required mesh for complex geometries can be an extremely complicated and time consuming task, and consequently, new solution techniques that can reduce the required amount of human interaction are desirable. While the finite element method (FEM) and the boundary element method (BEM) have been developed to a mature stage such that they are now utilized routinely to model complex multi-physics problems, they require significant effort in formulation, mesh generation, and data management. NOMENCLATURE ( ), ( ) j j x x   [-] Multiquadrics Radial-basis functions (RBF) ( ) j r x [m] Euclidean distance between points x and xj c [m] RBF shape parameter , j j   [-] RBF expansion coefficients  [m ] Domain of interest R  [-] Domain residual ( ) w x [-] Weight function   L [-] Derivative interpolation vector ( ) T x [K] Temperature field ( ) V x [m/s] Velocity Field  [kg/m] Density p c [J/kgK] Specific heat capacity k [W/mK] Thermal conductivity