Moving least squares method and its improvement: a concise review

Abstract

The concise review systematically summarises the state-of-the-art variants of Moving Least Squares (MLS) method. MLS method is a mathematical tool which could render cogent support in data interpolation, shape construction and formulation of meshfree schemes, particularly due to its flexibility to form complex arithmetic equation. However, the conventional MLS method is suffering to deal with discontinuity of field variables. Varied strategies of overcoming such shortfall are discussed in current work. Although numerous MLS variants were proposed since the introduction of MLS method in numerical/statistical analysis, there is no technical review made on how the methods evolve. The current review is structured according to major strategies on how to improvise MLS method: the modification of weight function, the manipulation of discrete norms, the inclusion of iterative feature for residuals minimising and integration of these strategies for more robust computation. A wide range of advanced MLS variants have been compiled, summarised, and reappraised according to its underlying principle of improvement. In addition, inherent limitation of MLS method and its possible strategy of improvement is discussed too in this article. The current work could render valuable reference to implement and develop advanced MLS schemes, whenever complexity of the specific scientific problems arose.