meshfree methods for partial differential equations

In this paper, we discuss the assessment of the margins of safety (MOS) for engineered structures (buildings, bridges, machines and aircraft). Moreover, the problem of severely ill-conditioned matrices is completely avoided. No weakly or strongly singular integrals have to be evaluated. The resulting multi-level method is robust and significantly reduces the computational cost. The approximate solutions are calculated by enforcing the boundary conditions in the sense of least squares. Based on this classification, a multi-level technique is built up, where the smoothing procedure is defined by performing some familiar iterative technique e.g. The ‘far’ sources are interpreted to form a ‘coarse grid’, while the densely distributed ‘near-boundary’ sources are considered a ‘fine grid’ (despite they need not to have any grid structure). The sets of sources are defined by the quadtree/octtree subdivision technique controlled by the boundary collocation points in a completely automatic way, resulting in a point set, the density of the spatial distribution of which decreases quickly far from the boundary. Instead of using a single set of subtly located external sources, a special strategy of defining several sets of external source points is introduced. The classical form of the Method of Fundamental Solutions is applied. It continues the 2-year-cycled Workshops on Meshfree Methods for Partial Differential Equations.

This symposium aims to promote collaboration among engineers, mathematicians, and computer scientists and industrial researchers to address the development, mathematical analysis, and application of meshfree and particle methods especially to multiscale phenomena. While the fundamental theory of meshfree methods has been developed and considerable advances of the various methods have been made, many challenges in the mathematical analysis and practical implementation of meshfree methods remain. The numerical treatment of partial differential equations with meshfree discretization techniques has been a very active research area in recent years. They address various aspects of this very active research field and cover topics from applied mathematics, physics and engineering. This volume collects selected papers presented at the Ninth International Workshop on Meshfree Methods held in Bonn, Germany in September 2017.

Meshfree methods for partial differential equations