10:15
In many fields of science and engineering, such as fluid or structural mechanics and electric circuit design, large scale dynamical systems need to be simulated, optimized or controlled. They are often described by discretizations of systems of nonlinear partial differential equations yielding high-dimensional discrete phase spaces. For this reason, in recent decades, research has mainly focused on the development of sophisticated analytical and numerical tools to help understand the overall system behavior. During this time meshless methods have enjoyed significant interest in the research community and in some commercial simulators (e.g., LS-DYNA). In this talk I will describe a normalized-corrected meshless method which ensures linear completeness and improved accuracy. The resulting scheme not only provides first order consistency O(h) but also alleviates the particle deficiency (kernel support incompleteness) problem at the boundary. Furthermore, a number of improvements to the kernel derivative approximation are proposed.
To illustrate the performance of the meshless method, we present results for different problems from various fields of science and engineering (i.e. nano-tubes modelling, solid mechanics, damage mechanics, fluid mechanics, coupled interactions of solids and fluids). Special attention is paid to fluid flow in porous media. The primary attraction of the present approach is that it provides a weak formulation for Darcy's law which can be used in further development of meshless methods.