Strong and weak form meshless methods in computational biomechanics

Meshless methods (MMs) were introduced in the late 1970s to solve problems in astrophysics. In MMs the spatial domain is represented by a set of nodes (cloud of points) and not discretized by elements as in most of the mesh-based methods (finite difference method, finite element method, finite volume method); consequently, there is no need for predefined connectivity between the nodes. In this chapter we are going to give an overview of applications, advantages, and disadvantages of various MMs developed and applied in the context of computational biomechanics. Strong and weak formulations will be presented, focusing on the novel interpolation schemes such as modified moving least squares and discretization correction particle strength exchange method, along with the meshless total Lagrangian explicit dynamics method. The applicability of the methods in multiscale problems and their inherent parallelization will be depicted through various applications, along with their advantages over the traditional mesh-based numerical methods. MMs can be considered as mainstream numerical methods able to tackle demanding engineering applications. Intensive and rigorous research in the field will make MMs robust enough to be used by industry.