Fatigue failures occur in aerospace, marine, nuclear structures and automobile com ponents from initiation and propagation of cracks from holes, scratches or defects in the material. To design against these failures, crack propagation life and fracture strength need to be accurately predicted. It is reported in the literature, that these failures often initiate as surface cracks, corner cracks and cracks emanating from fastner holes. Such cracks are with elliptic or nearly elliptic in shapes. The deviation from elliptic shape is due to varying constraint effect along the crack front. Even in situations, when the cracks are through the thickness of the material, there would be thicknesswise variation of constraint effects leading to three dimensional nature of crack growth. Accurate predictions of the crack growth in these cases by numerical methods can be made only by solving three-dimensional boundary value problems. Empirical relationships have been developed 1] based on Linear Elastic Fracture Mechanics over years describing fatigue crack growth response. Some of these empirical relationships required modifications in the later stages, to meet the design applications. The Crack closure phenomenon discovered by Elber 2, 3] during the crack growth phase is mainly attributed to the local material yielding near the crack tip and the consequent residual plastic wake behind the crack tip. It helped considerably in understanding several aspects of fatigue crack growth and rewrite these relations.

The finite element and meshless methods are numerical simulation algorithms used to mathematically model physical phenomena. The finite element method relies on the creation of an underlying simplex structure in the region that is being modeled. By specifying these finite elements, a rough model can be developed that can be used to generate approximate solutions while requiring minimal processor time. The meshless method eliminates these finite elements and instead generates a dynamic model from a particle perspective using a complex matrix of integral equations to define the properties of the region being modeled. Without the limitations of the simplex structure, the model can account for large deformations, advanced materials, complex geometries, nonlinear behavior, and discontinuities. However, this utility results in a significant increase in the computational time that is require to generate a solution.
Application of these techniques extends to all areas of computational physics, engineering, and mechanics.

Although several textbooks exist on the topic of the finite element method and there are some that focus specifically on the meshless method, this is the first textbook to describe both modeling techniques in detail and evaluate their relative advantages.