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In this work, arbitrarily smooth, globally compatible, Im/Cn/Pk interpolation hierarchies are constructed in the framework of reproducing kernel element method (RKEM) for multi-dimensional domains. This is the first interpolation hierarchical structure that has been ever constructed with both minimal degrees of freedom and higher order continuity and reproducing conditions over multi-dimensional domains. The proposed hierarchical structure possesses the generalized Kronecker property, i.e., ∂αΨI/(β)/∂xα(xJ) = δIJδαβ, α, β ≤ m. The newly constructed globally conforming interpolant is a hybrid of global partition polynomials (C∞) and a smooth (Cn) compactly supported meshfree partition of unity. Examples of compatible RKEM hierarchical interpolations are illustrated, and they are used in a Galerkin procedure to solve differential equations. [ABSTRACT FROM AUTHOR]

For the past few decades, tremendous research efforts have been directed to the development of modeling and simulation approaches for fluid-structure interaction problems. Methods developed by Tezduyar and his coworkers are widely used in the simulations of fluid-particle (rigid) and fluid-structure interactions.428, 430 To accommodate the complicated motions of fluid-structure boundaries, we often use adaptive meshing or the arbitrary Lagrangian Eulerian (ALE) techniques.207, 216, 290-292 Recently, such approaches have also been adopted by Hu, et al. in the modeling of fluid-particle (rigid) systems.206 Nevertheless, mesh update or remeshing algorithms can be time consuming and expensive and a detailed discussion on this issue is presented in Ref.222 [ABSTRACT FROM AUTHOR]

To construct natural neighbor interpolation, the first step is to establish the natural neighbor coordinate. The concepts of the nearest neighbors and neighboring nodes are contained in the first-order Voronoi diagram. However, in the first-order Voronoi diagram, the nearest neighbors and neighboring nodes are only identified for nodal points, xI, I ∈ Λ, not for arbitrary point x ∈ IRd. To define the nearest neigbbor and neighboring nodes for an arbitrary point in a domain of interests, we need the second-order Voronoi diagram 1 [ABSTRACT FROM AUTHOR]

Due to the advance of nano-science and nanotechnology, computational nano-mechanics has emerged as a major research area in the field of computational mechanics. The main research activities include: developing algorithms for contemporary molecular dynamics, concurrent multiscale simulations, and bridging scale methods. In this chapter, some basic concepts and algorithmics of molecular dynamics are introduced first, and then the coupling between molecular dynamics and finite element methods or meshfree methods are discussed. [ABSTRACT FROM AUTHOR]

Interests in constructing versatile finite element interpolants, meshfree interpolants, or general partition of unity shape functions is the current trend in improving the-state-of-the-art finite element technology (see Belytschko, Liu and Moran49) and meshfree technology (see Li and Liu274 and Babuška, Banerjee, and Osborn25). In this chapter, we introduce and analyze a new method called the reproducing kernel element method (RKEM), which is constructed by combining the virtues of finite element approximations and reproducing kernel particle approximations (RKPM,96, 297, 298, 303). [ABSTRACT FROM AUTHOR]

In this Chapter, several aspects of applications of meshfree Galerkin methods are discussed. The materials presented here are mainly selected from the authors' research. On the other hand, a comprehesive survey for the important applications of meshfree methods is also presented. [ABSTRACT FROM AUTHOR]

The approximation theory of meshfree methods consists of several key elements. First how to distribute particles to represent a continua and to build a valid meshfree discretization; second how to measure the quality of meshfree interpolation, third, how to measure the convergence of a meshfree Galerkin weak form. Overall, we are dealing with the theoretical foundation or mathematical structure of meshfree interpolation. [ABSTRACT FROM AUTHOR]

In 1981, Lancaster and Salkauskas255 published their seminal work in construction of smooth interpolation functions on a set of scattered or disordered data, which is called the Moving Least Square Interpolant (MLS). Until 1992, the methodology had remained a curve/surface fitting algorithm. [ABSTRACT FROM AUTHOR]

In 1977, Lucy284 and Gingold and Monaghan176 simultaneously formulated the so-called Smoothed Particle Hydrodynamics, or SPH. Early contributions have been reviewed in several articles (e.g.55, 331, 338). [ABSTRACT FROM AUTHOR]

Almost half a century ago, Turner, Clough, Martin, and Topp (Turner et al.433) published their seminal work in finite element (FE) approximation in structural mechanics. Today, finite element method (FEM) based computational mechanics, as a far-reaching field ranging from basic science over applied research to applications, plays a prominent role in technology advancement. Model-based simulations are changing the face of scientific and engineering inquiry. [ABSTRACT FROM AUTHOR]