C. A. Duarte

Generalized finite element methods for three-dimensional structural mechanics problems

Abstract The present paper summarizes the generalized finite element method formulation and demonstrates some of its advantages over traditional finite element methods to solve complex, three-dimensional (3D) structural mechanics problems. The structure of the stiffness matrix in the GFEM is compared to the corresponding FEM matrix. The performance of the GFEM and FEM in the solution of a 3D elasticity problem is also compared. The construction of p-orthotropic approximations on tetrahedral meshes and the use of a-priori knowledge about the solution of elasticity equations in three-dimensions are also presented.

A generalized finite element method for the simulation of three-dimensional dynamic crack propagation

This paper is aimed at presenting a partition of unity method for the simulation of three-dimensional dynamic crack propagation. The method is a variation of the partition of unity finite element method and hp-cloud method. In the context of crack simulation, this method allows for modeling of arbitrary dynamic crack propagation without any remeshing of the domain. In the proposed method, the approximation spaces are constructed using a partition of unity (PU) and local enrichment functions. The PU is provided by a combination of Shepard and finite element partitions of unity. This combination of PUs allows the inclusion of arbitrary crack geometry in a model without any modification of the initial discretization. It also avoids the problems associated with the integration of moving least squares or conventional Shepard partitions of unity used in several meshless methods. The local enrichment functions can be polynomials or customized functions. These functions can efficiently approximate the singular fields around crack fronts. The crack propagation is modeled by modifying the partition of unity along the crack surface and does not require continuous remeshings or mappings of solutions between consecutive meshes as the crack propagates. In contrast with the boundary element method, the proposed method can be applied to any class of problems solvable by the classical finite element method. In addition, the proposed method can be implemented into most finite element data bases. Several numerical examples demonstrating the main features and computational efficiency of the proposed method for dynamic crack propagation are presented.

hp-Meshless cloud method

A methodology to build discrete models of boundary-value problems (BVP) is presented. The method is applicable to arbitrary domains and employs only a scattered set of nodes to build approximate solutions to BVPs. A version of moving least-square interpolation and the collocation method are used to discretize BVP equations, which results in a truly meshless method (i.e. without a background mesh of integration points). h- and p-adaptive strategies are tested and very good convergence of the method was observed. The improvements in the methodology, in particular the introduction of spectral degrees of freedom, result in a fast and accurate method, significantly more efficient than the Finite Element Method or Element Free Galerkin Method. Several practical applications of the method to solve various engineering problems are presented.

Patterned Three-Dimensional Encapsulation of Embryonic Stem Cells using Dielectrophoresis and Stereolithography

Controlling the assembly of cells in three dimensions is very important for engineering functional tissues, drug screening, probing cell-cell/cell-matrix interactions, and studying the emergent behavior of cellular systems. Although the current methods of cell encapsulation in hydrogels can distribute them in three dimensions, these methods typically lack spatial control of multi-cellular organization and do not allow for the possibility of cell-cell contacts as seen for the native tissue. Here, we report the integration of dielectrophoresis (DEP) with stereolithography (SL) apparatus for the spatial patterning of cells on custom made gold micro-electrodes. Afterwards, they are encapsulated in poly (ethylene glycol) diacrylate (PEGDA) hydrogels of different stiffnesses. This technique can mimic the in vivo microscale tissue architecture, where the cells have a high degree of three dimensional (3D) spatial control. As a proof of concept, we show the patterning and encapsulation of mouse embryonic stem cells (mESCs) and C2C12 skeletal muscle myoblasts. mESCs show high viability in both the DEP (91.79% ± 1.4%) and the no DEP (94.27% ± 0.5%) hydrogel samples. Furthermore, we also show the patterning of mouse embryoid bodies (mEBs) and C2C12 spheroids in the hydrogels, and verify their viability. This robust and flexible in vitro platform can enable various applications in stem cell differentiation and tissue engineering by mimicking elements of the native 3D in vivo cellular micro-environment.

Analysis of Interacting Cracks Using the Generalized Finite Element Method With Global-Local Enrichment Functions

This paper presents an analysis of interacting cracks using a generalized finite element method (GFEM) enriched with so-called global-local functions. In this approach, solutions of local boundary value problems computed in a global-local analysis are used to enrich the global approximation space through the partition of unity framework used in the GFEM. This approach is related to the global-local procedure in the FEM, which is broadly used in industry to analyze fracture mechanics problems in complex three-dimensional geometries. In this paper, we compare the effectiveness of the global-local FEM with the GFEM with global-local enrichment functions. Numerical experiments demonstrate that the latter is much more robust than the former In particular, the GFEM is less sensitive to the quality of boundary conditions applied to local problems than the global-local FEM. Stress intensity factors computed with the conventional global-local approach showed errors of up to one order of magnitude larger than in the case of the GFEM. The numerical experiments also demonstrate that the GFEM can account for interactions among cracks with different scale sizes, even when not all cracks are modeled in the global domain.

The role of cohesive properties on intergranular crack propagation in brittle polycrystals

We analyze intergranular brittle cracking of polycrystalline aggregates by means of a generalized finite element method for polycrystals with cohesive grain boundaries and linear elastic grains. Many random realizations of a polycrystalline topology are considered and it is shown that the resulting crack paths are insensitive to key cohesive law parameters such as maximum cohesive strength and critical fracture energy. Normal and tangential contributions to the dissipated energy are thoroughly investigated with respect to mesh refinement, cohesive law parameters and randomness of the underlying polycrystalline microstructure.

Analysis of Reflective Cracks in Airfield Pavements using a 3-D Generalized Finite Element Method

ABSTRACT Prediction and simulation of load-related reflective cracking in air field pavements require three-dimensional models in order to accurately capture the effects of gear loads on crack initiation and propagation. In this paper, we demonstrate that the Generalized Finite Element Method (GFEM) enables the analysis of reflective cracking in a three-dimensional setting while requiring significantly less user intervention in model preparation than the standard FEM. As such, it provides support for the development of mechanistic-based design procedures for airfield overlays that are resistant to reflective cracking. Two gear loading positions of a Boeing 777 aircraft are considered in this study. The numerical simulations show that reflective cracks in airfield pavements are subjected to mixed mode behavior with all three modes present. They also demonstrate that under some loading conditions, the cracks exhibit significant channeling.

Extraction of Stress Intensity Factors for the Simulation of 3-D Crack Growth with the Generalized Finite Element Method

Two methods for the extraction of Stress Intensity Factors (SIFs) from three-dimensional (3-D)problems are presented: the Contour Integral Method and the Cutoff Function Method. The formula-tions are tailored for the Generalized Finite ElementMethod andmixed-mode 3-D propagating cracks.The case of crack faces loaded by prescribed tractions is also considered. Another contribution of thispaper is a procedure to control the noise of extracted SIFs based on theMoving Least SquaresMethod.The proposed approach provides a continuous and smooth approximation of 3-D SIF functions foreach fracture mode. Numerical experiments demonstrating the accuracy and robustness of the pro-posed methodology are presented. They include 3-D mixed-mode fatigue crack growth simulationsand the case of a pressurized crack.

Effects of Nonuniform and Three-Dimensional Contact Stresses on Near-Surface Cracking

Near-surface cracking, sometimes referred to as top-down cracking, is one of the predominant distress types in flexible pavements. The incidence of near-surface cracking has increased in recent years with the increased construction of relatively thick (hot-mix asphalt layer > 200 mm) flexible pavements. However, understanding the mechanisms of near-surface cracking and its integration into pavement design protocols remains a challenge. Analysis of this problem can be complex because of multiaxial stress states in the vicinity of tires. The near-surface response to nonuniform tire contact stresses is investigated, and the potential for crack occurrence near the surface is analyzed in a typical relatively thick flexible pavement. The generalized finite element method (GFEM) is used to analyze pavement structure. This method provides a computational framework for the arbitrary orientation of cracks in a finite element mesh that is particularly useful for mixed-mode fracture problems. A three-dimensional (3-D) model for a typical pavement structure with a thick bituminous layer is created, and 3-D and nonuniform tire–pavement contact stresses are applied to the pavement surface. Aggregate-scale cracks are inserted at various locations and orientations in the pavement. Results of this numerical study indicate that complex stress states in the presence of strong mode mixity may cause shear or tensile fracture in flexible pavements. The importance of novel computational methods such as the GFEM to the discovery and understanding of mechanisms governing the premature failure of pavements is highlighted.

A three-dimensional generalised finite element analysis for the near-surface cracking problem in flexible pavements

Near-surface cracking is one of the major distress types which results in reducing pavement service life. Heavy traffic loads, construction deficiencies, and surface mixture characteristics are among the predominant factors contributing to near-surface cracking. In addition, non-uniform tire-pavement contact stresses have a potential to generate extremely complex stress states near the surface. Prediction of crack initiation under these conditions requires high accuracy in the computation of state variables in pavement structure such as stresses, strains and displacements in the pavement. The generalised finite element method (GFEM) provides a computational framework in which arbitrary orientation of cracks in a finite element mesh is possible when using an enrichment strategy. The enrichment strategy in the GFEM can also increase the accuracy of the solution using higher-order polynomial approximations. A 3D analysis of near-surface cracking is performed using the GFEM. A 3D large-scale model of a long-lasting pavement is built, and cracks at various locations near the surface are introduced. Numerical experiments of a long-lasting pavement structure with defects at the aggregate scale illustrate the complex fracture conditions on and near the surface in the vicinity of a dual tire configuration.