SUMMARYIn this paper, conforming finite elements on polygon meshes are developed. Polygonal finite elements provide greater flexibility in mesh generation and are better-suited for applications in solid mechanics which involve a significant change in the topology of the material domain. In this study, recent advances in meshfree approximations, computational geometry, and computer graphics are used to construct different trial and test approximations on polygonal elements. A particular and notable contribution is the use of meshfree (natural-neighbour, nn) basis functions on a canonical element combined with an affine map to construct conforming approximations on convex polygons. This numerical formulation enables the construction of conforming approximation on n-gons (n 3), and hence extends the potential applications of finite elements to convex polygons of arbitrary order. Numerical experiments on second-order elliptic boundary-value problems are presented to demonstrate the accuracy and convergence of the proposed method.

In this paper, we present mesh-independent modeling of discontinuous fields on polygonal and quadtree finite element meshes. This approach falls within the class of extended and generalized finite element methods, where the partition of unity framework is used to introduce additional (enrichment) functions within the classical displacement-based finite element approximation. For crack modeling, a discontinuous function and the two-dimensional asymptotic crack-tip fields are used as enrichment functions. Linearly complete partition of unity approximations are adopted on polygonal (convex and nonconvex elements) and quadtree meshes. Excellent agreement with reference solution results is obtained for mixed-mode stress intensity factors on benchmark crack problems, and crack growth simulations without remeshing are conducted on polygonal and quadtree meshes to reveal the potential of the proposed techniques in computational failure mechanics.

In this paper, the quadtree data structure and conforming polygonal interpolants are used to develop an h-adaptive 7 finite element method. Quadtree is a hierarchical data structure that is computationally attractive for adaptive numerical simulations. Mesh generation and adaptive refinement of quadtree meshes is straight-forward. However, 9 finite elements are non-conforming on quadtree meshes due to level-mismatches between adjacent elements, which results in the presence of so-called hanging nodes. In this study, we use meshfree (natural-neighbor, nn) basis 11 functions on a reference element combined with an affine map to construct conforming approximations on quadtree meshes. Numerical examples are presented to demonstrate the accuracy and performance of the proposed h-adaptive 13 finite element method. ᭧

A new technique is presented to study fracture in nanomaterials by coupling quantum mechanics (QM) and continuum mechanics (CM). A key new feature of this method is that broken bonds are identified by a sharp decrease in electron density at the bond midpoint in the QM model. As fracture occurs, the crack tip position and crack path are updated from the broken bonds in the QM model. At each step in the simulation, the QM model is centered on the crack tip to adaptively follow the path. This adaptivity makes it possible to trace paths with complicated geometries. The method is applied to study the propagation of cracks in graphene which are initially perpendicular to zigzag and armchair edges. The simulations demonstrate that the growth of zigzag cracks is self-similar whereas armchair cracks advance in an irregular manner. The critical stress intensity factors for graphene were found to be 4.21 MPa √ m for zigzag cracks and 3.71 MPa √ m for armchair cracks, which is about 10% of that for steel.

In this paper, a conforming polygonal finite element method is applied to problems in linear elasticity. Meshfree natural neighbor (Laplace) shape functions are used to construct conforming interpolating functions on any convex polygon. This provides greater flexibility to solve partial differential equations on complicated geometries. Closed-form expressions for Laplace shape functions on pentagonal, hexagonal, heptagonal, and octagonal reference elements are derived. Numerical examples are presented to demonstrate the accuracy of the method in two-dimensional elastostatics.

Molecular dynamics (MD) modeling is used to study the fracture toughness and crack propagation path of monolayer molybdenum disulfide (MoS(2)) sheets under mixed modes I and II loading. Sheets with both initial armchair and zigzag cracks are studied. The MD simulations predict that crack edge chirality, tip configuration and the loading phase angle influence the fracture toughness and crack propagation path of monolayer MoS(2) sheets. Furthermore, under all loading conditions, both armchair and zigzag cracks prefer to extend along a zigzag path, which is in agreement with the crack propagation path in graphene. A remarkable out-of-plane buckling can occur during mixed mode loading which can lead to the development of buckling cracks in addition to the propagation of the initial cracks.

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