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A four-node quadrilateral shell element with smoothed membrane-bending based on Mindlin-Reissner theory is proposed. The element is a combination of a plate bending and membrane element. It is based on mixed interpolation where the bending and membrane stiffness matrices are calculated on the boundaries of the smoothing cells while the shear terms are approximated by independent interpolation functions in natural coordinates. The proposed element is robust, computationally inexpensive and free of locking. Since the integration is done on the element boundaries for the bending and membrane terms, the element is more accurate than the MITC4 element for distorted meshes. This will be demonstrated for several numerical examples.
We present an extended finite element formulation for dynamic fracture of piezo-electric materials. The method is developed in the context of linear elastic fracture mechanics. It is applied to mode I and mixed mode-fracture for quasi-steady cracks. An implicit time integration scheme is exploited. The results are compared to results obtained with the boundary element method and show excellent agreement.
This work describes an algorithm and corresponding software for incorporating general nonlinear multiple-point equality constraints in a implicit sparse direct solver. It is shown that direct addressing of sparse matrices is possible in general circumstances, circumventing the traditional linear or binary search for introducing (generalized) constituents to a sparse matrix. Nested and arbitrarily interconnected multiple-point constraints are introduced by processing of multiplicative constituents with a built-in topological ordering of the resulting directed graph. A classification of discretization methods is performed and some re-classified problems are described and solved under this proposed perspective. The dependence relations between solution methods, algorithms and constituents becomes apparent. Fracture algorithms can be naturally casted in this framework. Solutions based on control equations are also directly incorporated as equality constraints. We show that arbitrary constituents can be used as long as the resulting directed graph is acyclic. It is also shown that graph partitions and orderings should be performed in the innermost part of the algorithm, a fact with some peculiar consequences. The core of our implicit code is described, specifically new algorithms for direct access of sparse matrices (by means of the clique structure) and general constituent processing. It is demonstrated that the graph structure of the second derivatives of the equality constraints are cliques (or pseudo-elements) and are naturally included as such. A complete algorithm is presented which allows a complete automation of equality constraints, avoiding the need of pre-sorting. Verification applications in four distinct areas are shown: single and multiple rigid body dynamics, solution control and computational fracture.
The lattice dynamics properties are investigated for twisting bilayer graphene. There are big jumps for the inter-layer potential at twisting angle θ=0° and 60°, implying the stability of Bernal-stacking and the instability of AA-stacking structures, while a long platform in [8,55]° indicates the ease of twisting bilayer graphene in this wide angle range. Significant frequency shifts are observed for the z breathing mode around θ=0° and 60°, while the frequency is a constant in a wide range [8,55]°. Using the z breathing mode, a mechanical nanoresonator is proposed to operate on a robust resonant frequency in terahertz range.