TY - JOUR A1 - Areias, Pedro A1 - Rabczuk, Timon A1 - Dias-da-Costa, D. A1 - Piresh, E.B. T1 - Implicit solutions with consistent additive and multiplicative components JF - Finite Elements in Analysis and Design N2 - 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. KW - Angewandte Mathematik KW - Strukturmechanik Y1 - 2012 U6 - http://dx.doi.org/10.1016/j.finel.2012.03.007 SP - 15 EP - 31 ER - TY - JOUR A1 - Chau-Dinh, T. A1 - Zi, Goangseup A1 - Lee, P.S. A1 - Song, Jeong-Hoon A1 - Rabczuk, Timon T1 - Phantom-node method for shell models with arbitrary cracks JF - Computers & Structures N2 - A phantom-node method is developed for three-node shell elements to describe cracks. This method can treat arbitrary cracks independently of the mesh. The crack may cut elements completely or partially. Elements are overlapped on the position of the crack, and they are partially integrated to implement the discontinuous displacement across the crack. To consider the element containing a crack tip, a new kinematical relation between the overlapped elements is developed. There is no enrichment function for the discontinuous displacement field. Several numerical examples are presented to illustrate the proposed method. KW - Angewandte Mathematik KW - Strukturmechanik Y1 - 2012 U6 - http://dx.doi.org/10.1016/j.compstruc.2011.10.021 ER - TY - JOUR A1 - Chen, Lei A1 - Rabczuk, Timon A1 - Liu, G.R. A1 - Zeng, K.Y. A1 - Kerfriden, Pierre A1 - Bordas, Stéphane Pierre Alain T1 - Extended finite element method with edge-based strain smoothing (ESm-XFEM) for linear elastic crack growth JF - Computer Methods in Applied Mechanics and Engineering N2 - This paper presents a strain smoothing procedure for the extended finite element method (XFEM). The resulting “edge-based” smoothed extended finite element method (ESm-XFEM) is tailored to linear elastic fracture mechanics and, in this context, to outperform the standard XFEM. In the XFEM, the displacement-based approximation is enriched by the Heaviside and asymptotic crack tip functions using the framework of partition of unity. This eliminates the need for the mesh alignment with the crack and re-meshing, as the crack evolves. Edge-based smoothing (ES) relies on a generalized smoothing operation over smoothing domains associated with edges of simplex meshes, and produces a softening effect leading to a close-to-exact stiffness, “super-convergence” and “ultra-accurate” solutions. The present method takes advantage of both the ES-FEM and the XFEM. Thanks to the use of strain smoothing, the subdivision of elements intersected by discontinuities and of integrating the (singular) derivatives of the approximation functions is suppressed via transforming interior integration into boundary integration. Numerical examples show that the proposed method improves significantly the accuracy of stress intensity factors and achieves a near optimal convergence rate in the energy norm even without geometrical enrichment or blending correction. KW - Angewandte Mathematik KW - Strukturmechanik Y1 - 2012 U6 - http://dx.doi.org/10.1016/j.cma.2011.08.013 ER - TY - JOUR A1 - Jiang, Jin-Wu A1 - Park, Harold S. A1 - Rabczuk, Timon T1 - Enhancing the mass sensitivity of graphene nanoresonators via nonlinear oscillations: The effective strain mechanism JF - Nanotechnology N2 - Enhancing the mass sensitivity of graphene nanoresonators via nonlinear oscillations: The effective strain mechanism KW - Angewandte Mathematik KW - Strukturmechanik Y1 - 2012 ER - TY - JOUR A1 - Jiang, Jin-Wu A1 - Wang, Bing-Shen A1 - Rabczuk, Timon T1 - Acoustic and breathing phonon modes in bilayer graphene with Moire-acute patterns JF - Applied Physics Letters N2 - 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. KW - Angewandte Mathematik KW - Strukturmechanik Y1 - 2012 U6 - http://dx.doi.org/10.1063/1.4735246 ER - TY - JOUR A1 - Jiang, Jin-Wu A1 - Zhao, Jun-Hua A1 - Zhou, K. A1 - Rabczuk, Timon T1 - Superior thermal conductivity and extremely high mechanical strength in polyethylene chains from ab initio calculation JF - Journal of Applied Physics N2 - The upper limit of the thermal conductivity and the mechanical strength are predicted for the polyethylene chain, by performing the ab initio calculation and applying the quantum mechanical non-equilibrium Green’s function approach. Specially, there are two main findings from our calculation: (1) the thermal conductivity can reach a high value of 310 Wm−1 K−1 in a 100 nm polyethylene chain at room temperature and the thermal conductivity increases with the length of the chain; (2) the Young’s modulus in the polyethylene chain is as high as 374.5 GPa, and the polyethylene chain can sustain 32.85%±0.05% (ultimate) strain before undergoing structural phase transition into gaseous ethylene. KW - Angewandte Mathematik KW - Strukturmechanik Y1 - 2012 U6 - http://dx.doi.org/10.1063/1.4729489 ER - TY - JOUR A1 - Natarajan, S. A1 - Chakraborty, S. A1 - Thangavel, M. A1 - Bordas, Stéphane Pierre Alain A1 - Rabczuk, Timon T1 - Size dependent free flexural vibration behavior of functionally graded nanoplates JF - Computational Materials Science N2 - Size dependent free flexural vibration behavior of functionally graded nanoplates KW - Angewandte Mathematik KW - Strukturmechanik Y1 - 2012 SP - 74 EP - 80 ER - TY - JOUR A1 - Nguyen-Vinh, H. A1 - Bakar, I. A1 - Msekh, Mohammed Abdulrazzak A1 - Song, Jeong-Hoon A1 - Muthu, Jacob A1 - Zi, Goangseup A1 - Le, P. A1 - Bordas, Stéphane Pierre Alain A1 - Simpson, R. A1 - Natarajan, S. A1 - Lahmer, Tom A1 - Rabczuk, Timon T1 - Extended Finite Element Method for Dynamic Fracture of Piezo-Electric Materials JF - Engineering Fracture Mechanics N2 - 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. KW - Angewandte Mathematik KW - Stochastik KW - Strukturmechanik Y1 - 2012 U6 - http://dx.doi.org/10.1016/j.engfracmech.2012.04.025 SP - 19 EP - 31 ER - TY - JOUR A1 - Nguyen-Xuan, Hung A1 - Rabczuk, Timon A1 - Nguyen-Thoi, T. A1 - Tran, T. A1 - Nguyen-Thanh, Nhon T1 - Computation of limit and shakedown loads using a node-based smoothed finite element method JF - International Journal for Numerical Methods in Engineering N2 - This paper presents a novel numerical procedure for computing limit and shakedown loads of structures using a node-based smoothed FEM in combination with a primal–dual algorithm. An associated primal–dual form based on the von Mises yield criterion is adopted. The primal-dual algorithm together with a Newton-like iteration are then used to solve this associated primal–dual form to determine simultaneously both approximate upper and quasi-lower bounds of the plastic collapse limit and the shakedown limit. The present formulation uses only linear approximations and its implementation into finite element programs is quite simple. Several numerical examples are given to show the reliability, accuracy, and generality of the present formulation compared with other available methods. KW - Angewandte Mathematik KW - Strukturmechanik Y1 - 2012 U6 - http://dx.doi.org/10.1002/nme.3317 SP - 287 EP - 310 ER - TY - JOUR A1 - Simpson, R. A1 - Bordas, Stéphane Pierre Alain A1 - Trevelyan, J. A1 - Kerfriden, Pierre A1 - Rabczuk, Timon T1 - An Isogeometric Boundary Element Method for elastostatic analysis JF - Computer Methods in Applied Mechanics and Engineering N2 - The concept of isogeometric analysis, where functions that are used to describe geometry in CAD software are used to approximate the unknown fields in numerical simulations, has received great attention in recent years. The method has the potential to have profound impact on engineering design, since the task of meshing, which in some cases can add significant overhead, has been circumvented. Much of the research effort has been focused on finite element implementations of the isogeometric concept, but at present, little has been seen on the application to the Boundary Element Method. The current paper proposes an Isogeometric Boundary Element Method (BEM), which we term IGABEM, applied to two-dimensional elastostatic problems using Non-Uniform Rational B-Splines (NURBS). We find it is a natural fit with the isogeometric concept since both the NURBS approximation and BEM deal with quantities entirely on the boundary. The method is verified against analytical solutions where it is seen that superior accuracies are achieved over a conventional quadratic isoparametric BEM implementation. KW - Angewandte Mathematik KW - Strukturmechanik Y1 - 2012 U6 - http://dx.doi.org/10.1016/j.cma.2011.08.008 ER -