A smoothed finite element method for plate analysis

University of Glasgow, Civil Engineering, Rankine building, G12 8LT, United Kingdom
Computer Methods in Applied Mechanics and Engineering (Impact Factor: 2.96). 02/2008; 197(13-16):1184-1203. DOI: 10.1016/j.cma.2007.10.008


A quadrilateral element with smoothed curvatures for Mindlin–Reissner plates is proposed. The curvature at each point is obtained by a non-local approximation via a smoothing function. The bending stiffness matrix is calculated by a boundary integral along the boundaries of the smoothing elements (smoothing cells). Numerical results show that the proposed element is robust, computational inexpensive and simultaneously very accurate and free of locking, even for very thin plates. The most promising feature of our elements is their insensitivity to mesh distortion.

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Available from: Stéphane Pierre Alain Bordas,
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    • "Interested readers are referred to the literature [24] [25] and references therein. Nguyen-Xuan et al. [32] employed CS- FEM for Mindlin-Reissner plates. The curvature at each point is obtained by a nonlocal approximation via a smoothing function. "
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    • "Clearly this technique of smoothing is a powerful tool and it has already been applied to a wide range of practical mechanics problems, e.g, [39] [46] [47] [49] [57]. Nevertheless, if the displacement is approximated only by ES-FEM or FS-FEM i.e. without enrichment by bubble functions, these methods violate the inf-sup condition and uniform convergence. "
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    ABSTRACT: We present in this paper a rigorous theoretical framework to show stability, convergence and accuracy of improved edge-based and face-based smoothed finite element methods (bES-FEM and bFS-FEM) for nearly-incompressible elasticity problems. The crucial idea is that the space of piecewise linear polynomials used for the displacements is enriched with bubble functions on each element, while the pressure is a piecewise constant function. The meshes of triangular or tetrahedral elements required by these methods can be generated automatically. The enrichment induces a softening in the bilinear form allowing the weakened weak (W2) procedure to produce a high-quality solution, free from locking and that does not oscillate. We prove theoretically that both methods confirm the uniform inf–sup and convergence conditions. Four numerical examples are given to validate the reliability of the bES-FEM and bFS-FEM.
    Computer Methods in Applied Mechanics and Engineering 03/2015; 285:315–345. DOI:10.1016/j.cma.2014.10.022 · 2.96 Impact Factor
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    • "However it is well-known that naïve numerical implementations of the standard FSDT using low-order Lagrangian shape functions typically suffer from shear-locking in the thin-plate or Kirchhoff limit resulting in totally incorrect solutions. Special techniques such as the MITC family of elements [4], assumed strain method [5], field consistent approach [6], smoothed finite element [7] with strain smoothing stabilization technique, are often applied to solve the shear-locking problem, but with additional expense and implementation complexity. However, with the introduction of numerical methods relying on basis functions with natural C 1 continuity such as NURBS in an isogeometric analysis framework (IGA) [8] and meshfree methods [9] [10] we believe that the physical accuracy and straightforward numerical implementation are no longer at odds. "
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    ABSTRACT: An effective, simple, robust and locking-free plate formulation is proposed to analyze the static bending, buckling, and free vibration of homogeneous and functionally graded plates. The simple first-order shear deformation theory (S-FSDT), which was recently presented in Thai and Choi (2013) [11], is naturally free from shear-locking and captures the physics of the shear-deformation effect present in the original FSDT, whilst also being less computationally expensive due to having fewer unknowns. The S-FSDT requires C1-continuity that is simple to satisfy with the inherent high-order continuity of the non-uniform rational B-spline (NURBS) basis functions, which we use in the framework of isogeometric analysis (IGA). Numerical examples are solved and the results are compared with reference solutions to confirm the accuracy of the proposed method. Furthermore, the effects of boundary conditions, gradient index, and geometric shape on the mechanical response of functionally graded plates are investigated.
    Composite Structures 12/2014; 118(1):121–138. DOI:10.1016/j.compstruct.2014.07.028 · 3.32 Impact Factor
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