Hourglass control in linear and nonlinear problems. Comput Methods Appl Mech Eng

Department of Civil and Mechanical/Nuclear Engineering, Northwestern University, Evanston, IL 60201, U.S.A.
Computer Methods in Applied Mechanics and Engineering (Impact Factor: 2.96). 05/1984; 43(3):251-276. DOI: 10.1016/0045-7825(84)90067-7

ABSTRACT Mesh stabilization techniques for controlling the hourglass modes in under-integrated hexahedral and quadrilateral elements are described. It is shown that the orthogonal hourglass techniques previously developed can be obtained from simple requirements that insure the consistency of the finite element equations in the sense that the gradients of linear fields are evaluated correctly. It is also shown that this leads to an hourglass control that satisfies the patch test. The nature of the parameters which relate the generalized stresses and strains for controlling hourglass modes is examined by means of a mixed variational principle and some guidelines for their selection are discussed. Finally, effective means of implementing these hourglass procedure in computer codes are described. Applications to both the Laplace equation and the equations of solid mechanics in 2 and 3 dimensions are considered.

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    • "A suitable choice of s E thus ensures the stability of the method by guaranteeing that discrete bilinear form inherits the coercivity of the exact bilinear form 5 . As mentioned before, for hexahedral elements, H is the space of hourglass modes, and so, in light of (42), s E essentially prescribes the " hourglass " stiffness (see, for example, [13]). By virtue of the decomposition, the choice of s E does not affect the polynomial consistency of a "
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    ABSTRACT: We explore the recently-proposed Virtual Element Method (VEM) for numerical solution of boundary value problems on arbitrary polyhedral meshes. More specifically, we focus on the elasticity equations in three-dimensions and elaborate upon the key concepts underlying the first-order VEM. While the point of departure is a conforming Galerkin framework, the distinguishing feature of VEM is that it does not require an explicit computation of the trial and test spaces, thereby circumventing a barrier to standard finite element discretizations on arbitrary grids. At the heart of the method is a particular kinematic decomposition of element deformation states which, in turn, leads to a corresponding decomposition of strain energy. By capturing the energy of linear deformations exactly, one can guarantee satisfaction of the engineering patch test and optimal convergence of numerical solutions. The decomposition itself is enabled by local projection maps that appropriately extract the rigid body motion and constant strain components of the deformation. As we show, computing these projection maps and subsequently the local stiffness matrices, in practice, reduces to the computation of purely geometric quantities. In addition to discussing aspects of implementation of the method, we present several numerical studies in order to verify convergence of the VEM and evaluate its performance for various types of meshes.
    Computer Methods in Applied Mechanics and Engineering 11/2013; 282. DOI:10.1016/j.cma.2014.05.005 · 2.96 Impact Factor
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    • "In fact, the one-point integration in the elements has its own drawback which is a mesh instability known as hourglassing. But the hourglass modes can be eliminated by well-developed hourglass control schemes (see [8]). Kosloff and Frazier [9] had showed that a one-point integration implementation coupled with a stiffness hourglass control scheme can produce a more accurate flexural response than fully integrated elements. "
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    ABSTRACT: In this paper, we will use the explicit finite element to compute ground motion due to Tangshan earthquake. The expli-cit finite-element method uses one integration point and an hourglass control scheme. We implement the coarse-grain method in a structured finite-element mesh straightforwardly. At the same time, we also apply the coarse-grain method to a widely used, slightly unstructured finite-element mesh, where unstructured finite elements are only used in the ver-tical velocity transition zones. By the finite-element methods, we can compute the ground velocity with some distance to the seismogenic fault in Tangshan earthquake. Through the computation, we can find the main character of ground motion for the strike slip earthquake events and the high frequency vibration motion of ground motion.
    Journal of Applied Mathematics and Physics 11/2013; 01(06). DOI:10.4236/jamp.2013.16003
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    • "To account for compressible immersed solids, the conservation of mass equation in the background fluid domain will be suitably modified to take into consideration the presence of a volumetric deformation. In the incompressibility limit, the IFEM approach, based on a Finite Element discretisation of the solid phase, suffers from well reported locking effects [1]. A solution to this deficiency, based on the use of enhanced elements (see [30] [28] [29]), will be implemented and tested. "
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    ABSTRACT: Continuum immersed strategies are widely used these days for the computational simulation of Fluid–Structure Interaction problems. The principal characteristic of such immersed techniques is the representation of the immersed solid via a momentum forcing source in the Navier–Stokes equations. In this paper, the Immersed Finite Element Method (IFEM), introduced by Zhang et al. (2004) [41] for the analysis of deformable solids immersed in an incompressible Newtonian viscous fluid, is further enhanced by means of three new improvements. A first update deals with the modification of the conservation of mass equation in the background fluid in order to account for non-isochoric deformations within the solid phase. A second update deals with the incompressibility constraint for the solid phase in the case of isochoric deformations, where an enhanced evaluation of the deformation gradient tensor is introduced in a multifield Hu-Washizu variational sense in order to overcome locking effects. The third update is focussed on the improvement of the robustness of the overall scheme, by introducing an implicit one-step time integration scheme with enhanced stability properties, in conjunction with a consistent Newton–Raphson linearisation strategy for optimal quadratic convergence. The resulting monolithic methodology is thoroughly studied for a range of Lagrangian and NURBS based shape finite element functions for a series of numerical examples, with the purpose of studying the effect of the spatial semi-discretisation in the solution. Comparisons are also established with the newly developed Immersed Structural Potential Method (ISPM) by Gil et al. (2010) [7] for benchmarking and assessment of the quality of the new formulation.
    Computer Methods in Applied Mechanics and Engineering 11/2012; s 247–248:51–64. DOI:10.1016/j.cma.2012.07.021 · 2.96 Impact Factor
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