Hourglass control in linear and nonlinear problems
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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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.11/2013;
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ABSTRACT: A stabilised second order finite element methodology is presented for the numerical simulation of a mixed conservation law formulation in fast solid dynamics. The mixed formulation, where the unknowns are linear momentum, deformation gradient and total energy, can be cast in the form of a system of first order hyperbolic equations. The difficulty associated with locking effects commonly encountered in standard pure displacement formulations is addressed by treating the deformation gradient as one of the primary variables. The formulation is first discretised in space by using a stabilised Petrov–Galerkin (PG) methodology derived through the use of variational (work-conjugate) principles. The semi-discretised system of equations is then evolved in time by employing a Total Variation Diminishing Runge–Kutta (TVD-RK) time integrator. The formulation achieves optimal convergence (e.g. second order with linear interpolation) with equal orders in velocity (or displacement) and stresses, in contrast with the displacement-based approach. This paper defines a set of appropriate stabilising parameters suitable for this particular formulation, where the results obtained avoid the appearance of non-physical spurious (zero-energy) modes in the solution over a long term response. We also show that the proposed PG formulation is very similar, and under certain conditions identical, to the well known Two-step Taylor Galerkin (2TG). A series of numerical examples are presented in order to assess the performance of the proposed algorithm. The new formulation is proven to be very efficient in nearly incompressible and bending dominated scenarios.Computer Methods in Applied Mechanics and Engineering 09/2013; · 2.62 Impact Factor
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ABSTRACT: In this work, solid-shell NURBS elements are developed in order to address static problems of slender structures under small perturbations. A single layer of elements is considered through the thickness of the shell, and the degree of approximation in that direction is cho-sen to be equal to two. A full 3D constitutive relation is assumed. The objective is to obtain highly accurate low-degree elements to be used in coarse meshes. In order to do that, we propose a mixed method from which we derive a B-projection to deal with locking. The main idea is to modify the interpolation of the average stresses and strains through the thickness. More precisely, we develop two finite elements. The first element, which is based on the mixed method or, equivalently, on the B-projection, is extremely accurate, but leads to a fully-populated global stiffness matrix. To improve the efficiency, the second element uses a local least-squares-type procedure to define a new B-projection, leading to a sparse global stiffness matrix. The quality and efficiency of the methods are assessed through several usual test cases and by comparison with other published techniques.Computer Methods in Applied Mechanics and Engineering 12/2013; 267:86-110. · 2.62 Impact Factor