Hourglass Modes Research Articles

Three‐dimensional finite element for analysing thin plate/shell structures

AbstractA three‐dimensional (3‐D) hexahedron finite element is presented for the analysis of thin plate/shell structures. The element employs an explicit algebraic definition of six uniform (continuum) strains, six rigid body modes and classical Lagrange‐Germain‐Kirchhoff thin plate bending modes. Nine additional stiffness factors are used to control higher‐order hourglass modes. The element may be used for plate/shell analyses where the flat plate assumptions are appropriate. Also it can easily be adapted to form transition elements to lower order 2‐D elements, or to higher‐order 3‐D continuum elements.The stiffness matrix satisfies the geometric isotropy requirement, passes the patch test, and gives essentially identical response to either applied transverse corner forces or to twisting moments applied on the corner, a requirement of Kirchhoff's corner conditions for a classical thin plate. Several examples are presented to demonstrate the performance of this finite element.

Unified and mixed formulation of the 4‐node quadrilateral elements by assumed strain method: Application to thermomechanical problems

AbstractIn this paper, a class of ‘assumed strain’ mixed finite element methods based on the Hu–Washizu variational principle is presented. Special care is taken to avoid hourglass modes and shear locking as well as volumetric locking. An unified framework for the 4‐node quadrilateral solid and thermal as well as thermomechanical coupling elements with uniform reduced integration (URI) and selective numerical integration (SI) schemes is developed. The approach is simply implemented by a small change of the standard technique and is applicable to arbitrary non‐linear constitutive laws including isotropic and anisotropic material behaviours. The implementation of the proposed SI elements is straightforward, while the development of the proposed URI elements requires ‘anti‐hourglass stresses’ which are evaluated by classical constitutive equations. Several numerical examples are given to demonstrate the performance of the suggested formulation, including static/dynamic mechanical problems, heat conduction and thermomechanical problems.

Improvement Techniques for Bilinear Element in Elastoplastic Large-Deformation Analysis. Effect of Plastic Constraint on Numerical Solutions.

A bilinear 4-node element, widely used in finite-element numerical analysis, yields poor solutions in some cases of large elastoplastic deformation. The locking phenomenon and hourglass modes are observed when applying the exact and reduced integration techniques, respectively, for the displacement-based method. Even though the mixed method can overcome these shortcomings, the cpu time increases rapidly for a large-scale problem, because the pressure field is added to the matrix formulation and the exact integration technique becomes indispensable. On the other hand, several improved methods, such as the selective reduced integration technique, the so-called B approach and the hourglass stiffness stabilization method, have been proposed to overcome these difficulties. We here give the finite-element formulation of rate types in large deformation fields for these improvement techniques for a bilinear 4-node element and discuss the differences among these methods using the numerical simulations.

A second-order Godunov algorithm for two-dimensional solid mechanics

The second-order Godunov method is extended to dynamic wave propagation in two-dimensional solids undergoing nonlinear finite deformation. It is shown that this explicit method is linearly stable for timesteps satisfying the standard CFL condition, does not support the development of hourglass modes, and handles non-reflecting boundaries very naturally. The computational cost is essentially one evaluation of the kinetic equation of state per cell and timestep, the same as explicit finite element methods employing reduced quadrature.

A diffusion equation with hourglass control in an axisymmetric geometry

A one-point integration scheme which analytically integrates the element volume and the gradients of linear fields for general quadrilaterals in an axisymmetric geometry is described. A method for isolating proper hourglass modes is presented. The procedure for the control of hourglass modes has been developed from an algebraic and from a variational viewpoint. The hourglass control obtained satisfies the patch test. Bounds on the quadrilateral eigenvalues, that provide stable time step for explicit time integration algorithms, are evaluated considering the influence of hourglass control. Several linear numerical examples demonstrate the effectiveness of the developed technique for finite element codes.

Eigenvalue analysis of compatible and incompatible rectangular four‐node quadrilateral elements

AbstractExact analytical expressions for the eigenvalues of the elastic stiffness matrix are obtained for the four‐node, rectangular, quadrilateral element. A procedure is given for identifying alternative hourglass modes and eigenvalues which render the element incompatible but with non‐monotonic convergence assured. A convergence study confirms that for the special case of when the hourglass modes coincide with beam bending the element can serve as a beam element. Analytical expressions are given for the resulting element stiffness matrix.

A one-point integration quadrilateral with hourglass control in axisymmetric geometry

A one-point integration scheme which analytically integrates element volume and uniform strain modes for general isoparametric quadrilateral in axisymmetric geometry is described. A method for isolating proper hourglass modes is presented. It is shown that these modes rather differ from hourglass modes in plane Cartesian geometry, especially near the axis of symmetry. A simple elastic and viscous hourglass control scheme is presented and discussed. Bounds on the quadrilateral eigenvalues that provide stable time steps for explicit time integration algorithms are obtained considering isotropic hypoelastic material. Numerical examples demonstrate the effectiveness of the developed technique for finite element code.

Elastoplastic Shell Element Based on Assumed Covariant Strain Interpolations

A curved, nine‐node C0 shell finite element for elastic analysis that is free from serious locking problems, does not possess hourglass modes, and provides solutions which are quite insensitive to mesh distortion has recently been proposed. The element is based on assumed interpolations of all covariant (nonphysical) strains referred to the element natural coordinate system. This paper extends the elastic formulation to include an elastoplastic constitutive model with nonlinear isotropic and linear kinematic hardening and a von Mises yield condition. An unconditionally stable return mapping algorithm is introduced for the integration of the constitutive equations under the zero normal stress shell hypothesis. Transverse shear stresses are included in the plasticity formulation. The algorithm is consistently linearized to define the shell elastoplastic tangent moduli. Numerical results for convergence of transverse shear in elastic plates and an elastoplastic example are presented.

An assumed covariant strain based 9‐node shell element

AbstractA new shell element, which is free from serious locking problems and which does not possess hourglass modes, is proposed. Solutions obtained with this element exhibit good convergence and are satisfactorily insensitive to mesh distortion. The element also exhibits good convergence for plates with variable thickness. This element is based on the use of assumed covariant strains which are obtained from the covariant strain field defined with respect to the element natural co‐ordinate system. Only the linear version for thin shell cases is considered. The element performance is tested by application to several standard plate and shell problems. A problem involving variable thickness is also presented.

Snap‐through and snap‐back response in concrete structures and the dangers of under‐integration

AbstractSnap‐through and snap‐back responses are usually associated with the buckling of shells. However, it is shown in this paper that they can also be expected with the cracking of concrete structures. It is also demonstrated that, for such structures, the common use of 2 × 2 Gaussian integration with the 8‐noded isoparametric element can lead to spurious responses associated with ‘truncated hour‐glass modes’.

Analysis of hourglass instabilities and control in underintegrated finite element methods

Belytschko et al. (1981, 1984) has developed stabilization methods for the treatment of underintegrated FEM problems; these methods involve the computation of an underintegrated stiffness matrix, which is rank-deficient, and the addition of a stabilization matrix which effectively eliminates the spurious modes. An attempt is presently made to give this a priori stabilization method a mathematical means of support. Attention is also given to an a posteriority stabilization method for hourglass control, in which an approximate solution of the underintegrated system is obtained and then subjected to a special projection in order to eliminate the hourglass modes. A proof is obtained for the convergence of this stabilized underintegrated approximation to the exact solution of a model problem at almost the same rate (as the mesh is refined) as the fully integrated solutions.

Hourglass control in linear and nonlinear problems

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.

Finite element methods with user-controlled meshes for fluid-structure interaction

A quasi-Eulerian finite element formulation for the analysis of transients in a fluid with a pressurized bubble is developed in both two and three dimensions. In these methods the mesh can be programmed to move independent of the material, so that the method lends itself well to the treatment of fluid-structure problems. Time integration is performed by an explicit method, and one point quadrature is used in the fluid elements. To eliminate hourglass modes, an hourglass control which is orthogonal to straining and rigid body modes is used. Several examples are described.