Locking-free Formulation Research Articles

A locking-free formulation for three-dimensional isogeometric analysis

Non-uniform rational B-splines (NURBS)-based isogeometric analysis (IGA) suffers from locking while analyzing highly slender geometries or problems dominated by incompressibility or near incompressibility constraint. The authors have recently proposed a class of reliable and efficient NURBS-based hybrid elements to alleviate locking in two-dimensional linear elasticity regime. Nevertheless, in several practical situations, the problem often necessitates the three-dimensional elements to analyze an accurate behavior of the domain. In the present work, novel stress-based elements are introduced to eliminate the adverse effects of locking in three-dimensional linear elastic NURBS-based IGA. The comprehensive assessment of several benchmark numerical examples shows the method’s capabilities.

Leveraging spectral analysis to elucidate membrane locking and unlocking in isogeometric finite element formulations of the curved Euler–Bernoulli beam

In this paper, we take a fresh look at using spectral analysis for assessing locking phenomena in finite element formulations. We propose to “measure” locking by comparing the difference between eigenvalue and mode error curves computed on coarse meshes with “asymptotic” error curves computed on “overkill” meshes, both plotted with respect to the normalized mode number. To demonstrate the intimate relation between membrane locking and spectral accuracy, we focus on the example of a circular ring discretized with isogeometric curved Euler–Bernoulli beam elements. We show that the transverse-displacement-dominating modes are locking-prone, while the circumferential-displacement-dominating modes are naturally locking-free. We use eigenvalue and mode errors to assess five isogeometric finite element formulations in terms of their locking-related efficiency: the displacement-based formulation with full and reduced integration and three locking-free formulations based on the B-bar, discrete strain gap and Hellinger–Reissner methods. Our study shows that spectral analysis uncovers locking-related effects across the spectrum of eigenvalues and eigenmodes, rigorously characterizing membrane locking in the displacement-based formulation and unlocking in the locking-free formulations.

On the locking free isogeometric formulations for 3-D curved Timoshenko beams

Locking-free isogeometric formulations of 3-D curved Timoshenko beams are studied. In particular, the global B¯ projection method and a mixed formulation are examined and compared, showing equivalently optimal convergence with regard to displacement solutions. These two formulations are proved to be equal when the beam cross-sectional properties are uniform. For beams with non-uniform cross-sectional properties, the mixed formulation is superior in terms of stress recovery. In addition to these two methods, an alternative locking-free formulation, a C0 NURBS element with selectively reduced integration is suggested in this study, making use of the traditional selective reduced integration (SRI) rule designed for Lagrangian elements. This SRI C0 NURBS element is simple to implement, preserves the sparsity of the global stiffness matrix and requires fewer quadrature point evaluations. Most importantly, due to the use of NURBS basis functions, the exact curve geometry is preserved with all three locking-free isogeometric elements. Benchmark problems and illustrative examples with complex curved geometries are examined for a detailed investigation of the considered locking-free elements.

An accurate singularity-free and locking-free formulation of a three-dimensional shear-deformable beam using Euler parameters

An accurate singularity-free and locking-free formulation of a three-dimensional shear-deformable beam with large deformations and large rotations is developed. The position of the centroid line of the beam is integrated from its slope that is related to the rotation of a corresponding cross-section and stretch and shear strains. The rotation is parametrized by Euler parameters, which can naturally avoid the singularity problem that Euler angles or the rotation vector can have. While stretch and shear strains are interpolated using polynomial functions, Euler parameters are interpolated using a C1-continuous interpolation function to guarantee that curvatures of the centroid line of the beam are continuous at nodes of its elements. Governing equations of the beam are obtained using Lagrange’s equations for systems with constraints, and several examples are simulated to show the performance of the current formulation. Results show that the current formulation does not suffer from shear locking and Poisson locking. Results from the current formulation for a planar static case are compared with its exact solutions, and they are in excellent agreement, which verifies accuracy of the current formulation. Results from the current formulation for some other cases are also in excellent agreement with those from commercial software ABAQUS and ADAMS.

A New Locking-Free Formulation of a Three-Dimensional Shear-Deformable Beam

A new locking-free formulation of a three-dimensional shear-deformable beam with large deformations and large rotations is developed. The position of the centroid line of the beam is integrated from its slope that is related to the rotation of a corresponding cross section and stretch and shear strains. The rotation is parameterized by a rotation vector, which has a clear and intuitive physical meaning. Taylor polynomials are used for certain terms that have zero denominators to avoid singularity in numerical implementation. Since the rotation vector can have singular points when its norm equals 2mπ, where m is a nonzero integer, a rescaling strategy is adopted to resolve the singularity problem when there is only one singular point at a time instant, which is the case for most engineering applications. Governing equations of the beam are obtained using Lagrange's equations for systems with constraints, and several benchmark problems are simulated to show the performance of the current formulation. Results show that the current formulation does not suffer from shear and Poisson locking problems that the absolute nodal coordinate formulation (ANCF) can have. Results from the current formulation for a planar static case are compared with its exact solutions, and they are in excellent agreement with each other, which verifies accuracy of the current formulation. Results from the current formulation are compared with those from commercial software abaqus and recurdyn, and they are in good agreement with each other; the current formulation uses much fewer numbers of elements to yield converged results.

Elimination of self-straining in isogeometric formulations of curved Timoshenko beams in curvilinear coordinates

Isogeometric formulations of curved Timoshenko beams in curvilinear coordinates often result in self-straining of membrane and shear strains, due to the discretization of the local displacement field. Self-straining means a failure in exact representation of rigid body motions, which consequently deteriorates response quality. To overcome the difficulty of self-straining, we propose an invariant formulation that discretizes the global displacement field. It turns out that the approximated membrane, shear, and bending strain measures are invariant regardless of initial geometry in the proposed formulation. For effective applications to any arbitrarily curved structures and locking-free formulations to alleviate membrane and shear locking, the proposed invariant formulation is combined with selective reduced integration (SRI) and B̄ projection method. Numerical examples demonstrate the effectiveness and applicability of the proposed invariant formulation, which gives much more accurate results together with both SRI and B̄ projection method, in comparison to the existing isogeometric formulations.

Order reduction method for locking free isogeometric analysis of Timoshenko beams

The order reduction (OR) method is proposed to provide an effective and efficient locking-free formulation in the isogeometric analysis of plane curved Timoshenko beams. The shear and membrane strains are represented by the reduced order B-spline basis functions, and the locking phenomenon can be prevented. The modified strains are equivalent to the original ones in the sense of the least square approximation. Different order reduction strategies are discussed, and it is suggested to use multiple sets of lower order B-spline basis functions to improve the accuracy. The theoretical analysis of the formulation and the computational cost is carried out to assess the efficiency of the OR method. The performance of the proposed method is tested by the straight beam, curved beams with constant and varied curvatures problems, and compared with other methods such as the selective and reduced integration method, the local discrete shear gap method, and the local B̄ method (without strain smoothing). The numerical results show that the proposed method is competitive, which leads to an efficient locking free formulation.

A new locking-free formulation for planar, shear deformable, linear and quadratic beam finite elements based on the absolute nodal coordinate formulation

Many widely used beam finite element formulations are based either on Reissner’s classical nonlinear rod theory or the absolute nodal coordinate formulation (ANCF). Advantages of the second method have been pointed out by several authors; among the benefits are the constant mass matrix of ANCF elements, the isoparametric approach and the existence of a consistent displacement field along the whole cross section. Consistency of the displacement field allows simpler, alternative formulations for contact problems or inelastic materials. Despite conceptional differences of the two formulations, the two models are unified in the present paper. In many applications, a nonlinear large deformation beam element with bending, axial and shear deformation properties is needed. In the present paper, linear and quadratic ANCF shear deformable beam finite elements are presented. A new locking-free continuum mechanics based formulation is compared to the classical Simo and Vu-Quoc formulation based on Reissner’s virtual work of internal forces. Additionally, the introduced linear and quadratic ANCF elements are compared to a fully parameterized ANCF element from the literature. The performance of the respective elements is evaluated through analysis of conventional static and dynamic example problems. The investigation shows that the obtained linear and quadratic ANCF elements are advantageous compared to the original fully parameterized ANCF element.

A physically stabilized and locking-free formulation of the (SHB8PS) solid-shell element

In this work, the formulation of the SHB8PS finite element is reviewed in order to eliminate some persistent membrane and shear locking phenomena. This is a solid-shell element based on a purely three-dimensional formulation. In fact, the element has eight nodes as well as five integration points, all distributed along the “thickness” direction. Consequently, it can be used for the modeling of thin structures, while providing an accurate description of the various through-thickness phenomena. The reduced integration has been used in order to prevent some locking phenomena and to increase computational efficiency. The spurious zero-energy modes due to the reduced integration are efficiently stabilized, whereas the strain components corresponding to locking modes are eliminated with a projection technique following the Enhanced Assumed Strain (EAS) method.

A physically stabilized and locking-free formulation of the (SHB8PS) solid-shell element

In this work, the formulation of the SHB8PS finite element is reviewed in order to eliminate some persistent membrane and shear locking phenomena. This is a solid-shell element based on a purely three-dimensional formulation. In fact, the element has eight nodes as well as five integration points, all distributed along the thickness direction. Consequently, it can be used for the modeling of thin structures, while providing an accurate description of the various through-thickness phenomena. The reduced integration has been used in order to prevent some locking phenomena and to increase computational efficiency. The spurious zero-energy modes due to the reduced integration are efficiently stabilized, whereas the strain components corresponding to locking modes are eliminated with a projection technique following the Enhanced Assumed Strain (EAS) method.

Meshless analysis of Timoshenko beams based on a locking-free formulation and variational approaches

A meshless method is developed, based on subdomain variational formulations and a local Petrov–Galerkin approximation, along with the locking-free formulation, for the analysis of bending of a thick beam. The local point interpolation method is employed to construct both trial and test functions. The present method is a truly meshless method based only on a number of randomly located nodes. No global background integration mesh is needed, no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. Effects of the sizes of local subdomain and interpolation domain on the performance of the present method are investigated. Problems of thick and thin beams under various loading, and boundary conditions are analysed by the proposed method and the numerical results are compared with analytical solutions. The contact problem for beam by means of subdomain variational inequalities and the meshless method has also been investigated.

On the Parametrization of the Three-Dimensional Rotation Group

Parameterization of group of rotations of Euclidean three-dimensional space applied to integration of rigid body motion equations