A large deformation, rotation-free, isogeometric shell

doi:10.1016/j.cma.2010.12.003

Computer Methods in Applied Mechanics and Engineering

Volume 200, Issues 13–16, 1 March 2011, Pages 1367-1378

https://doi.org/10.1016/j.cma.2010.12.003 Get rights and content

Abstract

Conventional finite shell element formulations use rotational degrees of freedom to describe the motion of the fiber in the Reissner–Mindlin shear deformable shell theory, resulting in an element with five or six degrees of freedom per node. These additional degrees of freedom are frequently the source of convergence difficulties in implicit structural analyses, and, unless the rotational inertias are scaled, control the time step size in explicit analyses. Structural formulations that are based on only the translational degrees of freedom are therefore attractive. Although rotation-free formulations using C⁰ basis functions are possible, they are complicated in comparison to their C¹ counterparts. A C^k-continuous, k ⩾ 1, NURBS-based isogeometric shell for large deformations formulated without rotational degrees of freedom is presented here. The effect of different choices for defining the shell normal vector is demonstrated using a simple eigenvalue problem, and a simple lifting operator is shown to provide the most accurate solution. Higher order elements are commonly regarded as inefficient for large deformation analyses, but a traditional shell benchmark problem demonstrates the contrary for isogeometric analysis. The rapid convergence of the quadratic element is demonstrated for the NUMISHEET S-rail benchmark metal stamping problem.

Introduction

Structural elements, like solid elements, have traditionally used C⁰ Lagrange polynomials as basis functions. The development of C¹ shell elements based on Kirchhoff–Love theory proved very difficult, and the structural analysis community eventually adopted the C⁰ shear deformable Reissner–Mindlin theory. Since the fiber vector is no longer defined as being normal to the reference surface, two rotational coordinates are needed to define its orientation, but three rotational coordinates are commonly used in practice to simplify the implementation.

Using six degrees of freedom per node doubles the number of degrees of freedom in comparison to the requirements of thin shell theory. Doubling the degrees of freedom increases the stiffness matrix size by a factor of four, and the cost of the matrix factorization by a factor of eight. Since structural elements are typically very ill-conditioned, a direct solver is needed. Sparse direct solvers currently scale poorly beyond a few dozen processors, and therefore the size of structural problems that may be efficiently solved implicitly is capped. Additionally, the rotational degrees of freedom are frequently the source of convergence problems in large deformation analyses. Even large scale quasi-static problems are currently solved using explicit time integration methods to achieve scalability on massively parallel computers. The stable time step is determined by the maximum eigenvalue of the system, which is invariably a mode dominated by the rotational degrees of freedom. Unless the rotational inertias are scaled up, the stable time step size for explicit time integration methods is therefore governed by the thickness of the shell, making explicit thin shell calculations impractical. Shell formulations that use only translational degrees of freedom are therefore attractive for many reasons.

Isogeometric analysis is a new computational method that is based on geometry representations (i.e., basis functions) developed in computer-aided design (CAD), computer graphics (CG), and animation, with a far-reaching goal to bridge the existing gap between CAD and analysis [17], [18], [30]. For the first instantiation of the isogeometric methodology, non-uniform rational B-splines (NURBS) were chosen as a basis, due to their relative simplicity and ubiquity in the worlds of CAD, CG, and animation. It was found that not only were NURBS applicable to engineering analysis, they were better suited for many applications, and were able to deliver accuracy superior to standard finite elements (see, e.g., [1], [5], [6], [7], [8], [18], [31], [39]). Subdivision surfaces [14], [15], [16] and, more recently, T-Splines [4], [19], were also successfully employed in the analysis context. It should be emphasized that isogeometric refers only to using the same basis functions for analysis as for the geometry, and is not restricted to NURBS [30].

Rotation-free isogeometric Bernoulli–Euler beams and Poisson–Kirchhoff plates were introduced and studied in [18], [17]. Kiendl et al. [33] have studied shell branching and joining patches for rotation-free elements.

A rotation-free isogeometric shell formulation is presented here using C^k-continuous, k ⩾ 1, NURBS. The efficiency of the element is enhanced by interpolating the normal from the control points. The normal directions are not uniquely defined and the effects of different choices are explored. A simply lifting operator is found to be a good option. The accuracy and efficiency of the proposed methodology is illustrated with numerical examples.

Section snippets

Kinematics

The translational degrees of freedom of the control points define the motion of the reference surface of the shell. Without additional degrees of freedom, the orientation of the fiber with respect to the reference surface is fixed. The simplest assumption regarding the fiber is that it is normal to the reference surface, i.e., the Kirchhoff–Love hypothesis. In the context of a finite element formulation, the Kirchhoff–Love hypothesis may be imposed at a number of different levels, ranging from

Implementation

The element is implemented using the generalized element formulation [12], and therefore may use any of the CAD basis functions that are, at the minimum, C¹ continuous. The example calculations are performed with NURBS [17], [36] but T-splines [4], [19], for example, would work as well.

The first term of B in Eq. (21) has the usual form of the discrete gradient in solids and membranes, while the second involves a summation over all the control points in the element. The number of operations

Dynamics and the mass matrix

The first variation of x is $δ x = \sum_{A} N_{A} (δ q_{A} + \frac{h_{A}}{2} s_{3} \sum_{C} \frac{\partial n_{A}}{\partial q_{C}} δ q_{C})$ and the acceleration, obtained by differentiating the velocity, is $\ddot{x} = \sum_{B} N_{B} ({\ddot{q}}_{B} + \frac{h_{B}}{2} s_{3} \sum_{D} \{\frac{\partial n_{B}}{\partial q_{D}} {\ddot{q}}_{D} + \sum_{E} \frac{\partial^{2} n_{B}}{\partial q_{D} q_{E}} {\dot{q}}_{D} {\dot{q}}_{E}\}) .$ The inertial contribution to the principle of virtual work therefore is $\int_{s} δ x \cdot \ddot{x} ρ \det (J) d s = \int_{s} \sum_{A, B} N_{A} N_{B} δ q_{A} \cdot {\ddot{q}}_{B} ρ \det (J) d s + \int_{s} \sum_{A, B} N_{A} N_{B} \frac{h_{A}}{2} s_{3} A ρ \det (J) d s + \int_{s} \sum_{A, B} N_{A} N_{B} \frac{h_{A} h_{B}}{4} s_{3}^{2} B ρ \det (J) d s,$ $A = \{(\sum_{C} \frac{\partial n_{A}}{\partial q_{C}} δ q_{C}) {\ddot{q}}_{A}\} + \{δ q_{B} \sum_{D} (\frac{\partial n_{A}}{\partial q_{D}} {\ddot{q}}_{D} + \sum_{E} \frac{\partial^{2} n_{A}}{\partial q_{D} q_{E}} {\dot{q}}_{D} {\dot{q}}_{E})\},$ $B = \{\sum_{C} \frac{\partial n_{A}}{\partial q_{C}} δ q_{C}\} \{\sum_{D} (\frac{\partial n_{B}}{\partial q_{D}} {\ddot{q}}_{D} + \sum_{E} \frac{\partial^{2} n_{B}}{\partial q_{D} q_{E}} {\dot{q}}_{D} {\dot{q}}_{E})\} .$ The first term on the right hand side has the same form as the

Example calculations

In this section we present one linear elastic and two nonlinear elastic–plastic computational examples. Unless otherwise specified, there are p + 1 integration points in each in-plane direction between knots where p is the polynomial order of the NURBS basis. We denote by E, ν, ρ, σ_y the Young’s modulus, Poisson’s ratio, density, and initial yield stress, respectively.

All calculations were performed in double precision on a Dell laptop computer with an Intel Core Duo using a single processor.

Conclusions

An accurate rotation-free isogeometric shell element has been developed.

The accuracy of the element is largely a function of the NURBS basis functions. Since the basis functions are not interpolatory, the notion of evaluating the normals at the control points is ambiguous. As demonstrated by the eigenvalue problem, a response that accurately approximates thin shell theory is obtained by using a simple lifting operator.

Efficiency for explicit calculations is obtained by using reduced

Acknowledgments

This research was supported by NSF grant 0700204 (Dr. Joycelyn S. Harrison, program officer) and ONR grant N00014-08-1-0992 (Dr. Luise Couchman, contract monitor).

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View full text

A large deformation, rotation-free, isogeometric shell

Abstract

Introduction

Section snippets

Kinematics

Implementation

Dynamics and the mass matrix

Example calculations

Conclusions

Acknowledgments

Computer Methods in Applied Mechanics and Engineering

Computer Methods in Applied Mechanics and Engineering

Computer Methods in Applied Mechanics and Engineering

Computer Methods in Applied Mechanics and Engineering

Computer Methods in Applied Mechanics and Engineering

Computer-Aided Design

Computer Methods in Applied Mechanics and Engineering

Computer Methods in Applied Mechanics and Engineering

Computer Methods in Applied Mechanics and Engineering

Computer Methods in Applied Mechanics and Engineering

Computer Methods in Applied Mechanics and Engineering

Computer Methods in Applied Mechanics and Engineering

Computer Methods in Applied Mechanics and Engineering

The role of continuity in residual-based variational multiscale modeling of turbulence

Computational Mechanics

Isogeometric collocation methods

Mathematical Models and Methods in Applied Sciences

Isogeometric fluid–structure interaction: theory, algorithms, and computations

Computational Mechanics

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Computational Mechanics

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Computer Methods in Applied Mechanics and Engineering