23rd Nordic Seminar on Computational Mechanics NSCM-23 A. Eriksson and G. Tibert (Eds) KTH, Stockholm, 2010c
ISOGEOMETRIC FINITE ELEMENT METHODS
Knut Morten Okstad, Kjell Magne Mathisen and Trond Kvamsdal
solid/structural mechanics problems involve objects where part of the geometry is described by circles or circle segments, and traditionally this has been represented inaccurately by means of low order Lagrange polynomials, whereas by using NURBS these inaccuracies may be eliminated.
Furthermore, in elasticity we do have continuous stresses and strains except for at certain singular points, lines or surfaces, i.e. the displacement field is C1-continuous away from singularities.
Classical finite elements based on Lagrange polynomials are only C0-continuous and this lack of regularity shows up in discontinuous (along inter-element boundaries) finite element stress and strain fields, whereas NURBS (p ≥ 2) may represent this behavior qualitatively correct.
NURBS-based and classical Lagrange, finite deformation, displacement-based, solid elements of any order have been implemented into our nonlinear FE solver. Both the material and spatial formulation, based on the reference and the current configuration, respectively, have been implemented. However, as pointed out in standard references on the subject3,4, the spatial formulation is more computational efficient on the elemental level due to the standard sparse structure of the B-matrix, that coincides with the sparsity of the B-matrix of the linear theory, compared to the full B-matrix for the material formulation.
Also as pointed out by Miehe5, the formulation and the finite element implementation of finite deformation isotropic elasticity turns out to be more compact and carried out with lower computational effort when adopting the spatial configuration. As demonstrated in5, the elas-ticity equations may be formulated exclusively based on the left Cauchy-Green tensor, often referred to as the Finger tensor; b = FFT, that may be be computed directly and with low computational effort for a given deformation gradient F. For this reason the spatial formulation is adopted in the present study.
2 NUMERICAL RESULTS
The isogeometric nonlinear solver has been tested on a geometry that cannot be represented exactly by Lagrange polynomials. It involves finite deformation analysis of a thick hollow cylin-der, for which the exact initial geometry can be obtained with quadratic NURBS. Due to symme-try only one quarter of the whole problem is discretized and analyzed, and boundary conditions at the two symmetry planes are set accordingly, as shown in Figure 1. This problem was first studied by B¨uchter and co-workers6 using shell elements, and later by others7,8 considering it as a 3D continuum. The material model applied is a standard compressible Neo-Hookean model with strain energy function
W (J, b) = 1
2µ(trb − 3) − µ ln J +1
2λ(ln J )2 (1)
where µ and λ are Lame’s constants that may be derived from Young’s modulus E and Poisson’s ratio ν, J is the determinant of the deformation gradient F and b is the Finger tensor. Since Poisson’s ratio ν = 0.4, no volumetric locking is expected.
Herein, the cylinder is parameterized by a single NURBS patch and analyzed using quadratic and cubic NURBS as basis functions, and compared with standard quadratic Lagrange elements.
Gaussian integration is used throughout all analyses applying 3×3×3 Gauss points for quadratic NURBS as well as Lagrange basis functions and 4 × 4 × 4 Gauss points for NURBS functions of order 3. Even though it is sufficient with quadratic basis functions in the thickness direction to capture the bending behavior, herein NURBS functions of order 3 are used in the thickness
Knut Morten Okstad, Kjell Magne Mathisen and Trond Kvamsdal
? ? ? ? ?
−p0· ex
PP P
i ux= uy = 0
uy = 0 (symmetry)
uy = 0 (symmetry)
)
uz= 0 (symmetry)
Half length : L = 15.0 Outer radius : Ro= 10.0 Inner radius : Ri = 8.0 Young’s modulus : E = 16800 Poisson’s ratio : ν = 0.4 Load intensity : p0 = 500.0
Figure 1: Compression of a thick cylinder: Geometry and properties.
direction as well for the cubic NURBS elements. In order to avoid singularities due to the concentrated loading, an equivalent constant traction is applied to the reference configuration of the top symmetry cross-section of the cylinder and kept fixed during deformation, such that the total load applied on the cylinder is 30 × 103. Therefore, we cannot expect full compliance with the results obtained in7,8 where the load is applied as a line load along the top edge.
The results are presented in Figure 2. As expected, cubic NURBS gives a somewhat softer solution than quadratic NURBS. However it is also interesting to note that the quadratic NURBS seems to give a slightly stiffer solution than the Lagrange elements. This is believed to be an effect of inter-element continuity, since this is one of the main differences between NURBS finite elements and Lagrange finite elements. While standard Lagrange finite elements provide C0-continuous interpolation between elements, Cp−1-continuity may be achieved with NURBS throughout an entire patch. In general it is interesting to investigate further whether such improved inter-element continuity will improve the quality of the approximate solution or not, particularly in cases where standard finite elements tend to lock.
REFERENCES
[1] T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement. Computer Methods in Applied Mechanics and Engineering, 194:4135–4195, 2005.
[2] J. A. Cottrell, T. J. R. Hughes, and Y. Bazilevs. Isogeometric Analysis: Toward Integration of CAD and FEA. John Wiley & Sons, Chichester, England, 2009.
[3] O. C. Zienkiewicz and R. L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Elsevier Butterworth and Heinemann, Oxford, England, 6th edition, 2005.
Knut Morten Okstad, Kjell Magne Mathisen and Trond Kvamsdal
Figure 2: Compression of a thick cylinder: Deformed configuration with σxz Cauchy stress for the 32 × 16 × 1 mesh with quadratic NURBS (left), and maximum vertical deflection for all meshes (right).
[4] P. Wriggers. Nonlinear Finite Element Methods. Springer, Berlin, Germany, 2008.
[5] C. Miehe. Aspects of the Formulation and Finite Element Implementation of Large Strain Isotropic Elastcity. International Journal for Numerical Methods in Engineering, 37:1981–
2004, 1994.
[6] N. B¨uchter, E. Ramm, and D. Roehl. Three-dimensional Extension of Non-linear Shell Formulation based on the Enhanced Assumed Strain Concept. International Journal for Numerical Methods in Engineering, 37:2551–2568, 1994.
[7] S. Reese, P. Wriggers, and B. D. Reddy. A New Locking-free Brick Element Technique for Large Deformation Problems in Elasticity. Computers and Structures, 75:291–304, 2000.
[8] T. Elguedj, Y. Bazilevs, V. M. Calo, and T. J. R. Hughes. B and F Projection Methods for Nearly Incompressible Linear and Non-linear Elasticity and Plasticity using Higher-order NURBS Elements. Computer Methods in Applied Mechanics and Engineering, 197:2732–
2762, 2008.
23rd Nordic Seminar on Computational Mechanics NSCM-23 A. Eriksson and G. Tibert (Eds)
KTH, Stockholm, 2010