Mats Ander* and Alf Samuelsson Department of Structural Mechanics
Chalmers University of Technology, Göteborg, Sweden e−mail: matsan@sm.chalmers.se
ABSTRACT
Summary A computer code for geometrically nonlinear analysis of thin structures using a p-hierarchical solid element approach has been developed using the MATLAB software. The p-hierarchical element formulation using integrals of Legendre polynomials as basis functions together with a Gauss-Lobatto quadrature has been implemented efficiently by aid of the MATLAB built in features for polynomials and multidimensional arrays. Numerical results for slender beams and thin shells will be presented
1.1 Introduction
A family of p-hierarchical solid elements has been implemented for geometrically nonlinear analysis. By use of integrals of Legendre polynomials as element basis functions [1] the conditioning properties of the element tangent stiffness matrix allows for analysis of thin shells.
The key feature of the method is that only translational variables are employed on element and global structural level and that the finite deformations are expressed in terms of the displacement gradients on continuum level in a total Lagrange formulation [2]. The so called core equations where the displacement gradients act as 'material point' degrees of freedom are exact including geometrical nonlinearities. With this formulation a tangent stiffness matrix is deduced already at the core level. The core equations are independent of the element geometry and the choice of element basis functions. The core equations are transformed into finite element equations by use of a transformation matrix and by integration over the element volume in the reference configuration. In this transformation phase the displacement field approximation and the mapping of the element geometry enters. The transformation matrix consists of derivatives of the element basis functions. It follows that the transformation matrix for each integration point in the element is computed only once. Only a few elements are employed in the interior of the region. At the boundaries an exponential h-refinement is used.
1.2 p-hierarchical elements
In the present work the use of a p-hierarchical Legendre type solid element for thin shells is investigated. In the through thickness a low order polynomial approximation of the
displacement field is tested. The solid element displacement field u = [u,v,w]T is approximated
by u hj functions respectively. The nodal degree of freedom number i is denoted by vibwhile vhj is the j:th hierarchical degree of freedom. Note that the hierarchical base functions all have zero values at the basic nodes and that the hierarchical degrees of freedom do not associate to physical nodal points, but should simply be interpreted as unknown amplitudes of the hierarchical functions.
In the p-hierarchical formulation for the 3D solid element the basis functions can be
distinguished into 27 groups. The first 8 includes the 8 standard tri-linear basis functions for the 8 vertices. The following 12 groups contain edge functions for the 12 edges, the next 6 groups base functions for the 6 faces and, finally, there is one group containing the internal volume basis functions. The number of base functions in the last 19 groups depends on the choice of polynomial degrees and hierarchical space. For the coded 3D solid element the polynomial degree is fixed to q=2 in the through thickness direction while it is varied from p=2 to p=9 in the other directions. This means that the implemented elements are restricted to model thin structures where a linear strain can be assumed through the thickness.
1.2.1 p-hierarchical shape functions for hexahedral elements
We define the cube {-1 ≤ ξ, η, ς ≤ 1}, Figure 1, as being the standard element domain Ωst.
Figure 1. Standard hexahedral element Ωst
The shape functions Nm are obtained as products of φ-functions, Figure 2, as )
The first two φ-functions are the simple linear functions.
)
Higher order φ-functions are defined as integrals of the Legendre polynomials
−
∫
Here Pi-1 are the Legendre polynomials which can be generated from Rodrigues' formula:
{
n}
n n
n n x
dx d x n
P ( 1)
2
! ) 1
( 2 −
= ⋅ ( 4)
It follows that the φ-functions (except the linear ones) vanish at the boundaries ξ ± 1 and that the derivatives of the φ-functions are orthogonal on the interval [-1,1].
Figure 2. φ-functions on the interval -1 ≤ ξ ≤ 1.
The base functions and their derivatives are deduced numerically using the MATLAB features for polynomials.
1.3 Numerical integration and continuation method
A Gauss-Lobatto quadrature procedure is adopted for the numerical integration [1] and a dominant displacement component driven continuation method is used for tracing the nonlinear structural response [3]. Secondary branches on the equilibrium path can be triggered by introducing imperfections in the structural geometry or in the load.
1.4 Numerical examples
A geometrically nonlinear analysis of a transversally loaded cantilever beam modelled with p-hierarchical Legendre type solid elements shows excellent agreement with an analytical non-linear beam solution. The developed code has been verified to work well for thin shells and beams in linear analysis. Numerical tests on the so called Scordelis and Lo cylindrical shell roof show that the present formulation gives accurate results in good agreement with analytical deep shell theory as well as with numerical results reported by other authors, see fig. and table below.
L = 6.0 m T =0.03 m R = 3.0m θ = 40°
Scordelis and Lo cylindrical shell roof loaded by a uniformly distributed vertical load fZ = 0.625×104 Pa.
Graded 3-element mesh of one quarter of the shell.
Reference values for the vertical displacements wB = -0.0361m, wC = 0.00541m from [4]
Polynomial degree p
NDOF Displacement wB
Displacement wC
Potential
energy π Conditioning number of K 1-element with q = 2 and p = 5, 7, 9
5 138 -0.035286 0.004714 -138.894 8.696367×107
7 249 -0.036136 0.005429 -144.468 9.817559×107
9 396 -0.036149 0.005438 -144.548 1.037179×108
3 -element graded mesh with q = 2 and p = 5, 7, 9
5 414 -0.036049 0.005163 -142.311 1.100866×108
7 747 -0.036135 0.005437 -144.504 1.213742×108
9 1188 -0.036149 0.005440 -144.550 1.254000×108
REFERENCES
[1] B.A. Szabo and I. Babuska. Finite Element Analysis. Wiley, New York, (1991).
[2] M.K.A. Ander and A.G. Samuelsson. Finite element analysis of geometrically non-linear structures using translational variables. Int. J. Num. Meth. Eng., 46, 1367−1383, (1999).
[3] M. Ander. On postbuckling analysis of thin-walled structures, numerical and statistical approaches. Publication 99:16, Department of Structural Mechanics, Chalmers University of Technology, S-412 96 Göteborg, Sweden, (1999).
[4] J.-L. Batoz and G. Dhatt. Modélisation des structures par éléments finis. volume 3, Coques, Hermes, (1992).