As a starting point for this class of elements we may consider a standard functional of Reissner type given by
Il = - 2 1 sn (LB)TD LO dR - z 1 /fl ST Q-’ S dR + In ST( V w - 8) dR - I w q dR + IT,,,
= stationary (5.47)
in which approximations for w , 8 and S are required.
Three triangular elements designed by introducing ‘bubble modes’ for rotation parameters are found to be robust, and at the same time excellent performers.
None of these elements is ‘obvious’, and they all use an interpolation of rotations that is of higher or equal order than that of w. Figure 5.14 shows the degree-of-free- dom assignments for these triangular elements and the second part of Table 5.2 shows again their performance in patches.
The quadratic element (T6S3B3) was devised by Zienkiewicz and L e f e b ~ r e ’ ~ starting from a quadratic interpolation for w and 8. The shear S is interpolated by
Elements with rotational bubble or enhanced modes 197
Fig. 5.14 Three robust triangular elements (a) the T6S3B3 element of Zienkiewicz and Lefebvre,’’ (b) the T6S161 element of Xu,*’ (c) the T3SIBlA element of Arnold and Falk.30
a complete linear polynomial in each element, giving here six parameters, S. Three hierarchical quartic bubbles are added to the rotations giving the interpolation
6 3
0 = c N , ( L k ) 8 , + c a N J ( L k ) a 8 ,
I = I / = I
where N l ( L k ) are conventional quadratic interpolations on the triangle (see Sec. 8.8.2 of Volume l), and shape functions for the quartic bubble modes are given as
A N J ( L k ) = 1‘ ( L , L 2 L 3 )
Thus, we have introduced six additional rotation parameters but have left the number of w parameters unaltered from those given by a quadratic interpolation. This element has very desirable properties and excellent performance when the integral for Kh in Eq. (5.25) is computed by using seven points (see Table 9.2 in Chapter 9 of Volume 1) and the other integrals are computed by using four points. Slightly improved answers can be obtained by using a mixed formulation for the bending terms. In the mixed form the bending moments are approximated as discontinuous quadratic interpolations for each component and a u - as form as described in Sec. 11.4.2 of Volume 1 employed. Using this approach we replace the first integral in Eq. (5.47) as follows
(5.48)
7 1 /n(LO)TDLOdR +
All other terms in Eq. (5.47) remain the same. The mixed element computed in this way is denoted as T6S3B3M in subsequent results.
Since the T6S3B3 type elements use a complete quadratic to describe the displace- ment and rotation field, an isoparametric mapping may be used to produce curved- sided elements and, indeed, curved-shell elements. Furthermore, by design this type passes the count test and by numerical testing is proved to be fully robust when used to analyse both thick and thin plate Since the w displacement interpolation is a standard quadratic interpolation the element may be joined compat- ibly to tetrahedral or prism solid elements which have six-noded faces.
A linear triangular element [T3SlBl - Fig. 5.14(b)] with a total of 9 nodal degrees of freedom adds a single cubic bubble to the linear rotational interpolation and uses linear interpolation for M? with constant discontinuous shear. This element satisfies all count conditions for solution (see Table 5.2); however, without further enhancements it locks as the thin plate limit is a p p r ~ a c h e d . ~ ~ As we have stated previously the count condition is necessary but not sufficient to define successful elements and numerical testing is always needed. In a later section we discuss a ‘linked interpolation’ modifi- cation which makes this element robust.
A third element employing bubble modes [T3SlBlA - Fig. 5.14(c)] was devised by Arnold and Falk.30 It is of interest to note that this element uses a discontinuous (non-conforming) w interpolation with parameters located at the mid-side of each triangle. The rotation interpolation is a standard linear interpolation with an added cubic bubble. The shear interpolation is constant on each element. This element is a direct opposite of the triangular element of Morley discussed in Chapter 4 in that location of the displacement and rotation parameters is reversed. The location of the displacement parameters, however, precludes its use in combination with standard solid elements. Thus, this element is of little general interest.
The introduction of successful bubble modes in quadrilaterals is more difficult. The first condition examined was the linear quadrilateral with a single bubble mode (Q4SlB1). For this element the patch count test fails when only a single element is considered but for assemblies above four elements it is passed and much hope was placed on this condition.27 Unfortunately, a singular mode with a single zero eigen- value persists in all assemblies when the completely relaxed support conditions are considered. Despite this singularity the element does not lock and usually gives an excellent p erf ~r man ce.~’
Linked interpolation - an improvement of accuracy 199 T o avoid, however, any singularity it is necessary to have at least three shear stress
components and a similar number of rotation components of bubble form. No simple way of achieving a three-term interpolation exist but a successful four-component form was obtained by Auricchio and Taylor.28 This four-term interpolation for shear is given by
(5.49)
The Jacobian transformation J& is identical to that introduced when describing the Pian-Sumihara element in Sec. 11.4.4 of Volume 1 and is computed as
T o satisfy Eq. (5.27) it is necessary to construct a set of four bubble modes. An appropriate form is found to be
(5.51) in which j is the determinant of the Jacobian transformation J (Le. not the determinant of Jo) and Nh = ( 1 - E 2 ) ( 1 - v 2 ) is a bubble mode. Thus, the rotation parameters are interpolated by using
8 = N;G + ANh (5.52)
where N j are the standard bilinear interpolations for the four-noded quadrilateral.
The element so achieved (Q4S2B2) is fully stable.