Stress recovery and error estimation for shell structures

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng. 2000; 47:1825–1840

Stress recovery and error estimation for shell structures A. A. Yazdani1 , H. R. Riggs1;∗;† and A. Tessler 2 1 Department 2

of Civil Engineering; University of Hawaii at Manoa; Honolulu; HI 96822; U.S.A. NASA Langley Research Center; Computational Structures Branch; Hampton; VA 23681; U.S.A.

SUMMARY The Penalized Discrete Least-Squares (PDLS) stress recovery (smoothing) technique developed for twodimensional linear elliptic problems [1–3] is adapted here to three-dimensional shell structures. The surfaces are restricted to those which have a 2-D parametric representation, or which can be built-up of such surfaces. The proposed strategy involves mapping the nite element results to the 2-D parametric space which describes the geometry, and smoothing is carried out in the parametric space using the PDLS-based Smoothing Element Analysis (SEA). Numerical results for two well-known shell problems are presented to illustrate the performance of SEA=PDLS for these problems. The recovered stresses are used in the Zienkiewicz–Zhu a posteriori error estimator. The estimated errors are used to demonstrate the performance of SEA-recovered stresses in automated adaptive mesh re nement of shell structures. The numerical results are encouraging. Further testing involving more complex, practical structures is necessary. Copyright ? 2000 John Wiley & Sons, Ltd. KEY WORDS:

stress recovery; shell structures; error estimation; nite element

1. INTRODUCTION The nite element method is used extensively to model a variety of industrial problems. Although powerful computers, relative to those available just a few years ago, are readily available, the economic aspects of these problems are still important. To minimize the computational cost, an eective and reliable technique of post-processing is necessary for use in adaptive mesh re nement, such that we have a reasonable distribution of error throughout the computational domain. Recent interest in improved stress recovery procedures is driven in large measure by the eort to implement automated mesh re nement based on a posteriori error estimators. These error estimators, such as that proposed by Zienkiewicz and Zhu [4], require a more accurate stress eld to compare with the consistent nite element stress eld. Numerous contributions to stress recovery

∗ Correspondence

to: H. R. Riggs, Department of Civil Engineering, University of Hawaii at Manoa, 2540 Dole St., Holmes Hall 383, Honolulu, HI 96822-2382. U.S.A. [email protected]

† E-mail:

Contract=grant sponsor: NASA; contract=grant number: NAG-1-1850

Copyright ? 2000 John Wiley & Sons, Ltd.

Received 5 January 1999 Revised 14 June 1999

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and error estimation for two-dimensional linear elliptic problems can be found [1–16], while there are relatively few applications to shell structures [17]. Stress recovery procedures can be classi ed as (a) local (i.e. element level), (b) patch-based, and (c) global. The widely used projection schemes, in which optimal stresses within an element are used to obtain more accurate nodal stresses, are an example of a local procedure. To obtain a smooth stress eld, averaging of either projected or consistent nite element nodal stresses is an example of a patch-based scheme. A more recent patch-based procedure is the Superconvergent Patch Recovery (SPR) technique introduced by Zienkiewicz and Zhu [16] and subsequently re ned by others [18–21]. The procedures proposed by Belytschko and co-workers [17; 22; 23] are also patch-based procedures. An alternative stress recovery procedure is SEA=PDLS, which has been developed over the last several years by the second two authors and co-workers. SEA=PDLS has a variational formulation based on the minimization of an error functional. Its numerical implementations is accomplished naturally by a nite element methodology (hence the name Smoothing Element Analysis). It was developed originally to lter, smooth, and interpolate spatially distributed experimental data [24]. It was then adapted and applied to improved stress recovery in nite element analysis (see [1–3; 25; 26]). The formualtion admits both global and patch-based recovery procedures. A distinguishing feature of SEA=PDLS is that it recovers a nearly C 1 continuous stress eld over the domain. It has been applied to 1-D and 2-D problems in elasticity, in which it has been shown to be both robust and capable of recovering a superconvergent stress eld when optimal sampling points are used. An order-of-accuracy argument has been proposed to provide a basis on which to develop a smoothing element mesh from the underlying nite element mesh [2]. The resulting mapping from nite element meshes to smoothing element meshes is based on the relative order of the nite elements and smoothing elements. A stabilization procedure, based on the addition of a curvature control term to the error functional, has been developed for use when an insucient number of sampling points would otherwise cause singularity [3]. Recently, the method has also been applied to built-up structures, such as might occur in the structural analysis of airframes [27]. In the latter application, the stresses in the built-up structure were recovered using a ‘patch’ approach, wherein each planar section of the structure constituted a patch. The 2-D smoothing element developed by Tessler is a three-node triangle, incorporating quadratic interpolation for the stress and linear interpolations for the two stress derivatives [1–3]. The error functional includes a penalty-constraint term to impose the C 1 continuity. Anisoparametric interpolation functions are used, and locking is avoided even for very large values of the penalty parameter [1]. The method is especially robust and eective when the smoothing element mesh is constructed with quadrilateral macro-elements. Each 5-node macro-element consists of four triangular elements in a cross-diagonal pattern. Recently, an equivalent, but more ecient, four-node smoothing element has been developed based on this macro-element [27]. The extension and application of SEA= PDLS to curved structures in three-dimensional space has not been accomplished to date; this is the topic of the present paper. It is assumed that the reader is familiar with the general problem of stress smoothing and error estimation. The basic formulation of SEA= PDLS will be reviewed in Section 2. Additional details of the method can be found in References [1; 3]. A strategy to apply the procedure to curved shell structures is then proposed. In Section 3 error measures are discussed. These measures are used to evaluate the accuracy of the recovery procedure and for use in adaptive mesh re nement. Numerical results are presented in Section 4 for two common, benchmark shell problems. Copyright ? 2000 John Wiley & Sons, Ltd.

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2. SMOOTHING ELEMENT ANALYSIS (SEA) 2.1. Standard technique Let = {x ∈ R d } denote the problem domain, where d is the spatial dimension of the problem (i.e. 1, 2 or 3) and x = {xi }; i = 1; : : : ; d, is a position vector in Cartesian coordinates. We assume the nite element stress eld, represented by b h (x) in vector notation, is de ned over and has been obtained by means of a discretization of with charateristic element size h. The smoothed stress eld, b h (x), is to be recovered from b h (x) via a penalized discrete-least-squares variational formulation. The error functional involves scalar quantities only, and so each component of bh (x) is recovered independently. The nite element stress eld is sampled at xq ; q = 1; 2; : : : ; N , to obtain the set of stresses {qh (xq )}. To minimize the error functional, we adopt the nite element Snel e methodology and therefore discretize with nel smoothing nite elements such that = e=1

, where e is the domain of smoothing element e. Within our smoothing element model, we use element-based interpolation functions to obtain C 0 continuity for the primary variables: stress, s , and the independent quantities is ; i = 1; : : : ; d, whose mathematical interpretation will soon be established. The error functional can be written as   Z nel N d P P 1P h s 2 e (2−d)=d s s 2 ( ) wq [q −  (xq )] +  (x)(; i − i ) d

(1) =

q=1 e=1

e i=1 in which wq is a discrete weight for the sample point xq ; is a normalization factor, which would typically be the sum of the discrete weights;  is a dimensionless penalty parameter; (x) is a weight density function; and ;si is the partial derivative of s with respect to xi . The rst term in equation (1) represents the error between the smoothed stress eld and the sample data. The second term represents a penalty functional which, for  suciently large, enforces the derivatives of the smoothed stress eld to be equal to the is . Because is are interpolated with continuous functions, the stress eld is (nearly) C 1 continuous. The discrete weights wq are introduced in equation (1) so that sample data known to be of higher accuracy can be assigned more weight than less accurate data. Similarly, the weight density (x) allows the enforcement of C 1 continuity to be relaxed in certain regions of and more strictly enforced in others. The factor ( e )(2−d)=d reduces to hes , 1, and (approximately) 1=hes for the one-, two- and three-dimensional cases, respectively, where hes represents the size of element e. If wq = 1; (x) = 1, and = N , then the error functional for the two-dimensional case (d = 2; i = x; y) is  Z nel N P 1 P h s 2 s s 2 s s 2 [ −  (x)] +  [(; x − x ) + (; y − y ) ] d

(2) = N q=1 q e=1

e A two-dimensional smoothing element based on equation (2) has been proposed and discussed in detail in [1; 2]. Numerical results show that the smoothed stress eld is: (1) superconvergent when optimal sampling stresses are used; (2) more accurate than the underlying nite element stress eld; and (3) C 1 continuous. Results also indicate that the proposed smoothing technique can recover more accurate stresses than does the Superconvergent Patch Recovery (SPR) technique. Because the recovery is done independently for each stress component, there are only 3 degreesof-freedom per node. The coecient (‘stiness’) matrix depends only on the co-ordinates of the sampling points, and each stress component corresponds to a separate ‘load’ case. It has been shown that the recovery procedure can be computationally ecient, because the smoothing of a Copyright ? 2000 John Wiley & Sons, Ltd.

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coarse mesh nite element analysis can give results similar in accuracy to a much more re ned nite element analysis [3; 27].

2.2. Obstacles to extension of SEA=PDLS to 3-D shell structures Application of the smoothing technique presented in the previous section to surfaces in threedimensional space involves two main obstacles. The rst obstacle is that in general the actual structure and the discretized mesh will not be geometrically identical. Hence, the smoothing mesh and the nite element mesh may not describe the identical curve or surface. The two meshes will be identical only if there is a 1-to-1 mapping between nite elements and smoothing elements and if the two elements have the same shape functions. Unless there is a 1-1 correspondence between nite elements and smoothing elements, mapping points with arbitrary co-ordinates between the meshes is dicult if the meshes do not de ne the identical geometric body, i.e. there are two separate surfaces with no direct link. In this case, a mapping procedure is required; in general, such a mapping will neither be simple nor unique. For example, one ‘simple’ approach might be to use a normal vector projection scheme, where the normal vector at a point on one mesh is used to project to a point on the other mesh. Although the normal projection approach oers the potential for mapping complete surfaces, it has major diculties. Obvious problems occur when such a procedure is applied to faceted surfaces, which involve discontinuous surface normals. Another mapping approach is to assume that the actual surface is described, perhaps piecewise, by 2-D parametric coordinates, and that the nite element mesh and the smoothing element mesh are both based on the same 2-D parametric description. Hence, any point in both meshes can be related to a common coordinate (the parametric coordinate). The stresses can then be smoothed in the 2-D parametric space, and the smoothing technique is identical to the 2-D procedure described in the previous section. That is, if both meshes can be mapped to the same parametric space, the smoothing procedure explained previously can be used. The second obstacle is that even if there is a 1-1 correspondence between FEA and SEA meshes, the issue of the gradients, and their meaning, leaves some ambiguity. That is, the meaning of a continuous stress gradient is not obvious when the surface is not smooth, e.g. a curve with a kink or a faceted shell. Consider the circular cantilever, subjected to an end load, which is modeled with two straight nite elements and smoothing elements (1-to-1 mapping), as shown in Figure 1. In the actual problem, the gradient of the moment is continuous with respect to the arc length. However, it is not continuous with respect to the directions of elements 1 and 2. That is, the gradient in moment of element 1 along the direction of element 1 is not continuous with the gradient in moment of element 2 along the direction of element 2. A similar situation occurs for faceted surfaces. This is a particular problem for SEA=PDLS, because it attempts to recover a C 1 continuous stress eld.

2.3. Adapted technique for 3-D shell structures In this paper we smooth in a 2-D parametric space, such that the SEA procedure explained previously is directly applicable. The procedure consists of four steps: (1) calculate the nite element solution in the real space which is a 3-D shell structure; (2) map the nite element Copyright ? 2000 John Wiley & Sons, Ltd.

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Figure 1. Circular arc cantilever modeled by two straight nite elements.

solution to the corresponding 2-D parametric space; (3) apply the standard smoothing technique on the 2-D space; and (4) map the smoothed solution from 2-D parametric space to the real space. Three-dimensional curves can be represented in a 1-D parametric space by the arc length. Presumably, a mapping can be obtained then between the arc length and the FEA and SEA meshes. For surfaces, a 2-D parametric space must be de ned. If the surface is de ned (piecewise) analytically, such a representation is available, at least for patches using the same parametric de nition. For example, a circular cylindrical section can be represented by cylindrical co-ordinates, and the two-parameter space may be de ned by the two parameters z and r (r is the radius). A spherical section can be represented by spherical co-ordinates, and the two-parameter space may be de ned by  and  (see Figure 2). Note that for many 3-D shell structures, the geometry is de ned by such 2-D parametric spaces. If the surface is described piecewise, then smoothing can be carried out independently over ‘patches’. If so, the recovered stress eld would be discontinuous along patch boundaries. Alternatively, global smoothing can be carried out if the  degrees of freedom between patch boundaries are disconnected to allow discontinuous gradients between patches. In this case, the recovered stress eld would be continuous between patches, and C 1 continuous within each patch. For shell problems, in principle, one can smooth either stress resultants or actual stresses. In the following we are concerned with smooth surfaces and we focus on stress resultants. However, for structures with actual ‘kinks’, such as faceted shells, some of the stress resultants may not be continuous along the patch interfaces, and it would be inappropriate to enforce continuity. This situation, which is beyond the scope of this paper, requires further investigation and development.

2.4. Shell element The shell problems considered herein have been analysed with a ve-node, quadrilateral Mindlin shell element. The element is actually a macro-element consisting of four triangular MIN3S elements in a cross diagonal pattern, which is the preferred meshing scheme for this element (Figure 3). The theory of MIN3S can be found in References [28–30]. The fth node of the quadrilateral is an interior node, common to the four triangles and located at the intersection of Copyright ? 2000 John Wiley & Sons, Ltd.

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Figure 2. Mapping from the 3-D real spaces to the 2-D parametric spaces.

Figure 3. A patch of four triangular shells (MIN3S) elements in a cross-diagonal pattern.

Figure 4. Illustration of the two sampling strategies.

the diagonals. In each triangle, the in-plane stress resultants and bending moments are constant, while the transverse shear forces vary linearly. 2.5. Optimal stress points If the geometric shape of the ve-node quadrilateral element is rectangular, then the average of the stresses from the four constant stress triangles, assigned to the location of the fth node, are optimal in the sense of Barlow [31]. This is readily shown following Barlow’s procedure. It is demonstrated here for the in-plane stresses; the results are directly applicable to the bending moments as well. De ne a general quadratic displacement eld by )   ( 2 ax + bxy + cy2 u = (3) v dx2 + exy + fy2 Copyright ? 2000 John Wiley & Sons, Ltd.

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where x and y are the co-ordinates; a; b; c; d; e and f are constant coecients. The corresponding strain eld is       2ax + by     x   u; x   y v; y ex + 2fy = = (4)        

xy u; y + v ; x (b + 2d)x + (2c + e)y For a three-node triangular element the nite element strain eld is calculated by {} = [B]{un } or

   1  x  1  y = 0   3 2 −2 3

xy 1

0 1 1

2 0 2

0 2 2

3 0 3

(5)  0  3  h u1 v1 u2 v2 u3 v3 iT 3

(6)

where i = xk −xj and i = yj −yk with a cyclic permutation of the indices (i = 1; 2; 3; j = 2; 3; 1; k = 3; 1; 2). For a rectangle with length L and height H; the average of the strain elds in the four triangular elements is     aL + bH=2     x   y fH + eL=2 = (7)      

xy dL + eH + (bL + eH )=2 It is easily veri ed that equation (7) corresponds to the exact strain at the fth (interior) node, calculated from equation (4). Hence, this post-processing strategy gives us an optimal stress which is one order more accurate than the stress at an arbitrary point. It can be readily shown that the above strategy for obtaining the optimal stress at the cross diagonal is also valid for a parallelogram. However, it does not hold for a general quadrilateral; in this case the procedure must be considered approximate. It also does not hold for the transverse shears, which are not constant within the triangles. Nevertheless, the procedure is used herein to approximate ‘optimal’ values of all stresses in a macro-element. 2.6. Sampling strategies Two strategies for sampling the nite element stress eld on a shell structure are proposed. First, we use the above averaging procedure to obtain optimal stresses (SEAOPT in Figure 4), resulting in one point per quadrilateral. The second strategy is to sample the stresses at the centre of each triangular element (SEACTR in Figure 4), resulting in four points per quadrilateral.

3. ERROR MEASURES To reduce engineering time spent in nite element model development, automatic adaptive niteelement mesh re nement procedures have been developed. These procedures involve an iterative sequence of solution, error estimation, and mesh re nement to improve accuracy where it is required. Error measures have been developed to indicate when solutions need improved accuracy Copyright ? 2000 John Wiley & Sons, Ltd.

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and hence a ner mesh. Error indicators point to those regions that need a ner discretization. We review here the main error measures used in this study. The energy error norm between the reference (exact) stress eld, b, and the nite element stress eld, bh , may be de ned as [4] Z kek =

−1

h

(b−b )D

1=2



h

(b−b ) d

=

nel P

1=2 kei k

2

(8)

i=1



where nel denotes the total number of elements and the error in element i is kei k. For practical problems, the exact solution is unknown, and the error in the nite element stress eld can be approximated as 1=2

Z

(bs −bh )D−1 (bs −bh ) d

kek ˆ =



nel P

=

1=2 keˆi k2

(9)

i=1



in which the recovered eld bs replaces the exact stress eld b. Similarly, the estimated error in element i is keˆi k. However, for non-uniform meshes (the element areas are not the same through the whole domain), the average element error density is useful:  ˆ = i

1 Ai

Z

1=2

(bs −bh )D−1 (bs −bh ) d

(10)

i

To assess the global accuracy of a smoothing procedure, the energy norm of the error between smoothed (estimated) stress eld, bs , and the reference (exact) stress eld, b, is used: Z kek ˜ =

1=2

(bs −b)D−1 (bs −b) d



 =

nel P

1=2 ke˜i k2

(11)

i=1

where the error in element i is ke˜i k. Note that for perfect stress recovery, kek ˜ will be zero. To facilitate error visualization, following [32] it is convenient to use the natural logarithm of the normalized estimated error density within an element as follows:    keˆi k2 A (12) ˆ i = Loge Ai kek ˆ 2 + kuh k2 where kuh k denotes the global strain energy of the FE stresses.

4. NUMERICAL RESULTS The objectives of this section are to (1) evaluate the adapted smoothing technique for 3-D shell structures; (2) compare the performance of the two sampling strategies discussed in Section 2.6 (SEAOPT and SEACTR ); and (3) evaluate the performance of SEA-recovered stresses in error estimators for adaptive mesh re nement. Two common benchmark shell problems are used, the Scordelis–Lo roof and the pinched hemisphere [33]. For each of the following problems, the ‘exact’ (reference) stress resultants at the centre of each triangle of an initial mesh are obtained from a much more re ned nite element mesh as Copyright ? 2000 John Wiley & Sons, Ltd.

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follows: (i) Map the centre of each triangle of the initial mesh to the two-dimensional parametric space; (ii) Map the stress resultants at the centre of each triangle of the re ned mesh to the twodimensional parametric space; (iii) Find the nearest point of the re ned mesh to the desired point of the initial mesh in parametric space; (iv) Approximate the exact stress resultants with the values obtained from the nearest point found in (iii). Although this procedure is fairly simple (e.g. no interpolation is done), the reference meshes were so ne that the reference solution so obtained is considered to be adequate. In the following we focus on the membrane components (i.e. Nx ; Ny ; Nxy ) because for both problems these components are much more signi cant than the bending components (i.e. Mx ; My ; Mxy ; Qx ; Qy ). However, the recovery procedure is identical for all quantities, and both membrane and bending stress resultants have been recovered and used to evaluate the errors (equations (8) – (12)). 4.1. Scordelis–Lo roof subjected to self-weight This standard test problem [33; 34] has been discussed numerous times in the context of shell elements [35; 36]. It consists of a reinforced concrete, uniform, single span, cylindrical roof as shown in Figure 5. It is simply supported by walls at each end and free along the sides. The roof is loaded by its own weight. The geometric and material properties of the roof used in this example are E = 4:32 × 108 ; R = 25;

 = 36:7347;

t = 0:25;

L = 50;

 = 0:0  = 40◦

where t is the thickness of the roof and L is the total span. As indicated in Figure 5, double symmetry is exploited and only 1=4 of the roof is modelled. The kinematic boundary conditions are ux = uy = z = 0

on AB

ux = y = z = 0

on BC

uz = x = y = 0

on CD

The initial nite element solution (denoted FEA) is obtained from a regular 6 × 6 mesh of 5-node quadrilateral elements, while the reference solution (REF) is computed from a 35 × 35 mesh. The stress resultants recovered by the smoothing procedure (SEAOPT and SEACTR ) are compared with the initial nite element and the reference solutions in Plates 1a and 1b. Five components of the stress resultants (Nx ; Ny ; Nxy ; Mx ; Qx ) are shown, where the subscripts refer to the local element co-ordinates shown in Figure 5. These values are used as an approximation of the values in cylindrical co-ordinates. The smoothing element analysis was carried out in the parametric space R × ; z. Following the order-of accuracy argument discussed in Reference [2], one quadratic smoothing Copyright ? 2000 John Wiley & Sons, Ltd.

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Figure 5. Scordelis–Lo roof problem—geometry.

macro-element was used per four quadrilateral shell elements. Speci cally, a 3×3 mesh of smoothing elements was used. In Plate 1a, comparison of the FEA solution with the REF solution reveals strange behaviour for the FE membrane force in the x direction (Nx ). We think the reason for this behaviour is most likely the relatively weak performance of constant stress membrane elements. From Plates 1a and 1b, we can conclude that the rst sampling strategy, SEAOPT , is slightly better than the second one, SEACTR . We can make approximately the same conclusion for those stress resultants which are not shown. From equations (8) and (9), we have computed the global energy norm of the real and the estimated error. The global energy norm of the error between the reference solution and nite element solution is ||e|| = 16:89 (REF). For the rst sampling strategy, ||e|| ˆ = 14:28 (SEAOPT ), while for the second strategy, ||e|| ˆ = 13:44 (SEACTR ). So, globally we can see again the better performance by using optimal stresses, because it gives a better estimate of the actual error: ˆ SEAOPT ¡||e|| ||e|| ˆ SEACTR ¡||e|| The distribution of the element error calculated from equations (8) and (9) is presented in Plate 2 and one can observe better agreement between smoothing using optimal stresses and the reference solution. To assess the global accuracy of the two proposed strategies, the energy norm of the error between smoothed (estimated) stress eld, bs , and the exact stress eld, b, is computed from equation (11). ||e|| ˜ = 9:97

(SEAOPT )

||e|| ˜ = 10:38

(SEACTR )

while

Again SEAOPT is better than SEACTR . Finally, the distribution of the element error calculated from Equation (11) is presented in Plate 3. We can observe that these values are much less than the values shown in Plate 2; that is, the SEA-recovered stress eld is more accurate than the nite element stress eld. Copyright ? 2000 John Wiley & Sons, Ltd.

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Figure 6. Spherical shell problem—geometry.

Regarding the computational time, the SEA analysis is generally faster than the FEA analysis because it involves a much smaller problem. For example, in this example the FEA analysis involved 510 degrees of freedom while the SEA analysis involved 75 degrees of freedom. The eciency of the mapping of stress data points to SEA elements is primarily an implementation issue. A production code would have a mapping between FEA to SEA elements, and therefore the mapping should be relatively fast. 4.2. Hemispherical shell problem This benchmark, which consists of a hemispherical shell (Figure 6), was proposed in [33] and it has been widely used [37–42]. The equator is a free edge so that the problem represents a hemisphere with four-point loads alternating in sign at 90◦ intervals on the equator. Both membrane and bending strains contribute signi cantly to the radial displacement at the loads [33]. An 8 × 8 mesh is used for a quarter of the hemisphere, as shown in Figure 6. Symmetry boundary conditions are used: ux = y = z = 0

on AB

uz = x = y = 0

on CD

The geometric and material properties are E = 6:825 × 107 ; R = 10;

t = 0:04;

 = 0:3  = 18◦ ;

F =1

where t is the shell thickness. Plate 4 compares the performance of the two sampling strategies. Note that the local quadrilateral coordinate system in which the results are reported is shown by x-y in Figure 6, and these are used to represent the values in spherical coordinates. The membrane components of the stress resultants (i.e. Nx ; Ny ; Nxy ) from the two proposed strategies (denoted by SEAOPT and SEACTR ) are compared with the initial nite element and the reference solutions. Again, a 4 to 1 ratio of nite elements and smoothing macro-elements was used; i.e. a 4 × 4 smoothing mesh in spherical co-ordinates was used. The reference solution is computed from a re ned 40 × 40 mesh of nite elements. The results for Nx seem to be good, and the magnitudes of Nxy are small and hence they Copyright ? 2000 John Wiley & Sons, Ltd.

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are relatively uninteresting. The results for Ny are not as good as expected; from these results, we cannot conclude very well which sampling strategy is better (SEAOPT or SEACTR ). To assess the global accuracy of the smoothed stresses, the energy norm of the error between smoothed (estimated) stress eld, bs , and the exact stress eld, b, is computed from equation (11): kek ˜ = 0:13751

(SEAOPT )

kek ˜ = 0:08092

(SEACTR )

while

The rst strategy, which is based on using the sampling points at the optimal points, is globally less accurate than the second strategy, which is based on using the center of gravity of each triangular element as a sampling point. The distribution of the element error calculated from equation (11) is presented in Plate 5. The results demonstrate clearly that the stresses are highly localized under the concentrated loads. It has been demonstrated previously that, in the neighbourhood of a stress concentration, using sampling points nearer to the location of the local maximum results in better performance than the use of optimal points which are located farther away [3]. This result is con rmed here by the better performance of SEACTR compared to SEAOPT . The performance of the smoothing procedure for the roof is better than for the sphere. The reason for the relatively weaker performance for the latter problem is that the stress eld is too localized and the initial (coarse) mesh simply did not pick up adequately the high gradients. To illustrate the localization issue further, and to demonstrate the eectiveness of SEA in the context of adaptive mesh re nement, in the next section we use the smoothing procedure in conjunction with an adaptive mesh re nement strategy to analyse the hemispherical shell. 4.3. Mesh re nement of the hemispherical shell Previously we used an 8 × 8 mesh of 64 quadrilateral elements for one-quarter of the hemisphere. Based on the estimated error of the nite element solutions, we applied an adaptive mesh re nement strategy to obtain a sequence of re ned meshes. It should be noted that the initial 8 × 8 mesh was re ned; i.e. the 64 element faceted surface was re ned rather than the exact hemisphere. Both nite element and smoothing meshes with the total number of elements, nel , for four consecutive steps are presented in Figure 7. The SEACTR sampling strategy was used. As a local error measure we have used the element error density de ned by equation (10). At each step we have re ned 10 per cent of the total number of elements with the largest error densities. At the same time we have re ned the corresponding part of the smoothing mesh. Note that in the previous sections, we worked with uniform meshes, so we used the element error de ned in equation (9). However this problem involves non-uniform meshes, and so we applied equation (10) to obtain the element error densities. Plate 6 shows the distributions of the normalized error density. Note that the maximum local error value of mesh 3 is slighly higher than mesh 2; however the global error of mesh 3 is much less than mesh 2 (see Figure 8). This phenomenon may be explained by the presence of the distorted elements in mesh 3. That is, the larger local errors are a result of the distorted elements in mesh 3. Copyright ? 2000 John Wiley & Sons, Ltd.

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Plate 1a. Scordelis-Lo roof problem – membrane components of the stress resultants (Nx, Ny and Nxy)

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Plate 1b. Scordelis-Lo roof problem – bending components of the stress resultants (Mx and Qx)

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Plate 2. Scordelis-Lo roof problem – distribution of the energy norm of the element error (Equations (8) and (9))

Plate 3. Scordelis-Lo roof problem – distribution of the energy norm of the element error (Equation (11))

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Plate 4. Spherical shell problem – membrane components of the stress resultants (Nx, Ny and Nxy)

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Plate 5. Spherical shell problem – distribution of the energy norm of the element error (Equation (11))

Plate 6. Spherical shell problem – distribution of the natural logarithm of the normalised error density (Equation (12))

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Figure 7. Spherical shell problem—adaptive mesh re nement strategy.

In Figure 8 the eectiveness of adaptive mesh re nement relative to uniform mesh re nement is compared. This gure demonstrates the good performance of the proposed stress recovery and error estimating technique in the context of adaptive mesh re nement. Signi cantly, the proposed error estimator has identi ed the most critical parts of the mesh at the zone of the stress concentration. Copyright ? 2000 John Wiley & Sons, Ltd.

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Figure 8. Spherical shell problem—convergence of the global energy norm of the error (equation (9)).

5. CONCLUSIONS Based on the numerical results presented, the following conclusions may be made: (1) The proposed method of stress recovery for 3-D shell structures (i.e. applying SEA = PDLS in a 2-D parametric space) is quite promising. Application to additional, more complex problems is required to evaluate the approach more completely. (2) In general, the use of optimal FEA stresses result in higher accuracy than non-optimal stresses (SEAOPT vs. SEACTR ). However, in the neighbourhood of a high local stress gradient, SEACTR performs better than SEAOPT because more data points closer to the local maximum are provided. (3) The use of SEA-recovered stresses in error estimators for adaptive mesh re nement results in a substantial improvement in computational eciency compared to uniform mesh re nement. ACKNOWLEDGEMENTS

The rst two authors gratefully acknowledge the nancial support provided by NASA grant NAG-1-1850. REFERENCES 1. Riggs HR, Tessler A, Chu H. C 1 -Continuous stress recovery in nite element analysis. Computer Methods in Applied Mechanics and Engineering 1997; 143:299 –316. 2. Tessler A, Riggs HR, Macy SC. A variational method for nite element stress recovery and error estimation. Computer Methods in Applied Mechanics and Engineering 1994; 111:369 –382. 3. Tessler A, Riggs HR, Freese CE, Cook GM. An improved variational method for nite element stress recovery and a posteriori error estimation. Computer Methods in Applied Mechanics and Engineering 1998; 155:15–30. Copyright ? 2000 John Wiley & Sons, Ltd.

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