- Email: [email protected]
local finite element analysis
Computers & Strucrures Printed in Great Britain.
Vol.40,No 4,
pp. 1027-1031,
ITERATIVE
cws-7949/91 s3.00 + 0.00 Pergmm Press plc
1991
GLOBAL/LOCAL FINITE ANALYSIS
ELEMENT
J. D. WHITCOMB Aerospace Engineering Department, Texas A&M University, College Station, TX 77843-3141, U.S.A. (Received 26 June 1990) Abstract-In spite of the advances in computer technology, there is still a need for more computationally efficient methods for performing stress analysis. One approach which is receiving increasing attention is global/local analysis. Such analyses can take a variety of forms. The form described herein uses two distinct meshes (one global and one local), but retains the same level of accuracy as one would obtain if one was to use a single refined global mesh. The accuracy is retained by using an iterative procedure to enforce equilibrium between the global and local regions. The procedure was tested for three configurations. The good performance observed indicates that the iterative global/local procedure warrants further examination.
INTRODUCTION of supercomputers and ‘super workstations’ has made it possible to analyze many problems which were not tractable just a few years ago. And one can confidently expect that computational power will continue to increase at a rapid pace. In spite of the advances which have been made and those which are expected in the near future, there is a need to reduce the computational effort required to obtain reliable solutions. By increasing the efficiency of an analysis one can either perform the analysis on a less powerful and less expensive computer or make fewer simplifying assumptions in the analysis. One approach which has been proposed for improving efficiency is global/local analysis. The basic concept is that a coarse global model can be used to obtain appropriate boundary conditions for some local region where a much more refined mesh is required. Many strategies have been proposed for implementing this process. References [l-3] discuss some of the possibilities. Of course, engineers have been performing global/local analysis of some form for many years. For example, detailed analysis of a joint in an airplane would not include an entire airplane in the analytical model. One of the most difficult aspects of global/local analysis in assessing the accuracy of such an analysis as compared to the accuracy of a model which explicitly includes local refinement in the global model. These two analyses are shown schematically in Fig. 1 for a plate with a circular hole. Only one fourth of the plate is modeled because of symmetry. Global/local analysis involves two distinct finite element meshes (Fig. 1a). Figure 1b shows the finite element model when the local refinement is included as part of the global model. The objective of this paper is to describe a procedure for global/local analysis which gives the same degree of accuracy as one would obtain if the local The advent
refinement was explicitly included in the global model. For this initial study only linear problems will be considered. Also, local refinement in only one subregion will be discussed. The theoretical basis for the algorithm will be described first. Then the configurations used to test the algorithm will be described. Finally a few results will be presented which demonstrate the performance of the algorithm. It should be noted that the algorithm described herein is similar to one described in ref. [4]. The approaches differ primarily in terms of derivation of the algorithm, convergence criteria, and guidelines for stability of the algorithm.
THEORY Before introducing any equations, an example problem will be discussed to highlight the basic problem. Suppose the global model for a sheet is that
shown in Fig. la. The model is analyzed and nodal displacements are obtained. Now suppose that, in fact, there is a hole in the center of the sheet which is not included in the global model. A local model is developed for the region around the hold and displacements are specified along the boundary of the local model. The magnitudes of these displacements are those calculated using the global model. The local model is then analyzed and displacements and stresses are obtained. This is probably the simplest of global/local procedures. But there is a potential problem due to the differences in the stiffness of the global model and the local model in the region near the hole. If the differences in the stiffnesses are significant, the displacements from the global model (which were used as boundary conditions on the local model) are not accurate. The problem is to make the global model deform as though the local refinement was present.
1027
1028
J. D.
WHITCOMB
/’ Kg2 Global Model (b)LocaUy refined global model (traditional)
Local Model (a) Global/local
models
Fig. 1. Global/local and traditional models of sheet with hole.
The effects of local refinement can be included by using a Raphson iterative procedure. In analysis the governing equations
on global response modified Newtonlinear finite element can be expressed as
K6 =F.
(1)
For the global model in Fig. 1, the governing equations are (Kg’ + Kg*)6 = F,
displacements 6 obtained are not those in eqn (3), but 6 + S,. The 6, is the error in the displacements. An iterative procedure is proposed next which permits elimination of the error. Even though there are two finite element models one can consider the governing equations to be eqns (3). Because global/local analysis is being used, the matrix Kg’ + K’ is never formed. Instead, we use Kg’ + Kg2 as an approximation to Kg’ + K’ in the following modified Newton-Raphson procedure:
(2)
where Kg’ and Kg* are stiffness matrices for the subregions defined in Fig. la. The governing equations for the case in which local refinement is explicitly included in the global model are
1. Solve global model. 2. Solve local model using displacements global model as boundary conditions. 3. Calculate residuals + as
$ = (Kg’ + K’)6 - F. (Kg’ + K?6 = F,
from
(4)
(3)
where K’ is the stiffness matrix for the local model. If the displacements from a global analysis are used as boundary conditions for a local analysis, the
This is an element-by-element calculation, so the matrix Kg’ + K’ is never actually assembled. 4. Check for convergence (stop if tj sufficiently small).
1029
Iterative global/local finite element analysis 5. Calculate correction to global displacements as A6 = -(Kg’ + Kg*)-I$.
(5)
6. Go to step 2. This procedure has several attractive features. As promised, the solution obtained is that for the case in which the local refinement is explicitly included in the global model. The stiffness matrices for the global and local models only need to be decomposed once. Also, there is no need to anticipate where local refinement will be required when assemblying the global model; there can be one or more arbitrarily located refined regions. This algorithm is very similar to that often used in finite element analysis of elastic-plastic materials. In fact, the algorithm was initially conceived as one for a special kind of ‘material nonlinearity’. Refining a mesh by increasing the number of elements (in a displacement formulated finite element analysis) ‘softens’ the region being refined. This is directly analogous to yielding of an elastic-plastic material. A similar analogy can also be drawn for locking materials and selective stiffening of a finite element model. For elastic-plastic analysis, one popular approach is to use the linear global stiffness matrix in a modified Newton-Raphson procedure to iteratively drive the residuals to zero. This method is successful because the residuals are calculated accurately even though the stiffness matrix is only approximate. In the iterative global/local (IGL) analysis the residuals are calculated using eqn (4) which is accurate. As described earlier, the IGL method also uses an approximate stiffness matrix. Because of the similarity to material nonlinear analysis (which has been studied for many years), one can anticipate the response of the IGL procedure based on its material nonlinear analogue. For example, refining a mesh (which causes local softening) is expected to cause no significant numerical problems since the related material problem (yielding) has been well behaved. In contrast, experience has shown that the type of modified NewtonRaphson procedure described above has problems with locking (i.e. stiffening) materials. Hence, one can expect divergence problems with the global/local procedure when a local region is stiffened. In this paper under-relaxation is used to improve the convergence when a local region is stiffened. An under-relaxation factor of 0.5 was used. Hence, only half of the correction AS [eqn (S)] was used on each iteration. The IGL procedure has been described as though there was only one region in which local analysis is desired. Just as material yielding can occur in multiple disconnected subregions without causing numerical or conceptual difficulties, the IGL procedure can easily handle multiple disconnected local models. CAS 40,-
Fig. 2. Extreme test case for checking convergence of iterative global/local algorithm. Also, the procedure described above requires that the boundary nodes of the local model match up with nodes in the global model. However, the addition of smoothing tranformations would permit models which do not exactly match up.
CONFIGURATIONS
Three configurations were analyzed to evaluate the iterative global/local algorithm. The first configuration is that shown in Fig. 1, a sheet with a circular hole. The finite element meshes used are those in Fig. 1. Only one fourth of the plate is modeled. The quarter plate is 2 x 5 and the hole radius is 0.2. The load is parallel to the long axis of the plate. The second configuration is strictly for the purpose of testing the convergence properties of the algorithm. The global mesh is the same as that for the first configuration (Fig. la). Instead of a hole in the local mesh (as was the case for the first configuration), the entire local mesh is a ‘hole’. That is, the global mesh in Fig. la will be used to obtain the solution for the configuration in Fig. 2. This a fairly severe test of the algorithm. The third configuration is a plate with local stiffening. The global mesh is like that in Fig. la. The local model is identical to the shaded region in Fig. la. Hence, the mesh refinement is the same as the global and local models. The difference is the Young’s
1030
J. D. WHITCOMB Table 1. Comparison of displacements and stresses for a sheet with circular hole. Remote uY= 0.198 x IO-*
Displacementt
Non-iterative global/local 0
-0.15 -0.3 -0.3 -0.3 “1 “6 u9 08 01
Max. CT, Kr No. of iterations
kX < 10-d
*,
0
< 1o-5
CL, < lo-’
0
0
0
x x x x
IO-3 10-3 10-j lo-)
-0.1317 x -0.2962 x -0.3598 x -0.4266x
IO-’ IO-? IO-’ 1O-3
-0.1276x -0.3003 x -0.3847 x -0.4729 x
1O-3 1O-3 1O-9 10-j
-0.1267 x -0.3034 x -0.3977x -0.4955 x
1O-3 1O-3 lo-’ 1O-3
0.1 x 0.1 x 0.1 x 0.5 x 0
10-Z 10-r IO-2 10-j
0.1284 x 0.1187 x 0.1069 x 0.5334 x 0
1O-2 lo-* IO-* IO-3
0.1360 x 0.1234 x 0.1085 x 0.5399 x 0
10-r 1O-2 1O-2 10-j
0.1389x 0.1251 x 0.1090 x 0.5416 x 0
1O-2 IO-* 10-2 IO-’
0.4676 x 1O-2 2.12
0.5749 x 10-l 2.61 1
0.6033 x lO-2 2.74 2
$ IMx< 10-10
0.6144 x 1O-2 2.79 6
-0.1267 -0.3034 -0.3978 -0.4957
Locally refined global mesh 0
x x x x
IO-’ 1O-3 1O-3 IO-’
0.1390 x 0.1251 x 0.1090 x 0.5413 x 0
IO-2 1O-2 10-r IO-’
0.6144 x 1O-2 2.79 12
-0.1267 -0.3035 -0.3979 -0.4958
x x x x
1O-3 IO-3 10-j 1O-3
0.1390 x 0.1251 x 0.1090 x 0.5416 x 0
10-J 1O-2 10-2 1O-3
0.6145 x 1O-2 2.79 -
t Subscripts refer to node numbers, which are defined in Fig. 2. The u and v displacements are transverse and parallel to the load direction, respectively.
moduli. The global model has E = 1.0 throughout. The local model has a larger value for E. Hence, the
model was also analyzed. The results to be compared are displacements along the boundary of the local model and the peak crustress. The IGL algorithm was used with several tolerances on the maximum allowable residual $. The results are tabulated in Table 1. The non-iterative global/local solution is quite inaccurate in terms of both the displacements and the peak a,,. A specified tolerance of 10m4 for the IGL analysis required a single iteration and resulted in a.’ much improved estimate of displacements and peak stress. The results for tolerance of lo-’ are almost identical to those for a tolerance of 10 -lo and those for the locally refined global model. Thus, with six iterations (which is what was required for a tolerance of lo-‘) the IGL analysis gave essentially the same results as a locally refined global model. The configuration in Fig. 2 is a more severe test of the IGL algorithm, since the change in compliance of the local region between the global and local models is very severe. Although the convergence was slower than for the first configuration, the solution did converge. The slower convergence is not surprising, since the perturbation due to the ‘local refinement’ is much more severe. The calculated displacements for several different tolerances are presented in Table 2.
local model accounts for some locally increased thickness which is not included in the global model. The elements used were four node isoparametric quadrilaterals. Selective reduced numerical integration was used. Four point quadrature was used for the terms related to normal strains and single point quadrature was used for shear-strain related terms. The stresses presented herein were obtained by calculating the stresses at the quadrature points and extrapolating to the nodes using the procedure described in ref. [5]. The material in all cases was assumed to be isotropic. The Young’s modulus was 1.0 and the Poisson’s ratio was 0.3 for all models except the local models for configuration 3. For configuration 3 several different values for the Young’s modulus were used. They will be listed when the results are discussed. RESULTS AND
DISCUSSION
The sheet with a circular hole was analyzed using the IGL algorithm. For comparison a refined global
Table 2. Comparison of displacements for a sheet with a square cut-out Displacementt “3 ‘6 u9 “8 4 03 v6 09 08 01
No. of iterations
Non-iterative global/local 0
-0.15 x 10-J -0.3 x IO-’ -0.3 x lo-’ -0.3 x 10-j 0.1 x 10-Z 0.1 x IO-2 0.1 x IO-2 0.5 x 10-J 0 -
I),,, < low4 0
-0.2832 x -0.5258 x -0.7042x -0.7507 x 0.2837 x 0.2701 x 0.1754 x 0.8795 x 0 5
$,., < toe5
0
0
IO-’ IO-’ lo-’ IO-) lO-2 1O-2 lo-2 lo-’
-0.3056x -0.5443 x -0.6744x -0.7247 x 0.3096 x 0.2936 x 0.1822 x 0.9006 x 0 11
$m,,, < IO-’
10-j lo--’ lO-3 lO-3 lO-2 lO-2 IO-* lO-3
-0.3063 x -0.5410 x -0.6533 x -0.7031 x 0.3128 x 0.2965 x 0.1828 x 0.9011 x 0 26
k,,, < lo-“’
0
0
1O-3 lO-3 lo-’ lO-3 lo-* 1O-2 lO-2 IO-2
-0.3063 x -0.5409 x -0.6529 x -0.7027 x 0.3128 x 0.2965 x 0.1828 x 0.9010 x 0 50
Locally refined global mesh
IO-’ lo-’ IO-’ lo-’ lO-2 10m2 lO-2 10-J
-0.3064x -0.5410 x -0.6529 x -0.7028 x 0.3128 x 0.2966 x 0.1828 x 0.9011 x 0 -
lo-? lO-3 IO-’ lO-3 lO-2 lO-2 IO-’ IO-3
t Subscripts refer to node numbers, which are defined in Fig. 2. The u and v displacements are transverse and parallel to the load direction, respectively.
1031
Iterative global/local finite element analysis Table 3 Local mesh Young’s modulus
Iterations for convergence
Under-relaxation?
2 2.5 2.15 3.00 2.5 2.15 5
11 31 122 Diverged 8 8 20
No No No No Yes Yes Yes
The configuration in Fig. 3 tests the behavior of the IGL procedure when the local model has a larger stiffness than the corresponding region in the global model. Table 3 summarizes the results of the analysis. The convergence tolerance was set at lo-’ for all cases. As shown in Table 3, without under-relaxation the IGL is slow to converge or divergent when the local stiffness is more than twice the global stiffness. Use of under-relaxation greatly improves the stability of the IGL procedure. A converged solution was obtained with 20 iterations when the local stiffness was five times the global stiffness. CONCLUSIONS
An iterative algorithm was developed for global/local analysis which has essentially the same level of accuracy as one would obtain if local refine-
ment was explicitly included in the global mesh. Tests of the algorithm showed that it is quite stable and will converge even for very severe changes in local compliance. Since the location of local refinement can be selected after the analysis of the global model, the results of the global model can be used as a guide in deciding where local refinement is required. The purpose of this paper was to demonstrate a concept. The problems examined do not prove that it will be efficient for realistic problems. Further work is needed in which the procedure is applied to complicated configurations to determine the actual efficiency.
REFERENCES I. Hirai, Y. Uchiyama, Y. Mizuta and W. Pilkey, An exact zooming method, pp. 61-69. Finite Elements in Analysis and Design 1. Elsevier (1985). 2. C. Jara-Alamonte and C. Knight, The specified bound-
1.
ary stiffness/force SBSF method for finite element subregion analysis. Int. J. Numer. Meth. Engng 26, 1567-1578 (1988). 3. J. B. Ransome, Global/local stress analysis of composite structures. Master’s thesis, Old Dominion University (1989). 4. M. Panthaki and W. Gerstle, A new integrated, computer graphical design tool. Topical report RSI-0339, REjSPEC Inc., Albuquerque, New Mexico (1988). 5. E. Hinton, F. C. Scott and R. E. Ricketts, Local least squares stress smoothing for parabolic isoparametic elements. Int. J. Numer. Merh. Engng 9,235-256 (1975).
Vol.40,No 4,
pp. 1027-1031,
ITERATIVE
cws-7949/91 s3.00 + 0.00 Pergmm Press plc
1991
GLOBAL/LOCAL FINITE ANALYSIS
ELEMENT
J. D. WHITCOMB Aerospace Engineering Department, Texas A&M University, College Station, TX 77843-3141, U.S.A. (Received 26 June 1990) Abstract-In spite of the advances in computer technology, there is still a need for more computationally efficient methods for performing stress analysis. One approach which is receiving increasing attention is global/local analysis. Such analyses can take a variety of forms. The form described herein uses two distinct meshes (one global and one local), but retains the same level of accuracy as one would obtain if one was to use a single refined global mesh. The accuracy is retained by using an iterative procedure to enforce equilibrium between the global and local regions. The procedure was tested for three configurations. The good performance observed indicates that the iterative global/local procedure warrants further examination.
INTRODUCTION of supercomputers and ‘super workstations’ has made it possible to analyze many problems which were not tractable just a few years ago. And one can confidently expect that computational power will continue to increase at a rapid pace. In spite of the advances which have been made and those which are expected in the near future, there is a need to reduce the computational effort required to obtain reliable solutions. By increasing the efficiency of an analysis one can either perform the analysis on a less powerful and less expensive computer or make fewer simplifying assumptions in the analysis. One approach which has been proposed for improving efficiency is global/local analysis. The basic concept is that a coarse global model can be used to obtain appropriate boundary conditions for some local region where a much more refined mesh is required. Many strategies have been proposed for implementing this process. References [l-3] discuss some of the possibilities. Of course, engineers have been performing global/local analysis of some form for many years. For example, detailed analysis of a joint in an airplane would not include an entire airplane in the analytical model. One of the most difficult aspects of global/local analysis in assessing the accuracy of such an analysis as compared to the accuracy of a model which explicitly includes local refinement in the global model. These two analyses are shown schematically in Fig. 1 for a plate with a circular hole. Only one fourth of the plate is modeled because of symmetry. Global/local analysis involves two distinct finite element meshes (Fig. 1a). Figure 1b shows the finite element model when the local refinement is included as part of the global model. The objective of this paper is to describe a procedure for global/local analysis which gives the same degree of accuracy as one would obtain if the local The advent
refinement was explicitly included in the global model. For this initial study only linear problems will be considered. Also, local refinement in only one subregion will be discussed. The theoretical basis for the algorithm will be described first. Then the configurations used to test the algorithm will be described. Finally a few results will be presented which demonstrate the performance of the algorithm. It should be noted that the algorithm described herein is similar to one described in ref. [4]. The approaches differ primarily in terms of derivation of the algorithm, convergence criteria, and guidelines for stability of the algorithm.
THEORY Before introducing any equations, an example problem will be discussed to highlight the basic problem. Suppose the global model for a sheet is that
shown in Fig. la. The model is analyzed and nodal displacements are obtained. Now suppose that, in fact, there is a hole in the center of the sheet which is not included in the global model. A local model is developed for the region around the hold and displacements are specified along the boundary of the local model. The magnitudes of these displacements are those calculated using the global model. The local model is then analyzed and displacements and stresses are obtained. This is probably the simplest of global/local procedures. But there is a potential problem due to the differences in the stiffness of the global model and the local model in the region near the hole. If the differences in the stiffnesses are significant, the displacements from the global model (which were used as boundary conditions on the local model) are not accurate. The problem is to make the global model deform as though the local refinement was present.
1027
1028
J. D.
WHITCOMB
/’ Kg2 Global Model (b)LocaUy refined global model (traditional)
Local Model (a) Global/local
models
Fig. 1. Global/local and traditional models of sheet with hole.
The effects of local refinement can be included by using a Raphson iterative procedure. In analysis the governing equations
on global response modified Newtonlinear finite element can be expressed as
K6 =F.
(1)
For the global model in Fig. 1, the governing equations are (Kg’ + Kg*)6 = F,
displacements 6 obtained are not those in eqn (3), but 6 + S,. The 6, is the error in the displacements. An iterative procedure is proposed next which permits elimination of the error. Even though there are two finite element models one can consider the governing equations to be eqns (3). Because global/local analysis is being used, the matrix Kg’ + K’ is never formed. Instead, we use Kg’ + Kg2 as an approximation to Kg’ + K’ in the following modified Newton-Raphson procedure:
(2)
where Kg’ and Kg* are stiffness matrices for the subregions defined in Fig. la. The governing equations for the case in which local refinement is explicitly included in the global model are
1. Solve global model. 2. Solve local model using displacements global model as boundary conditions. 3. Calculate residuals + as
$ = (Kg’ + K’)6 - F. (Kg’ + K?6 = F,
from
(4)
(3)
where K’ is the stiffness matrix for the local model. If the displacements from a global analysis are used as boundary conditions for a local analysis, the
This is an element-by-element calculation, so the matrix Kg’ + K’ is never actually assembled. 4. Check for convergence (stop if tj sufficiently small).
1029
Iterative global/local finite element analysis 5. Calculate correction to global displacements as A6 = -(Kg’ + Kg*)-I$.
(5)
6. Go to step 2. This procedure has several attractive features. As promised, the solution obtained is that for the case in which the local refinement is explicitly included in the global model. The stiffness matrices for the global and local models only need to be decomposed once. Also, there is no need to anticipate where local refinement will be required when assemblying the global model; there can be one or more arbitrarily located refined regions. This algorithm is very similar to that often used in finite element analysis of elastic-plastic materials. In fact, the algorithm was initially conceived as one for a special kind of ‘material nonlinearity’. Refining a mesh by increasing the number of elements (in a displacement formulated finite element analysis) ‘softens’ the region being refined. This is directly analogous to yielding of an elastic-plastic material. A similar analogy can also be drawn for locking materials and selective stiffening of a finite element model. For elastic-plastic analysis, one popular approach is to use the linear global stiffness matrix in a modified Newton-Raphson procedure to iteratively drive the residuals to zero. This method is successful because the residuals are calculated accurately even though the stiffness matrix is only approximate. In the iterative global/local (IGL) analysis the residuals are calculated using eqn (4) which is accurate. As described earlier, the IGL method also uses an approximate stiffness matrix. Because of the similarity to material nonlinear analysis (which has been studied for many years), one can anticipate the response of the IGL procedure based on its material nonlinear analogue. For example, refining a mesh (which causes local softening) is expected to cause no significant numerical problems since the related material problem (yielding) has been well behaved. In contrast, experience has shown that the type of modified NewtonRaphson procedure described above has problems with locking (i.e. stiffening) materials. Hence, one can expect divergence problems with the global/local procedure when a local region is stiffened. In this paper under-relaxation is used to improve the convergence when a local region is stiffened. An under-relaxation factor of 0.5 was used. Hence, only half of the correction AS [eqn (S)] was used on each iteration. The IGL procedure has been described as though there was only one region in which local analysis is desired. Just as material yielding can occur in multiple disconnected subregions without causing numerical or conceptual difficulties, the IGL procedure can easily handle multiple disconnected local models. CAS 40,-
Fig. 2. Extreme test case for checking convergence of iterative global/local algorithm. Also, the procedure described above requires that the boundary nodes of the local model match up with nodes in the global model. However, the addition of smoothing tranformations would permit models which do not exactly match up.
CONFIGURATIONS
Three configurations were analyzed to evaluate the iterative global/local algorithm. The first configuration is that shown in Fig. 1, a sheet with a circular hole. The finite element meshes used are those in Fig. 1. Only one fourth of the plate is modeled. The quarter plate is 2 x 5 and the hole radius is 0.2. The load is parallel to the long axis of the plate. The second configuration is strictly for the purpose of testing the convergence properties of the algorithm. The global mesh is the same as that for the first configuration (Fig. la). Instead of a hole in the local mesh (as was the case for the first configuration), the entire local mesh is a ‘hole’. That is, the global mesh in Fig. la will be used to obtain the solution for the configuration in Fig. 2. This a fairly severe test of the algorithm. The third configuration is a plate with local stiffening. The global mesh is like that in Fig. la. The local model is identical to the shaded region in Fig. la. Hence, the mesh refinement is the same as the global and local models. The difference is the Young’s
1030
J. D. WHITCOMB Table 1. Comparison of displacements and stresses for a sheet with circular hole. Remote uY= 0.198 x IO-*
Displacementt
Non-iterative global/local 0
-0.15 -0.3 -0.3 -0.3 “1 “6 u9 08 01
Max. CT, Kr No. of iterations
kX < 10-d
*,
0
< 1o-5
CL, < lo-’
0
0
0
x x x x
IO-3 10-3 10-j lo-)
-0.1317 x -0.2962 x -0.3598 x -0.4266x
IO-’ IO-? IO-’ 1O-3
-0.1276x -0.3003 x -0.3847 x -0.4729 x
1O-3 1O-3 1O-9 10-j
-0.1267 x -0.3034 x -0.3977x -0.4955 x
1O-3 1O-3 lo-’ 1O-3
0.1 x 0.1 x 0.1 x 0.5 x 0
10-Z 10-r IO-2 10-j
0.1284 x 0.1187 x 0.1069 x 0.5334 x 0
1O-2 lo-* IO-* IO-3
0.1360 x 0.1234 x 0.1085 x 0.5399 x 0
10-r 1O-2 1O-2 10-j
0.1389x 0.1251 x 0.1090 x 0.5416 x 0
1O-2 IO-* 10-2 IO-’
0.4676 x 1O-2 2.12
0.5749 x 10-l 2.61 1
0.6033 x lO-2 2.74 2
$ IMx< 10-10
0.6144 x 1O-2 2.79 6
-0.1267 -0.3034 -0.3978 -0.4957
Locally refined global mesh 0
x x x x
IO-’ 1O-3 1O-3 IO-’
0.1390 x 0.1251 x 0.1090 x 0.5413 x 0
IO-2 1O-2 10-r IO-’
0.6144 x 1O-2 2.79 12
-0.1267 -0.3035 -0.3979 -0.4958
x x x x
1O-3 IO-3 10-j 1O-3
0.1390 x 0.1251 x 0.1090 x 0.5416 x 0
10-J 1O-2 10-2 1O-3
0.6145 x 1O-2 2.79 -
t Subscripts refer to node numbers, which are defined in Fig. 2. The u and v displacements are transverse and parallel to the load direction, respectively.
moduli. The global model has E = 1.0 throughout. The local model has a larger value for E. Hence, the
model was also analyzed. The results to be compared are displacements along the boundary of the local model and the peak crustress. The IGL algorithm was used with several tolerances on the maximum allowable residual $. The results are tabulated in Table 1. The non-iterative global/local solution is quite inaccurate in terms of both the displacements and the peak a,,. A specified tolerance of 10m4 for the IGL analysis required a single iteration and resulted in a.’ much improved estimate of displacements and peak stress. The results for tolerance of lo-’ are almost identical to those for a tolerance of 10 -lo and those for the locally refined global model. Thus, with six iterations (which is what was required for a tolerance of lo-‘) the IGL analysis gave essentially the same results as a locally refined global model. The configuration in Fig. 2 is a more severe test of the IGL algorithm, since the change in compliance of the local region between the global and local models is very severe. Although the convergence was slower than for the first configuration, the solution did converge. The slower convergence is not surprising, since the perturbation due to the ‘local refinement’ is much more severe. The calculated displacements for several different tolerances are presented in Table 2.
local model accounts for some locally increased thickness which is not included in the global model. The elements used were four node isoparametric quadrilaterals. Selective reduced numerical integration was used. Four point quadrature was used for the terms related to normal strains and single point quadrature was used for shear-strain related terms. The stresses presented herein were obtained by calculating the stresses at the quadrature points and extrapolating to the nodes using the procedure described in ref. [5]. The material in all cases was assumed to be isotropic. The Young’s modulus was 1.0 and the Poisson’s ratio was 0.3 for all models except the local models for configuration 3. For configuration 3 several different values for the Young’s modulus were used. They will be listed when the results are discussed. RESULTS AND
DISCUSSION
The sheet with a circular hole was analyzed using the IGL algorithm. For comparison a refined global
Table 2. Comparison of displacements for a sheet with a square cut-out Displacementt “3 ‘6 u9 “8 4 03 v6 09 08 01
No. of iterations
Non-iterative global/local 0
-0.15 x 10-J -0.3 x IO-’ -0.3 x lo-’ -0.3 x 10-j 0.1 x 10-Z 0.1 x IO-2 0.1 x IO-2 0.5 x 10-J 0 -
I),,, < low4 0
-0.2832 x -0.5258 x -0.7042x -0.7507 x 0.2837 x 0.2701 x 0.1754 x 0.8795 x 0 5
$,., < toe5
0
0
IO-’ IO-’ lo-’ IO-) lO-2 1O-2 lo-2 lo-’
-0.3056x -0.5443 x -0.6744x -0.7247 x 0.3096 x 0.2936 x 0.1822 x 0.9006 x 0 11
$m,,, < IO-’
10-j lo--’ lO-3 lO-3 lO-2 lO-2 IO-* lO-3
-0.3063 x -0.5410 x -0.6533 x -0.7031 x 0.3128 x 0.2965 x 0.1828 x 0.9011 x 0 26
k,,, < lo-“’
0
0
1O-3 lO-3 lo-’ lO-3 lo-* 1O-2 lO-2 IO-2
-0.3063 x -0.5409 x -0.6529 x -0.7027 x 0.3128 x 0.2965 x 0.1828 x 0.9010 x 0 50
Locally refined global mesh
IO-’ lo-’ IO-’ lo-’ lO-2 10m2 lO-2 10-J
-0.3064x -0.5410 x -0.6529 x -0.7028 x 0.3128 x 0.2966 x 0.1828 x 0.9011 x 0 -
lo-? lO-3 IO-’ lO-3 lO-2 lO-2 IO-’ IO-3
t Subscripts refer to node numbers, which are defined in Fig. 2. The u and v displacements are transverse and parallel to the load direction, respectively.
1031
Iterative global/local finite element analysis Table 3 Local mesh Young’s modulus
Iterations for convergence
Under-relaxation?
2 2.5 2.15 3.00 2.5 2.15 5
11 31 122 Diverged 8 8 20
No No No No Yes Yes Yes
The configuration in Fig. 3 tests the behavior of the IGL procedure when the local model has a larger stiffness than the corresponding region in the global model. Table 3 summarizes the results of the analysis. The convergence tolerance was set at lo-’ for all cases. As shown in Table 3, without under-relaxation the IGL is slow to converge or divergent when the local stiffness is more than twice the global stiffness. Use of under-relaxation greatly improves the stability of the IGL procedure. A converged solution was obtained with 20 iterations when the local stiffness was five times the global stiffness. CONCLUSIONS
An iterative algorithm was developed for global/local analysis which has essentially the same level of accuracy as one would obtain if local refine-
ment was explicitly included in the global mesh. Tests of the algorithm showed that it is quite stable and will converge even for very severe changes in local compliance. Since the location of local refinement can be selected after the analysis of the global model, the results of the global model can be used as a guide in deciding where local refinement is required. The purpose of this paper was to demonstrate a concept. The problems examined do not prove that it will be efficient for realistic problems. Further work is needed in which the procedure is applied to complicated configurations to determine the actual efficiency.
REFERENCES I. Hirai, Y. Uchiyama, Y. Mizuta and W. Pilkey, An exact zooming method, pp. 61-69. Finite Elements in Analysis and Design 1. Elsevier (1985). 2. C. Jara-Alamonte and C. Knight, The specified bound-
1.
ary stiffness/force SBSF method for finite element subregion analysis. Int. J. Numer. Meth. Engng 26, 1567-1578 (1988). 3. J. B. Ransome, Global/local stress analysis of composite structures. Master’s thesis, Old Dominion University (1989). 4. M. Panthaki and W. Gerstle, A new integrated, computer graphical design tool. Topical report RSI-0339, REjSPEC Inc., Albuquerque, New Mexico (1988). 5. E. Hinton, F. C. Scott and R. E. Ricketts, Local least squares stress smoothing for parabolic isoparametic elements. Int. J. Numer. Merh. Engng 9,235-256 (1975).