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Erratum to “Analysis and adaptive modeling of highly heterogeneous elastic structures” [Comput. Methods Appl. Mech. Engrg. 148 (1997) 367–391]
s/.__
__ ll!B
ELSEVIER
Computer methods in applied mechanics and englneerlng Comput. Methods Appl. Mech. Engrg. 163 (1998) 395-396
Erratum
Erratum to “Analysis and adaptive modeling of highly heterogeneous elastic structures’ ’ [Comput. Methods Appl. Mech. Engrg. 148 (1997) 367-3911’ J. Tinsley Oden, Tarek I. Zohdi Texas Institute for Computational
and Applied Mathematics,
The Universiry of Texas at Austin, Austin, TX 78712, US4
On p. 384, in the Acknowledgement
section, the ONR grant number
should read “NO0014-95-1-0401.
The following
omitted and should be inserted
on p. 388 directly below Eq. (C.l).
l
text was inadvertently
The finite-dimensional statement:
approximation
of the coarse-scale
solution
is characterized
ei l
(59)
where H is the largest element diameter in the coarse-scale mesh. The finite-dimensional approximation of the local HDPM solutions, which is characterized statement:
eI
’ PI1 of original
by the following
(60)
where the displacements on r,, are prescribed external tractions, t, are prescribed. The following
by the following
result is established
as follows:
in [l 11:
article: SOO45-7825(97)00032-7
0045.7825/98/$19.00 0 1998 Elsevier PII: SO0457825(98)00169-8
Science S.A. All rights reserved.
oil,,,
= ZJ’*~]~~~.As before, on r,, the given
Erratum
396
I Comput. Methods Appl. Mech. Engrg. 163 (1998) 395-396
- 2{LB(u0,U”,H- uO) - 5F(u”,H- u”>> * residual-uO-error +
where
~~~9”(uo2”
-
@2(oo’h, u”~, tK)~f2(,(~o.h)
E~=~’ EK(tii”
u”>/lf,,,
2((.90Vuo.H,4ovz40.H- uON)E(n,
+
_ ~(uO.H)) + ,J%, ~“:fU”J’
+ ~2~;)
Furthermore, and where ~H2~flf~~oV~o’HII(~~11).
-u:“)
(61)
(qo;
_ U;H),
o”*h zfUOJ’ +
if U’ is known exactly, then
N(9) fJ2(u”OSh,UO, 5) =rj2(ti0, ” calculated
UO, t> + c ’ ,K=l
IIG$h - ti;II;(@$ /
model
numerical N(9)
3
lb
~“ll;(f2, + z, IIG”- ~%9~,
-
= llu- zi”q;cn,
(62)
- ( 9(U0)) + 5’. where $‘(zZ’*~, UO, 5) “gf 2( 4;(P) Numerical experiments investigating the relationship presented in a forthcoming paper [8].
between
numerical
and modeling
errors
are to be
Appendix D. Determination of improved homogenized materials for HDPM We note that this is a function of the partition of the domain, external loads and microstructure. The following is an algorithm to determine E”‘*: The basic idea is to find the components of E ’ , in which the augmented elastic potential is minimized. The general minimization steps are as follows: zf {EY$}, 1 C i , j, k, 1s 3. Initially, for simplicity, we start with the l Step 0: Choose an initial vector E O.O relation between averages or an approximation to it for the microstructure present in the material being used. In the case of randomly oriented particulate composites, where the relation between averages is approximately isotropic, one can make a reasonable approximation to E”” by taking the average between bounds on the effective the optimal6 upper and lower bounds for Eoxo, the celebrated Hashin-Shtrikman isotropic properties [4].7 For two phase composites these take the form of KI’!) =
v2 K,
+
3(1 - v2>
1 K2 -
0.0
KI
+
Kc+)
=
/-q
+
1
the averages K
0.0 =
K;!)
+
2
K2
+
3% 3K2+4,!4 (1
0.0
II(+) =P2+
1
-
v2)
6~2(K2 + 2/+)
’
(64)
PI - I4 + 5,$(3& +4&j
5&(3’5 +4/J.,)
I-5 - t-+ Taking
1
K2
KI -
+ 6( 1 - uz)(K, + ~CL,)
(63)
+
3K, + 4/L, v2
p;!)
1 -v*
=
we obtain a first guess KY:,
(0.0) 0.0 ~ -PC+)
lu
6 The Hashin-Shtrikman bounds are optimal ’ u2 is the volume fraction of the particulate
2
(65)
.
if only volumetric constituents.
information
is known.
__ ll!B
ELSEVIER
Computer methods in applied mechanics and englneerlng Comput. Methods Appl. Mech. Engrg. 163 (1998) 395-396
Erratum
Erratum to “Analysis and adaptive modeling of highly heterogeneous elastic structures’ ’ [Comput. Methods Appl. Mech. Engrg. 148 (1997) 367-3911’ J. Tinsley Oden, Tarek I. Zohdi Texas Institute for Computational
and Applied Mathematics,
The Universiry of Texas at Austin, Austin, TX 78712, US4
On p. 384, in the Acknowledgement
section, the ONR grant number
should read “NO0014-95-1-0401.
The following
omitted and should be inserted
on p. 388 directly below Eq. (C.l).
l
text was inadvertently
The finite-dimensional statement:
approximation
of the coarse-scale
solution
is characterized
ei l
(59)
where H is the largest element diameter in the coarse-scale mesh. The finite-dimensional approximation of the local HDPM solutions, which is characterized statement:
eI
’ PI1 of original
by the following
(60)
where the displacements on r,, are prescribed external tractions, t, are prescribed. The following
by the following
result is established
as follows:
in [l 11:
article: SOO45-7825(97)00032-7
0045.7825/98/$19.00 0 1998 Elsevier PII: SO0457825(98)00169-8
Science S.A. All rights reserved.
oil,,,
= ZJ’*~]~~~.As before, on r,, the given
Erratum
396
I Comput. Methods Appl. Mech. Engrg. 163 (1998) 395-396
- 2{LB(u0,U”,H- uO) - 5F(u”,H- u”>> * residual-uO-error +
where
~~~9”(uo2”
-
@2(oo’h, u”~, tK)~f2(,(~o.h)
E~=~’ EK(tii”
u”>/lf,,,
2((.90Vuo.H,4ovz40.H- uON)E(n,
+
_ ~(uO.H)) + ,J%, ~“:fU”J’
+ ~2~;)
Furthermore, and where ~H2~flf~~oV~o’HII(~~11).
-u:“)
(61)
(qo;
_ U;H),
o”*h zfUOJ’ +
if U’ is known exactly, then
N(9) fJ2(u”OSh,UO, 5) =rj2(ti0, ” calculated
UO, t> + c ’ ,K=l
IIG$h - ti;II;(@$ /
model
numerical N(9)
3
lb
~“ll;(f2, + z, IIG”- ~%9~,
-
= llu- zi”q;cn,
(62)
- ( 9(U0)) + 5’. where $‘(zZ’*~, UO, 5) “gf 2( 4;(P) Numerical experiments investigating the relationship presented in a forthcoming paper [8].
between
numerical
and modeling
errors
are to be
Appendix D. Determination of improved homogenized materials for HDPM We note that this is a function of the partition of the domain, external loads and microstructure. The following is an algorithm to determine E”‘*: The basic idea is to find the components of E ’ , in which the augmented elastic potential is minimized. The general minimization steps are as follows: zf {EY$}, 1 C i , j, k, 1s 3. Initially, for simplicity, we start with the l Step 0: Choose an initial vector E O.O relation between averages or an approximation to it for the microstructure present in the material being used. In the case of randomly oriented particulate composites, where the relation between averages is approximately isotropic, one can make a reasonable approximation to E”” by taking the average between bounds on the effective the optimal6 upper and lower bounds for Eoxo, the celebrated Hashin-Shtrikman isotropic properties [4].7 For two phase composites these take the form of KI’!) =
v2 K,
+
3(1 - v2>
1 K2 -
0.0
KI
+
Kc+)
=
/-q
+
1
the averages K
0.0 =
K;!)
+
2
K2
+
3% 3K2+4,!4 (1
0.0
II(+) =P2+
1
-
v2)
6~2(K2 + 2/+)
’
(64)
PI - I4 + 5,$(3& +4&j
5&(3’5 +4/J.,)
I-5 - t-+ Taking
1
K2
KI -
+ 6( 1 - uz)(K, + ~CL,)
(63)
+
3K, + 4/L, v2
p;!)
1 -v*
=
we obtain a first guess KY:,
(0.0) 0.0 ~ -PC+)
lu
6 The Hashin-Shtrikman bounds are optimal ’ u2 is the volume fraction of the particulate
2
(65)
.
if only volumetric constituents.
information
is known.