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Finite element approximation of an evolution problem modeling shear band formation
ELSEVIER
Comput. Methods Appl. Mech. Engrg. 118 (1994) 153-163
Computer metllods in applied mechanics and engineering
,
Finite element approximation of an evolution problem modeling shear band formation Donald A. French a,,, Sonia M.E Garcia b a Department of Mathematical Sciences (ML 25), University of Cincinnati, Cincinnati, OH 45221, USA b Department of Mathematics, U.S. Naval Academy, Annapolis, MD 21402, USA
Abstract An error analysis is provided for the spatial approximation by the finite element method of a nonlinear time-dependent problem which models the antiplane shear deformations of a thermoplastic material. This problem consists of a system of two coupled partial differential equations with velocity and temperature unknowns. Optimal order error estimates in the L 2 norm are given for each of these.
1. Introduction Consider the following initial boundary value problem: Find v = v(x, t) and 0 = 0(x, t) such that g~- ST. ( i z ( O ) V v ) - f ,
in 12 x [0,T],
# - A0 - / z ( 0 ) IXTul2 -- g,
v=O=O,
i n / 2 × [0, T],
(2)
(3)
onF×[0,T],
v ( . , 0 ) = r e and 0 ( . , 0 ) =00
(13
in/2,
(4)
w h e r e / 2 is a bounded domain in R d, d = 1,2 with smooth boundary F and fv = Ow/Ot. The functions v and 0 are real valued, T is a fixed positive constant and f , g, vo and 00 are given data. The function/z is smooth on IR and satisfies 0 < / z ( . ) < oo although it is possible/z --+ oo as 0 ~ oo. We are especially interested in the case w h e r e / z ( 0 ) = e -'~° for a > 0. The study of problem ( 1 ) - ( 4 ) was suggested to us by R. Malek-Madani. He and J.H. Maddocks considered the one-dimensional version of this shear band model in [ 1 ]. The two-dimensional case models antiplane shear deformations of a thermoplastic material. The quantity ¢r - - / t ( O ) V v is the stress. Primarily to simplify the analysis we study the case with homogeneous Dirichlet conditions. Shear bands are thin regions in a thermoplastic material where the strain rate is very high due to the applied stresses. This phenomenon has received increasing attention in recent years and is believed to be a crucial part of the deformation processes in metal forming, ballistic impact, and penetration. We note the Ballistic Research Laboratory report [2] which points to a need to obtain a more complete understanding of these phenomena. * Corresponding author. 0045-7825/94/$07.00 (~ 1994 Elsevier Science S.A. All rights reserved SSDi 0045-7 8 2 5 ( 94 ) 00021 -E
D.A, French, S.M.E Garcia/Comput, Methods AppL Mech. Engrg. 118 (1994) 153-163
154
This paper is a part of a larger study aimed at understanding the qualitative aspects of the time-dependent model, ( i )-(4), for the deformations of a thermoplastic material in one and two dimensions. Aside from the paper [ I ] other relevant reports include Batra [3], Drew and Fl~erty [4], and xd,~Jter [5]. We shall analyze a semidiscrete version of (1)-(4). We discretize 12 into finite elements, use the Galerkin method, and thus obtain a system of ordinary differential equations. Let A4h be a family of finite dimensional spaces depending on a parameter h > 0 which represents the diameter associated with the triangles of this discretization. We assume that the functions in .A4n satisfy the Dirichlet boundary conditions. Once again for simplicity we will search for approximations for the velocity and the temperature in the same space, ,Mr,. Multiplying (1) and (2) by test functions X E .A4h and ,~ E .Mh, integrating over /2, and applying Green's Theorem we obtain the variational formulation of the finite element approximation: Find Vh : [0,Th] ~ .Mh and 0h : [0,Th] ~ A4h such that
(Oh,X) + (/z(O)VVh,VX) = (f,x), Vx ~ Mh, (Oh,A) + (VOh, VA) - (/z(O)lVUh[2,A) -- (g, A), VA E .A~h,
(5) (6)
where vh(.,O) = voj,,
(7)
Oh(.,O) = Ooh, in /2,
and (w,z) = f wz dA. /]
The functions VOh E .A4h and 00h E A4h are given approximations to v0 and 0o, respectively. After introducing a basis for A4h, (5)-(6) becomes a system of nonlinear ordinary differential equations. From the theory of ordinary differential equations it is clear that there exists a unique solution on some interval [0,Th] where Th > 0 (See, for instance, [6], Theorem 6.2). We will show in Section 3 that, in fact, Th/> T. Our main result is the following optimal order error estimates in the L2-norm;
max Iio(', t) - uh(,, t)lit
Ch
(8)
max I[0(., t) -
Ch I'+l,
(9)
o.<.r~<7' O<~t<~r
t)ll0
where p will typically be the degree of the piecewise polynomials in .Mr,. We refer to Theorem 3 for details. We assume problem ( 1)-(4) has a unique solution which satisfies the following uniform bounds.
(It) j--O,I I
L~(IIxlO,TI)
j--0,1 I
L~°(tlxlO,TI)
,
where ot is a multi-index. The outline of this paper is as follows: In Section 2 we give a complete description of the finite element spaces and certain required results. In Section 3 we prove the main theorem. We finish this section with some notation, Let n!
where m is a nonnegative integer. We will frequently use Young's inequality,
ab ~ ea 2 + C~b2, where a, b, e, and C~ are positive constants.
(12)
D.A. French, S.M.E GarcialComput. Methods Appl. Mech. Engrg. 118 (1994) 153-163
155
Throughout the paper, C will denote a positive constant not necessarily the same at each occurrence, which depends on the norms of v and 0. However, it is always independent of h, Vh, and Oh.
2. Preliminaries In this section we describe our assumptions about the spaces .Adh and the Ritz projection which we use in our estimation in Section 3. We will take advantage of the results discussed in [7] and [8]. The space A4h is a set of piecewise polynomial functions defined on a partition Oh C /2. We assume if X E .Adh then due to the Dirichlet boundary conditions X = 0 on / 2 - / 2 h - Under specific assumptions on uniformity in size and shape of the element domains [9] and the proximity of O/2h to F the following approximation property can be shown: For w E Wq,Oo(/2) n Hol(/2) there exists a X E A'lh such that
IIw - xll,~(a)
+ hllw -
xllw,.~(a) ~< Chq,
(13)
where q is an integer, 1 ~< q ~< p + 1 and
Ilwllw-.oo(a) = max{llO~wll,~(a)},
( m = 0 , 1,2).
lal=m
For w E HP+I(/2) n H~(/2) the following inverse properties also hold [9]
VX E J~h,
IlXllm <~ d-'llxllt,
(14)
where 0 ~< l <~ m <~ p are integers and
llxllw,,,.ooca) <~ Ch-illXllm, Vx ~ ~,,,
(15)
with m = 0, I. The Ritz projection TCh : H~(1"2) ---, A4h defined by (V(w-
T~hW),V)() -_O,
(16)
VX E .A/Ih,
will be used frequently. From [8] we have the following results for this operator: IIw - rChwllo + hllw - re/,wll, <~ Ch q,
w E Hq(12) I"l H~(12),
(17)
where q is an integer, I ~< q <~ p + 1. We will also need the following modified Ritz projection R~ ' H~(12) --+ A4h defined by (~.L(O)V(W--
"R,°hW),VX)--0, VX ~..Mh,
(18)
for each t ~ [0,T]. The estimate (17) is also valid for the operator ~ , as well as the following specialized result which is proved in [8], where A(t) = - V . ( ~ ( O ( . , t ) ) V ) :
IIw- 7¢°wllw,.,(a,.) <
{Ceh I-el2 Ch"
if p = 1, if p 1> 2,
(19)
where 0 < e < 1 and
IIn~zll~,(~).
llzll~,.., =
llo,ll=l Note that using (13), (15) and (17) it follows that V'~hW is uniformly bounded if w E W2'~(12) n/-/o!(/2).
117¢hwllw,.~(a) <~ ll~hW- XlIw'.°~<~> + IIx -- Wllw'.oo(~> + llwllw'.oo(~) <<.Ch-d/211~hW-- xlI' + Ch + C <<.ch-d/2(ll~hW wll~ + IIw- xIll) + C -
~
(20)
D.A.French, S.M.E GarcidCornpplrl. Methods Appl. Mech. Engrg. 118 (1994) I53-163
156
3. The error estimate
this section we and prove main theorem. first collect the proof. assume that problem defined (I has a particular, the bounds 10) and 11). and 0h the unique UO/,=
main assumptions will need smooth solution and 8 in of (5)-( with (21)
and @Oh
Also, assume the to state main theorem.
results
19)
for the approximation space
THEOREM 1. Under the hypotheses above uh and 8h are solutions of (S)-(7) constants C = C (u, 8, T) and ho = ho( u, 0, T) such that
We are
in a
on [O,T] and there exists
for 0 < h < ho and 0 < t < i? PROOF This argument benefits from ideas found in [ to]. We start with the standard decomposition of the
errors. Let o-Vh=(O-fi)+(fi-Uh)=q+v
(23)
e-eh=((e-e;)+(B-e*)=5+5,
(24)
and
where 0 = R$J and 8 = ‘&&I.From ( 17) it follows that
llrlllo+ h(lrlllI G Ch”‘“‘,
(25)
lItill -I-Irll7illI G Ck”?
(26)
Il6lla+ hllfll I G CX”“‘,
(27)
lllllo
(28)
-I- qllll
I G a”+‘*
and from ( 19) we have C,h’-d2
lIrlllW~*J(fI,,n,,) G Ch”
ifp = I, if p 2 2.
(2%
Note from (20) and the uniform bounds ( IO) and ( I 1) ; (30) These estimates also hold for 8 and t. Due to the estimates (25) and (27) and the error decomposition (23) and (24) we only need to estimate the quantities r and I. To do so we make the following induction hypothesis which we will justify after completing the main estimates:
Observe that Vu/, and V&, are bounded in Loo( ~2 x [ 0, Th] ) by the triangle inequality, (30), (31)) and (32). We now form two error equations. From (5)
D.A. French,S.M.E Garcia/Comput. MethodsAppl. Mech. Engrg. 118 (1994) 153-163
157
(#, x) + (~(Oh) V~r, VX) = (v, X) + (~(Oh) V~, VX) - (f, X)
=-(~, X) + (O,X) + ( (~(Oh) - ~(o) )v~, Vx) +(lz(O) Vv, V x ) - (f, x), where the projection equation (18) was used on the last step. From ( l ) we have ( # , X ) + ( ~ ( 0 h ) V ~ , V,v) = - ( ~ , X) + ((~(0h) -- ~(0))VO, VX).
(33)
Let X = # and observe that d
d-'~(1~(Oh)V~r, V~r) = 2(/z(Oh) VTr, V # ) -- (/z' ( Oh)OhVT/',
(34)
V'/r)
SO (33) becomes ld
II#llg + ~-~ (~( Oh) VTr, VTr) -- - (
~, 7r) -t- ((~(Oh) -- 1.~(0) ) VO, r e ) -- 1 (]Zt ( Oh)OhVTr ' VTr)
= - - ( ~ , '/r) -4- (([./,(Oh) - - / Z ( 0 ) ) V b , r e ) !
-~(~'
-~- 1 ( / ~ ' ( 0 h ) ~ V ~ ' ,
V'/r)
( Oh)~V~r, V~r).
(35)
Integrating (35) from 0 to t where t E (0,Th) we obtain using (21) t
t
t
f ll~rl12ds-}-Cll~71rll--- f ( ~ , T r ) ds-b f ( ( l . ~ ( O h ) - I . ~ ( O ) ) ~ u , V # ) d s o o o ! t
,/
,/
0
0
= l~ + h + 13 + / 4 .
(36)
From (26) and Young's inequality we have t
I~,1 ~< c,,h 2'+2 + a, /I1#11o~ ds.
(37)
0
We use integration by parts in s on 12 f
h= ((~(Oh) -- ~(0))V~, V~r)l~ -/((~'(Oh)Oh
-
-
~'(O)O)W.'~7~r) ds
0 t -
w)
-
ds.
(38)
0
From (21) we have ~r(., 0) = 0. By the meanvalue theorem, (32), and (11) we have in any norm, I I II, ll~(0h) -
~(o)11 ~
and
11/~'(0h) -- ~'(o)11
~
Also
[.t,'( Oh)Oh --/J,1(0)0
--
[.l,t( Oh) ( Oh --0) "t- (I.tt ( Oh) -- IJ,'( O) )0.
So
iltz'(Oh)Oh -- ~'(O)Oli ~< C(liOh -- Oil + lid'(Oh) - ~'(O)ii).
158
D.A. French, S.M.E Garcia/Comput. Methods Appi, Mech. Engrg. 118 (1994) 153-163
By the triangle inequality II0h - 01l ~< 116ll+ Ilg'll and
[[/Th- O{I <~ 11611+ II~'l[,
so from (30) equation (38) becomes
'
),/2/,
0 t (/ I/2 +(f(ll~ll2o + 11¢1120)d$) I/2 IlWrll2ods) o o
),/2
0
Terms I3 and/4 are estimated in a similar way. Applying Young's inequality and (27), (36) becomes t
t
f
f IL~llgd, + cllwltg -
0
o
t
*c,2 f II~rl120ds + 8311Wr1120+ C8~11¢1120. 0 Choosing 81 and 83 sufficiently small this becomes t
t
f I1#1120d,+
C,,V~r,120,Ch2t'+2+S2/(1[6"[12o
I'
+ "~"lo2) d, + C.~.f HV-II2Od,+ CI'"[2o •
0
0
(39'
0
We can estimate the last term as follows: t
!
[o
11~1120- Tsll¢(.,s)ll~ds=2 0
f (~'(,,s),~'(,,s))
t
ds.< e
0
t
f II~'(.,s)llo2 d s + c , f Ili(.,s)ll~. 0
0
So (39) becomes I
t
t
f ,iei,ds+C, f
f ,,,:oo,+ 0
0
We will complete the estimation of (40)
t
ds+ C,
0
f,l,,io
ds.
(40)
0
after we develop a similar inequality for g'. From (2),
(6), and (16)
we have
(~', a) + (v~',~Ta)= (~,,~) + (v~,va) - [(Oh,,~) + (voh, V,~)] - ( L a ) + [(0, a) + (Vo, va)l - (uCO,,)lVvhlL a) - (g,a) =
= - ( 6 , a) + (/~(O)IVvl 2 - t~(0n)IVvt, I2, A).
(41)
Let A = g', Id
2
11~'112o+ ~-~ll~Zg'llo=-(6,~')
+ (Ivol2(~(o) - ~(oh)),~') +
=Ti + r2 + r3. The first two terms are routine after the earlier estimates
IT,I <~¢,,F "+2 + ~111~'12, 1o
(~(o,,)(l~7vl a-
IVv,12),~) (42)
D.A. French. S.M.E Garcia/Comput. Methods Appl. Mech. Engrg. 118 (1994) 153-163
159
and
1721 < •211~'II~ + c,,(ll~ll 2 + llg'll 2) ~< C,~2h2p+2 + •211~'II02+ c,2 llg'll 2. We now estimate the third term,
T3 = ((~(0h) -- ~(0))(IVoI 2 - IVVhI2),~ ") + (~(0)(IV~I ~ IVVhI2),~ ") =Q, +Q2.
(43)
The estimate of the term QI follows from (31 ),
IQ, I <~ •311~'II02+ C~,h 2"+2 + C,,II~'II 2. For Q2 we use the following identity
IVvl 2 -IVohl 2 - (IVvl 2 - IVVl 2) + (IVVl 2 -IVVhl 2) = (2V~ . V ~ --IV,~I 2) + V(V + Oh)" V,r. So, Q~---2(/zCO)Vv. Vz/, ~') -
~,~=(o)lV~l~,~) +
(~(o)v(v
(44)
+ O h ) " VTr, ~') = e ! -I- P2 -~- P3.
We apply the product rule on the space and time derivatives to rewrite the Pl term.
~,~ = - ( v . ( ~ ( O ) V v ) ~ , ~ ) + ~ (~(o),7;.,. 7 ~ , ~ ) -
(~(O)Vv)~,v~
- (~(O)Vv. v~,n)
where we used the fact that v/= ~" = 0 on F. Using the smoothness of v and 0, and the estimates (25) and (26) we obtain t
t
0
t
0
0
where we used the fact that ~'(.,0) = 0. Using the smoothness of 0, uniform bounds on ~, and (31) we obtain
~ c,, IIWrllo2 + ",ll~'ll~, Finally, from the Schwarz and Young inequalities,
P, ~< CllV,TIl~,¢~)ll~'ll0 ~< C,,llV~ll~,(n) + ~811~'ll0~ Combining our estimates for Ti, T2, T3, QI, Q2, Pi,/'2, and/'3 in (42) we have t
t
0
0 t
0
t
t
0
t
0
0
Adding this last result to (40), choosing • and 8 small so the corresponding terms can be subtracted from the left side, and applying the Poincar6 inequality (that is, II~'lto~< cIIv~'llo) we obtain t
0
From Gronwall's inequality and (19)
2
0
t
0
t
0
D.A. French.S.M.E GarcialComput. MethodsAppl. Mech. Engrg. 118 (1994) 153-163
160 t
f
cil.llo~ + 11¢1120)ds + IlWrll2o + IIV(llo2 <
(CEh 4-2~
Ch2p+2
i f p = 1, if p ~> 2,
(45)
0
for 0 < h <~ ho and 0 <~ t <~ th. From (25) and (27) we now have
{C~h 2-~ Iio - o.11o + hollV(v - vh)ll0 ~<
ChP+l
if p - - 1, i f p ~>2,
(46)
for 0 < h ~< ho and 0 ~< t ~< th and
IiO - o.11o + IIV(O - on)Iio <
C,h -"
Chp+l
if p = 1, i f p >~2,
(47)
for 0 < h ~ ho and 0 <~ t ~< th. We will now justify the a priori hypotheses (31) and (32). We follow the standard argument found in Thom6e [ 11 ] (p. 176). From (10), ( 11 ), the fact that or(., 0) = ( ( . , 0) = 0, and the continuity of vh, Oh, ~, 0, there exists th > J such that max (IIo~,(.,s)IIL~¢.~ + IlWr(', s)llL=¢a>) ~< M + 1
(48)
max (]lOh(., s)IIL~.~ ÷ IIV((., s)IlL=m)) ~< M -I- 1
(49)
O<~s<~t
and 0<~s.<.r
for t E [0, th). Thus (46)-(47) hold for t (= [0, th). We now make the assumption that either (48) or (49) fails to hold for some 0, ~< T. Our goal from here on is to contradict this hypothesis. From the approximation property (13), the inverse property (15), the estimate (45) and the bound (10) we have for t E [0, th)
IIv,,llL,',.,.,~ ~< IlvllL~¢,,,~ + IlL:- vhll,..~.,~ M + C F + Ch='~/2(II x - vllo + IIv oo v,,llo) <~M + ChL where 7 = 2 - d/2 - ~. Also, by the same argument
IIOhllL~,,,~ <~M + Ch ~. for 0 ~< t < th. From (45) it follows that
llV~ll~c,,~~ h-~/211V~llo~
I1V¢11c=¢,,~ < h-'~/21lV~'llo ~< ch~. From these estimates we conclude that the left sides of (48) and (49) are bounded by M + Ch y. Since vh, Oh, It, ( are continuous and uniformly hounded on [0, th] they have a continuous limit at th and 3h0 > 0 such that for 0 < h ~< h0 the assumption that th is a "blowup" point is violated. Thus the finite element solution can be extended past th, in fact, it is well-defined and satisfies (31), (32), (45), (46), and (47) on [O,T]. Acknowledgments Research of the first author was partially supported by the Army Research Office thru grant 28535-MA. Research of the second author was supported by ONR grant DN N0001492WR24029.
D.A. French, S.M.E Garcia/Comput. Methods Appl. Mech. Engrg. i18 (1994) 153-163
161
References [l] J.H. Maddocks and R. Ma!ek-Madani, Steady-state shear-bands in thermo-plasticity, i: Vanishing yield stress, Int. J. Solids Struct. 29 (1992) 2039-2061. [21 J. Walter (Ed.), Material modeling for terminal ballistic simulation, BRL, Technical Report BRL-TR-3392, 1-83. [3! R.C. Batra, Analysis of shear bands in simple shearing deformations of nonpolar and dipolar viscoplastic materials, Appl. Mech. Rev. 45 (1992) 123-131. 141 D.A. Drew and J.E. Flaherty, Adaptive finite element methods and the numerical solution of shear band problems, in: Phase Transformations and Material Instabilities in Solids (Academic Press, New York, 1984) 3?-60. [5] J.W. Walter, Jr., Numerical experiments on adiabatic shear band formation in one dimension, BRL Technical Report BRL-TR-3381 ( 1991 ) 1-67. [6] K. Yosida, Lectures on Differential and Integral Equations (Interscience, 1960). [7] D.A. French and L.B. Wahlbin, On the numerical appoximation of an evolution problem in nonlinear viscoelasticity, Comput. Methods Appl. Mech. Engrg. 107 (1993) 101-116. [81 Y. Lin, V. Thomee and L.B. Wahlbin, Ritz-Volterra projections to finite element spaces and applications to integro-differential and related equations, SIAM J. Numer. Anal. 28 (1991) 1047-1070. [9] P.G. Ciarlet, Basic Error Estimates for the Elliptic Problem, Handbook of Numerical Analysis, Vol. II (North-Holland, Amsterdam, 1991) 19-351. [101 J. Douglas Jr. and J.E. Roberts, Numerical methods for a model for compressible miscible displacement in porous media, Math. Comput. 41 (1983) 441-459. [ l i l V. Thom~e, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Mathematics 1054 (Springer Verlag, Berlin, 1984).
Comput. Methods Appl. Mech. Engrg. 118 (1994) 153-163
Computer metllods in applied mechanics and engineering
,
Finite element approximation of an evolution problem modeling shear band formation Donald A. French a,,, Sonia M.E Garcia b a Department of Mathematical Sciences (ML 25), University of Cincinnati, Cincinnati, OH 45221, USA b Department of Mathematics, U.S. Naval Academy, Annapolis, MD 21402, USA
Abstract An error analysis is provided for the spatial approximation by the finite element method of a nonlinear time-dependent problem which models the antiplane shear deformations of a thermoplastic material. This problem consists of a system of two coupled partial differential equations with velocity and temperature unknowns. Optimal order error estimates in the L 2 norm are given for each of these.
1. Introduction Consider the following initial boundary value problem: Find v = v(x, t) and 0 = 0(x, t) such that g~- ST. ( i z ( O ) V v ) - f ,
in 12 x [0,T],
# - A0 - / z ( 0 ) IXTul2 -- g,
v=O=O,
i n / 2 × [0, T],
(2)
(3)
onF×[0,T],
v ( . , 0 ) = r e and 0 ( . , 0 ) =00
(13
in/2,
(4)
w h e r e / 2 is a bounded domain in R d, d = 1,2 with smooth boundary F and fv = Ow/Ot. The functions v and 0 are real valued, T is a fixed positive constant and f , g, vo and 00 are given data. The function/z is smooth on IR and satisfies 0 < / z ( . ) < oo although it is possible/z --+ oo as 0 ~ oo. We are especially interested in the case w h e r e / z ( 0 ) = e -'~° for a > 0. The study of problem ( 1 ) - ( 4 ) was suggested to us by R. Malek-Madani. He and J.H. Maddocks considered the one-dimensional version of this shear band model in [ 1 ]. The two-dimensional case models antiplane shear deformations of a thermoplastic material. The quantity ¢r - - / t ( O ) V v is the stress. Primarily to simplify the analysis we study the case with homogeneous Dirichlet conditions. Shear bands are thin regions in a thermoplastic material where the strain rate is very high due to the applied stresses. This phenomenon has received increasing attention in recent years and is believed to be a crucial part of the deformation processes in metal forming, ballistic impact, and penetration. We note the Ballistic Research Laboratory report [2] which points to a need to obtain a more complete understanding of these phenomena. * Corresponding author. 0045-7825/94/$07.00 (~ 1994 Elsevier Science S.A. All rights reserved SSDi 0045-7 8 2 5 ( 94 ) 00021 -E
D.A, French, S.M.E Garcia/Comput, Methods AppL Mech. Engrg. 118 (1994) 153-163
154
This paper is a part of a larger study aimed at understanding the qualitative aspects of the time-dependent model, ( i )-(4), for the deformations of a thermoplastic material in one and two dimensions. Aside from the paper [ I ] other relevant reports include Batra [3], Drew and Fl~erty [4], and xd,~Jter [5]. We shall analyze a semidiscrete version of (1)-(4). We discretize 12 into finite elements, use the Galerkin method, and thus obtain a system of ordinary differential equations. Let A4h be a family of finite dimensional spaces depending on a parameter h > 0 which represents the diameter associated with the triangles of this discretization. We assume that the functions in .A4n satisfy the Dirichlet boundary conditions. Once again for simplicity we will search for approximations for the velocity and the temperature in the same space, ,Mr,. Multiplying (1) and (2) by test functions X E .A4h and ,~ E .Mh, integrating over /2, and applying Green's Theorem we obtain the variational formulation of the finite element approximation: Find Vh : [0,Th] ~ .Mh and 0h : [0,Th] ~ A4h such that
(Oh,X) + (/z(O)VVh,VX) = (f,x), Vx ~ Mh, (Oh,A) + (VOh, VA) - (/z(O)lVUh[2,A) -- (g, A), VA E .A~h,
(5) (6)
where vh(.,O) = voj,,
(7)
Oh(.,O) = Ooh, in /2,
and (w,z) = f wz dA. /]
The functions VOh E .A4h and 00h E A4h are given approximations to v0 and 0o, respectively. After introducing a basis for A4h, (5)-(6) becomes a system of nonlinear ordinary differential equations. From the theory of ordinary differential equations it is clear that there exists a unique solution on some interval [0,Th] where Th > 0 (See, for instance, [6], Theorem 6.2). We will show in Section 3 that, in fact, Th/> T. Our main result is the following optimal order error estimates in the L2-norm;
max Iio(', t) - uh(,, t)lit
Ch
(8)
max I[0(., t) -
Ch I'+l,
(9)
o.<.r~<7' O<~t<~r
t)ll0
where p will typically be the degree of the piecewise polynomials in .Mr,. We refer to Theorem 3 for details. We assume problem ( 1)-(4) has a unique solution which satisfies the following uniform bounds.
(It) j--O,I I
L~(IIxlO,TI)
j--0,1 I
L~°(tlxlO,TI)
,
where ot is a multi-index. The outline of this paper is as follows: In Section 2 we give a complete description of the finite element spaces and certain required results. In Section 3 we prove the main theorem. We finish this section with some notation, Let n!
where m is a nonnegative integer. We will frequently use Young's inequality,
ab ~ ea 2 + C~b2, where a, b, e, and C~ are positive constants.
(12)
D.A. French, S.M.E GarcialComput. Methods Appl. Mech. Engrg. 118 (1994) 153-163
155
Throughout the paper, C will denote a positive constant not necessarily the same at each occurrence, which depends on the norms of v and 0. However, it is always independent of h, Vh, and Oh.
2. Preliminaries In this section we describe our assumptions about the spaces .Adh and the Ritz projection which we use in our estimation in Section 3. We will take advantage of the results discussed in [7] and [8]. The space A4h is a set of piecewise polynomial functions defined on a partition Oh C /2. We assume if X E .Adh then due to the Dirichlet boundary conditions X = 0 on / 2 - / 2 h - Under specific assumptions on uniformity in size and shape of the element domains [9] and the proximity of O/2h to F the following approximation property can be shown: For w E Wq,Oo(/2) n Hol(/2) there exists a X E A'lh such that
IIw - xll,~(a)
+ hllw -
xllw,.~(a) ~< Chq,
(13)
where q is an integer, 1 ~< q ~< p + 1 and
Ilwllw-.oo(a) = max{llO~wll,~(a)},
( m = 0 , 1,2).
lal=m
For w E HP+I(/2) n H~(/2) the following inverse properties also hold [9]
VX E J~h,
IlXllm <~ d-'llxllt,
(14)
where 0 ~< l <~ m <~ p are integers and
llxllw,,,.ooca) <~ Ch-illXllm, Vx ~ ~,,,
(15)
with m = 0, I. The Ritz projection TCh : H~(1"2) ---, A4h defined by (V(w-
T~hW),V)() -_O,
(16)
VX E .A/Ih,
will be used frequently. From [8] we have the following results for this operator: IIw - rChwllo + hllw - re/,wll, <~ Ch q,
w E Hq(12) I"l H~(12),
(17)
where q is an integer, I ~< q <~ p + 1. We will also need the following modified Ritz projection R~ ' H~(12) --+ A4h defined by (~.L(O)V(W--
"R,°hW),VX)--0, VX ~..Mh,
(18)
for each t ~ [0,T]. The estimate (17) is also valid for the operator ~ , as well as the following specialized result which is proved in [8], where A(t) = - V . ( ~ ( O ( . , t ) ) V ) :
IIw- 7¢°wllw,.,(a,.) <
{Ceh I-el2 Ch"
if p = 1, if p 1> 2,
(19)
where 0 < e < 1 and
IIn~zll~,(~).
llzll~,.., =
llo,ll=l Note that using (13), (15) and (17) it follows that V'~hW is uniformly bounded if w E W2'~(12) n/-/o!(/2).
117¢hwllw,.~(a) <~ ll~hW- XlIw'.°~<~> + IIx -- Wllw'.oo(~> + llwllw'.oo(~) <<.Ch-d/211~hW-- xlI' + Ch + C <<.ch-d/2(ll~hW wll~ + IIw- xIll) + C -
~
(20)
D.A.French, S.M.E GarcidCornpplrl. Methods Appl. Mech. Engrg. 118 (1994) I53-163
156
3. The error estimate
this section we and prove main theorem. first collect the proof. assume that problem defined (I has a particular, the bounds 10) and 11). and 0h the unique UO/,=
main assumptions will need smooth solution and 8 in of (5)-( with (21)
and @Oh
Also, assume the to state main theorem.
results
19)
for the approximation space
THEOREM 1. Under the hypotheses above uh and 8h are solutions of (S)-(7) constants C = C (u, 8, T) and ho = ho( u, 0, T) such that
We are
in a
on [O,T] and there exists
for 0 < h < ho and 0 < t < i? PROOF This argument benefits from ideas found in [ to]. We start with the standard decomposition of the
errors. Let o-Vh=(O-fi)+(fi-Uh)=q+v
(23)
e-eh=((e-e;)+(B-e*)=5+5,
(24)
and
where 0 = R$J and 8 = ‘&&I.From ( 17) it follows that
llrlllo+ h(lrlllI G Ch”‘“‘,
(25)
lItill -I-Irll7illI G Ck”?
(26)
Il6lla+ hllfll I G CX”“‘,
(27)
lllllo
(28)
-I- qllll
I G a”+‘*
and from ( 19) we have C,h’-d2
lIrlllW~*J(fI,,n,,) G Ch”
ifp = I, if p 2 2.
(2%
Note from (20) and the uniform bounds ( IO) and ( I 1) ; (30) These estimates also hold for 8 and t. Due to the estimates (25) and (27) and the error decomposition (23) and (24) we only need to estimate the quantities r and I. To do so we make the following induction hypothesis which we will justify after completing the main estimates:
Observe that Vu/, and V&, are bounded in Loo( ~2 x [ 0, Th] ) by the triangle inequality, (30), (31)) and (32). We now form two error equations. From (5)
D.A. French,S.M.E Garcia/Comput. MethodsAppl. Mech. Engrg. 118 (1994) 153-163
157
(#, x) + (~(Oh) V~r, VX) = (v, X) + (~(Oh) V~, VX) - (f, X)
=-(~, X) + (O,X) + ( (~(Oh) - ~(o) )v~, Vx) +(lz(O) Vv, V x ) - (f, x), where the projection equation (18) was used on the last step. From ( l ) we have ( # , X ) + ( ~ ( 0 h ) V ~ , V,v) = - ( ~ , X) + ((~(0h) -- ~(0))VO, VX).
(33)
Let X = # and observe that d
d-'~(1~(Oh)V~r, V~r) = 2(/z(Oh) VTr, V # ) -- (/z' ( Oh)OhVT/',
(34)
V'/r)
SO (33) becomes ld
II#llg + ~-~ (~( Oh) VTr, VTr) -- - (
~, 7r) -t- ((~(Oh) -- 1.~(0) ) VO, r e ) -- 1 (]Zt ( Oh)OhVTr ' VTr)
= - - ( ~ , '/r) -4- (([./,(Oh) - - / Z ( 0 ) ) V b , r e ) !
-~(~'
-~- 1 ( / ~ ' ( 0 h ) ~ V ~ ' ,
V'/r)
( Oh)~V~r, V~r).
(35)
Integrating (35) from 0 to t where t E (0,Th) we obtain using (21) t
t
t
f ll~rl12ds-}-Cll~71rll--- f ( ~ , T r ) ds-b f ( ( l . ~ ( O h ) - I . ~ ( O ) ) ~ u , V # ) d s o o o ! t
,/
,/
0
0
= l~ + h + 13 + / 4 .
(36)
From (26) and Young's inequality we have t
I~,1 ~< c,,h 2'+2 + a, /I1#11o~ ds.
(37)
0
We use integration by parts in s on 12 f
h= ((~(Oh) -- ~(0))V~, V~r)l~ -/((~'(Oh)Oh
-
-
~'(O)O)W.'~7~r) ds
0 t -
w)
-
ds.
(38)
0
From (21) we have ~r(., 0) = 0. By the meanvalue theorem, (32), and (11) we have in any norm, I I II, ll~(0h) -
~(o)11 ~
and
11/~'(0h) -- ~'(o)11
~
Also
[.t,'( Oh)Oh --/J,1(0)0
--
[.l,t( Oh) ( Oh --0) "t- (I.tt ( Oh) -- IJ,'( O) )0.
So
iltz'(Oh)Oh -- ~'(O)Oli ~< C(liOh -- Oil + lid'(Oh) - ~'(O)ii).
158
D.A. French, S.M.E Garcia/Comput. Methods Appi, Mech. Engrg. 118 (1994) 153-163
By the triangle inequality II0h - 01l ~< 116ll+ Ilg'll and
[[/Th- O{I <~ 11611+ II~'l[,
so from (30) equation (38) becomes
'
),/2/,
0 t (/ I/2 +(f(ll~ll2o + 11¢1120)d$) I/2 IlWrll2ods) o o
),/2
0
Terms I3 and/4 are estimated in a similar way. Applying Young's inequality and (27), (36) becomes t
t
f
f IL~llgd, + cllwltg -
0
o
t
*c,2 f II~rl120ds + 8311Wr1120+ C8~11¢1120. 0 Choosing 81 and 83 sufficiently small this becomes t
t
f I1#1120d,+
C,,V~r,120,Ch2t'+2+S2/(1[6"[12o
I'
+ "~"lo2) d, + C.~.f HV-II2Od,+ CI'"[2o •
0
0
(39'
0
We can estimate the last term as follows: t
!
[o
11~1120- Tsll¢(.,s)ll~ds=2 0
f (~'(,,s),~'(,,s))
t
ds.< e
0
t
f II~'(.,s)llo2 d s + c , f Ili(.,s)ll~. 0
0
So (39) becomes I
t
t
f ,iei,ds+C, f
f ,,,:oo,+ 0
0
We will complete the estimation of (40)
t
ds+ C,
0
f,l,,io
ds.
(40)
0
after we develop a similar inequality for g'. From (2),
(6), and (16)
we have
(~', a) + (v~',~Ta)= (~,,~) + (v~,va) - [(Oh,,~) + (voh, V,~)] - ( L a ) + [(0, a) + (Vo, va)l - (uCO,,)lVvhlL a) - (g,a) =
= - ( 6 , a) + (/~(O)IVvl 2 - t~(0n)IVvt, I2, A).
(41)
Let A = g', Id
2
11~'112o+ ~-~ll~Zg'llo=-(6,~')
+ (Ivol2(~(o) - ~(oh)),~') +
=Ti + r2 + r3. The first two terms are routine after the earlier estimates
IT,I <~¢,,F "+2 + ~111~'12, 1o
(~(o,,)(l~7vl a-
IVv,12),~) (42)
D.A. French. S.M.E Garcia/Comput. Methods Appl. Mech. Engrg. 118 (1994) 153-163
159
and
1721 < •211~'II~ + c,,(ll~ll 2 + llg'll 2) ~< C,~2h2p+2 + •211~'II02+ c,2 llg'll 2. We now estimate the third term,
T3 = ((~(0h) -- ~(0))(IVoI 2 - IVVhI2),~ ") + (~(0)(IV~I ~ IVVhI2),~ ") =Q, +Q2.
(43)
The estimate of the term QI follows from (31 ),
IQ, I <~ •311~'II02+ C~,h 2"+2 + C,,II~'II 2. For Q2 we use the following identity
IVvl 2 -IVohl 2 - (IVvl 2 - IVVl 2) + (IVVl 2 -IVVhl 2) = (2V~ . V ~ --IV,~I 2) + V(V + Oh)" V,r. So, Q~---2(/zCO)Vv. Vz/, ~') -
~,~=(o)lV~l~,~) +
(~(o)v(v
(44)
+ O h ) " VTr, ~') = e ! -I- P2 -~- P3.
We apply the product rule on the space and time derivatives to rewrite the Pl term.
~,~ = - ( v . ( ~ ( O ) V v ) ~ , ~ ) + ~ (~(o),7;.,. 7 ~ , ~ ) -
(~(O)Vv)~,v~
- (~(O)Vv. v~,n)
where we used the fact that v/= ~" = 0 on F. Using the smoothness of v and 0, and the estimates (25) and (26) we obtain t
t
0
t
0
0
where we used the fact that ~'(.,0) = 0. Using the smoothness of 0, uniform bounds on ~, and (31) we obtain
~ c,, IIWrllo2 + ",ll~'ll~, Finally, from the Schwarz and Young inequalities,
P, ~< CllV,TIl~,¢~)ll~'ll0 ~< C,,llV~ll~,(n) + ~811~'ll0~ Combining our estimates for Ti, T2, T3, QI, Q2, Pi,/'2, and/'3 in (42) we have t
t
0
0 t
0
t
t
0
t
0
0
Adding this last result to (40), choosing • and 8 small so the corresponding terms can be subtracted from the left side, and applying the Poincar6 inequality (that is, II~'lto~< cIIv~'llo) we obtain t
0
From Gronwall's inequality and (19)
2
0
t
0
t
0
D.A. French.S.M.E GarcialComput. MethodsAppl. Mech. Engrg. 118 (1994) 153-163
160 t
f
cil.llo~ + 11¢1120)ds + IlWrll2o + IIV(llo2 <
(CEh 4-2~
Ch2p+2
i f p = 1, if p ~> 2,
(45)
0
for 0 < h <~ ho and 0 <~ t <~ th. From (25) and (27) we now have
{C~h 2-~ Iio - o.11o + hollV(v - vh)ll0 ~<
ChP+l
if p - - 1, i f p ~>2,
(46)
for 0 < h ~< ho and 0 ~< t ~< th and
IiO - o.11o + IIV(O - on)Iio <
C,h -"
Chp+l
if p = 1, i f p >~2,
(47)
for 0 < h ~ ho and 0 <~ t ~< th. We will now justify the a priori hypotheses (31) and (32). We follow the standard argument found in Thom6e [ 11 ] (p. 176). From (10), ( 11 ), the fact that or(., 0) = ( ( . , 0) = 0, and the continuity of vh, Oh, ~, 0, there exists th > J such that max (IIo~,(.,s)IIL~¢.~ + IlWr(', s)llL=¢a>) ~< M + 1
(48)
max (]lOh(., s)IIL~.~ ÷ IIV((., s)IlL=m)) ~< M -I- 1
(49)
O<~s<~t
and 0<~s.<.r
for t E [0, th). Thus (46)-(47) hold for t (= [0, th). We now make the assumption that either (48) or (49) fails to hold for some 0, ~< T. Our goal from here on is to contradict this hypothesis. From the approximation property (13), the inverse property (15), the estimate (45) and the bound (10) we have for t E [0, th)
IIv,,llL,',.,.,~ ~< IlvllL~¢,,,~ + IlL:- vhll,..~.,~ M + C F + Ch='~/2(II x - vllo + IIv oo v,,llo) <~M + ChL where 7 = 2 - d/2 - ~. Also, by the same argument
IIOhllL~,,,~ <~M + Ch ~. for 0 ~< t < th. From (45) it follows that
llV~ll~c,,~~ h-~/211V~llo~
I1V¢11c=¢,,~ < h-'~/21lV~'llo ~< ch~. From these estimates we conclude that the left sides of (48) and (49) are bounded by M + Ch y. Since vh, Oh, It, ( are continuous and uniformly hounded on [0, th] they have a continuous limit at th and 3h0 > 0 such that for 0 < h ~< h0 the assumption that th is a "blowup" point is violated. Thus the finite element solution can be extended past th, in fact, it is well-defined and satisfies (31), (32), (45), (46), and (47) on [O,T]. Acknowledgments Research of the first author was partially supported by the Army Research Office thru grant 28535-MA. Research of the second author was supported by ONR grant DN N0001492WR24029.
D.A. French, S.M.E Garcia/Comput. Methods Appl. Mech. Engrg. i18 (1994) 153-163
161
References [l] J.H. Maddocks and R. Ma!ek-Madani, Steady-state shear-bands in thermo-plasticity, i: Vanishing yield stress, Int. J. Solids Struct. 29 (1992) 2039-2061. [21 J. Walter (Ed.), Material modeling for terminal ballistic simulation, BRL, Technical Report BRL-TR-3392, 1-83. [3! R.C. Batra, Analysis of shear bands in simple shearing deformations of nonpolar and dipolar viscoplastic materials, Appl. Mech. Rev. 45 (1992) 123-131. 141 D.A. Drew and J.E. Flaherty, Adaptive finite element methods and the numerical solution of shear band problems, in: Phase Transformations and Material Instabilities in Solids (Academic Press, New York, 1984) 3?-60. [5] J.W. Walter, Jr., Numerical experiments on adiabatic shear band formation in one dimension, BRL Technical Report BRL-TR-3381 ( 1991 ) 1-67. [6] K. Yosida, Lectures on Differential and Integral Equations (Interscience, 1960). [7] D.A. French and L.B. Wahlbin, On the numerical appoximation of an evolution problem in nonlinear viscoelasticity, Comput. Methods Appl. Mech. Engrg. 107 (1993) 101-116. [81 Y. Lin, V. Thomee and L.B. Wahlbin, Ritz-Volterra projections to finite element spaces and applications to integro-differential and related equations, SIAM J. Numer. Anal. 28 (1991) 1047-1070. [9] P.G. Ciarlet, Basic Error Estimates for the Elliptic Problem, Handbook of Numerical Analysis, Vol. II (North-Holland, Amsterdam, 1991) 19-351. [101 J. Douglas Jr. and J.E. Roberts, Numerical methods for a model for compressible miscible displacement in porous media, Math. Comput. 41 (1983) 441-459. [ l i l V. Thom~e, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Mathematics 1054 (Springer Verlag, Berlin, 1984).