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A Finite Element Error Estimate for Regularized Compressible Flow
Trends in the Theory and Practice of Non-Linear Analysis
V. Lakshmikantham (Editor)
127
0 Elsevier Science Publishers B.V. (North-Holland), 1985
A FINITE ELEMENT ERROR ESTIMATE FOR REGULARIZED COMPRESSIBLE FLOW Hung Dinh and Graham F. Carey Aerospace Engineering/Engineering Mechanics Department The University of Texas at Austin Austin, Texas 1I.S.A.
INTRODUCTION A wide class of compressible flow problems can be described by the full potential equation. This governing equation is nonlinear and may be of mixed type -elliptic in the subsonic flow region and hyperbolic in any supersonic regions Many numerical studies have (see, for example, Bers [1959], von Mises [1958]). been made using both finite difference and finite element methods for approximate solution of full potential problems. However, there appear to be no error analyses of the approximate methods to date. In the case of the mixed subsonicsupersonic flow the operator is non-monotone and standard techniques of finite element error analysis are not applicable. Even in the case where the flow is entirely subsonic the problem is difficult. since the analysis must incorporate the constraint that the flow remain suhsonic. In the present analysis, we further restrict the problem formulation by considering a class of regularized flows: this may be associated with the choice of a fictitious gas that still presents a viable approximation of the real gas flow. The idea of regularizing the flow by introducing a fictitious gas stems from the early analytical studies of Chaplygin [1902] and later by von Karman [1941]. It has recently been applied also in numerical schemes for shock-free airfoil design using both finite difference methods (Fung et al. [1980]) and finite element tiere we present finite element error estimates methods (Pan and Carey [1984]). for this regularized problem and the most comonly used low-degree triangular elements. FORMULATION Let p be the density of the gas and 9 the velocity with q = y@ for potential @. Conservation of mass implies y-pq = 6 where p = p ( q ) can-be determined from the equation of state and momentum equation. Using the adiabatic equation of state for the gas, we obtain 2 1h-1
1
!-[(I
yl
=
0
(1 1
where y is the gas constant (y = 1.4 for air). Note that the choice y =-!, yields the well-known minimal surface equation and is also the Chaplygin gas or tangent" gas of von Karman. As such it represents (asymptotically in Mach number) an accurate approximation to the real gas density relation. A weak formulation corresponding to (1) is obtained from the stationary condition for the variational functional J(v)
=
[1
a
-
(Y)(!V) ] 2 YlY-1
dx
H Dinh and G.E Carey
128
over admissible f u n c t i o n v, We have shown elsewhere {Dinh and Carey, 1984) t h a t J i s w e l l d e f i n e d over W7sp if y > l o r y < -1 and on H i f y 5 -1. irloreover, we can a l s o v e r i f y s t r i c t c o n v e x i t y and e s t a 6 1 i s h e x i s t e n c e and uniqueness of a solut i o n m i n i m i z i n g J provided y 5 -1.
hc i
I n t h e approximate problem we d ' s c r e i z e R and fi and c o n s t r u c t an a p p r o p r i a t e piecewise-polynomial subspace H H where h dehotes t h e mesh parameter. Then f o r y 5 -1 we may demonstrate e x i s t e n c e and uniqueness o f a s o l u t i o n t o t h e approximate problem. ERROR ANALYSIS
Lemma.
L e t ph = P(!@~). f o r approximation @h and
I f @ e W1 Irn(w)f) Hr(Q) w i t h r constant C = c(@).
Proof:
k
2 k+l where k is t h e element degree, then Eh 5 Ch ,
L e t vh be an a r b i t r a r y f u n c t i o n i n H
h
. w i t h vh = @hon aQh.
Then, using
t h e v a r i a t i o n a l equation and i t s approximation, we can show
where vh- @ i s extended by zero on Q t o o b t a i n (w&h v ( @ ) < 1 . c o n s t a n t )
5 Eh
I@- 'hl1,Qh
+
oh.
The i n t e g r a l s i n ( 4 ) can be bounded
'('1
EhIEh
+
I @-
'hl1,Qh'
whence
and from i n t e r p o l a t i o n t h e o r y we o b t a i n t h e d e s i r e d r e s u l t . Theorem,
IIO$dlrn,Qh Proof:
I
Under t h e r e g u l a r i t y assumptions i n t h e above Lemna and provided > 0 constant, we o b t a i n t h e o p t i m a l e s t i m a t e
5 C, f o r C
1 D i r e c t expansion o f t h e H -seminorm and some elementary c a l c u l u s y i e l d
Using t h e above Lemma, we o b t a i n t h e s t a t e d estimate.
m
< C f o r t h e standard elements. It remains f o r us t o v e r i f y t h e c o n d i t i o n llO@ 11 h '3ah For b r e v i t y we s t a t e here w i t h o u t p r o o f a r e l a t e d i n e q u a l i t y (obtained using t h e For @ s a t i s f y i n g Lemma and s t a t e d r e g u l a r i t y o f @ ) see Dinh and Carey [1984]: t h e above r e g u l a r i t y c o n d i t i o n , t h e r e e x i s t s a constant C = C($) > 0 such t h a t f o r element Re and r e g u l a r d i s c r e t i z a t i o n s
-
Error Estimate for Regularized Compressible Flow
129
I t i s then s t r a i g h t f o r w a r d t o v e r i f y t h a t t h e boundedness c o n d i t i o n used i n t h e theorem holds f o r l i n e a r elements and a s i m i l a r r e s u l t can be shown t o h o l d f o r q u a d r a t i c elements. These estimates have been c o r r o b o r a t e d i n numerical s t u d i e s (Dinh and Carey, [1984]).
ACKNDWLEDGMEllTS: T h i s research has been supported i n p a r t by t h e Department o f Energy. REFERENCES c11
Bers, L . , Mathematical Aspects o f Subsonic and Transonic Gas Dynamics, I n t e r s c i e n c e , New York. 1958.
c21
Chaplygin, S.A., On Gas Jets, Sci. Mem., Moscow Univ. Math. Phys. Sec. 21, pp. 1-121, 1902 (trans.: NACA Tech. Note 1063, 1944).
c 31
Dinh, H. and G.F. Carey, Approximate A n a l y s i s o f Regularized Compressible Flow Using a F i c t i t i o u s Gas Approach, J. Nonlinear A n a l y s i s (submitted Jan., 1984).
141 Fung. K.Y., Sobieczky, H. and Seebass, R., 10, 1153-1158, 1980. 151
Shock-Free Wing Design, Vol
. 18,
Karman, Th. von, C o m p r e s s i b i l i t y E f f e c t s i n Aerodynamics, J, f o r Aero. Sci., 8, 337, 356, 1941.
161 Mises, R. von, Mathematical Theory o f Compressible F l u i d Flow, Academic Press, New York, 1958. ~ 7 1Pan, T.T. and G.F. Carey, " F i n i t e Element C a l c u l a t i o n o f Shock-Free A i r f o i l Design, I n t . J. Numer. Fleth. F l u i d s ( i n press), 1934.
The f i n a l ( d e t a i l e d ) v e r s i o n o f t h i s paper has been submitted f o r p u b l i c a t i o n elsewhere.
V. Lakshmikantham (Editor)
127
0 Elsevier Science Publishers B.V. (North-Holland), 1985
A FINITE ELEMENT ERROR ESTIMATE FOR REGULARIZED COMPRESSIBLE FLOW Hung Dinh and Graham F. Carey Aerospace Engineering/Engineering Mechanics Department The University of Texas at Austin Austin, Texas 1I.S.A.
INTRODUCTION A wide class of compressible flow problems can be described by the full potential equation. This governing equation is nonlinear and may be of mixed type -elliptic in the subsonic flow region and hyperbolic in any supersonic regions Many numerical studies have (see, for example, Bers [1959], von Mises [1958]). been made using both finite difference and finite element methods for approximate solution of full potential problems. However, there appear to be no error analyses of the approximate methods to date. In the case of the mixed subsonicsupersonic flow the operator is non-monotone and standard techniques of finite element error analysis are not applicable. Even in the case where the flow is entirely subsonic the problem is difficult. since the analysis must incorporate the constraint that the flow remain suhsonic. In the present analysis, we further restrict the problem formulation by considering a class of regularized flows: this may be associated with the choice of a fictitious gas that still presents a viable approximation of the real gas flow. The idea of regularizing the flow by introducing a fictitious gas stems from the early analytical studies of Chaplygin [1902] and later by von Karman [1941]. It has recently been applied also in numerical schemes for shock-free airfoil design using both finite difference methods (Fung et al. [1980]) and finite element tiere we present finite element error estimates methods (Pan and Carey [1984]). for this regularized problem and the most comonly used low-degree triangular elements. FORMULATION Let p be the density of the gas and 9 the velocity with q = y@ for potential @. Conservation of mass implies y-pq = 6 where p = p ( q ) can-be determined from the equation of state and momentum equation. Using the adiabatic equation of state for the gas, we obtain 2 1h-1
1
!-[(I
yl
=
0
(1 1
where y is the gas constant (y = 1.4 for air). Note that the choice y =-!, yields the well-known minimal surface equation and is also the Chaplygin gas or tangent" gas of von Karman. As such it represents (asymptotically in Mach number) an accurate approximation to the real gas density relation. A weak formulation corresponding to (1) is obtained from the stationary condition for the variational functional J(v)
=
[1
a
-
(Y)(!V) ] 2 YlY-1
dx
H Dinh and G.E Carey
128
over admissible f u n c t i o n v, We have shown elsewhere {Dinh and Carey, 1984) t h a t J i s w e l l d e f i n e d over W7sp if y > l o r y < -1 and on H i f y 5 -1. irloreover, we can a l s o v e r i f y s t r i c t c o n v e x i t y and e s t a 6 1 i s h e x i s t e n c e and uniqueness of a solut i o n m i n i m i z i n g J provided y 5 -1.
hc i
I n t h e approximate problem we d ' s c r e i z e R and fi and c o n s t r u c t an a p p r o p r i a t e piecewise-polynomial subspace H H where h dehotes t h e mesh parameter. Then f o r y 5 -1 we may demonstrate e x i s t e n c e and uniqueness o f a s o l u t i o n t o t h e approximate problem. ERROR ANALYSIS
Lemma.
L e t ph = P(!@~). f o r approximation @h and
I f @ e W1 Irn(w)f) Hr(Q) w i t h r constant C = c(@).
Proof:
k
2 k+l where k is t h e element degree, then Eh 5 Ch ,
L e t vh be an a r b i t r a r y f u n c t i o n i n H
h
. w i t h vh = @hon aQh.
Then, using
t h e v a r i a t i o n a l equation and i t s approximation, we can show
where vh- @ i s extended by zero on Q t o o b t a i n (w&h v ( @ ) < 1 . c o n s t a n t )
5 Eh
I@- 'hl1,Qh
+
oh.
The i n t e g r a l s i n ( 4 ) can be bounded
'('1
EhIEh
+
I @-
'hl1,Qh'
whence
and from i n t e r p o l a t i o n t h e o r y we o b t a i n t h e d e s i r e d r e s u l t . Theorem,
IIO$dlrn,Qh Proof:
I
Under t h e r e g u l a r i t y assumptions i n t h e above Lemna and provided > 0 constant, we o b t a i n t h e o p t i m a l e s t i m a t e
5 C, f o r C
1 D i r e c t expansion o f t h e H -seminorm and some elementary c a l c u l u s y i e l d
Using t h e above Lemma, we o b t a i n t h e s t a t e d estimate.
m
< C f o r t h e standard elements. It remains f o r us t o v e r i f y t h e c o n d i t i o n llO@ 11 h '3ah For b r e v i t y we s t a t e here w i t h o u t p r o o f a r e l a t e d i n e q u a l i t y (obtained using t h e For @ s a t i s f y i n g Lemma and s t a t e d r e g u l a r i t y o f @ ) see Dinh and Carey [1984]: t h e above r e g u l a r i t y c o n d i t i o n , t h e r e e x i s t s a constant C = C($) > 0 such t h a t f o r element Re and r e g u l a r d i s c r e t i z a t i o n s
-
Error Estimate for Regularized Compressible Flow
129
I t i s then s t r a i g h t f o r w a r d t o v e r i f y t h a t t h e boundedness c o n d i t i o n used i n t h e theorem holds f o r l i n e a r elements and a s i m i l a r r e s u l t can be shown t o h o l d f o r q u a d r a t i c elements. These estimates have been c o r r o b o r a t e d i n numerical s t u d i e s (Dinh and Carey, [1984]).
ACKNDWLEDGMEllTS: T h i s research has been supported i n p a r t by t h e Department o f Energy. REFERENCES c11
Bers, L . , Mathematical Aspects o f Subsonic and Transonic Gas Dynamics, I n t e r s c i e n c e , New York. 1958.
c21
Chaplygin, S.A., On Gas Jets, Sci. Mem., Moscow Univ. Math. Phys. Sec. 21, pp. 1-121, 1902 (trans.: NACA Tech. Note 1063, 1944).
c 31
Dinh, H. and G.F. Carey, Approximate A n a l y s i s o f Regularized Compressible Flow Using a F i c t i t i o u s Gas Approach, J. Nonlinear A n a l y s i s (submitted Jan., 1984).
141 Fung. K.Y., Sobieczky, H. and Seebass, R., 10, 1153-1158, 1980. 151
Shock-Free Wing Design, Vol
. 18,
Karman, Th. von, C o m p r e s s i b i l i t y E f f e c t s i n Aerodynamics, J, f o r Aero. Sci., 8, 337, 356, 1941.
161 Mises, R. von, Mathematical Theory o f Compressible F l u i d Flow, Academic Press, New York, 1958. ~ 7 1Pan, T.T. and G.F. Carey, " F i n i t e Element C a l c u l a t i o n o f Shock-Free A i r f o i l Design, I n t . J. Numer. Fleth. F l u i d s ( i n press), 1934.
The f i n a l ( d e t a i l e d ) v e r s i o n o f t h i s paper has been submitted f o r p u b l i c a t i o n elsewhere.