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On numerical computations in mixed boundary problems
Compufm & Structures. Voi. 4. pp. 281-291. Pcrgamon P,
UN NUMERICAL
1974. PfbOd in m
Britain
~~M~~TATI~~S IN MIXED BOUNDARY PROBLEMS
IS,F. GoodrichResearch Center, BrecksviEe, Ohio 44141, U.S.A. Abstract-A technique of solving mixed boundary value problems is formulated. The problem is essentially reduced to solving a set of liwar algebraic equations, which can be solved efikiently on a computer. The technique is ilhxstrated by two examples. The first desk with tbe ektro&atic potential ~~0~ on a partially grounded ant&us. The ather conarns with a contact of holiow cylinder, bonded inside to a r&id core.
THEANALYSISaE mixed boundary value problems has received considerable attention. As a result of mixed boundary values, the use of integral transform reduces the problem in many cases to solving a set of singular or dual integral equations. These types of equations have been successfully solved in the past fl]. Another hrtportant group of mixed boundary value problems are those which involve finite boundaries and domains, domains of rnul~p~~~ and/or their combinations. Typical examples of these are problems of heat conduction in partially insulated cylinder, contact stress dis~bution for hollow cylinders, etc. The solutions to the governing equations in these cases can invariably be expressed in terms of orthogonal eigen functians f2]. As a result, the problem finally reduces to determining an infinite set of unknown functions which satisfy different unctions on different parts of the boundary. In the following, it will be shown that these types of problems can be readily solved. Through proper transformations the sets of equations can be reduced to a simple system of algebraic equations which can be solved efficiently on a computer. The technique is illustrated by two examples. The first deals with a two-dimensional electrostatic probiem for an ammlus, The mnulus is partially insulated at the outer bounWe are interested dary and given a unset el~~os~~~ potential to the inner funds. in determining the flux and the potential distribution throughout the domain. The second example illustrates an elastostatic problem which arises when a hollow elastic cylinder is brought in contact with a rigid rough surface. The cylinder is bonded to a rigid core at the inner boundary. We are interested in the normal and shear stresses in the contact area. We will consider only a two dimensional problem.
Consider the following mixed boundary problem: Y41=0 with the ~OXII&Q
in D
(1)
conditions &f#]=%)
on S,
(2)
S. R. MOQHE
282
where L represents a Laplace or biharmonic differential operator in a two or three dimensional domain D. S1 and S, represent parts of the boundary such that S= 5’,+S,. L1 and Lz are linear differential operators of lower order than L. We will assume that all the functions are suthciently smooth so that we will deal only with computation of the solution. For many of the mechanics problems, a general solution to equation (1) can be written on the basis of separation of variables in the form [2]: (3) Here )1and s are coordinates which are normal and tan#ential to the boundary. At are the unknown CoogfIultB,$&) represent surface harmonics whereas R&) arc functions of normal coordinate n. $&(a)are generally chosen so as to form an orthogonal set over the boundary surfs with respect to a weighting function q(s). Then
Substituting (3 into 2) we get on s,
(5)
on Sz
(6)
and k$o kkAk@k(d
= F(s)
where Hk and hk depend upon k and normal coordinate n at the boundary. The purpose of this investigation is to determine set of constants Ak from equations (5 and 6). In order to do this, we introduce a new function cc(s)such that
cr(d=
kgo hdkh@)
on Sr.
Then through the orthogonality condition (4) and equations (6 and 7) we get
1
ds)F(s)tik(s)ds
s2
(k-0,
1,2.. . ).
(8)
Substituting (8 into 5) we obtain
This is an integral equation for p(s) defined over a region sl. Let us now define another set of functions @k(r)so that they form orthogonal set over the region with respect to the weighting function q(s). Then by definition
k
(k=m)
On NumericalComputationsin Mii BoundaryProblems
283
where R, are constants. Now since e&) and p(s) are well defined functions over a region sr we can express W)=
&(k,
m)&As) on S,
(11)
where
Through the use of these relations, equation (9) becomes
Multiply both sides by q(s)$,(s) and integrate over the region sl we get
-
k$o “;;;;p’$2 &)F(x)$k(x)dx
(p--0,1,2...).
(13)
These form a system of linear algebraic equations for ,u,,,with constant coefficients such as i m=O
Ati, mh,,=b,;
(p=O,1,2...).
(14)
A(p, m)= i2B(k, k=onkhk
p)B(k m)
(15)
where
and b, represent the right hand side of (12). Equation (13) can be solved by taking finite matrix provided proper convergence exists. The convergence requirements on such a system are [3]. (1) b, must be bounded for all p.
(2) must exist for all p. Here &I, m) is the usual Kronecker delta function.
284
S. R. MOOHE
It may be intuitively said that these requirements will always hold in all physical problems where the method of se+parationof variables can be justified. The adequate sire of the matrix A@, m) for satisfactory numerical results depends upon the rate of convergence of A@, m) as well as 6,. If b, converges faster than A(p, m), a relatively small number of equations are sufhcient for computation. It is clear from (15) that the matrix elements A(p, m) depend upon the relative sires of S, and S, and orthogonal functions. A sufkient number of terms must be taken in the summation in (15) in order to achieve proper convergence for each element of the matrix [4]. Once p,,, are obtained from (14) it is a routine matter to compute other quantities. This method of solution is illustrated through the following two examples which are formulated in terms of Laplace and biharmonic equations respectively. Example 1 Consider a two-dimensional electrostatics problem defined as
d2V 1 av arz+;
1 a’v -+Tr-r”o ar r 36
(R,
Odk2a)
(16)
with the boundary conditions
v-v,
(r-R,,;
0<8<2rc)
v-o
(r=R,;
-a
av z=o
(r=R,;
a<0<27r-a).
(17)
The general solution is of the form
(18)
Applying the boundary conditions we get O=l+B,,lnR+
2 H&cosntI; X3=1
(O
(a-ze
(19)
On NumtricalComputationsin Mixed Boundary Problems
285
Let
(21)
2’ X”=- p(O)ws &de. 110 I
(22)
Finally we obtain
B(m, n)=
II: n -=a m-l
( >
a 2 na2sin(na +m-ln’ n2a2-(m-l)2n2
A(m, P)= g_ 2$(m, b =asin(p-In) p (p-1)71
’
n)B(p, n)
*
(23)
Here lb,,14 a, IA@, p)/ behaves as (JJ- 1)-2, (m- l)-’ and converges properly. Therefore the solution to the algebraic equations converge properly. Computations are made for the total flux at the surface r = R,. Then
which is shown in Fig. 1 for various values of a when R,= 1, RI =4. AS can be seen, when a-m the flux approaches to a known limit lr/ln4.
FICL
1. Flux for partially grouaded annulus.
S. R. Momm
286 Example 2
Consider now an elastic contact problem when a hollow circular cylinder with perfectly bonded rigid core is brought in contact with a rigid surface. The resulting mixed boundary problem is defined in terms of Airy stress function (p as follows: v4f#J=o
&
(25)
The stress and displacement fields are
and
(27)
where
a2 I a I a2
V2-=~+~a~+~a~
Assuming that the contact area is symmetrical about B=O axis, the boundary conditions become tl,=t4,-0
(r=R,;
U,F G(6),
u,=c(:(8);
V, -me- -0
(r=R,;
0<8<2n) (r=Ri,
-a
a<8<2n+)
(28)
and G and G are known functions depending upon the rigid surface. This problem can be solved by using the method outlined earlier. We will avoid many details except the bare essentials. The biharmonic equation (25) has a solution suitable to this problem in the form
where a,, b, . . . K, are constants to be determined from the boundary conditions.
Here the cosine and sine functions are associated with the upper and lower sets of constants respectively.
On Numerical Computationsin Mixed Boundary Problems
287
FIG.2. Schematicdiagram. Applying the boundary conditions (28) we get b,=2(1-2u)C,
(30) where
5m=-
( > 1+J m
;
Bm=_- l
m
1.,
3-4v 4m=7n.'
Also -a
(31) and
(32)
S. R. MOCSHE
288
where 07 w”;‘m(m-l)~~R”-2~m(m+l)~~R-m-2f
m-2(,+l)R” m
0;,,,=m(m-l)~,,,R”‘-2fm(m+l)&R~~-2f,
m+2(m- l)R-”
@4 z3=
~m~~m-‘-m~~R~m~‘f~~~~~~~+2Rmi’
z=
~rnr]~~-‘+rn&R-“-’
+~[:~:~$~~+2R-m’1
R=R,IR,.
(33)
To solve for unknowns C,,,, D,,,, H,,, and K, from equations (31 and 32) we introduce four functions Is&?), (i= 1, . . . 4) defined over (-or
(34)
and a E --WWu s ..,1+v
On Numerical Computations in Mixed Boundary Problems
289
where a+@)=
k~l~~k)cGs~ (i=l, E
axe) =
kilt#$“)sin?!
P(m, k)=
Qh
k)=
3)
(i=2, 4)
= s -01
s
k-lne sin me sin 70 O1 -LI
co’;;= ol;w’; - O~loJm.
(36)
It can be shown that the set of equations forms a convergent system. Once +j”) are known it is a simple matter to compute the entire stress and displacement field.
Half an(#e of contact a FIG. 3. Contact with rigid rolkr.
S. R. MOOHE
290
l.Z-
l.O-
0.20 -L--~o_L_
0.10
Mcximum
dlsplocement
FIG. 4. Contact with rigid roller.
Numerical results are illustrated for a roller of inner and outer radii of one and four inches. Rigid roller of eight inch radius is brought in contact at various contact widths. Results are shown for ~=O~OOland 0.495. These correspond to open and closed cell foam rubbers. Figure 3 shows total force required to maintain different contact widths. Figure 4 shows a typical radial stress distribution inside the contact area. Here circles represent computer results based upon 15 x 15 matrix. Figure 5 shows a typical shear stress distribution in the contact area.
’ o”---o
“‘, O\ \ O.OL 0.0
I 0.2
I a6
I 0.4
a8
1.0
0/a RG.
5.
Normal stress distribution
inside the contact region.
un Numerical Computations in Mixed Boundary Problems
291
FIG. 6. Shear stress distribution inside contact region. In co&usion, this technique can be used in reducing the mixed boundary problems to a system of linear algebraic equations which can be solved efficiently on any computer.
~~~~~~~~-T~~
computer program was written by Mr. R. MacMiftan. It is a pleasure to acknowIedge helpful discussions with Mr. H. F. Neff.
REJXRENCES [I] I, N. Sr+armo& &fix& ~~d~ry
Vake Pmbkms in Putential Theory. North Holland Publ. Amsterdam
[2] x%uam Fourier Sprles and Boundary Value Problems. McGraw-Hill New York. [3] H. T. D~vta, f&ov ofLinear Operators, Princeton Press, New York (19363. [4] S. R. MOOHE,B. F. Goodrich Research Report (1969). (Received 18 Ad.. 1972)
UN NUMERICAL
1974. PfbOd in m
Britain
~~M~~TATI~~S IN MIXED BOUNDARY PROBLEMS
IS,F. GoodrichResearch Center, BrecksviEe, Ohio 44141, U.S.A. Abstract-A technique of solving mixed boundary value problems is formulated. The problem is essentially reduced to solving a set of liwar algebraic equations, which can be solved efikiently on a computer. The technique is ilhxstrated by two examples. The first desk with tbe ektro&atic potential ~~0~ on a partially grounded ant&us. The ather conarns with a contact of holiow cylinder, bonded inside to a r&id core.
THEANALYSISaE mixed boundary value problems has received considerable attention. As a result of mixed boundary values, the use of integral transform reduces the problem in many cases to solving a set of singular or dual integral equations. These types of equations have been successfully solved in the past fl]. Another hrtportant group of mixed boundary value problems are those which involve finite boundaries and domains, domains of rnul~p~~~ and/or their combinations. Typical examples of these are problems of heat conduction in partially insulated cylinder, contact stress dis~bution for hollow cylinders, etc. The solutions to the governing equations in these cases can invariably be expressed in terms of orthogonal eigen functians f2]. As a result, the problem finally reduces to determining an infinite set of unknown functions which satisfy different unctions on different parts of the boundary. In the following, it will be shown that these types of problems can be readily solved. Through proper transformations the sets of equations can be reduced to a simple system of algebraic equations which can be solved efficiently on a computer. The technique is illustrated by two examples. The first deals with a two-dimensional electrostatic probiem for an ammlus, The mnulus is partially insulated at the outer bounWe are interested dary and given a unset el~~os~~~ potential to the inner funds. in determining the flux and the potential distribution throughout the domain. The second example illustrates an elastostatic problem which arises when a hollow elastic cylinder is brought in contact with a rigid rough surface. The cylinder is bonded to a rigid core at the inner boundary. We are interested in the normal and shear stresses in the contact area. We will consider only a two dimensional problem.
Consider the following mixed boundary problem: Y41=0 with the ~OXII&Q
in D
(1)
conditions &f#]=%)
on S,
(2)
S. R. MOQHE
282
where L represents a Laplace or biharmonic differential operator in a two or three dimensional domain D. S1 and S, represent parts of the boundary such that S= 5’,+S,. L1 and Lz are linear differential operators of lower order than L. We will assume that all the functions are suthciently smooth so that we will deal only with computation of the solution. For many of the mechanics problems, a general solution to equation (1) can be written on the basis of separation of variables in the form [2]: (3) Here )1and s are coordinates which are normal and tan#ential to the boundary. At are the unknown CoogfIultB,$&) represent surface harmonics whereas R&) arc functions of normal coordinate n. $&(a)are generally chosen so as to form an orthogonal set over the boundary surfs with respect to a weighting function q(s). Then
Substituting (3 into 2) we get on s,
(5)
on Sz
(6)
and k$o kkAk@k(d
= F(s)
where Hk and hk depend upon k and normal coordinate n at the boundary. The purpose of this investigation is to determine set of constants Ak from equations (5 and 6). In order to do this, we introduce a new function cc(s)such that
cr(d=
kgo hdkh@)
on Sr.
Then through the orthogonality condition (4) and equations (6 and 7) we get
1
ds)F(s)tik(s)ds
s2
(k-0,
1,2.. . ).
(8)
Substituting (8 into 5) we obtain
This is an integral equation for p(s) defined over a region sl. Let us now define another set of functions @k(r)so that they form orthogonal set over the region with respect to the weighting function q(s). Then by definition
k
(k=m)
On NumericalComputationsin Mii BoundaryProblems
283
where R, are constants. Now since e&) and p(s) are well defined functions over a region sr we can express W)=
&(k,
m)&As) on S,
(11)
where
Through the use of these relations, equation (9) becomes
Multiply both sides by q(s)$,(s) and integrate over the region sl we get
-
k$o “;;;;p’$2 &)F(x)$k(x)dx
(p--0,1,2...).
(13)
These form a system of linear algebraic equations for ,u,,,with constant coefficients such as i m=O
Ati, mh,,=b,;
(p=O,1,2...).
(14)
A(p, m)= i2B(k, k=onkhk
p)B(k m)
(15)
where
and b, represent the right hand side of (12). Equation (13) can be solved by taking finite matrix provided proper convergence exists. The convergence requirements on such a system are [3]. (1) b, must be bounded for all p.
(2) must exist for all p. Here &I, m) is the usual Kronecker delta function.
284
S. R. MOOHE
It may be intuitively said that these requirements will always hold in all physical problems where the method of se+parationof variables can be justified. The adequate sire of the matrix A@, m) for satisfactory numerical results depends upon the rate of convergence of A@, m) as well as 6,. If b, converges faster than A(p, m), a relatively small number of equations are sufhcient for computation. It is clear from (15) that the matrix elements A(p, m) depend upon the relative sires of S, and S, and orthogonal functions. A sufkient number of terms must be taken in the summation in (15) in order to achieve proper convergence for each element of the matrix [4]. Once p,,, are obtained from (14) it is a routine matter to compute other quantities. This method of solution is illustrated through the following two examples which are formulated in terms of Laplace and biharmonic equations respectively. Example 1 Consider a two-dimensional electrostatics problem defined as
d2V 1 av arz+;
1 a’v -+Tr-r”o ar r 36
(R,
Odk2a)
(16)
with the boundary conditions
v-v,
(r-R,,;
0<8<2rc)
v-o
(r=R,;
-a
av z=o
(r=R,;
a<0<27r-a).
(17)
The general solution is of the form
(18)
Applying the boundary conditions we get O=l+B,,lnR+
2 H&cosntI; X3=1
(O
(a-ze
(19)
On NumtricalComputationsin Mixed Boundary Problems
285
Let
(21)
2’ X”=- p(O)ws &de. 110 I
(22)
Finally we obtain
B(m, n)=
II: n -=a m-l
( >
a 2 na2sin(na +m-ln’ n2a2-(m-l)2n2
A(m, P)= g_ 2$(m, b =asin(p-In) p (p-1)71
’
n)B(p, n)
*
(23)
Here lb,,14 a, IA@, p)/ behaves as (JJ- 1)-2, (m- l)-’ and converges properly. Therefore the solution to the algebraic equations converge properly. Computations are made for the total flux at the surface r = R,. Then
which is shown in Fig. 1 for various values of a when R,= 1, RI =4. AS can be seen, when a-m the flux approaches to a known limit lr/ln4.
FICL
1. Flux for partially grouaded annulus.
S. R. Momm
286 Example 2
Consider now an elastic contact problem when a hollow circular cylinder with perfectly bonded rigid core is brought in contact with a rigid surface. The resulting mixed boundary problem is defined in terms of Airy stress function (p as follows: v4f#J=o
&
(25)
The stress and displacement fields are
and
(27)
where
a2 I a I a2
V2-=~+~a~+~a~
Assuming that the contact area is symmetrical about B=O axis, the boundary conditions become tl,=t4,-0
(r=R,;
U,F G(6),
u,=c(:(8);
V, -me- -0
(r=R,;
0<8<2n) (r=Ri,
-a
a<8<2n+)
(28)
and G and G are known functions depending upon the rigid surface. This problem can be solved by using the method outlined earlier. We will avoid many details except the bare essentials. The biharmonic equation (25) has a solution suitable to this problem in the form
where a,, b, . . . K, are constants to be determined from the boundary conditions.
Here the cosine and sine functions are associated with the upper and lower sets of constants respectively.
On Numerical Computationsin Mixed Boundary Problems
287
FIG.2. Schematicdiagram. Applying the boundary conditions (28) we get b,=2(1-2u)C,
(30) where
5m=-
( > 1+J m
;
Bm=_- l
m
1.,
3-4v 4m=7n.'
Also -a
(31) and
(32)
S. R. MOCSHE
288
where 07 w”;‘m(m-l)~~R”-2~m(m+l)~~R-m-2f
m-2(,+l)R” m
0;,,,=m(m-l)~,,,R”‘-2fm(m+l)&R~~-2f,
m+2(m- l)R-”
@4 z3=
~m~~m-‘-m~~R~m~‘f~~~~~~~+2Rmi’
z=
~rnr]~~-‘+rn&R-“-’
+~[:~:~$~~+2R-m’1
R=R,IR,.
(33)
To solve for unknowns C,,,, D,,,, H,,, and K, from equations (31 and 32) we introduce four functions Is&?), (i= 1, . . . 4) defined over (-or
(34)
and a E --WWu s ..,1+v
On Numerical Computations in Mixed Boundary Problems
289
where a+@)=
k~l~~k)cGs~ (i=l, E
axe) =
kilt#$“)sin?!
P(m, k)=
Qh
k)=
3)
(i=2, 4)
= s -01
s
k-lne sin me sin 70 O1 -LI
co’;;= ol;w’; - O~loJm.
(36)
It can be shown that the set of equations forms a convergent system. Once +j”) are known it is a simple matter to compute the entire stress and displacement field.
Half an(#e of contact a FIG. 3. Contact with rigid rolkr.
S. R. MOOHE
290
l.Z-
l.O-
0.20 -L--~o_L_
0.10
Mcximum
dlsplocement
FIG. 4. Contact with rigid roller.
Numerical results are illustrated for a roller of inner and outer radii of one and four inches. Rigid roller of eight inch radius is brought in contact at various contact widths. Results are shown for ~=O~OOland 0.495. These correspond to open and closed cell foam rubbers. Figure 3 shows total force required to maintain different contact widths. Figure 4 shows a typical radial stress distribution inside the contact area. Here circles represent computer results based upon 15 x 15 matrix. Figure 5 shows a typical shear stress distribution in the contact area.
’ o”---o
“‘, O\ \ O.OL 0.0
I 0.2
I a6
I 0.4
a8
1.0
0/a RG.
5.
Normal stress distribution
inside the contact region.
un Numerical Computations in Mixed Boundary Problems
291
FIG. 6. Shear stress distribution inside contact region. In co&usion, this technique can be used in reducing the mixed boundary problems to a system of linear algebraic equations which can be solved efficiently on any computer.
~~~~~~~~-T~~
computer program was written by Mr. R. MacMiftan. It is a pleasure to acknowIedge helpful discussions with Mr. H. F. Neff.
REJXRENCES [I] I, N. Sr+armo& &fix& ~~d~ry
Vake Pmbkms in Putential Theory. North Holland Publ. Amsterdam
[2] x%uam Fourier Sprles and Boundary Value Problems. McGraw-Hill New York. [3] H. T. D~vta, f&ov ofLinear Operators, Princeton Press, New York (19363. [4] S. R. MOOHE,B. F. Goodrich Research Report (1969). (Received 18 Ad.. 1972)