# User manual | Chapter 5 Convergenc e ## Convergenc

e

In Section 3.4 we presented three general problems, Problems A, Band C.

In Section 4.1 we formulated the problems for the Galerkin approximations.

T hese are Problems AG , BG and CG.

## 5.1 Equilibrium problem

In thi s section we consider the convergence of the s olution of Problem BG to the soluti o n of Problem B.

Assume that

### u

h

E

Sh

i s the solution of

b(u h

, v)

=

(j ,

v)

for a ll

v

E

Sh

(5.1.1

) and

u

E

V is the solution of

b ( u,

### v)

=

(j,

v )

f o r all

v

E

V.

( 5.1.2)

In the proof of the theorem, we will use the pr o je c tion

P ,

defined in Se c ­ tion 4.6.

5.1. 1

1.

II

u h

-

u"

E

-+

0

i

h

-+ o.

74

5.

ON

2. If

U

E

H4

n

V ; then and

75

P

### roof

If

(5.l.2) is subtracted from (5.l.1), it follow s that

### b(

u

-

u\ v)

=

0 for all

v

E

Sh

This means that uk = Pu. The first part of the theorem follows directly from Lemma 4.6.2

. The estimates in the second part of the theorem follow from Lemma 4.6

.

1 and Lemma 4.6.3. 0

H A P

5.

NV

e

e

o

e

## m

76

Our approach is based on the ideas of [BDSW] , [BF] and [SF] and we fol­ low the presentation in [SF]. The theory in the book of Strang and Fix ,

[SF, Section 6.3], concerns eigenvalue problems for general symmetric elliptic operators. Most of the presentation is written in a style which encourage abstraction. In collaboration with others, [ZVGV2], we verified that the theory is valid for abstract eigenvalue problems such as Problem

C.

In this thesis we present this abstract version, and also offer a number of modest improvements.

The rate of convergence for eigenvalue problems also depends on the regula­ rity of the eigenvectors. In the absence of such theory for interface problems, we pose the following assumption which we showed to be true in the one­ dimensional case. (See Section 3.5.)

Regularit y Ass umption

The eigenvectors of the eigenvalue problem , Pro­ blem C, are in

n

for

### k

=

4 or 6 , and there exists a constant

C b

­

depending on th bilinear forms band (')' ) such that for e a ch eigenvector

y

T

e

e

q

t

e

### Minmax principle

To analyse the convergence of eigenvalues and eigenvectors, some preparation is necessary. First we establish bounds for the approximate eigenvalues using the Rayleigh quotient and the minimax principle .

It is well-known that the eigenvalues are the stationary values of the Rayleigh quotient. However, the following result gives a more convenient characteri­ zation of the eigenvalues. See [SF, p 221].

Lemma 5.2.1

Minmax principle

Let T denote the class of subspaces of V having dimens'ion J' , then

= min max

### R(v)

.

SET vES

CHAPTER

5. CO

VERGENCE

W e may assume that the eigenvalues are ordered

77

For some in teg er m, consider the eigenvalues

)'1,

### A2, ... , A

m

and corre s ­ ponding eigenvectors Yl, Y2, ... ,

Ym'

Equal eigenvalues are possible but we assume that

=I-

### Am

for each

J'

> m.

In the finite dimensional subspace

h

we have

### A7,

A~,

.. . ,

A~

(a lso ordered) and corresponding eigenvectors

### y7,

y~,

... , y~.

( Equal eigenvalues do not matter. In the case of multipliCity,

yJ

is not uniquely determined, but it does not influence any proof.)

e

### eigenvalues

The minimax principle yields lower bounds for the approximate eigenvalues.

L

5.2.2

A]

2::

Aj for

### Proof

The minmax principle is also true for the space

Any subspace of

is a subspace of

.

0

### Notation

Yi

will be used to denote the normalised eigenvectors, i.e. II Yi l1 = 1.

For

j

=

I, 2, ... , m, let

E j

denote the subspace of

V

spanned by {Yl,

Y 2

, ... ,

### Yj}.

Consider t he subspaces

Sj

where

Sj

=

P E j

for

j

=

1 , 2, ... , m.

=

### Ph

is the projection defined in Section 4.6.

An upper bound for some approximate eigenvalue depends on the construc­ tion of

### Sh.

Clearly the construction of

### Sh

must be s uch that dim

=

### N

> m. However, it is st ill possible that dim

< m .

As

### sumption

The construction of

is such that dim

### Sm

= m.

We define a quantity f.L~ to measure the " distortion " of the projection of the unit ball

=

{y

E

IIYII

=

I} : Se t

fL':n

= inf{IIPyl1

2

:

E

### Em}

.

CHAPTER 5. CONVERGENCE 78

t

1

M':n

>

0

~f

and only if

dim

Sm

= m.

### Proof

The function

M':n

IIPyI1

2 ha s a minimum on the compact set

Bm.

Hence

>

0 if and only if

Py

-=/:.

0 for each

y

E

Bm·

But this is so if and onl y if the vectors

PYl, PY 2,

.. . ,

pYm

form a linearly independent set. D

P

5

A':n

:::;

m ax{R( Py )

:

y

E

B m}.

P

### roof

Since dim

Sm

= m, it follows from the minimax principle that

A':n :::;

max{R(v)

:

v

E

Sm}.

Consider any

v

E

Sm, v-=/:.

O. There exists a vector

y

E

Em

such that

Py

=

v.

N ote that

R(P(ay))

Consequently,

=

R(av )

=

R (v).

Choose a such that

ay

E

B m. max{R(P z

) : z

E

B m }

=

m ax{R(v)

:

v

E

Sm}·

D

The following result is crucial.

5

A':n

:::;

A:.

Mm

### Proof

If

y

=

2:::':1

CiYi,

then

b(y, y)

= m

L

i=l orthonormal set. Hence c; \, since

{Yl, Y2,

.

.

. , Ym} is an m i=l

Since

P

is a projection with respect to the inner product

b, b(Py, Py )

:::; Am for each

y

E

Bm.

From the definition of

M':n,

we h a ve

R(P )

y

=

b(Py , Py)

<

Am

IIPyl12 -

M':n '

ow use Proposition 5.2.2. D

CHAPTER

5.

CO N VERGENCE

C orollary 5.2.1

J-lr:n :::;

1.

79

This is a direct consequence of Lemmas 5.2.2 and 5.2.3. It i s convenient to formulate error estimates in terms of the quantity 1 -

J-l':n.

N otati o n

CJ':n

=

1 -

J-l':n.

Corollar y 5.2.2

0:::;

CJ~

<

1

and

A~

-

Am

:::;

A~P~,

Since the eigenvalue error is bounded by and p rove that

CJ!

---->

O.

CJ!, it is sufficient to estimate

CJ!

5.2.3 Estimates

Proposition 5.2.

3

CJ!

= max{2(y,

y

-

Py) II Y

Pyl12 : y

E

Em}.

P roof

IIPyll2

+ lIyll 2 -

2(y, Py)

IIPyll2

+

2(y, y)

-

2(y,

Py) -

1 (since lIyll2

=

1 ).

As a consequence 1 -

IIPyll2

=

2(y , y

- Py

)

-

Ily -

p Y 11 2 .

The result follows from the definition of

CJ!.

0

Remark

In [SF , Section 6 .

3]

CJ! is defined by

CJ~

= max{12(y,

y

- Py

)

lIy pYll21 : y

E

Em}.

The absolute value is not necessary since max{2(y,

y

Py) -

Ily

Pyl12 : y

E

Em}

~

O.

The assumption is then made that

CJ!

<

1 , and they prove tha t dim

Sm

= m.

W e proved the fact that dim

Sm

= m is equivalent to

CJ!

<

1 and we believe that it is important to take note of this equivalence.

N

NC

### Proposition

5.2

.4

For any y

E

Em !

(y,y

- Py

)

=

b(y* - Py*,y - Py

) .

### Proof

b(Yi - PY

i, Y

- Py)

=

b(Yi, Y

- Py

)

since

b(y

- Py, PY

i)

=

O.

H ence,

Multiply by c i A;-l and sum over i. We have

(y,

y - Py)

= m

L

i=l

Ci\-lb(Yi

- P

i,

Y - Py)

80 o

The following result also differ from [SF].

L

### emma

5.2

.4

(J~ ~ max{2 1Iy * -

pY*IIE IIY

- pYllE :

Y

E

E m }·

P

### roof

Consider the result of Proposition 5.2.3. We have demonstrated that the quantity

2(y, Y

_ py) _

Ily -

pyll2

must have a non negative maximum (Corollary 5.2.2) . Cons e quently

(J~ ~ max{2(y,

y

- Py ) :

y

E

Em}.

Use Proposition 5.2.4 and the Schwartz inequality for the inner product

b.

0

o

ti

5.2.5

For

any c >

0

there exists a

<5

>

0

such that for h

<

<5!

Ily* -

Ily -

Py *I IE pYllE

< c

for each y

E

E m,

< c

for each y

E

E m·

CHAPTER

### 5.

C O

VERGENCE

81

P roof

From Lemma 4.6.2 there exist po s itive numb e rs

0 1) 02 )

.

.

. )

On

such that for each i

N ow, suppose

h

< min i

Oi,

then m i =l

The same arguments are valid for

PY*//E'

5.2.5

For any

c

>

0

there exists a 0

>

0

such that

(J~

< c

i f h

<

O.

Proof

For any c

>

0 there exists a

0

>

0 su c h that if

h

<

0 ,

then

Ilu

PUIlE

< c for each

u

E

Em

### ·

The r e sult follows from Lemma 5.2.4 and Proposition 5 .

2.5. o o

### Proposition

5.2.6

If Problem

C

satisfies the regularity assumption, then for any

Y

E

Em and

Proof

W e may assum e t hat C i

~

0 for each i.

First estimate:

IIY* -

Py*IIE

<

<

< m

L

c) , .

;-lIlYi -

pYiliE

i = l m c

c)\i

1

IYil k · h

P

-

2 i=l

c/ ]

' m

L

Ci A f-

1

I1Yill hk

*-2 i =l

<

CCbAr:n1 h k '

-2.

CHAPTER

5.

C O NVERGENCE

82

Second estimate: m

<

i = l

CG b

f

### hY-2,

using the same arguments as for the first estimate.

L e mma 5 .2.6

If Problem

the

### regularity assumption) then

CT~

:::;

CG

A;::-lh 2(kO-2) b

P roof

Use Lemma 5.2.4 and Proposition 5.2.6. o o

5.2

.4 C onv e rg e n ce of e i g e n v a l ues

W e may now use the results of the previous subsection to establish the con­ vergence of

A~ to

L e mma 5.2.7

6

>

0

such

### that for h

<

6,

A~

- Am :::;

2AmCJ~.

Proof

Choose 6 such that

CJ~

<

~.

Consequently

A ~

<

2A m.

o

T he o rem 5.

2 .1

1.

A~

-

--t

0

as

--t

O.

2. If Probl em

,

.

A~

-

15G b

A;:

### h

2

(k "

2) .

Proof

1. This is a direct consequence of Lemmas 5.2.5 and 5.2

.

7.

2. Use Lemmas 5.2.6 and 5.2.7. o

5.

CO N

### VERGENCE

83

5.2.5 Convergence o f eig e nv e c t ors

To estimate the error

"ym-y~ll, we need to estimate the difference

IIPYm-y~ll·

It is necessary to consider the possibility that

Am

has multiplicity more than one. Suppose that the multiplicity of

A

=

1, 2, ... , m -

r,

m

Am

is

r

and let

+

1, ... ,

N.

From Theorem 5.2.1 it follows that there exist real numbers

p

>

0 and

0

>

0 such that if

h

<

0 ,

then lAm -

AJI

>

p

for each

j

E

A.

As sumption

Assume that

h

is sufficiently small for

(5.2.1) to hold.

(5.2.1)

Suppose

{y~-r+l' y~-r+2 '

... , corresponding to

A~ r+l' A~-r+2 ' y~}

... is

, an orthonormal set

A~.

The strategy of eigenvectors now is to estimate the distance between y~-r+ i and some (uniquely defined) vector in

E

Am' the eigenspace corresponding to

Am.

We define a projection

Pm

with domain

P(EArn ):

PmW

= m

L

(w, yJ)yJ

for each

w

E

P(EArn)'

=

### m-r+l

This projection enables us to deal with the case of a repeated eigenvalue.

Here we differ from [SF]. Although most of the computations are the same , we believe that our construction of the projection

Pm

is a worthwhile contri­ bution. W e will show that

PmP

(and hence

Pm)

is invertible for

### h

sufficiently small.

Propo s ition 5.2.7

For

j

E

A

y

E

E

AmJ

- Am)(Py,

### yJ)

= Am(Y - Py

, yJ).

Proof

It is only necessary to prove that

AJ(Py, yJ)

=

Am(Y, yJ)

since the term -

Am

### (Py,

yJ)

appears on both sides of the equation.

Since

yJ

and yare eigenvectors, it follows that

A7(Py, yJ)

=

b(Py,

and

Am(Y, yJ)

=

b(y, yJ).

But

b(Py - y, yJ)

=

0 for each

### j,

thus ( 5.2.2) follows.

(5.2.2) o

5 .

N

### VERGENCE

Lemma 5.2.8

84

P roof

From the assumption we have the estimate

Am

IAJ -

Ami

:=:; Pm for each

j

E

1\ , where

Pm

=

A m p-1.

The set y~ ,

yq ,

... ,

Y'N

form an orthonormal basis for

Sh ,

h e nc e

N

Py

=

L( Py, yJ)yJ. j= l

Cons e qu e ntly ,

Py - PmPy

=

I)py, yJ)yJ. j Ei\

If

y

E

E Arn

' then

L (Py , yJ?

jEi\

We now us e Proposition 5.2.7

.

CAJ

## ~mAml)

2

(y Py , yj)'

<

p~

L(y

- Py,

yJ)2

(Inequality (5.2.3)) jEi\

N

<

p~

'L) y

-

Py, yJ)2 j=l

(5.2.3)

D

CH A PTER

5 .

CONVERGE N CE

L e mma 5.2

.9

85

PmPy11

<

Pyll

+

IIPy -

<

(1

+

Amp-I)

o

C

### orollar

y 5.2.3

PmP is invertible for h sufficiently small.

Let

E

n

E

Am'

Then for

### h

sufficiently small. Since it follows that

IIPmPyl1

>

~. Consequently

1

for each

E

E

Am .

o

C

### orollary

5.2.

4

If h is s'ufficiently small , then for each

### j,

there exists a unique

Xj

E

EAm

w~th

=

1

such that

.i

=

1, 2 , ... , r

### Proof

There exists a unique

E

EAm

such that

y

=

### (PmP)-lYm_r+j.

Hence ,

Let

{3 be a real number such that

1{31

= lIyll and let

Xj

=

### {3-1y.

'vVe can choose

Xj

such that

{3

>

O. As a consequence lIyll

=

{3.

C H APTER

5.

CO N VERGENCE

86

Hence

Xj

-

<

IIXj - yll

+

-

<

1

m

)

### lly-Pyll·

o

It is important to realise that one compute the approximation

y':n-r+ j .

The result above guarantees the existence of an exact eigenvector, with norm one, close to the approxi mate one.

The following result from [S F] shows that an error es timate in the energy norm depends on error estimates in the norm

II

### ·II

and eigenvalue errors. We modified it slightly to m ake it useful for th e case of repeated eigenvalues.

L emma 5 .2.10

Proof

b(Ym

-

yj, Ym

-

yj) b(Ym, Ym )

- 2b(Ym, yj)

+

b(yj , yj)

2

AmllYmll

- 2A

m(Ym, yj)

+

AJIlY~1I2

Am - 2Am(Ym, y~)

+

A J

Am[2

-

2(Ym,

y~)]

+

AJ

-

A m

Am [IIYmIl

2

-

2 ( Ym, yj)]

+

lIyJ II

2

]

+

AJ - Am

AmilY m yj ll2 +

AJ

-

Am.

o

T heorem 5.2.2

1. Let

c

>

0

be arbitrary. If h is sufficiently small ) then for each j, j

=

1, 2 , ..

. ,

r there exists a unique

Xj

E

EArn wi th IIx j

II

= 1

such that

CHAPTER

5.

CO N VERGENCE

87

2.

Suppose Problem

C

If

IS

j, j

=

I , 2, ... ,

r

e.'LZsts

E

EArn

IIxj

=

1

that

.

-

h

E

:s;

C~C

\ a:

### Proof

1. Use Corollary 5.2.4: There exists a unique

### Xj

E

EArn

such that

But

(Lemma 5.2.10). Hence

Ilxj

-

Y~-r+jll~

:s;

+

- PXjl12

+

A~_r+j

-

### Am

(5.2.4)

N ow use Proposition 5.2.5 and Theorem 5.2.1.

2. Consider the Inequality (5.2.4). We have the estimates

(5.2.5) from Proposition 5.2.6 and from Theorem 5.2.1. The result follows from (5.2.5) and (5.2.6).

(5.2.6) o

CH A

PTER 5. CO

N VERGENCE

5.3

## Vibration probl

e m

88

Our concern is the difference between the solution u of Problem

A

and the solution

Uh

of Problem

AG.

It is possible to estimate this error in terms of the projection error (Section 4.6) and errors for the initial conditions. See

[SF , Section 7.3]. This is called a projection method and was first used for parabolic problems. For second order hyperbolic problems , it appear that credit is due to [DJ, [De] and [SF]. Research in this direction was also done by [Ba].

After this it appear that abstract methods became popular. See for example

[Sh, Section

6.4].

In Section 3 of an invited paper, [BIt], very general results are given. (Incidentally they use results in [Sh].)

A general approximation theory , using functional analysis, is obviously im­ portant. However, we found that the basic error inequality mentioned before

([SF, Section

7.3] and [D , Lemma 1]) is valid for an abstract problem as general as P roblem

A.

As a final remark we mention the paper [FXX] where the autho rs also use what they term a "partial projection " me t hod to obtain

£ 2 error estimates.

5.3.

1

Di s c

### retizat

ion e rror

In this section we show that the convergence proof sketched by Strang and Fix

[SF , Section 7.3] can be applied to Problem A in Section 3.4 and Problem AG in Section 4.l. In this proof the projection operator P defined in Section 4.6 is used to find an estimate on the discretization error

Ilu(t) - uh(t)IIE for

t

E

[0 ,

(0 ) . vVe also use the symbol

P

to denote the " projection"

Pu

of the solution

u

of

Problem A , i.e.

(Pu)(t)

=

Pu(t)

for each

t

?:

O.

Let

e(t)

=

Pu(t) - Uh(t) and

ep(t)

=

u(t) - Pu(t).

Then

Ilu(t) - uh(t)IIE::;

+

### Ile(t)IIE'

(5.3.1)

The following result is required for the main result of this section. N ote that differentiability with respect to the energy norm is required to prove that the projection function

Pu

is differentiable. This regularity requirement is not stated by [SF].

CHAPTER 5. CO

NV ERGENCE

89

Lemma 5.3.1

ffu

E

C

2

([O,

00) ,

V), then Pu

E C2([O , OO), V)

wdh

(Pu)'(t)

=

Pu'(t) and (Pu)"(t)

=

Pu//(t).

P roof

As the projection operator

P

is a bounded linear operator with norm less than one, it follows that

### II(ot)-l

(Pu(t

+ c5t) Pu(t)) - Pu'(t)IIE ::;

11(c5t)-l(u(t

+ c5t) -

u(t))

-

u'(t)IIE'

This implies that

Pu

E

C

1

([0,00),

V) and

(Pu)'(t)

=

Pu'(t).

In exactly the same way we prove that

(Pu)//(t)

=

Pu//(t).

(Pu)'

E Cl

([0 , 00),

V) and o

Since we already have an estimate for the projection error

e p(t)

### ,

it is only necessary to estimate the other part of the error.

In the next proof the following " energy" expression will be c onvenient:

E(t)

=

~(e'(t),

e'(t) )

+

~b(e(t),

e(t))

1 1

- 211

e' ( t )

+

211

e

( t )

### II

~.

(5.3

.

2)

Lemma 5 .3.2

Assume that u

E

C

2

([O, 00) ,

V). Then, for any t

2 0 ,

liE::; IIPcx

-

CXhllE

+

liP;]

- ;]h

+

it

Ile~(s )

### II

+

~[ Ile ~( s )

110

ds.

Proof

From P roblem A and the Galerkin approximation (Problem AG) we deduce that for any v

E

Sh ,

(u//(t)

u~(t) ,

v)

+

a(u'(t) u~(t) ,

v)

+

b(u(t) - Uh(t), v

)

=

O. (5.3.3)

Since

P

is a projection, we have

b(u(t) - Pu(t)

, v)

=

b(u'(t) -

Pu'(t), v)

=

0 for all

v

E

Sh.

Using the fact that

Pu//(t)

=

(Pu)//(t) ,

(5.3.3) can be written as

(el/(t), v)

+

b(e(t), v)

= -(e~(t),

v) k(e~(t),

v ) o

-

k(e'(t) , v)o

-J-ib(e'(t),v)

for all

v

E

Sh. (5.3.4)

CHAPTER 5. CO

N VERGENCE

( ote that

a(u, v)

=

fJb(u, v)

+

k(u, v)o

where

fJ

or

k

or both can be z ero.)

W e will use the fact that

E'(t)

=

(el/(t), e'(t))

+

b(e(t), e'(t)).

As

e(t)

E

Sh it follows that

e'(t)

E

Sh. Choose

v

=

e'(t)

in (5.3.4), then

E'(t)

-

-(e~(t),

e'(t))

-

(ke~(t),

e'(t))o - k(e'(t),

e'(t))o

- fJb(e'(t)

, e'(t))

~ (1/e~(t)11

+

~llle~(t)llo)

### Ile'(t)ll·

From (5.3.2),

1/e'(t)11 ~

J2E(t).

Thus

E'(t)

~

J2E(t)

(1Ie~(t)11

+

~I

II

~(t)llo) and consequently

90

This yields that

JE(i5

~

JE(O)

+

~

1t

(1Ie~(s)11

+

~Ille~(s)llo)

ds. (5.3 .

5)

As

E

(

0)

=

1

2

1

2

P' -

P'hll

+

QhllE and

### Ile(t)IIE

~

J2E(t) ,

again from (5.3.2) , the result follows from (5.3.5). 0

Theorem 5 .3

.1

Assume that u

E

C

2

([0 ,

(0),

V). Then ,

JOT any t

~

O .

)

~

+

+

### IIPp' - P'hll

+

1t

(1Ie~(s)11

+

~I Ile~(s)llo)

ds.

Proof

Use L emma 5.3.2 and Equation (5.3.1). o

To prove the convergence results, it is now necessary to consider the terms on the right side of the inequality in this theorem.

C HA PTER

5. CO

N VERGENCE

5.3.2

### Convergence

The main factor that determines the rate of convergence of the solution

Uh

of Problem AG to the solution

U

of Problem A as

h

tends to zero, is the regularity of the weak solution u. The regularity of u depends on the regularity of the initial values

Q

and

f3,

as we pointed out in Section 3.4. [Raj gave an example to show that the regularity of the solution is necessary to obtain optimal order convergence.

Th e rate of convergence is also directly influenced by the choice of the initial values

Qh

and

f3h

for the solution

Uh

of Problem AG. V·..,re will c onsider two cases, i.e.

Qh

=

ITQ, f3h

=

ITf3

and

Qh

=

PQ , f3h

=

Pf3.

In the following result we show that the rate of convergence in the en e rgy norm is of order

h

2

if certain regularity c onditions ar e satisfied. The estimates are expressed in terms of the constants

C

J

and

C

E

d e fined in Section 3 .4 as well a s

C d e fin ed in Section 4.5.

Theorem 5.3.2

LetQh

=

ITQ andf3h

=

ITf3. A ssu me thatu

E C

2

([0 , 00 ),

V) and that u(t)! u'(t) and ul/(t) are in H4

n

V

for t

2:

O.

Then ,

Ilu(t) - uh(t)IIE::;

C

### (IQI4

+

C e

1 1f314

+

lu(t)14

+

k(CECJ)-lt

max

lu'(s)14 sE[a,t[

+Ce1t

max

lul/(s)14) h

2

SE[a,f)

for

t

E [0 , 00).

Proof

From Theor e m 5.3.1,

Ilu(t) -

uh(t)IIE

::;

II

E i

f3 -

) I

### I

+ it

( 1Ie~ (s ) 1 1

+

~J Il e ~(s)lla)

ds.

All that remains to be done is to apply the a ppro x imation results from

### Co­

rollary 4.6.1 to ea c h of the terms in this expression:

91 and

CHAPTER

5.

CONVERGENCE

92

From Lemma 5.3

.1 ,

( Pu)'

=

Put

and hence e~(t)

=

u'(t) -Pu'(t).

This yields that and

Similarly, o

Under less strict regularity conditions we can still show that the solution

Uh

of P roblem AG converges to the solution

### u

of Problem A in the energy norm if

h

tends to zero .

e

### orem 5.3.3

Let O:h

=

and u

E

C

2

([0,

00),

V ) , th e n

ITo:

and Ph

=

ITp. Assum e that

0:

E

V ,

P

E

V lim

h-+O

Ilu(t) -

uh (t) II E

=

0

t

E

[0 ,

T].

### Proof

From Theorem 5.3.1,

u(t) -

uh(t )

IIE ::;

Il ep(t) II E

+

IIPo:

E

+

IIP

p -

TIp ll

+

## fo·t

( 1I e ~( s ) 1 1

+

~J Ile ~(s)l l o

dS) .

From the approximation results we know that for any

E

>

0 , each term is less th an

E,

provided that

h

is suffi c i e ntly small. 0

e

es

### timat

e

I n a final result we show that the Aubin-Nits c he tri ck can also be applied to this probl e m to find inertia norm estimates for th e dis c retization e rror.

CH A

PTER 5. CONVERGENCE 93

Theorem 5.3.4

ul/ (t) are all in

V

Let O'.h

=

PO'. and !3h

=

P!3. Assume that u(t) , u'(t) and

n

H4 for all t

;::

0.

Then ,

Ilu(t) uh(t)11 :::;

C

(lu(t)14

+

kt(CJCE)-l

m ax lu'(s)14

+

tCi/

max

lul/(S)14) h4

SE[O,tl

SE[O,tj

for t

E

[0,

(0).

Proof

From Theorem 5.3.2,

. I/u(t)

- Uh(t)/1

<

l/ep(t)/1

+

/Ie(t ) /1

<

/Ie p(t ) /1

+

Ci

< l/e p(t)

/1

+

Ci1

1

1I e(t)/IE l t

( lI e ~(s) 1I

+

~J ll e~(s ) ll o)

ds.

For a fixed

t;::

0, we consider

ep(t)

=

u(t)

### - Pu(t

) .

We conclude from Corollary 4 .6

.2 that

II

e

p (

t )

1/ :::;

C

u (

t)14

h

4

.

Simil ar arguments yield that

### lIe~(t)l/o

:::;

C!llle~(t)1I

:::;

Ci1Clu'(t)14h4

and lie~(t)1I

:::;

Clul/(t)14h4. (5.3.6)

o

A useful result is also obtained if the Aubin-Nits c he trick is us e d only for the terms containing the integrals .

T heorem 5 .3.

5

Let O'.h and ul/ (t) are all in

V

=

### ITO'.

and!3h

=

IT l)',

Assume that

0'..

.

;3 ,

u(t). u'(t)

n

H4 for all t. Then, lI u(t)

### -

Uh(t)

liE :::;

C

(10'.14 + Ci

1

1!314

+

lu(t)14) h2

+

C

(ktCi2

max

l u'(s)14

+

t

max lul/(s)14)

h4 for t

E [0,

(0).

SE[O , t] sE [O,tj

Proof

T he proof is exactly the same as the proof of Theorem 5.3.2. The estimates in (5.3

.

6) are used for the terms containing the integr a l. D

Remark

We consider thi s result to be significant. It is advantageous to have an error estimate in the energy norm, while the terms containing tare

" suppressed" by

h4.

CHA PT

ER 5. CO

N VERGENCE

4

e

## nces

94

In this section we consider the system of ordinary differential equations ,

Problem A D in Section 4.1, and the finite difference method for approxima­ ting the solution. The objective is to prove that the solution of the discre­ tized problem converges to the solution of the Galerkin approximation. This method has been extensively studied e ven in the context of finite difference methods for second order hyperbolic partial differential equations. However , one must be careful when matching the estimates. Although all norms are equivalent in the finite dimensional space

Sn,

the " c onstants" may depend on the dimension of Sh. Presenting error estimates for semi-discrete and fully discrete systems in the same presentation is a line also followed by others .

See for example [ D ], [Ea] and [FXX].

W e consider Problem AG in Section 4.1 and the finite difference scheme proposed in Section 4.4. In the first subsection we estimate the local error and then proceed to establish stability results.

.4

L

### ocal error

The first step is to derive finite difference formula.s similar to the Newma.rk schemes [Zi]. Since we need error estimates in terms of the unknown fun c tion or its derivatives, it is necessary to derive the formulas .

We will use Taylor's theorem in the following form:

g(t )

=

g(to)

(t t

) n -l

+

(t - to)g'

( to)

+ ... +

(~ _ o

1)1

g(n-l)(to)

+

R(t)

where

R ( t)

=

( n ~l)1

ft:(t

-

e)n-lg(n)(e ) de .

It is also true for

t

<

### to·

See [el, p 179] or [Ap, p 279].

The following notation is introduced for convenience.

1

N

### otation

R~(t)=

( _ )1

n

1 .

t j tHt

(t+6t-et-

1 g( n )(e)de a nd

The first proposition contains well-known results and the proofs are trivial.

CHAPTER

5. C

ON V ERGENCE

95

Proposition 5.4.1

l.

If the real valued function 9 is

m

C3 [t -

bt, t

+

bt], then g(t

+

bt)

g(t - bt)

=

2btg'(t)

+

Rj(t)

- R3(t). (5.4.1)

2. If the real valued function

9 is

in C4[t - bt, t

+

btl, then g(t

+

bt)

-

2g(t)

+

g(t

-

bt)

=

(bt)2 gl/(t)

+

R t(t)

+

Ri(t). (5.4.2)

### Proof

1.

Use Taylor 's theorem to get:

g(t

(bt

) 2

+

bt)

=

g(t)

+

bt g'(t)

+

2 -

gil ( t)

+

Rj (t)

and

g(t -

bt)

(bt)

2

=

g(t) -

btg'(t)

+

-2- gl/(t )

+

R3 (t).

Clearly

9 (t

+

bt)

-

9(t

-

bt)

=

2bt g' (t) +

Rj

(t

)

-

R3

( t ) .

2. A pproximate

9

by a polynomial of degree three and co mpute

g ( t

+

bt)

+

g(t

-

bt).

o

W e gave the proof of part one in detail b eca u se we us e the re s ult. in the n e xt proposition.

### Proposition

5.4.2

Let Po and

PI

be real numbers such that Po

+

2PI

= l. l.

If th e real valued function 9 is in C4[t

-

bt, t

+

bt], then g(t

+

bt) - g(t -

bt)

=

2bt (PIg'(t

+

bt)

+

Pog'(t)

+

PIg'(t - bt))

+

R

4(t),

(5.4.3)

CHAPTER

5.

CONVERGE N CE

96

2. Suppose the real valued function 9 is

m

C5[t - ot, t

+

M], then g(t

+

M) - 2g(t)

+

g(t - M)

=

(M)2 (Plg//(t

+

M)

+

Pog"(t)

+

Plg//(t - M))

+

R5(t), (5.4.4) where R5(t)

=Wl

{Rt(t) +Ri(t)} +W2{ Rt(t) +R5(t) ­

-

(M)2jt+6t

24

t

(t

+ M - 8)2g(5)(8) d8 - -

(M)2jt-c5t

24

t

}

(t

- M - 8)2g(5)(8) d8 .

### Proof

1.

Use Taylor's theorem to get:

9

(t

(M)2 (M)3

+

M)

=

9

(t)

+

Mg' (t)

+

2-

g"

(

t)

+

-6-

g'"

(

t)

+

R; (t

) and

9

(t

- M)

=

9

(

t)

- M g'

(t)

(ot)2 (ot) 3

+

-2 -

g// (t)

-

-6-

g'" (t)

+

Ri

(t ) .

This yields

g(t

(M)3

+

M) - g(t - M)

2M g'(t)

+ -

g"'(t)

+

Rt(t) - R

;

### (t).

3

(5.4.5)

Applying Taylor ' s theorem once more on

g' we obtain

g' (t

(M)2

+

M)

= g' (

t)

+

M g" (t)

+

-2-

g//' (t) +

11

2

t t 6t

-

+ (t

+

M

-

8)2g(4)(e) de

and

g' (t - M)

(M)2

=

g'

( t) -

M g"

(

t)

+

-2 -

g//'

(

t) +

-

1

2 t

1 t c5t

­ (t-M-8)2g(4)(8)d8.

The two equations yield

CHAPTER

5.

CO N VERGENCE g'(t

+

5t)

+

g'(t - 5t)

=

2g'(t)

+

(5t)2 gl/'(t)+

97

From this we get an expression for into (5 .4.5). The result is

(5t)2g//'(t)

which can substituted

g(t

+

bt)

-

g(t -

bt)

(5.4.6)

Finally we combine (5.4.1) and (5 .4.6

) with weights

WI

and

W2

to get the desired result.

2. This proof is similar to the proof in

(1) .

D

### Remark

The results above will also be used in the case where the function

9

is not defined for

t

<

O. In this case we ma y extend

9

by using the polynomial approximation on

[t -

5t ,

0). This will only influence the result in so far as there will be fewer remainder terms.

The second step is to apply the difference formulas to

Problem

AG a nd to estimate the errors.

A ss umption

We assume that

f

E

C3[O ,

T]

so that the solution

Uh

of Pro­ blem

AG is in

C

5

[0,T].

N otation

### Illuhlllf

max

=

,

L

5

k=O

max

tE[O,T ]

Ilu~k)(t)IIE'

Notation

In the rest of this section

C b

will denote a generic constant that depends on the bilinear forms, i.e.

C b

is a combination of

C

E

and

C

I .

Notation

Illflhmax

=

L

3

k=O

max

tE[O , T I

Ilf(k)(t)ll·

CHAPTER 5. CONVERGENCE

98

s

t

### ion

5.4.3

Suppose Ui

E

C

5

[t

-

M, t

{¢1 , ¢2, ... ,

+

Ml

faT

i

= 1, 2, ... ,

nand

¢n} is the basis faT Sh and let Uh(t)

=

L ~ = l

Ui(t)¢i' Suppose also that Po and P1 aTe Teal numbeTs such that Po

+

2P1

=

1 .

If Uh(t

+

M)

-

2Uh(t)

+

Uh(t

-

M)

=

(M?(P1U~(t

+

ot)

+

Pou~(t)

+

P1U~(t

-

M))

+ e~,

(5.4.7)

Uh(t

+

M)

-

Uh(t

-

M)

=

2M

(P 1U~(t

+

M)

+

Po ' u~(t)

+

P1U; 1(

t -

M))

+ e ~

(5.4.8)

and

(5.4.9)

then

lle~ll

S;

K(M)3 { max

eE[t,tHt j

I lu~3)(e)11

+ max

eE[t,tHtj

Ilu~4)(e ) 11

+ max

eE [ t,tHt)

Ilu~5)(e)ll }

and

P roof

Consider (5.4.7) as an example: Use (5.4.4) in P roposition 5.4.2 for denote the remainder by

U i

and

R

5i

(t),

N ow, each term in (5.4

.

7)can be written as a linear combination, for example,

Uh(t

+

M)

=

L n

i = l

Ui(t

+

M)¢i.

Consequently we have (5.4.7), if we set e~(t)

=

L~l

R

5i

(t)¢i'

CHAPTER 5.

CO N VERGENCE

It remains to estimate the error term e~(

t),

which is ac tually the sum of six error terms. Consider one of the terms: For any

(W I v

E

Sh t

[+01

(t

+

8t

-

e)3 u

;41 (e )t/>, de, v)

- 1 [+01

W ,

(t

+

8t

-

e)3 (

u~41

(e), v) del

<

I t +bt

< t

(5

llv

max

Ilu~4)(

)

### ll·

4

eE[t ,l + M)

H ence there exists a constant

K,

which depends only on the wei g hts

W 2

such that

( l e~ (t),v )1

:S

K (M)3

1Ivll { max

eE[t,t+6tj e El t , t +ot)

WI a nd

Ilu~3)(e)11

+ max

Ilu ~4)(e) 11

+ ma x

eElt,Hot)

lIu~5)(e)II}.

N ote that the worst of the errors are of order

(M)3.

Since

v

is a rbitr a ry, we have the desired result. The same procedure yields estimates in the energy norm. o

Lemma 5.4.

1

S up po se ' uh 'is the s olut ion of Problem

AG.

L e t Po and P I be real nu mb ers suc h that Po

+

2Pl

= l.

If u*(t ,

M )

i s defined by

(u*(t,

M) -

2Uh(t)

+

Uh

(t -

5t), v)

+

(M ) a(u *(t,

M) -

Uh(t

-

5t ) , v )

2 .

+ ( M )2 b(PIU *( t,

M)

+

POUh(t)

+

PIUh(t

- M

), v)

(M)2 (pI! ( t

+ M) +

Pof(t)

+

pI!(t

- M)

, v)o for ea x h v

E

S\

(5.4.10)

99

Proof

Using Proposition 5.4.3 we have

(Uh(t

+ M) -

2Uh(t)

+

Uh(t - M),

v)

+

-

(M)

2-

a(uh(t

+ M) -

Uh(t

-

M ), v)

(M)2

( PIU~ ( t + M ) + Pou~(t) + PIU~(t

-

M), v)

+

(e 7 , v)

+(M )2

a(pIu~(t

+ M) +

Po ' u~(t)

+

PIU~(t

-

M), v )

+

(~t) a(e~,

v).

CHAPTER

5.

CO N VERGENCE

100 ow use the fact that

Uh

is the solution of Problem AG to prove that

Uh

satisfies (5.4

.

10) with

u(t

+

M) in stead of

u*(t,

M) provided that the error terms

(e~,

v)

and

(~t)

a(

e~,

v)

are included.

Consequently,

(u(t

+

M) -

u*(t, M), v)

=

(e~,

v)

+

(~t) a(e~,

v)

for each

v

E

Sil.

Re place

v

by

u*(t,

M) -

Uh(t

+

M) to obtain the estimate.

0

Reconsider the semi discrete system in Section 4.4.

Mul/(t)

+

Lu'(t)

+

Ku(t)

=

J(t)

(5.4.11)

u(O)

=

Ci ,

u'(O)

=

/1

To estimate the local errors for a finite difference scheme, we consider a one-to-one correspondence between

Sh

and

lR n .

D

t

5

### .4.1

For uh

E

Sh, the vector u n

=

Quh has components Ui wh ere u h

=

### LUi(/Ji.

i = l

If we use the norm

1

=

(Mu· up

for

lR n ,

then

=

### Iluhll·

In our next result use the fact that

u

is a solution of (5.4.11) if and only if

Uh

i s a solution of Problem AG.

C

### orollary 5.4.1

If u is a solution of the system of differ-ential equatzons

(5.4.11) andu*(t,M) is defined by

M [u*(t ,

M) -

2u(t)

.

+

u(t

-

M) ]

+

(M) L [u* (t,

M) -

u(t

-

M)]

2

+(M)2 K [PIU*(t,

M)

+

Po u(t)

+

PIU(t

-

M ) ]

(M)2[PIJ(t

+

M)

+

PoJ(t )

+

PIJ(t

-

M)], then Ilu*(t,

u(t

+

b

(5.4.12)

### Proof

Consider the terms in (5.4. 10) .

If

fj

=

Qv h

)

then

(Uh( t

+

M), vh)

=

j '\1u(t

+

M) .

v.

In this way we can associate each term in (5.4

.12) with a corresponding term in (5.4.10). The resul t follows fom the fact that

u(t

+

M)

=

QUh(t

+

M). u*(t ,

M)

=

Qu*(t ,

M) and

0

C H

APTER 5. C

ONV ERGENCE

102

2. If we assume that

f

is merely conti nuous and hence

Uh

twice continu­ ous l y differentiable, we could still estimate the local errors but not obtain the same order. The results would be of the form:

Given

c

>

0 ,

there exists a real number

6.

>

0

such that the error will be less than

cOt

K for Ot

<

6..

(K

a constant depending on

Uh

and

f. )

5.4.2

### Transformation

Due to symmetry considerations , it will be more conve nient to consider a transformed system for stability ana l ysis. Since

!vI

is symmetric and posit­ ive definite, there exists a symmetric positive definite matrix

N

such that

N 2

=

M.

Set

v(t)

=

Nu(t),

then

v

is a solution of the problem

v"

+

N - 1LN-1v'

+

N- 1 K N 1v

=

N- 1

J

or

### v"

+

Lv'

+

Kv

=

.9. where

L

=

N - 1 LN1

,

K

=

N -l

K

N -

1 and

9

=

N -l

].

(5.4.14)

The advantage of the transformation is that the matrix

K

is symmetric , and hence has orthogon al eigenve ctors.

Let

y

=

Ni ,

then

Ki

=

A j Vff

if and on l y if

The eigenvalues of K are the eigenvalues of the eigenvalue problem Pro­ blem CG (See Sections 4.1 and 5.2.)

W e use the norm

IIi l12

=

(i . i)

~ , and in the remaining part of this sectio n

II . I I

will refer to

I I

.

11 2 unless stated otherwise.

Corollary 5.4.3

If v is a solution of the system of differential equations

(5.4.14), and v*(t, Ot) is defined by

[v * ( t , Ot)

- 2v(t)

(Ot)

~

+

v(t -

Ot)]

+

-2-

L [v * (t, c5t)

-

v(t

-

Ot)]

+(Ot f i( [ PIV*(t, Ot)

+

Pov(t)

+

PIV(t

-

Ot)]

(Ot)2 [ Plg(t

+

Ot )

+

POg(t)

+

Plg(t - Ot

) ],

CHAPTER

5.

CO NV

103

### Proof

Direct from Corollary 5.4.1, since

v*(t,M)

=

N u*(t,M).

0

C

y 5

### .4

.4

If v is a solution of the system of dIfferential equations

(5.4.14), and v* *(t,

M)

is defined by

2

[v** (t,M)

-

v(t )]

(

M)2

+

-2-K

[v**(t,

M) +

v(t)]

(6t)2

-2[g(t

(6t)3

+

M)

+

g(t)]

+

2Mv '(t)

-

(M)2 Lv'(t)

+

2-Kv'(t)

(M)3 _'(

2

t

)

'

then

Il v* *(t,

6t)

-

v(t

+

M)II ::;

C b

(6t)3

(1IIuhl

### l

l f,max

+

(M )311Ifl hmax) .

o

### f

See Corollary 5.4.3.

### Remark

The result remains true for

t = o. o

5.4

.

3 G l o bal e rror

v V e approximate the solution of (5.4

.14) on the interval

[0 ,

T ] .

Let M indicate the time step len gt h , i.e.

6t

=

T

/

N,

and let

Wk

denote the approxi mation for

v(t k )

.

W e use the difference sc heme (which co rrespond s to (5.4

.12))

(

-

l -

2-

+ )

+

(M)L-(-

-

Wk-l

+(M)2 K(PIWk+l

+

POWk

+

PIWk-d

=

(6t)2(Plgk+l

+

POgk

+

Plgk-d·

(5.4. 15)

The initi a l conditions for the system of differenti al equations are

v(O)

=

No:

and

v'(O)

=

N iJ,

and the initial conditions of the finite difference system are

C HA PTER

5.

C ON V E R GEN C E

To estimate local errors the following scheme will also be used:

104

To estimate the global error tions

w

N-

V

(T)

we introduce artificial numerical sol u­

wk')

For each i,

wk i

)

satisfies

(5.4.14) with

w?)

=

v(t i )

and

-(i) -

W i

l

WHl -

2

s:t-/(t)

U

V i .

N

0 t th t -

-

_ _

(0)

.

For the global error we have

Ilv(T) - WNII

~

Ilv(T) - wt-l)11

+

I l wtl

) w~V-2)11

+ ... +

Ilw~)

-

~wNII·

(5.4.17)

( N ote that the global error fo r the original system can be derived fro m this error. )

It is clearly necessary to estimate

Ilw~)

-

~w~-l)ll.

The next t w o subsections will be devoted to the estimation of the differences between "neighbouring numerical solutions" .

5.4.

4 Consistency

In this subsection we consider the differences

II

+

2 -

-(i+l)

.

. with the "starting" error. l"t d t

Ilw~21

-

v(ti+d

II and

-(t)

b Th fi t I d

1

L emma 5 .4.3

### Proof

This is a direct consequence of Corollary 5.4.4.

1 ext we have the error at the second step. o

C HA P T ER

5.

.4.4

105

### Proof

Combine the results of Corollary 5.4.3 and Lemma 5.4.3. 0

Lemma 5.4.3 provide an estimate for the difference

W~i)

-

W~21.

The following result provide an estimate for the difference at the second step.

C

5 .4.

5

w;~~l)11

b(bt)3

+

P

### roof

Use Lemmas 5.4.3 and 5.4.4.

WH2 -

W

H2

<

_(i

i

+

)

2

-

-

Vi+2

+

Vi+l -

.

o

5.4.5

### Stabilit

y

For the stability analysis we introduce the following matrices:

A

B

C

(bt)

­

1+--L+Pl(bt)2K ,

2

=

-21

+

PO(bt) 2

K,

1 -

(bt)

2

Z

+

Pl(bt)2

K.

The system (5.4.15 ) is now

(5.4.18)

As mentioned at the end of Subsection 5.4.3, we need to estimate the differ­ ence w~) _W~+l) for each i. Since both

W;i)

and

wy+l)

satisfy the system

(5.4.18) it follows that the error

ej

=

wJ i)

w;Hl) must satisfy

(5.4.19)

CHAPTER

5.

C ONV ERGE N CE

106 with the starting values, the local errors

ei+l

and

ei+2,

already estimated.

For the case

L

=

J.1K ,

we derive the eigenvalues of

A,

Band

C. If

Ky

=

).,y,

then

By

Cy

+

J.1--Ky

+

Pl(cSt)2Ky

=

(

1

+

2(J.1).,)

+

) y,

2

-2y

+

Po

Ky

=

(-2 +

).,)

(1-

~t(J.1).,)

+

### Pl(<5t)2).,)

y.

It is now possible to solve (5.4.19). Let Yl, Y2, ... , Yn denote the normalized eigenvectors of

K

and suppose

e i +l

=

.L~1

TJ(Y i

and

ei+2

=

.L~1 ~iYi'

Since the eigenvectors are orthogonal , it is sufficient to solve difference equa­ tions of the form where

ai,

{3i and

Ii

denote the eigenvalues of the matrices

A,

Band

C respecti vel y.

The following result can be obtained by elementary calculations. Note that we do not use the subscripts for the coefficients

a,

{3 and i. We take

Tl =

~ and

TO

=

TJ·

Solution of t he differenc e equation

C ase 1

(32

<

4ai.

N ote that in this case

I

>

O. The solution is of the form

Tk

=

pk(A

+

B

sinwk),

=

Vi/Ct.,

### cosw

=

-(3/(2...flYY), A

=

TJ

and

B

=

(~-

PTJCosw)/(sinw).

Cas e

2

{32

=

4Ct./, .

The solution is of the form

CHAPTER 5. CONVERGENCE

The solution is of the form

107 with

7'1 and

r2

the real roots of the equation

o:r2

+

;3r

+

r

=

O. The constants are

A

= (~

-1]T2)/(rl - r2) and

B

= (~

-1]Tr)/(r2 - rr).

We now prove the stability result. Bear in mind that

)..k

-----+

(X) as

k

-----+ 00.

Lemma 5.

4 .5

Stabzlzty

If Po ::; 2pl) then there exists a constant K -independent of the dimenszon

of Sh -

uch that

II

W N - W N

II

K

(II

::; 'Ui+l -

W i

+

1

II

+

II

(i) (i+l)

II)

.

o

### of

For the eigenvalues

1

M

+

Pl(M)

### 2

)..

+

2)..jJ-,

-2

+

Po(M? )..,

l+Pl(M)

2

M

)..-2)..jJ­

of

A,

Band

C, we get

-4Po(M)2)..

+

P6(c5t)4)..2 - 8Pl(M)2)"-4pi(M)4)..2

+

(M)2 jJ-2)..2

)"(M)2 [jJ-2).. -

4

+

(P6 - 4Pi)(M)2)..] .

Consider the different cases:

C a se

1

If

;32

<

40: r

then

rk

is bounded if "/ ::;

0:.

This case,

;32 -

40: r

<

0, is possible only for a finite number of small eigen­ values and only if

Po

~

2pl. Since

r/O:

<

I, the corresponding modes will not cause error growth.

C as e 2

If

;32

=

40: r

then

rk

is bounded if

1;3/0:1

<

2.

If

PO(M)2).. < 2, we have

1;31/10:1

<

## f·

If

Po(M)2)..

>

2, we have

Po ( c5

t )

2 )..

Po .

1/31/10:1

<

(c5

)2)..

= ::;

2 If

Po::;

2Pl'

PI

t

PI

CH AP TER

5.

C ONV ERGE N CE

108

C ase 3

If

13

2

>

4CYi

then

Tk

is bounded if both roots of

CYT

2

+

j3T

+ /'

=

0 are less than one in absolute value. Let and let

T

max denote the absolute value of the root largest in absolute value.

Tmax

<

13

+

J75,

2cy

PO(bt)2)...

+

)"'6t

VM2

+

(P6 - 4pi)(6t)2

2pl(6t)2)...

+

6t)...M

Po6t

+

VM2

+

(P6 - 4pi)(6t)2

2P1 6t

+

M

Po6t

+

M

<

2P1 6t

+

M

<

1 ( if

Po

S;

2pl)·

o

Remark

If damping is excluded, the difference system is considered to be unconditionally stable for

PI

=

~ and

Po

=

~, see [RM] or [Zi]. However, the bound may depend on the eigenvalues. cosw

= --'­

2y1CFi

- Po ( 6

t )

2 )...

+

2

2(1

+

PI (6t)2)...)

-po(bt?

+

2)",

-1

-Po

2Pl(bt)2

---7 -

+

2)...-1

2Pl

as )...

---7

00.

Consequently, sin

w

---7

0 as )...

---7

00 and sin

w

is present in the numerator of a constant.

R e marks

l. Exactly the same results hold if we assume that rotary inertia can be ignored and we have only viscous damping. In this case

L

=

kMo and

M=Mo·

2. The eigenvalues of

K i

=

)...j'vloi

are much larger than the eigenvalues of

Ki

=

)...j'vli

(with rotary inertia).

3. Rotary inertia and Kelvin-Voigt damping both enhance stability.

CHAPTER

5.

CO N VERGENCE

5.4.6

e

L

### emma

5.4.6

Global error

109

(w here

N

is the number of steps).

IleN11

<

I

w ~ ~ -lll

+ ... +

Ilw~ )

- wN11

<

K N

max{

2

Ilvi+l - W;2111

+

Ilv

### i+

l w;2211}.

N ow use Lemmas 5.4.3 and 5.4.4.

To the se quence of finite differ ence vectors Vk, correspond a sequence of ap­ proximations for

uh:

u~k )

=

(Q J)-lVk

E

Sh.

o

T

### heorem

5.4.1

If

Uh is the solution of Probl em AG. then

### Proof

Since

eN

=

QN(U n(

T) -

u;:),

we have

N ow use Lemma 5.4.6. o

### Remark

Error estimates for the fully discrete system is obtained by com­ binin g Theorem 5.4.1 with the results of Section 5.3. Note that the error estim ates are with respect to the inertia norm.

6

## Application. Damaged beam

c

### tion

We consider Problem 1 (from Section 2.3). This model for a damaged beam was proposed in [VV] . See also [JVRV].

The detection of damage in structures or materials is clearly of great im­ portance. Ideally it should be possible to infer the location and extent of damage from indirect measurements or signals. To facilitate such deduction , a mathematical model of the object or structure is necessary. See [ VV ] for details and numerous other references.

Viljoen

### et al.

[VV] use changes in the natural frequencies of the beam to loca te and quantify the damage. The natural angular frequencies for the damaged beam are calculated from the characteristic equation obtained from the as s ociated eigenvalue problem. As is well-known, only the first few n atu ral angular frequencies and modes are usually calculated with this method, because of computational difficulties with the hyperbolic functions. Due to this limitation, the need arises for a numerical method to simulate the dynamical behaviour of the beam.

In a joint paper, [ZVV], we developed a finite element method (FEM) to ap­ proximate the solution of the model problem for arbitrary initial conditions.

(Ironically we also found it possible to calc ulate eigenvalue s a nd eigenfunc­ tions more accurately with the FEM .

)

It wa s nec es sary to adapt sta nd ar d procedures to deal with the discontinuity in the derivative that arises a s a

110

CHAPTER

6.

APPLICATION. DAMAGED BEAM

111 result of the elastic joint. W e made the assumption that damping would not influence t he so lution significantly on a small time scale. We now inves t igate the validity of this assumption, and als o deem it prudent to include the effect of rotary inertia.

In the paper [ZVV], only He rmite piecewise cubics were used as b as i s func­ tions.

In

this investigation we a lso demonstrate the effectiveness of Hermite piecewi se quintics.

From Section 3.1 we have the variational formulation. The Galerkin appro x i­ mations for the eigenvalue and initial value problems a re given in Section 4.l. eW e do not cons ider the equilibrium problem.) In Section 4.2 we showed ho"v the stand ard basis functions are adapt ed to deal with the discontinuity

111 the deriv ative.

In

Section 6.2 we compute the n atura l angular frequencies and modes of v ibr ation from the characteristic equation for comparison purpos es. The computation of the matrices is discussed in Section 6.3.

I n

Sections 6.4 and 6.5 numeric al results are presented that demonstrate not only the effect of dama ge on the motion of a beam but also the effect of dampin g and rotary inertia. We a lso investi gate t he use of Hermite piecewise quintics as basis functions instead of Hermite piecewise cubics.

6 .

112

N

e

e

e

m

t

o

### n

One way to calculate the natural angular frequencies and modes of vibration for the damaged beam is to apply the method of separation of variables directly to Problem 1 (from Section 2.3).

For the case

T

=

0 (without rotary inertia), we have the following eigenvalue problem:

W(4) -

.Aw w(O) w(a+) w"(a

+)

w'"

(a +)

w" (a)

0,

0 < x

<

1,

x

I:a,

w'(O)

-

w"

(1)

w(a-), w

"( a

-)

,

w

### '"

( a

-)

,

= w"'(l)

0,

~(w'(a+)

- w'(a-)).

For this eigenvalue problem it is possible to find so called exact solutions .

It is convenient to introduce the positive real number

u,

with

.A sequently

u

2

= vf).. is a natural angular

=

u

4

.

Con­ frequency. Analogous to the case of the undamaged beam, the corresponding mod e is of the form

_ ( Asin(ux) - Asinh(ux)

+

Bcos(ux) - Bcosh(ux) for 0

<

x

< a ,

w(x) (C

+

A)

sin(ux)

+

(D - A)

sinh(ux)

+(E

+

B) cos(ux)

+

(F

- B)

cosh(ux)

for a

<

x

<

1.

N ote that the boundary conditions at

x

=

0 have already been taken into account.

From the continuity conditions and the jump condition at

x

= a, the con­ stants

C, D, E, and

can be expressed

in

terms of

and

### B.

Finally, from the two boundary conditions at x

=

1, the characteristic equation for u can be constructed from

(6.2.1)

CHAPTER

6.

APPLICATION. DAMAGED BEAM

113 where

-(sin

v

+ sinh

v)

+

ov (sin

va

+ sinh

va)

x

2

(sin

v cos va sinh

v cosh va cos

v sin va

+ cosh

v

sinh

va),

( cos

v ov

+ cosh

v)

+ ( cos

va

+ cosh va) x

2

(sin v cos va sinh v cosh va cos v sin va

+ cosh v sinh va)

-(cos

v ov

+ cosh

v)

+

-(sin

va

+ sinh

va)

x

2

(cos

v

cos

va

cosh v cosh

va

+ sin v sin va

+ sinh

v

sinh

va),

(sin

v

sinh

v ) ov

+

-(cos

va

+ cosh //a) x

2

(cos

v cos

va

cosh

v cosh va

+ sin

v sin va

+ sinh

v sinh va).

Solving equation (6.2.1) numerically using the Newton-Raphson method, yields the natural angular frequencies for the damaged beam. For each na­ tural angular frequency a corresponding mode can then be obtained. As is expected, only the first few natural angular frequencies and modes could be calculated, as it is difficult to handle the hyperbolic functions numerically for large values of

v.

N umerical results obtained using the finite element method using cubics as well as quintics as basis functions- are given in Section6.4.

114

CHAPTER

6.

AP PLICATI O N. DAMAGED BEAM

### 6.3 Computation of Matrices

The matrices

K ,

Land

M

are defined in Section 4.1 in terms of the bilinear forms defined in Section 3.1. The computation of the matrices is complicated by the interface conditions which results in non-standard basis elements.

In this section we give an indication of how we went about in computing th ese matric es. The first step is to reorder the basis elements constructed in

Subsection 4.2.2.

Consider the matrix

j \1 o:

[ J\!IoJi

j

= (¢i'

¢j)

=

l Q cPjl

+

1 1 o te that

### ¢i

(0,

cPi)

or

(cPi ,O)

except when we are dealing with a basis element associated with the node

xp

=

0: , the location of the damage. In general then, the entries will be those of the standard mass matrix for an undamaged beam. N ow suppose one of the basis elements are associated with

xp :

Again the result will be the same as in the standard case. The same for

(2)

cP P .

On the other hand, suppose

=

### ¢;

2 ) then

[j\1

0 l ij

=

let cPj1 cPi l

+

0 , which is not the same as for an undamaged beam. Similarly for

### ¢i

=

¢;~.

Thus the standard matrix h as to be modified for the damaged beam.

W e have the same situation for the matrix

M r

where we define

There is an additional complication for the

K

matrix:

CHAPTER

6.

APPLICATION. DAMAGED BEAM

115

Only four entries in the standard K-matrix will change due to the additional term,

Cu~(ex)

- 'U'l(ex))

(v~(ex)

v~(ex))

/6, in the bilinear form

b.

For greater clarity we will explain the procedure in another way. In the discussion that follows, we refer to

¢~k) as a Type k basis function.

In modifying the matrices for an undamaged beam to the matrices for a dam­ aged beam, we have to keep in mind that the Type 1 basis function associated with

xp

= ex, has changed. By replacing the row and column associated with the Type 1 basis function at

x

P'

in the matrix of the undamaged beam, by two rows and columns respectively, provision is made for the modified basis function. The values in the matrix in these two rows and columns have to be modified accordingly. For the

K -matrix one must also keep the additional term in mind.

Having computed M

o,

Mr and K we are done since L

p,K

M

=

j\!fo

+

Mr.

+

kMo

and

CHAP T ER

6.

APPLICATION. DAMAGED BEAM

116

N

e

s

e

e

## problem

Cubics as basis functions, are usually sufficiently accurate in solving one­ dimensional vibration problems with the finite element method. In a joint paper

[ ZV V] we discussed the use of cubics as basis functions for the damaged · beam.

In this section of the thesis we also consider numeri ca l converg e nce of th e eigenvalues. The order of convergence that is sugg e sted b y the numeri ca l results is also compared to the order obtained from the th eory. Additionally, quintics are considered as basis fun c tions. Th e main reason for this is that cubics are not compatible with reduced quintics in plate beam models . We also investigate the effe c t of rotary inertia.

C

### ubics

N atural angular frequencies and modes for the vibration problem are calcu­ lated by solving the eigenvalue problem with the FEM. 'vVe developed the code to construct the relevant matrices in Matlab and use standard Matlab subroutines to calculate the eigenvalues and ei ge nvectors of the generalised eigenvalue problem.

It i s possible to compare only the first few FEM eigenvalues to the so called exact eigenvalues calculated from the characteristic equation. Thereafter the exact values can not be computed accurately a nd the F E M is used to calculate the eigenvalues.

In Table 6.1 we list values for the eigenvalues obtained from the characteristic equation (see Section 6.2) and values obtained by the FE M using cubic s a s b as is functions with 20, 40, 80 and 160 subintervals respectively. This give approximations for respectivel y the first 40, 80 , 160 and 320 eigenvalues.

CHAPTER

6.

### APPLICATION.

DAMAGED BEAM

117

). (20) t

).

(80) t

1 11.81469 1l.81469

11.81469 11.81469 1l.81469

2 406.01614 406.01757 406.01623

406.01615 406.01615

3 3806.05283 3806.17742 3806.06067 3806.05332 3806.05287

4 12544.12940

12545.47137 12544.21439 12544.13473 12544.12972

5 39943.82322 39957.35763 39944.68387

39943.87724 39943.82661

Table 6.1:

,es

ics

c5

=

0.1

### and

Ct

=

0.5.

Throughout. t.his sed .in

n

n

clellntp. th>:> nu m be r of subintervals. (All of e quoJ length. )

To investigate the convergence of the FEM eigenvalues, we calculate the relative difference between

FEM approximations, that is ( ).

(2n)

).

(n))

/ ).

(2n).

These differences are calculated and listed in Table 6.2 for n

=

20, 40, 80 and 160 subintervals respectively. n

=

20 n

=

40

I

=

80 n

=

160

6

6.1

X

10-

4

3.9

X

10-

5

2.5

X

10-

6 l.6

X

10-

7

12 l.1 x 10

-'4

7.6 x 10

4

4.9 x 10

5

3.1 x 10

b

24 2.2 x

10-

1 l.2

X

10-

2

8 .6

X

10-

4

5.5

X

10-

5

48

-

2.2 x 10

1

1.3 x 10

2

9.2

X

10-

4

Table 6.2:

ive

### differences fOT FEM eigenvalues using cubics as basis

The tendency of the relative difFerence to decrease (by roughly a factor 10) each time that the number of subintervals is doubled , is empirical verifica­ tion that there is convergence of the FEM eigenvalues. Vve found th a t the eigenvalues computed from the characteristic equation were less dependable.

It is necessary to determine a relationship between the number of

FEM eigen­ values that is suffi c iently accurate (c riterion to be specified) and the number of subintervals used.

A relative difference strictly lCtls than 10-

3 is considered sufficiently accurate for our purpose. Using this as criterion, we find that approximately a sev­ enth of the 2n eigenvalues calculated using n subintervals, yields a relative difference,

(

).(2n) ).(n))/).(2n) , strictly less than 10-

3 , see Table 6.2.

6.

N.

### 118

The relative difference between the FEM eige nvalues with 160 an d 320 subin ­ tervals is an indi cation of the relative error between the exact eigenvalue an d the FEM e igenv a lu e using 320 subinterval s.

Since we u se

(>,(320) >-(160))/ >-(320) as measure of the relative error,

(>>-(3 20)) / >-, we conclude that the first 90 eigenvalues obta ined usmg

320 subintervals y ield a relati ve e rror that is sufficiently accurate.

A n indic at ion of the order of conve rgence of the FE M eige nvalue s can be obtained from the ratio of two success ive differences

>-(2n) _ >-(n)I/I>-(4n) _

1 t t t

### >-

(2n)

I t '

Typical results are listed in Table 6 .3

. z

1

3

6

12

24

### l>-i:3n

)

_

>-~n) I /I >;4n) _

>-?n)

I

=

20

=

40

=

80

10 .25

15.88

15.53

14.23

18.29

0.03

16.41

15.88

15.57

14 .47

0.14

6.57

15.72

15.90

15.62

Table 6.3

:

su ccessive

### relative

differences wz th cubzcs

### as basis functions.

These relative differen ces decrease b y roughly a factor 16 if the number of subintervals is doubled. F rom this it would appear tha t the co nver gence i s of order

### h4

whi ch matches the theory, Section 5.2.

It is observed that those differences not yieldin g a factor 16 typically occur in the ri ght top pa rt as well as the lef t bottom part o f Table 6.3. These deviations are illus trate d by the first , third and 24th eig enva lue s:

Fir st l y, the accuracy of an a pproximation can decr ease if the number of s ubin te rvals is increased. This i s due to an increase in the roundoff error and has significant effe cts in situations where the errors ar e alr eady small. For example, FEM approximations for the first eigenvalue y i eld

(>-i40) _ >-i20))

=

-1.1 x

### 10-

13 while

(>-i80 ) >-i40 ))

=

3.8 x

### 10-

12 .

From the theory, Se ct ion 5.2, we know that the FE M approxirnations of a n eigenvalue will decrease i f the number of subintervals is in crease d. This can

C H APTER

6.

A PPLICATION. DAMAGED BEAM

119 be used to detect cases where the effect of the roundoff error is greater than the advantageous effect of an increase in the number of subintervals used.

R ounding error also explain the decrease in the ratios for the third eigenvalue from roughly a factor 16 to 6.5. This situation differ from the first eigenvalue in that the decrease (improvement) in the relative difference was just partially cancelled by the increase in the roundoff error.

Secondly, as we have showed previously, there is a relationship between the number of FEM eigenvalues that can be calculated sufficiently accurately and the number of subintervals used. The 24th eigenvalue is such an example.

The effect of the poor approximation of

.\~~O ), is seen in Table 6.3 in that

18.29

>

14.47. This is expected as only the first six eigenvalues obtained , using 20 subintervals, yield relative errors less than

10-

3

6.4.2

### Quintic

s

W e now consider quintics as basis functions, and compare the results to the case where we used cubics.

In Table 6.4 we list values for the eigenvalues obtained from the characteristic equation and values obtained by the FEM using quintics as basis functions with 2, 4, 8 and 16 subintervals respectiv e l y . This gives approxim a tions for respectively the first 6 , 12, 24 and 48 eigenvalu e s .

1 11.81469 11.81469 11.81469 11.81469 11.81469

2 406.01614 406.01954 406.01618 406.01614 406.01614

3

3806.05283 3822.51900 3806 .

09344 3806.05297 3806 .

052 8 3

4

12544.12940 12844.88875 12544.53719 12544.13441

12544.12941

5 39943.82322

41569.18041 40042.72518 39944.02589 39943.82387

Table 6.4:

### Eigenvalues

from the characteristic equation as well as

ei­ genvalues

quintics as

### basis

functions with

<5

=

0.1

and

ex

=

0.5.

If these values are compared to those in Table 6.1, it seems as if the same accuracy can be obtained, using quintics as basis functions, with less subin­ tervals, than in the case where cubics were us e d as basis functions. For example, the fifth FEM eigenvalue using quintics as b a sis functions with

CH AP TER

6.

APPLICATION. DAlvIAGED BEAM

120

16 subintervals, already yields a better approximation than using cubics with

80 subintervals.

As in the case with cubics as basis functions, we investigate the convergence of the

F E M

eigenvalues by considering relative differences, PI

(2n) _)., (n)) /)., (2n).

These values are are list e d in Table 6.5 for 2, 4, 8 and 16 subintervals re­ spectively.

'/,

1

2.1

n =2 n=4 n = 8 n

=

16 x

10-

8

7.9

X

10-

11

1.2

X

10 -

11

1.7

X

10-

11

2 8

.

3

X

10-

6

4 2.4

X

10

### ­

2

9.5

X

10 ­

8

3.2

X

10-

5

3.4

X

10-

10

4.0

X

10-

7

1.3

X

10-

11

1.4

X

10-

9

8 -

3.9

X

10-

2

7.2

X

10-

5

8.2

X

10-

7

Table 6.5:

functions .

Relative differences for FEM eigenvalues using quintics as basis

The numerical results suggest convergence of the FEM eigenvalues s ince the relativ e error decreases (by roughly a factor 100) each time that the number of subintervals is doubled, see Table 6.5.

For approximately a third of the

3n

eigenvalues computed , u s ing

n

s ubinter­ vals , the relative differ e nce

().,(2n)

-

).,(n))/).,(2n)

is strictly les s than 10-

3

.

As with the cubics, we now consider the ratio of two successive differences to get an idea of the order of convergence. Typical results are listed in

Table 6.6.

As was the case in T able 6.3, the values in the top right of T able 6.6 exhibit effect of roundoff error and the values in the bottom left the result of eigen­ values not calculated sufficiently accurately. From this it would appear that the order of convergence is

h

8

which matches the theory, Section 5.2.

To compare the accuracy of the FEM eigenvalues using quintics as basis function s to the case using cubics as basis functions , we choose the number of subintervals in each of the cases such that the sizes of the matrices in the two cases are equal. For example, using 30 subintervals for cubics yie ld

61 x 61 matrices and 20 subintervals for quintics 62 x 62 matrices. We then

CHAPTER

6.

AP P LICATIO N. DAMAGED BEAM

121

~

IAr~n)

Ain)I/IAi4n)

### ­

Ai~n)1

n=2 n=4 n=8

1

265.60 6.75

0.69

2 87.21 278.22 25.45

3 405.90 278.67 319.70

4 745.69 80 .

59 291.89

5 15.43 488.57 311.65

6 16.24

330.71

307.27

7 304.98

166.95

286.96

8

302.37 540.94 87.70

Table 6.6:

### Relationship

between successive relative

### differen

ces with quintics basis functions.

compare the eigenvalues calculated in the two cases with the eigenvalues computed using cubics with 320 subintervals. (We use the first 90 FEM eigenvalues using cubics as basis functions with 320 subintervals as the FEM approximation to the first 90 exact eigen val ues.)

Our numerical experiments indicate that using quintics with

n

subintervals , yield at least double the number of eigenvalues to the prescribed accuracy

(relative error strictly le ss than 10-

3

) than when cubics a re used with

3n/2

subintervals. In Table 6.7 we give an example of results obtained.

In Table 6.7 we use the following notation:

• Let Ai denote the ith FEM eigenvalue that we use as approximation for the exact eigenvalue. (In this case those FEM eigenvalues obtained using cubics as basis fun ct ions with 320 subintervals.)

• To distinguish between the FEM eigenvalues computed using quintics and cubics as basis functions, we denote the ith FEM eigenvalue using cubics with 30 subintervals by

A~c) and using quintics with 20 subinter­ vals by

A~q ).

N ote that the FEM approximations for the first eigenvalue are identical in both cases .

From Table 6.7 we see that using cubics, the first 9 eigenvalues (app roxim­ ately a seventh of the number of eigenvalues calculated, 61/7

~

8.7) have

CHAPTER

6.

APPLICATION. DAMAGED BEAM

122

1

2.6

X

10-

6

5

6.8 x

10

-5

9

8.5

X

10-

4

10

1.2

X

10-

3

15

6.8 x 10-

3

20

1.9

X

10 ­

2

2.6

X lO b

9.6

X

10

9

4.3

X

10-

7

8.6

X

10-

7

4.6

X

10-

5

1.4

X

10-

4

Table 6.7:

Comparzng FEM ezg envalues us zng quintics wit h

20

subzntervals to FEM ei genvalu es using cubics wzth

30

subintervals.

relative difference less than 10-

3

.

Using quintics , the first 20 eigenvalues, that is approximately a third of the number of eigenvalues calculated , have relative difference less than 10-

3

.

In conclusion , for the same computational effort (same size of the matrices), quintics yield twice as many eigenvalues sufficiently accurate than when cu­ bics are used i.e. to obtain the first

k

FEM eigenvalues with relative differ e nce less than 10 -

3

)

7k/2

subintervals must be used with cubics as basis functions and

k

subintervals with quintics .

6.4

.3 T h e

t

rot a

ert

### ia

W e now consider the effect of rotary inertia on th e eigenvalues and use quintics as basis functions.

N ote that this eigenvalue problem differs from the one excluding rotary in e r­ tia, Section 3.5. The parameter

r

is a measure of the effect of rotary inertia,

Section 2.2.

W e start by establishing convergence of the FEM eigenvalues for the case where rotary inertia is included, thereafter, we investigate the effect of rotary inertia on the eigenvalues.

As for th e case without rotary inertia, the numerical re s ults indicate conver­ ge nce of the FE M eigenvalues. In Table 6.8 typical results for the relative differences,

(>-(2n)

- >-

(n))/ >(2n),

including rot ar y inertia , are listed for 2 , 4 , 8 and 16 subintervals respectively.

CHA P TER

6.

APPLICATI ON. DAMAGED BEAM

123

n=2 n=4

I

n=8

### n

=

16

1

2.1

X

10-

8

2 7.8

X

10-

6

7.8

8.9

X

X

1O-

10-

11

8

2.9

3.2

X

X

10-

10-

11

10

7.6

1.4

X

X

10-

10-

10

11

4 1.6 x

10

3

4.0 x

10

6

2.0 x 10

-tl

7 .

9

X

10-

11

8

-

1.6

X

10-

2

9.5

X

10-

5

3 .

3

X

10-

7

Table 6.8:

Relatwe differences for FEM eigenvalues including rotary inertza with 1 1 r

=

4800 .

The numeric al results again suggests convergence of the FEM eigenvalues .

The same pattern with respect to the order of convergence is observed as for the case without rotary inerti a.

The presen ce of rotary inertia decreases the values of corresponding eigenval­ ues in comparison to the case without rotary inertia. Furthermore, the bigger the parameter

r,

the greater the change in the eigenvalues in comparison to the case without rotary inertia. In Table 6.9 we list eigenvalues for different values of

r

as well as for the case without rotary inertia

(r

=

0). We use 32 subintervals for these approximations.

In Table 6.9

Ai

denotes the ith FEM eigenvalue.

I

i

I

A i with

r

=

0

I

11r

=

19200

I

11r

=

4800

I

11r

=

1200

I

1

2

11.81469

406.01614

11 .

81138

403.97925

11.80145

397.71100

11.76186

370.06960

4 12544.12940 11057.05461 5401.71657 3576.19925

8 273293.79309

169832.12061 158744.64019 125970.79578

Table 6.9:

FEM eigenvalues for different effects of rotar y inertia using 32 subintervals.

These results are for the dimensionless case. W here rotary inertia is included , two dimensionless constants ,

T

a nd

r,

must be calculated if the results is to be connected to a spec ific beam, Section 2.2.

M

### ode

s

In [ Z VV] we showed that only up to the seventh so called exact mode can be computed before computational difficulties are encountered. Therefore we

CHA P TER

6.

APPLICATION. DAMAGED BEAM

124 consider the convergence of the FEM modes using quintics as basis functions and include rotary inertia.

Let w~n) denote the F E M approximation for the ith mode using

n

elements, normalised with respect to the infinity norm ,

II .

1100'

The way in which we ord ere d our ba5i5 elements implies that the firsL

n

components of w~n) are associated with the function values at the

n+

1

+

1 nodes.

The next

n

+

2 values represent the values of the first order derivatives at the nodes. Two values are associated with the point where the damage occurs.

Quintics as basis functions also yield approximations for the values of the second order derivatives, and the last

n

+

1 values of w~n) represent the values of the second order derivatives at the nodes.

In

Table 6.10 the numerical convergence of the FEM modes are illustrated.

W e list the differences

Ilw ~ 2n) w ~ n)lloo, II(w?n

)y ( w~n)Ylloo and

II ( w?n))" -

(w~n))"l l oo for different values of

n.

CHAPTER

6. APPLICATION. DAMAGED BEAM 125 z

1

2

4

8

14

16

Ilw;LnJ - wtJ

1100

II(wi2n)), ­ (win))'lloo

II

(w?n))" - (win))"

II

00

n=4 n=8 n

=

16

6.83140

X

10-

6

4.56592

X

10-

7

2.578292

X

10-

8

9.65478

X

10

-0

6.45299

X

10

1 3.64876

X

10

-I)

2.48714

X

10

-0

6.33818

X

10-1) 8.43408

X

10-»

2.49236

X

10-

5

2.09579

X

10-

6

1.47615

X

10-

7

1.18092

X

10

-4

9.93009

X

10

-0

6.99423

X

10

1

1.96527

X

10

-'I

6.46058

X

10

-0

2.06282

X

10

-I

1.46719

X

10

5

6.47026

X

10-

6

1.61154

X

10-

4

7.01558

X

10-

5

5.44719

5.90054

X

X

10-

10-

7

0

8.16804

X

10

-3

2.53189

X

10

-4

8.2

8 3 69

X

10

-0

3.70776

X

10

4

6.16539

X

10

6

1. 75076

X

10-

0

7.90080

X

10-

3

1.30886

X

10-

4

3.71925

X lOb

6.26608

X

10

-L

1.00243

X

10

- j

2.90161

X

10

-b

3.20162

X

10-

4

9.48592

X

10-

5

9.65450

X

10-

7

1.68312

X

10

4.23303

X

10

· 3

4.29

8 24

X lOb

1.17602 1.65933

X

10

1 1.00174

X lO-

L

-

-

-

9.03027

X

10-

4.78628

X

10-

5

3

4.81429

X

10-

6

2.60210

X

10-

4

1.91004

X

10

-1

2.44992

X

10

-L

Table 6.10:

Convergence of FEM modes with 6

1/r

=

4800.

0.1,

Go

0.5 and

The rate at which convergence of the function values and the first order derivatives occur, differ from the convergence rate of the second order de­ rivatives, which is much slower. Those modes that are associated with the first eigenvalues, starting with the smallest, converges faster than the modes associated with later eigenvalues. (Convergence in the energy norm implies that the second order derivative converges in the mean.)

CHAPTER

6.

APPLICATION. DAMAGED BEAM

126

es

e

## m

Consider the initial value problem. From Section 4.1 we have the following system of differential equations

Mul/(t)

=

-Lu'(t) - Ku(t).

For the numerical experimentations ) we choose the following initial condi­ tions: u~(O)

=

0 and

Uh(O)

a quintic "solitary wave)).

To approximate the solution of this problem we use the difference scheme in

Section 5.4 with

Po

=

2Pl

=

1 / 2.

(

M L

M2 2M 4

1)

Uk+l+

(!VI

### bt

2

2

1 )

!VI L

(

M2 2M

1)

+

4

K

Uk-l

=

O .

Since t he initi a l velocity is ze ro , we have

U

1

=

U-l'

The results obtained for the eigenvalue problem motivated us to use quintics as basis functions.

C onvergence

The verify convergence) we choose a fixed spacial discretization and a fixed final time

I.

Then) starting with 10 time intervals) we increased the num­ ber of intervals until the relative difference is strictly less than

10-

3

.

This approximation is then considered as sufficiently accurate for the system of differential equations.

Decreasing the time step size , we found the first order derivative s needed approximately double the number of time steps to yield the same relative difference in

### II

.

1100 than the function values do.

It seems as if the the second order derivatives do not converge point wise, if they do) the convergence is very slow. This is not altogether surprising (see Section 5.4).

To establish the number of elements needed for our approximation , we choose a fixed final time,

I)

and time step size, M. Then the number of elements, starting with

10, is doubled until the relative difference satisfy our criterion.

CHAPTER

6.

APPLICATION. DAMAGED BEAM

127

e

bea

### m

W e are primarily concerned with the detection of damage. In this section we give an indication of the effect of respectively damage, damping and rotary inertia on the motion of a beam.

Our experiments indicate that measurable differences between the undam­ aged and damaged beams occur in displacements as well as gradients.

(Table 6.11.) Viscous damping has no significant effect on the motion. Look­ ing at the modal analysis this was expected, since it only effects the first few modes. Adding Kelvin-Voigt damping, the differences between the damaged and undamaged cases decrease , but is still clearly detectable. (Table 6.12.)

The presence of rotary inertia can have a more significant effect on the dif­ ference between the motion of the damaged and undamaged beams. (Tables

6.13 and 6.14).

To illustrate the above effects we compare the motion of an undamaged beam to that of a damaged beam where the initial velocity is zero and the initial position a 'solitary wave". For this simulation we choose

C\'

=

0.4, 6

=

0.1 which is ra t her excessive , 80 elements ,

T

=

0.02 and 400 time subintervals.

Almost immediately after the first wave front pass through the damaged point, measurable differences in displacements as well as gradients between the two cases occur. (See Figure 6.l.) In Table 6.11 we compare the dis­ placement of the damaged and undamaged beams on

T

=

0.02 at

x

=

0.3 and

x

=

0.7.

I

x

I

Undamaged beam

I

Damaged beam

I

% difference

I

0.3

0.7

2.325 x 10-

1

4 .

733 x 10-

1 l.505

X

10-

1

5.508

X

10-

1

8.2

7.8

Table 6.11:

Eff·ect

### oj damage

during motion where

6

=

0.1

,

C\'

=

0.4,

T

= 0.02.

CHAPTER

6 .

APPLICATION DAMAGED BEAM

128

1

Ini tial position

~

Q

Q)

E

Q) u

CIl

.~

Q 0

Damaged"",

### .....

..:.-

Undam a ed

0.4

Position

1

Figure 6.1:

Companng the moNon of an undamaged beam to that of a dam­ aged beam where

6

=

0.1,

Ct'

=

0.4

and

T

=

0.02. vVe now add Kelvin-Voigt damping to the same situation as in the previous case. We use

f-L

=

3.469

X

### 10

-

5

.

This value for

f-L

was obtained from [JVRVj.

I

x

I

Undamaged beam

I

Damaged beam

I

% difference

I

0.3

0.7

2.099 x

1

4.716 x

1

1.378

X

1

5.436

X

### 10-

1

7.2

7.2

Table 6.12:

Effect of KelvinVoigt damping on the damage during motion where

6

=

0.1,

Ct'

=

0.4

,

T

=

0.02

and f-L

=

3.469

X

### 10-

5

.

The presence of rotary inertia can make the differences more difficult to detect. An example is given in Tables 6.13 and 6.14.

CHAPTER

6.

APPLICATION. DAMAGED BEAM

129

x

0 .

2

0.6

I

Und ama ged beam

I

Damaged beam

I

% difference

I

-2.525 x 10

1

-2.065 x 10

1

4 .

762 x 10-

1

5.248

X

10-

1

4 .

6

4.9

Table 6.13 :

Rota r y

the

motion

o

=

0 .1, c¥

=

0.4

,

T

=

0.02

and 1 1r

=

19200.

I

x

I

Undamaged b ea m

I

Dama g ed beam

I

% diff er enc e

I

0.2

-2.236 x 10-

1

0.6

4.140 x 10-

1

-2.092 x 10

1

4.287

X

10-

1 l.4 l.5

T a bl e 6.14:

Effect

Rotary inertia

the

during

o

=

0.1

, c¥

=

0.4

,

T

=

0.02

11r

=

4800 .

c

te

## beam model

1

t

### ion

W e consider Problem 3 (from Section 2.6).

It is a mathematical model for a plate connected to two beams. Problems of this type are clearly of great practical importance. The plate can be rigidly connected to the beams or simply supported by the beams. The same model can be used for an I-shaped structural member (depending on the type of vibration) . For simplicity we restrict our investigation to the case of a plate supported by beams.

If the plate is rigidly connected to the beams, it may result in a problem with six unknown functions (excluding shear) due to dynamical effects. Even in our restricted case, one may easily encounter very large matrices.

In collaboration with others, [ZVGV1], we considered the equilibrium and eigenvalue problems of a rectangular plate supported by two beams at the boundary. In this thesis we extend the investigation and include the effect of rotary inertia.

The computation of the matrices is explained in Section 7.2. We use reduced quintics for the plate, which necessitates the use of quintics for the beams.

We treat the equilibrium problem in Section 7.3 and the eigenvalue problem

Section 7.4.

130

7.

P

## of matrices

131

F or the numerical experimentation we consider a square plate, 0, rigidly supported at two opposing sides and supported by identical beams at the remaining sides. The plate has thickness h and the beams are of square profile with thickness

### d.

Furthermore, we assume the plate and beams are of the same material. (These restrictions are evidently not necessary.)

The reference configuration 0 is the rectangle with 0

<

Xl

<

1 and o

<

X2

< l.

Thos e parts of the boundary where

Xl

=

0 and

Xl

=

1 are denoted by

~o and

~l respectively and correspond to the rigidly supported parts of the boundary . Those parts where

X2

=

0 and

X2

=

1 are denoted by

0 and

### r

1 respectively and correspond to the sections of the boundary supported by beams.

7

1 B

en

### t

s

For the plate we use only reduced quintics as basis functions. These functions are in

H2(0)

or fully conforming , in finite element language . They are defined on a triangular mesh. The mesh for the rectangle 0 is generated in the following way: The interval

[0 , 1] is divided into nl subintervals and the interval

[0 , 1] into

n2

subintervals. This partition of the intervals yields n l x n2 rectangles. The final triangular mesh is then obtained by dividing each of these rectangles into two triangles by connecting the lower left corner with the upper right corner. The rectangle 0 is divided into

2nl x

n2

triangles.

Consequently we have 2nl x n2 elements

### Oi.

Reduced quintics are defined in Section 4.3.l. The computation of the coeffi­ cients is not trivial and we describe it. in

ApppnniY

C The choice of reduced quintics "force" one to use quintics for the beams. Hermit.e piecewise quintics are defined in Section 4.2.l.

CHAPTER

7.

APPLICATIO N. PLATE BEAM MODEL

132

7

St

### andard Matr

ic es

First we compute standard matrices for the two beams with quintics. The procedure is the same as with cubics.

=

Y

C'Y

)

[MfO

L j

=

0cP

cP

,

[Mci1Lj

=

Y

cPi

j) ,

L j

=

1

C'Yl

e,Di)

C'YI

e,D

### j

) ' , as well as

N e xt we compute stand a rd m a trices for the plate. These computations are quite involved and we provide some detail in Appendix

C.

The bilinear forms are given in Section 3 .

3. Each basis element is of the form

i

=

i, 1

### 0cPi,

I lcP i )'

(The forced boundary conditions are satisfied by eliminating certain basis elements .) Now , consider for example

C

e,D

,

### cPj

)

involves the restriction of basis functions e,D i and

e,D

### j

to the bound ­ ar y roo

These restrictions are non-zero only for some of the basis functions associ a ted with nodes on roo

( The restriction of a reduced quintic on ro is an one-dimension a l quintic.

) The result is

Consequently

M

=

Mil.

+

{3

r o

+

{3

r l ,

=

lvtg

+

+

### {3Mrl.

where

The computation of the K-matrix is the similar ,

CH AP TER

7.

APPLICATION. PLATE BEAM MODEL

7.3

## Equilibrium probl

e m

133

To find the Galerkin approximation for the solutions of the equilibrium prob­ lem, we solve a system of linear equations.

BD

ii

=

where

### Fi

=

(f,

cPi)'

The parameter

a

gives an indication of the stiffness of the beams in c ompa­ rison to that of the plate . In c reasing the value of ex implies an increase in the stiffness of the beams and

a

=

0 corresponds to the case where two sides are free.

For different values of

a

we compare in Table 7.1 the FEN! approximations for the maximum displacement, to the so called exact solution. See [TW].

(Interesting historical remarks are found in [TW] .

)

Note that the maximum displacement occurs at the centre of the plate as a result of s ymmetry.

W e consider a square plate with the same number of equal intervals per side.

W e denote this number by

n ,

and use it to distinguish between different meshes.

Denote the maximum displacement obtained from th e so called exact solution by

U max

and the FEN! approximation of the maximum displacement where n subintervals are used, by u~2x.

Choose Poisson 's ratio

l/

= 0.3.

CHAPTER

7.

APPLICATION. PLATE BEAM MODEL

134

I

Q

I Exact

n=2

(U max

-

I u~L)

/

U max

n=4

I

n=8

100 4.09

X

10-

3

2.0421

X

10-

3

2.9308

X

10-

4

2.3653

X

10-

4

30

4.16

X

10-

3

1.0507

X

10-

3

6.5975

X

10-

4

7.1510

X

10-

4

10

4.34 x

10

. j

1. 7896 x lOj

1. 7460

X

10 -

4

1.2220

X

10-

4

6

4.54

X

10-

3

3.5724

X

10-

3

5.0933

X

10-

3

5.1428

X

10-

3

4

4.72 x

10

-j

2 .

9835 x

10-

3

1.547 x lOj 1.5006

X

10-

3

2

5.29

X

10-

3

3.2719

X

10-

3

2.0580

X

10-

3

2.0181

X

10-

3

1

6.24 x lOj 1.0228 x lOj

9.3231

X

10-

5

6.2112

X

10-

5

0.5

7 .5

6 x

10 -

3

2 .

2617

X

10-

3

1.6195

X

10 -

3

1.5973

X

10 -

3

0

1.309

X

10-

2

3.2828

X

10-

4

2.8584

X

10-

4

2.8129

X

10-

4

Ta .

ble 7.1:

Comparison oj exact v alues with FEM approximations oj ma :;; ­ imum displacement.

The fact that the relative error originally improves if we double the number of intervals from 2 to 4 and then r emains almost the same, suggests that the so called exact solution is not very accurate, as could be expected since only a f e w significant digits are given.

The relative difference between consecutive FEM approximations strenghtens this observation as can be seen in Table 7.2.

Q

(u~L

u~~ x)/u~~ (u~~x

­ u~~x)

/ u~~x

100 1.748505

X

10 -

3

5.653982

X

10-

5

30

1.711612 x

10

3

10

1.614713

X

10-

3

6 1.528720 x

10

3

4

2

1.433896

X

10-

3

1.211367

X

10-

3

9.294900

X

10-

4

1

0.5

6.412267 x

10-

4

0

4.242264

X

10-

5

5 .

539486

5.238781

X

10-

5

4.971930 x

10

5 x

10

5

4.677682

X

10-

5

3.987090 x

10

5

3.111738

X

10-

5

2.214781

X

10-

5

4 .5

50660

X

10 -

6

Table 7.2:

Compariso n oj FEM approximations Jar the max i mum displace­ ment.

CHAPTER

7.

APPLICATION. PLATE BEAM MODEL

E

e

e

e

### m

135

As mentioned before, Section 4.4, the occurence of eigenvalues has a highly irregular pattern in the two-dimensional case. We have an elementary ex­ ample to illustrate this, and also to show how difficult it can be to identify eigenvalues with multiplicity.

1

a

### lue

s

Consider the following eigenvalue problem ,

- \]2U

=

AU

on the unit square with

' U

=

0 on the boundary.

Clearly,

u( x, y)

= sin(mrx) sin(m7fY) is an eigenfunction, for nand m integers. The corresponding eigenvalue is

=

### n

2

+

m

2

.

The popular difference scheme is or where h is the length of a subinter va l.

Let

U i ,j

= sin(iwk) sin(Jwe), then

Hence

Ui ,j

We

satisfies the boundary conditions if

Wk

(br)/(n + 1) and

=

(t'7f)/(n+ 1). It satisfies the difference equations if

### Ae

=

h2(2 -2

cos

w e) .

Hence

Ui,j

is an eigenvector and every eigenvalue is of the form

Ak

+

Ae.

In

Table 7.3 we list the exact eigenvalues for this problem as well as the nu­ merical approximations obtained for different subinter va l lengths. We give

CHAPTER

7.

APPLICATION. PLATE BEAM MODEL

I i I Exact I

h

=

0.2 I

=

0 .

1 I

=

0.05 I

h

=

0.005 I

1

19 .

7 19.3

2 49.3 45.6

3

49.3 45.6

4 79.0 72

5 98.7 81.6

6 98.7 81.6

7 128

8 128

108

9

168

10

168

11 178

12

197

13 197

1 4 247

15

247

16

257

17 257

18 286

108

118

118

144

1

1

4 4

44

144

144

170

170

180

19 286

20 316

180

206

192

192

240

240

245

245

275

275

307

19.7

49 .

0

49.0

78.4

97.2

97.2

127

127

163

163

175

19.6

48 .

2

48.2

76.8

93.3

93.3

122

122

151

151

167

180

180

217

217

225

225

246

246

283

19.7

49 .3

196

196

245

245

254

254

283

283

313

49.3

78.8

98.3

98.3

128

128

167

167

177

136

T a ble 7.3:

.

### Comparison of finite

d~fference

eigenvalues to the exac

### t

eigenva­

only three significant digits as it is sufficient to illustrate difficulties of match­ ing exact eigenvalues and approximate eigenvalues.

Let i denote the number of the eigenvalue and

h

the length of a subinterval.

Interpreting numerical results with respect to multiplicity of eigenvalues is difficult . Great care should be taken to establish whether approximate ei­ genvalues that are close together are an indication of multiplicity of exact eigenvalues, or not.

For example, for

h

=

0.2 , the eleventh to fifteenth eigenvalues seem to be one eigenvalue with multiplicity five, while it actually approximates three diffe­ rent eigenvalues. Another example is the fifteenth and sixteenth eigenvalues for

### h

=

0.05. These eigenvalues seem close and one might expect them to approximate the same eigenvalue with multiplicity more th a n one.

CHAPTER

7.

APPLICATION. PLATE BEAM MODEL

137

7

.

P

t

### e beam

Th e ge ner a lized eigenvalue problem associated with the plate beam model is given by

Probl e m C D

KiD

=

AM ' w.

In a joint report [ZVGV1] we consider this eigenvalue problem for th e plate be am model excluding rotary inertia . In this subsection we investigate th e effect of rotary inertia if included in the model.

The ratio

CY./{3

of the dimensionle ss co nstants,

CY.

{3

=

(Ebh)/(aD)

an d

=

(Pb A)(pah ),

defined in Section 2.6 , with the plate of thickness hand the beams of square profile with thickness

d,

is a measure of the stiffness of the beams in comparison to that of the plate.

In

the special case whe re both the beam s and the plate are of the same material, we have

CY.

= -

d

{3

(

h

)

2

2

(1 -

l/

)

.

As the values of

d/ h

increase, i.e. the stiffness of the beams is increased, the s ituation approaches the plate problem where a ll four sides of the plate are rigidly supported. For this problem the eigenvalues an d eigenfunctions are known. The eigenvalues are of the form

((mr)2

+

(m7f)2)2 with corresponding eigenfunctions si n(n7f x) sin(m7fY).

Since the exact eigenvalue s for the plate beam problem are not available, the FEM approxim ati ons for the eigenvalues for l arg e values of

d/ h

can be compared to the eigenvalues of this limiting case, see Table 7.4.

Denote the ith eigenvalue for the case where a ll four sides are rigidly s up­ po rte d by

Ai.

The eigenvalues ar e ord ered ac cording to size. The FEM approximation of the ith eigenvalue i s denoted by

A~n) where

n

s ubintervals are used.

Throughout this subsection we use Poisson's ratio as

l/

=

0.3.

7. APPLICATION.

PLATE

'/,

### Ai

8

)

for different values of

=

1

=

10

=

100

=

200

### Ai

1 92.6654 386.3556 389.6361 389.6364 389.6364

2 250.7783 2359.4575 2435.2366 2435.2398 2435.2273

3

1264.1968 2433.5697 2435.2500 2435.2525 2435.2273

4 1514.0745 6221.0210 6234.2338 6234.2346 6234.1818

5 2142.1461 7345.9342 9741.4914 974l.5319 9740.9091

8 7725.6133 11308.1573 16463 .7086 16463.7127 16462.1364

10 11599.1799 16455 .3398 28158.9627 28159.0563 28151.2273

138

Table 7.4:

As

### d/ h

increas es, the FEM approximation of the eigenvalues approaches the eigenvalues of the plate rigidly supported on all four sides.

For values of

d/ h

that do not correspond to the limit case, the numerical convergence of he FEM eigenvalues are illustrated in Table 7.5.

=

10

(

A~2n)

_

A~n))

/

=

100

i

A~2n)

A~n))

/

i

1

6.94630 x

10

4

8.2136 x

10

-b

7.0601 x

10

4

8.3506

X

10-

6

2 2.1888

X

10-

2

4.0239

X

10-

4 l.2764

X

10-

2

2.9014

X

4

3

4.1506

4

0.6013 x

10

-2

4.2474 x

10

4

5.3144 x

10

2

5.6098

X

10-

4

10

:J

7.:32b6

lU

' l

0.0438 x

10

-'L

7.

3 537

X

10 -

4

5

2.3179

X

10-

2

9 .

5076

X

10-

4

10

2.3327 x

10

1

5.0927

X

10-

3

4.8512

X

2

5.7506

X

10-

1

3.1069

X

10-

4

1.3180

X

10-

2

Table 7.5:

### Remark

Choosing

n

=

8, yields (486 x 486) matrices which are already very time consuming to handle with our availab le computer hardware and software. Therefore we do not consider more than 8 subintervals.

y

e

### mod

e l

In addition to the joint report [ZVGV 1] we now establish the effect of rotary

7.

PLATE BEAM

### MODEL

139 inertia on the eigenvalues of the plate beam problem.

From Sections 2.5 and 2.6 we have the dimensionless constants

rb

and

=

hi

(a

2

2

) r

=

### I

I (a

2 h).

In the experimentation we work with a fixed plate , i .e.

a

and

h

are fixed, and modify the beams by changing

d.

Consequently

### r

depend on the relationship

rb

and

dl

### h and indicate the effect

of rotary inertia.

I n

Table 7.6 we illustrate numerical convergence of the FEM eigenvalues if rotary inertia is included. For

d l

=

50 we have

rb

=

2.083 x 10-

2 and

r

= 8.3~:3 x 10-

6

.

~

(

A~4)

_

A~2))

I

A~4)

1 7.0599

X

10-

4

5

4.8512

X

10-

2

10

5.7479

X

10-

1

(

A~8)

_

A;4))

I

A~8)

8.3271

X

10-

0

3.1350

X

10 -

3 l.312 x 10

2

Table 7.6:

convergence of

e~genvalues

d l h

=

50.

rotary inertia.

CHAPTER

7.

N.

### PLATE BEAM

MODEL

140

For the plate supported on all four sides, repeated eigenvalues are expected­ and indeed observed. For the plate beam problem the symmetry is partially lost, and the question arises if repeated eigenvalues will occur, and whether those F E M eigenvalues will be observed as repeated eigenvalues?

As with the case excluding rotary inertia, the exact solution is not avail­ able. Again, the exact eigenvalues for the plate rigidly supported on all four sides are used to give an indication of what can be expected of the FEM eigenvalues.

As is expected, the presence of rotary inertia decreases the eigenvalues in comparison to the case without rotary inertia. The effect of rotary inertia is illustrated in Table 7.7. For

d/ h

=

50 we have

rb r

=

8.333 x 10-

6

.

=

2 .

083 x 10-

2 and

6

7

8

9

3

4

5

10

11

Excluding rotary inertia

II

A

(8) t

1

389.6313

2 2435.1545

2435.2439

6234 .

2146

9740.7884

974l.5344

16463.2389

16463.6608

28154.6617

28158.7179

31564.1696

Including rotary inertia

A

( 8) t

389.5673

2434.1536

2434.2429

6230 .

1154

9732.7844

9733.5289

16445.6552

16446.0765

28115.3573

28119.4013

31517.5097

Exact value

Ai

389.6364

2435.2273

2435.2273

6234 .

1818

9740.9091

9740.9091

16462.1364

16462.1364

2815l.2273

2815l.2273

31560.5455

T able 7.7:

### Effect

of rotary inertia on the e7gen v alues for d / h

=

10.

In Table 7.7 the multiplicity of eigenvalues of the plate rigidly supported on all four sides are observed. These repetitions give reason to expect that the corresponding FEM eigenvalues for the plate beam problem may also be repeated eigenvalues.

The question arises whether the FEM approximation will yield repeated ei­ genvalues or will the eigenvalues only be close?

Was this manual useful for you?
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

advertisement