Basis transformation of trial function space in lagrange interpolation

COMPUTER METHODS IN APPLIED MECHANICS @ NORTH-HOLLAND PUBLISHING COMPANY

AND ENGINEERING

23 (1980) 85-99

BASIS TRANSFORMATION OF TRIAL FUNCTION SPACE IN LAGRANGE INTERPOLATION Masayuki

OKABE

Mitsui Mining and Smelting Company Ltd., Tokyo, Japan

Yoshiaki

YAMADA

and Isoharu

NISHIGUCHI

Institute of Industrial Science, University of Tokyo, Japan

Received 5 June 1979

Various transformation rules that are due to two different bases of the same trial function space kept over an arbitrary finite element are presented in simple matrix form. Such interpolation theory clarifies the relation between nodeless variables and conventional nodal ones. It is further demonstrated that the finite element solutions are theoretically independent of the basis choice unless the assembly admissible conditions are violated.

1. Introduction In finite element analysis there exists one kernel called elementology. It is well known that the elementology is partly but fundamentally based on interpolation theory as well as on the variational principle including the weighted residual scheme. Conventional interpolation theory utilizing the Lagrangian and Hermitian interpolation polynomials has produced a variety of useful finite elements [l], [2], but conversely the remarkable success of the finite element methods in various engineering fields has been stimulating interpolation theory. Such typical examples can be seen in the development of finite element families. For normalized rectangles and cuboids Argyris [l] proposed the regular Lagrange family with regularly placed nodes. Removing its face and internal nodes, Taylor [3] then presented the mid-edge Lagrange family. The Serendipity family proposed by Zienkiewicz [4] is also famous, but only the lower members have been derived intuitively. Furthermore, Wilson [S] developed the variable-node expressions for the shape functions of originally quadratic Serendipity members. On the other hand, for n-simplex Co elements, Argyris [6] presented the complete Lagrange family with regular nodal placement. Silvester [7] then formalized its interpolation bases in alternative expressions. Although all of the elements developed seem to be dependent on similar polynomial interpolation, the conventional theory is not a powerful unifier. This paper focuses on the trial function space [8-10] in Lagrange interpolation kept over an arbitrary finite element in order to deal unifiedly with the finite element families. Starting from the definition of the trial function space, two different bases of the same space are picked up, one of which is especially termed the interpolation basis, while the other is an arbitrarily chosen basis. Obviously, not only different basis function sets but also diverse variable sets arise corresponding to the basis choice. We shall present their transformation rules in simple

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M. Okabe et al., Basis transformation of hiai function space in Lagrange jnte~po~ation

matrix form. In particular, it is shown that any nodeless basis functions in the so-called nodeless variable scheme can be transformed into the traditional nodal shape functions by assuming appropriate side or face nodes. The arguments are then applied to the general finite element minimization. We shall demonstrate that two elemental matrices as well as different overall ones can be transformed each other only through constant matrix operations. Although the actual computation may suffer from the round-off variations due to different matrix conditions, the finite element solutions are theoretically independent of the basis choice, unless the assembly admissible conditions are violated.

2. Entrance monition and inte~olatio~ basis of trial function space The trial function of the unknown ordinary form as

4 within an arbitrary finite element is expressed in the

(1) where Ni is the prescribed function of position, and & denotes the discretized variable. This is the general Lagrange interpolation that we are concerned with. If 4i is accepted as the # value at node i in eq. (l), then each function Ni is to satisfy the interpolation condition such that Ni is unity at node i and zero at all other nodes; this can be written as Ni(j) = Sij,

(2)

where Ni(j) denotes the nodal value of the Ni function at j, and 8, is the Kronecker delta. Regarding 4 as an arbitrarily prescribed function and +i as its nodal value, we introduce here the trial entrance expression by

E(#) = Q,- 2 -N#i-

(3)

Then we define the trial function space as composed of all the functions which make E(4) = 0.

(4)

Note that any functions satisfying eq. (4) can be reproduced i6 the trial function (1). Hence, eq. (4) is termed the entrance (or reproducible) condition. The Ni function is usually called the shape function, and occasionally the interpolation or basis function. But in this paper we restrict ourselves to use the name of the basis function generally, and only a basis function that satisfies the interpolation condition (2) should be termed the shape function. Assume for instance the unknown function # to be identical to a shape function Ni. Then

M. Okabe et at., Basis transformationof trialfunction space in Lagrange interpolation

87

we have

Hence,

(6) which guarantees that all the shape functions pass the entrance examination mentioned above. Therefore, each shape function is surely contained in the concerned trial function space. In order to conclude that the shape functions can compose a typical basis of the concerned space, it is further necessary for us to prove that the shape functions are independent each other. Regarding $i as coefficient, we assume as usual that

If there exists at least one nonzero coefficient 4(. among the 4i, then we have

At node k, eq. (8) can be rewritten as

From the interpolation condition (2) the left-hand side of eq. (9) is unity, while the right-hand term is zero. Thus the shape functions are surely independent each other. It is now clear that the shape functions compose a basis of the trial function space, which is termed the interpolation basis 181. Naturally, the trial space can be defined uniquely by the interpolation basis, and any arbitrary components of the space can be expressed as a linear combination of the shape functions. In fact, eq. (1) can be regarded as such a combination of using C$icoefficients.

3. Basis transformation Consider a trial function space of dimension n, which is uniquely defined by IZ shape functions N,, . . . , IV,. Then arbitrary n components S1, . . . , S, of the same trial space can be expressed as Si =

5

f.ZijA$,

i=l

9 ***

9 n,

i=l

where aij denotes an appropriate

coefficient.

(10)

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M. Okabe et al., Basis transformation of trial function space in Lagrange interpolation

At node k we have immediately Si(k) = i

ai&

= aik.

j=l

Thus, eq. (10) can be rewritten as Si = i

(12)

Siti)47

j=l

or in matrix form S=TN,

(13)

where S = {SiIl,

N = {Ni}

(14)

and T = [S;(j)]. If the determinant

(15) of the constant matrix T is nonzero, i.e.

det Tf 0, then the interpolation functions {Si} by N = T-‘S.

(16) basis N can be uniquely

determined

from the set of IZ arbitrary

(17)

Therefore, these arbitrary functions can surely compose another basis of the trial function space. Eqs. (13) and (17) can thus be interpreted as the transformation rule between the interpolation basis N and the arbitrary one S unless the latter violates the necessary and sufficient condition (16) which defines the space uniquely. Usually we define the trial function space by prescribing the monomial basis S. Consider, for example, the well-known triangular element of three vertices as shown in fig. 1. We adopt here the monomial basis (1, x, y) as usual. Then the basis transformation (13) yields

where Xi and yi denote the coordinates of apex i. The determinant of the constant matrix T is twice the triangular area in this case, and hence this monomial basis surely satisfies the basis

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M. Okabe et al., Basis transformation of trial function space in Lagrange interpolation

Fig. 1. Linear 2-simplex element.

condition (16). Obviously, eq. (18) is the definition of the 2-dimensional volume (i.e. area) coordinates [4], and each shape function is identical to the volume coordinate associated with the concerning node. For monomial bases a similar basis transformation rule was presented by Dunne [ll] with some comments [4], [12]. It should however be emphasized that such a rule is indeed valid for any arbitrarily chosen bases. Consider, for instance, the quadratic case in fig. 2. Usually the monomial basis is of (1, X, y, x2, xy, y’), but notice that the same trial function space can be characterized by the products of all possible quadratic combinations of volume coordinates. Thus we have

= 100

1 0 1

0 @Z(4) 0)X4) W(4)03(4)

m?(5) 0 0 S(5) 0 @3(5)W(5)

0 where wi denotes the volume coordinate

w:(6) ’ 0~Z(6) 0 0 0 01(6)~2(6)

‘Nl’ N2 N3

,

N

(19)

N5 _Ne

_

due to vertex i.

(w, (6)

(O,w*(4) ,w3(4)) (O,l,O) Fig. 2. Quadratic triangular element with arbitrarily placed mid-edge

nodes in volume coordinates

(WI, ~2.03).

90

M. Okabe et al., Basis transformation of trial function space in Lagrange interpolation

The reader can easily verify that the shape functions obtained by eq. (19) are identical to those presented simultaneously by de Veubeke [13] and Argyris [14] if and only if all the side nodes are placed at the corresponding centers of edges. It shall be demonstrated later that the generalization of the placement of some nodes except for vertices is meaningless in the finite element analysis. However, such generalization is frequently and essentially needed in other surface generation fields.

4. Lagrangian interpolation polynomials A further example can be observed in the so-called Lagrangian interpolation polynomials in one dimension. If we prescribe the trial function space of an n-node bar element in fig. 3 by a monomial basis (1, x, . . . , x “-l), then from eq. (13) we have 1

I X

X

(20)

n-l

where xi stands for the x coordinate of node i. Obviously each shape function Ni is a polynomial in degrees n - 1 at most. Then from the interpolation condition (2), Ni should be zero at all other nodes, and therefore the roots of the concerned polynomial are the coordinates of other II - 1 nodes themselves. Furthermore, the polynomial should be unity at node i (which prescribes its amplitude also). Hence, these shape functions are identical to the well-known Lagrangian interpolation polynomials. It is interesting to note that the constant transformation matrix T in this case is equal to the famous Vandermonde matrix, and hence its determinant can be expressed as the simplest alternating function det T = fl

(21)

(xi - xi).

i>i

Consider now two 3-node elements normalized to the range (-1,l) in the 5 coordinate system in fig. 4 - one has a mid-edge node at the centre ,$ = 0, while the other has an arbitrarily placed node at 6 = &. The basis functions of the former can be easily obtained as s3 = 1 - 5’.

s2= 50 + 5)/Z

Sl = -50 - 5)/2,

Node 1

3

...

n

2 .--)X

*

Fig. 3. n-node segment element.

(22)

91

M. Okabe et al., Basis transformationof trial function space in Lagrange interpolation 2

3

1

.

c 5=-l

<=l

<=o

5=-l

(a)

2

3

1

.

l

(bl

C=l

5 = 53

Fig. 4. Different 3-node bar elements. (a) mid-edge node at center, (b) arbitrarily placed mid-edge node.

Then, using the basis transformation expressed as 1 0

--&(I-

rule (13), the shape functions

Nl Nz

5‘3)/2 1

[

N3

1 .

of the latter can be

(23)

The reader can easily verify that these shape functions are identical to the La~an~an interpolation polynomials associated with the concerned nodal placement. It is thus demonstrated that two Lagrangian interpolation polynomials due to the different placements of nodes can, in general, be transformed each other.

5. Variable-node trial function space In the preceding section it was shown that the basis transformation rule produces the conventional Lagrangian interpolation polynomials. Here we discuss the variable-node expressions combining different order Lagrangian polynomials. Wilson [S] has shown that the linear and quadratic Lagrangian polynomials can be unifiedly treated in variable-node expressions. His methodology can be more easily followed by introducing the existence parameter (which was associated with node 3 at 5 = 0 in our previous example)

K3

1

if node 3 exists,

0

if node 3 disappears.

Then the variable-node Nz

(24)

=

=

(1 -

.f)U-

interpolation ~3flf

basis can be simply written as

5))/2,

N2

=

(If

6X1-

K3(1-

The reader can easily verify that eq. (25) can be reexpressed

5)1/2,

N3 = ~(1 - 5”). (25)

as

where

s1= (I - 5)/Z,

s2

=

(1

f

5)/Z,

s3 = 1 -

5’.

(27)

M. Okabe et al., Basis transformation of hial function space in Lagrange interpolation

92

Eq. (26) is the modified basis transformation rule for the variable-node trial function space. Notice that the constant transformation matrix is independent of the existence parameter. Obviously the basis (S1, Sz, S,) yields the upper triangular matrix and is termed the hierarchy ranking basis [15]. Note that such hierarchy ranking basis produces a highly generalized Lagrange family, the details of which will be reported soon.

6. Nodeless, nodal and mixed bases Notice that the interpolation basis is tightly connected to the nodal position within a finite element, while the monomial basis is, in its original form, independent of the coordinates of nodes. Besides such nodal and nodeless bases, mixed bases have also been proposed in finite element analysis and are known as nodeless variables [4], [16]. In this category both the nodeless and nodal basis functions compose a basis of the trial function space. Consider for example the rectangular element normalized to (-1,l) and shown in fig. 5. We introduce here four nodal basis functions associated with vertex-type nodes of the linear form Si

=

(l + t&)(1 +

where & and ni denote internal mode by

71ir1)/4,

i = 1, . . . ,4,

the normalized

coordinates

of vertex i. We further

prescribe

ss = COS(T(/2) cos(?r77/2), which gives identically zero on the boundaries. defines the trial function by

an

(29) This scheme introduced

by Zienkiewicz

[4]

(30)

C-1,1)

(l,l)

(-1,-l)

(1,-l)

Fig. 5. 2-dimensional square element normalized (-1, 1).

M. Okabe et al., Basis transformation of trial function space in Lagrange interpolation

93

Obviously 4: does not correspond to any nodal value of 4. Hence 4: is termed a nodeless variable, and S, a nodeless basis function. Suppose, for instance, an internal node 5 at the baricentre 6 = n = 0. Then the basis transformation (13) can be used again and yields

Notice that thus prescribed basis S can be also regarded Solving eq. (31) we have immediately N5 = Sg,

Ni = Si - NJ49

i =

1, . . . ) 4.

as the hierarchy-ranking

basis.

(32)

Thus the mixed basis S can be transformed into the traditional interpolation basis N by eq. (32) by assuming a centroidal node. For the polynomial space Zienkiewicz et al. [16] have also proposed some mixed bases relating to Legendre’s polynomial. Naturally the applications of any other polynomials such as Chebyshev’s are straightforward, and the hierarchy functions proposed in the adaptive finite element procedures [17]-[20] fall within this category also. However, it can be easily verified that all the mixed bases mentioned above can be transformed into the interpolation bases by assuming appropriate mid-edge and face nodes. It will be demonstrated in the following sections that the mixed basis and its transformed interpolation basis are theoretically equivalent in the finite element analysis.

7. Incompatible mixed basis A further example of the mixed basis can be seen in an incompatible finite element developed by Wilson et al. [21]. The original basis functions for corner nodes in fig. 5 are just the same as those by eq. (28), while two internal modes are introduced by ss = 1 - t2,

sg = 1- g2,

(33)

which are not identically zero on the boundaries and hence are nonconforming. Assume here two internal nodal variables & at (&, r/*) and & at (&, n6). Then the basis transformation (13) yields -

s1S2 SJ

1 0 I =

SS si_

0 0

1

0

1

S4

_

0 0

0

S(5)

S(6)

S2(5)

S2(6)

S3(5) S(5)

S3(6) S(6)

1- 6: 1-v:

1- 5: 1-q;

(34)

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M. Okabe et al., Basis transformation of trial function space in Lagrange interpolation

Thus we have No =

{(I- s$‘)(l- q;)- (l- 5;)(1- q’))/A?

Ntj = ((1 - (:)(I - q*)- (I- (‘)(I - q:))/A

N = St-

C(1+ t&)(1 + viqj)A$/4,

i =

1, . . . ,4,

(35)

j=S

where

A = (l- ml-

qZ)- (l- &(l-

(36)

7:).

It should be noted however that the proposed modes in eq. (33) can be regarded as the appropriate mid-edge modes, where node 5 is assumed on edges q = 21, and node 6 on edges 6 = +l. Name1 y th ere exist two different types of the nodeless variables, one of which must be declared at the place where it contributes, while the contribution of the other is automatically clear from the definition.

8. Variable and elemental matrix transformations As shown in eq. (30) the trial function of the form (37)

4=kN#i i=l

can in general be rewritten (by using the arbitrary basis) as (38) where 4: is not always a nodal variable. Since the unknown 4 is to be single-valued

within the finite element, we have (39)

or in matrix form 4 = T’cf~”

(40)

where (41)

A4. Okabe et al., Basis ~ansfor~ati~~ of triutfunction space in Lagrange inte~ola~on

95

Eq. (40) is the variable transformation rule which guarantees that any nodeless variables can be interpreted as a linear combination of appropriate nodal variables. Let x” be an appropriate elementary functional. Then the minimization of the functional through the variables #i yields

a,y’la+ = Kdi -f,

(42)

where K denotes the elemental Similarly for #‘, we have &f/a+*

matrix,

and f stands for the elemental

= K”&* -j*,

loading

vector.

(43)

where K” and f* are the elemental matrix and loading vector due to +*. Noting the well-known partial differential formula

and applying the variable transformation

Substituting

(~), we have immediately

eqs. (42) and then @I), eq. (45) can be rewritten as

$y”ia~* = TKT’Qi* - Tf. Thus we have K” = TKT’,

f* = Tf.

(47)

IIence two elemental matrices K and K* due to the different bases can be always transformed to each other only through the constant matrix operations as suggested by Strang [ST and followed by others [203. Wowever, it is evident that the eigenvalues of these matrices are not necessarily identical, and hence practical computation cannot be free from the basis choice. In our experiences the nodeless basis functions yield rather well-conditioned finite element matrices. It is remarked that the constant transformation matrix T appears in all the element level transformation rules presented.

9. Overall matrix tra~formation

with assembly ~dmissibi~ conditions

The simple assembly process of the elemental matrices as well as the elemental loading vectors is the common and fundamental feature of finite element computations. Consider, for simplici~, such a 2-element problem as shown in fig. 6. It is quite obvious that the matrix assembly requires each variable to be clarified whether it contributes to the

M. Okabe et al.. Easis ~ansfor~ation of trial function space in Lagrange interpolation

96

Interface I

element 1

element 2

Fig. 6. 2-element problem.

interface I or not, as argued in the preceding incompatible example. ventional interpolation basis realizes this constraint automatically. Then the elemental matrices for element 1 can be written as

Notice that the con-

and for element 2 we have (2)

(2)

(3

trr

tIB

kII

c-3

12)

tBf

fBB

12)

b

(4%

(e)

where g!, k?. and E! stand for the partial matrices of K, K* and If the assembly-admissible conditions of the form

T,

respectively, for element e.

(1)

tAI =

0,

(2)

tBI =

0,

(1)

(2)

&I = &I =

(50)

trr

are satisfied, then we have immediately

the overall matrix transformation

rule, which can be

M. Okabe et al., Basis transformation of tial function space in Lagrange interpolation

97

expressed as (1) k:r

(1) k*AA (1)

k ;A

(1)

0 (2)

k;+k; (2)

0

k;r

(2) k;B (2) k;SB

(1) k AA

(1)

(1) kAI

(1) (2)

kA

krr + kn

0

kBI

(2)

0

(2) hB (2) kBB

II 1 (1)

or

tAA

tIA

0

0

;I

0

(2)'

(2)'

0

tZB

tBB

.

(51)

Thus two overall matrices due to different bases can be transformed into each other only through the constant matrix operations unless the assembly admissible conditions are violated. The arguments in the 2-element problem can be easily extended to the general finite element problems, and after assembly we have ---

Iii* = TKT’,

J* = Ff

(52)

where the bar denotes the assembled matrix. Notice that the overall variable vectors C$ and 3* can be also transformed into each other through the F matrix operation by

(53) Suppose the final simultaneous K*$*

equations are

+*

(54)

Then, using eq. (52) we have --_ TKT’c&* = i+f

(55)

If and only if det p is not zero, eq. (55) can be rewritten as

I@$*) Then, substituting

= f.

(56)

eq. (53) for eq. (56), we have an alternative final system of the form

&=f Hence, the finite element solutions are theoretically

(57) independent

of the basis choice unless

98

M. Okabe et al., Basis transformation of tial function space in Lagrange interpolation

the assembly admissible conditions are violated. It is easy to verify that all the mixed bases as well as the nodal ones in our examples are surely assembly-admissible. On the other hand, the monomial bases obviously violate the assembly-admissible conditions, and hence they cannot be accepted in the normal finite element computations with a simple matrix assembly. It is thus demonstrated that the generalization of the placement of mid-edge nodes or internal ones as shown in fig. 2 does not make any sense in the finite element analysis. However, notice again that the interpolation elementology is being watched with keenest interest in other surface generation fields also, where such generalization is essentially needed.

10. Conclusions It is thus shown that the elementology based on the interpolation theory in the finite element method should be examined through the trial function space. In Lagrange interpolation the trial space concept is demonstrated to produce the well-known Lagrangian interpolation polynomials on which the conventional interpolation theory is dependent. In particular, the basis and variable transformation rules presented in simple matrix form clarify the relation between the nodeless basis functions and the traditional nodal ones indeed. Obviously the basis transformation rule enables us to define any shape functions uniquely for arbitrary finite elements only by prescribing the appropriate basis functions, such as in monomial form. However, such rash procedures may generally yield extremely incompatible elements, which cannot be accepted in finite element analysis. Although the trial function space concept is a quite powerful tool to deal unifiedly with the finite element families developed so far, further techniques should undoubtedly be devised in order to produce a new wide range of possibilities in the interpolation elementology. In fact, the authors have already succeeded in developing a new family of Co continuity including the regular Lagrange family, the mid-edge Lagrange family and the Serendipity family as well as the variable-node elements, which will be reported soon. The matrix transformation rules associated with the general finite element minimization process show that the numerical solutions are theoretically independent of the basis choice unless the assembly-admissible conditions proposed are violated. The efforts to improve accuracy by changing only the placement of some nodes except for vertices are thus meaningless in the finite element analysis without refining the finite element modeling and without improving the trial function space. However, the actual computations cannot be free from the variation of the matrix conditions due to the basis choice, and hence the criterion in the selection of a desirable basis is whether it produces a sparse and well-conditioned finite element matrix or not [8].

Acknowledgments The authors are grateful to Professor N. Takenaka comments and suggestions.

of the Nihon University, for his useful

M. Okabe et al., Basis transformation

of trial function space in Lagrange interpolation

99

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