2.7 Transformations

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Transformations are an important tool for the finite element practitioner. For instance, if a

structure exhibits cylindrical symmetry, boundary conditions are more easily formulated in

a cylindrical coordinate system than in the global rectangular system. Another important

application is the definition of anisotropic material properties in cases in which the material

axes do not coincide with the global axes. In all these instances, it is advantageous to

introduce a local coordinate system. Here, we will concentrate on local rectangular and local

cylindrical systems. Both systems are orthogonal, that is, the covariant and contravariant

unit vectors coincide. If

I I

             := GI

            /_GI

           

I

             (2.183)

and similarly for the contravariant base vectors, one can write

I I

             = I I

           

, I

             = 1, 2, 3 (2.184)

GI

             · GJ

             = I I

             · I J

             = 0, I

             _= J

           

. (2.185)

98 LINEAR MECHANICAL APPLICATIONS

X1

X2

X3

a

X1      

X2      

X3      

I 1       

I 2       

I 3       

b

Figure 2.17 Local rectangular system

Let us characterize the global rectangular coordinate system by unit vectors I 1, I 2, I 3 and

coordinates X1, X2 and X3.

A local rectangular coordinate system X1       

-X2     

-X3     

can be defined by a point a on the

X1      

-axis and a second point b within the X1         

-X2     

plane excluding the X1

-axis (Figure 2.17).

For transformation purposes, it is important to determine unit base vectors in the local

coordinate system. The unit vector along the X1         

-axis is easily determined

I 1        = a

a. (2.186)

A vector on the X2     

-axis can be found by moving b in direction I 1 such that the resulting

vector is orthogonal to I 1        :

(b + λI 1          ) ⊥ I 1 (2.187)

or

(b + λI 1          ) · I 1  = 0 ⇒ λ = −b · I 1       . (2.188)

Consequently,

I 2        = b − (b · I 1   )I 1    

b − (b · I 1       )I 1     . (2.189)

LINEAR MECHANICAL APPLICATIONS 99

Finally,

I 3        = I 1   × I 2   (2.190)

where × symbolizes the vector product.

A local cylindrical coordinate system can be defined by two points on the cylindrical

axis (Figure 2.18). A local cylindrical system is also orthogonal, that is, the three unit

vectors are perpendicular to each other. However, the orientation of the local unit vectors

varies in space. In the finite element code CalculiX(CalculiX 2003), the first unit vector

is in radial direction, the second in tangential direction and the third in axial direction. Let

us determine a set of unit vectors in point p. From Figure 2.18 we have

I 3        = b − a

b − a. (2.191)

Point q is a point on the axis such that

(p − q) ⊥ (b − a) (2.192)

or, since a point on the axis can be written as a + λI 3  , λ ∈ R

(p − a − λI 3    ) · I 3  = 0 (2.193)

from which

λ = (p − a) · I 3           . (2.194)

X1

X2

X3

a

p

q

I 1       

I 2       

I 3       

b

Figure 2.18 Local cylindrical system

100 LINEAR MECHANICAL APPLICATIONS

Accordingly,

p − q = (p − a) − [(p − a) · I 3 ]I 3     (2.195)

and

I 1        = p − q

p − q. (2.196)

If p is on the axis, p − q = 0 and Equation (2.196) cannot be applied. In that case, any

direction perpendicular to I 3   can be taken for I 1     . Finally,

I 2        = I 3   × I 1   . (2.197)

This concludes the determination of local unit vectors for rectangular and cylindrical systems.

An arbitrary vector p can be expressed as a function of I 1, I 2 and I 3 or I 1  , I 2     and

I 3        :

p = X1I 1 + X2I 2 + X3I 3 (2.198)

= X1   

I 1        + X2  

I 2        + X3  

I 3        . (2.199)

Taking the scalar product of Equation (2.198) with I 1

we arrive at

p · I 1   = X1(I 1 · I 1 

) + X2(I 2 · I 1

) + X3(I 3 · I 1

) = X1 

(2.200)

and similarly,

X1(I 1 · I 2     

) + X2(I 2 · I 2

) + X3(I 3 · I 2

) = X2 

(2.201)

X1(I 1 · I 3     

) + X2(I 2 · I 3

) + X3(I 3 · I 3

) = X3 

. (2.202)

Notice that we have multiplied p by the contravariant unit vectors in the local coordinate

system, which, for rectangular and cylindrical coordinate systems happen to coincide with

the covariant unit vectors. Equations (2.200) to (2.202) can also be written as

XK

             = QK

           

LXL (2.203)

where

QK

           

L

= IK

             · IL. (2.204)

In a completely similar way, one arrives at

XK = T K

L          XL

           

(2.205)

where

T K

L

             = IK · IL

             . (2.206)

LINEAR MECHANICAL APPLICATIONS 101

T is the inverse of Q, that is, _QK

           

L_

−1 = _T L

K

             _. For orthogonal systems, where covariant

and contravariant unit vectors coincide, one can write

QK

           

L = IK

             · IL (2.207)

TLK

             = IL · IK

             (2.208)

which, in terms of matrix operations, means

T = QT. (2.209)

Consequently,

Q

−1 = QT (2.210)

that is, Q is an orthogonal matrix. Contravariant vectors satisfy

U = ULIL

= UL∂XK

           

∂XL

GK

           

=

3

_

K

            =1

UL ∂XK

           

∂XL

_GK

           

K

            IK

           

= UK

           

IK

             (2.211)

from which one obtains

UK

             = UL ∂XK

           

∂XL

_GK

           

K

             . (2.212)

Since (Equation (2.203))

UK

             = QK

           

LUL (2.213)

one finds

QK

           

L

= ∂XK

           

∂XL

_GK

           

K

             . (2.214)

For covariant tensors, we have

CK

           

L

             = CMN

∂XM

∂XK

           

∂XN

∂XL

             _GK

           

K

            _GL

           

L

             (2.215)

= CMNT M

K         T N

L

             . (2.216)

Notice that Q and T are not symmetric.

If boundary conditions or material orientations are expressed in local coordinate systems,

they have to be transformed into the global system used to formulate Equation (2.27),

usually a global rectangular system. In practice, the following situations occur:

102 LINEAR MECHANICAL APPLICATIONS

1. Single point constraints are formulated in local coordinates:

UK

             = a (2.217)

This is equivalent to

QK

           

LUL = a (2.218)

that is, an inhomogeneous single point constraint in local coordinates is transformed

into an inhomogeneous multiple point constraint in global coordinates.

2. Multiple point constraints are formulated in local coordinates. The same procedure

applies as under item 1: a homogeneous multiple point constraint in local coordinates

is transformed in a (usually longer) homogeneous multiple point constraint in global

coordinates, an inhomogeneous multiple point constraint in local coordinates is

expanded into an inhomogeneous multiple point constraint in global coordinates.

3. Forces are given in local coordinates:

FK

             = f. (2.219)

This transforms into

QK

           

LFL = f (2.220)

or

FL = T L

K

            f. (2.221)

Consequently, a force with one nonzero local component in direction K

             generally

results in three nonzero force components L in global coordinates.

4. The material orientation is given in local coordinates. The tangent stiffness matrix,

which is a generalization of the elasticity matrix, satisfies

dSKL = _KLMN dEMN. (2.222)

Since

dSP

           

Q

             = dSKLQP

           

KQQ

           

L (2.223)

dEMN = dER

           

S

            QR

           

MQS

           

N. (2.224)

Equation (2.222) can be transformed into

dSP

           

Q

             = _KLMNQP

           

KQQ

           

LQR

           

MQS

           

N dER

           

S

             . (2.225)

Hence,

_P

           

Q

           

R

           

S

             = _KLMNQP

           

KQQ

           

LQR

           

MQS

           

N. (2.226)

Similar relationships apply to other tensors such as the matrix of expansion coefficients.

In the CalculiXcode, (CalculiX 2003) all quantities expressed in local coordinates

are internally transformed into global coordinates using the previous relationships.

LINEAR MECHANICAL APPLICATIONS 103

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