7.1 Scalar product

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Small boldface letters denote vectors and capital bold letters matrices,

u =

_

ux

uy

_

E =

_

εxx εxy

εyx εyy

_

S =

_

σxx σxy

σyx σyy

_

(7.1)

with the exception of G0 which denotes the vector-valued Green’s function

(displacement field) of a plate.

The gradient of a scalar-valued function u is a vector, and the gradient of

a vector-valued function u = [u1, u2]T is a matrix,

∇u =

_

u,1

u,2

_

∇u =

_

u1,1 u1,2

u2,1 u2,2

_

ui,j := ∂ui

∂xj

(7.2)

while the operator div does the opposite. The divergence of a matrix-valued

function is a vector-valued function, and the divergence of a vector-valued

function q = [q1, q2]T is a scalar-valued function:

div S =

_

σ11,1 +σ12,2

σ21,1 +σ22,2

_

div q = q1,1 +q2,2 . (7.3)

The following identity that relates these two operators

_

Ω

divS •ˆu dΩ =

_

Γ

Sn•ˆu ds −

_

Ω

S •∇ˆu dΩ (7.4)

is fundamental for structural mechanics. Note that if S = ST, then

_

Ω

−divS •ˆu dΩ +

_

Γ

Sn•ˆu ds =

_

Ω

S •∇ˆu dΩ

=

_

Ω

S •

1

2

(∇ˆu + ∇ˆuT ) dΩ , (7.5)

which is just the statement that δWe = δWi if ˆu is considered to be a virtual

displacement field.

Vector fields u obey the same rule,

504 7 Theoretical details

_

Ω

divu ˆudΩ =

_

Γ

(u•n) ˆuds −

_

Ω

u •∇ˆudΩ, (7.6)

and in 1-D problems div = ()_ and ∇ = ()_ are the same:

_ l

0

u

_ ˆudx = [u ˆu]l

0

−

_ l

0

u ˆu

_

dx . (7.7)

By default all vectors are column vectors, and a dot indicates the scalar product

of two vectors:

f •u = fxux + fyuy . (7.8)

Occasionally the notation f •u = fTu is also used. The dot also denotes the

scalar product of the strain and stress tensor, as in

Wi =

1

2

_

Ω

E •S dΩ

=

1

2

_

Ω

[εxx σxx + εxy σxy + εyx σyx + εyy σyy]

_ _ _

scalar product

dΩ . (7.9)

Other notations used in the literature for the scalar product of matrices are

E •S = tr(E ⊗ S) = E : S (tr = trace) . (7.10)

where E ⊗ S is the direct product of the two tensors E and S. The direct

product of two vectors is a matrix

f ⊗ u =

_

fx

fy

_

⊗

_

ux

uy

_

=

_

fx · ux fx · uy

fy · ux fy · uy

_

= A (7.11)

where aij = fi · uj .

The scalar product of a strain and stress vector

ε =

⎡

⎣

εxx

εyy

γxy

⎤

⎦ σ =

⎡

⎣

σxx

σyy

τxy

⎤

⎦ γxy = 2εxy, τxy = σxy , (7.12)

is—because of the factor 2 in γxy = 2εxy—the same as the scalar product of

the tensors; E •S = ε •σ.

The scalar product

a(u,ˆu) =

_

Ω

S • ˆE dΩ =

_

Ω

C[E(u)] •E(ˆu) dΩ (7.13)

is called the strain energy product between two displacement fields. It is a

bilinear form, because for any numbers ci, di,

7.1 Scalar product 505

a(c1u1 + c2u2, d1ˆu1 + d2ˆu2) =

_2

i,j=1

ci dj a(ui,ˆuj) . (7.14)

The scalar product between the vector u and the vector f is the projection

of the vector u onto the vector f

u• f = |u| |f| cos ϕ . (7.15)

Because the projection of u onto f should be the same as the projection of

f onto u we expect the scalar product to be symmetric, cos(ϕ) = cos(−ϕ),

which the scalar product (7.13) is. According to Green’s first identity—here

in an abbreviated symbolic notation

G(u,ˆu) = p(ˆu) − a(u,ˆu) = 0, (7.16)

the strain energy product between u and ˆu is equivalent to the work done

by the load p acting through ˆu and because of the symmetry of the scalar

product this can also be expressed as

p(ˆu) = a(u,ˆu) = a(ˆu,u) = ˆp(u) (7.17)

which is Betti’s theorem.

The integral

_ l

0

p(x)w(x) dx =: (p,w) (7.18)

is called the L2 scalar product of p and w. The notations

(p,u) =

_

Ω

p •udΩ =

_

Ω

[pxux + pyuy + pzuz] dΩ (7.19)

and

(S,E) =

_

Ω

S •E dΩ

=

_

Ω

[σxx εxx + σxy εxy + σyx εyx + σyy εyy] dΩ (7.20)

are extensions of this concept to vector-valued and matrix-valued functions,

respectively. The expression

||f || 0 := (f, f)1/2 =

__ l

0

f(x)2 dx

_1/2

(7.21)

is the L2-norm of the function f(x). The space of all functions defined on (0, l)

with a finite L2-norm, ||f||0 < ∞, is called L2(0, l). Note that the function

506 7 Theoretical details

f(x) = 1/

√

x can be integrated but its L2-norm is infinite because of the

square in (7.21)

_ 1

0

1 √

x

dx = 2

_ 1

0

1

x

dx = ∞. (7.22)

On the other hand if two functions f and g lie in L2, then the scalar product

of f and g exists, it is bounded

||f||0 < ∞, ||g||0 <∞ ⇒

_ l

0

f g dx < ∞. (7.23)

Note that ||f||0 = ||g||0 does not imply that ||f − g||0 = 0. In the Euclidean

norm, for example, all unit vectors ei have the same length, ||ei|| = 1 but of

course their tips do not touch, so that ||e1 − e3||           = 0.

Hence, if the FE solution seems to converge, because the variations in the

strain energy a(uh,uh) = fT u come to a halt, then (theoretically at least)

this does not imply that two consecutive solutions are the “same”:

||uh(1) ||E ∼ ||uh(2) ||E ⇒ ||uh(1) − uh(2) ||E   1 . (7.24)

The inequality

|

_ l

0

f g dx| ≤

__ l

0

f2dx

_1/2 __ l

0

g2dx

_1/2

(7.25)

or

|(f, g)| ≤ ||f||0 ||g||0 (7.26)

is known as Cauchy-Schwarz inequality.

The extension of the space L2(Ω) to higher derivatives constitutes the

Sobolev spaces. Imagine that we form a one-dimensional array that contains

the function u and all its derivatives up to the order m, for example

u(1) := [u, u,x , u,y ]T m = 1. (7.27)

The Sobolev space Hm(Ω) then consists of all functions u for which the L2

scalar product of these vectors is bounded,

||u||2

m =

_

Ω

u(m) •u(m) dΩ :=

_

Ω

[uu + u,x u,x +. . . ] dΩ < ∞ (7.28)

i.e., u and all its derivatives up to order m are square integrable (they lie in

L2(Ω)):

||u||21

=

_

Ω

u(1) •u(1) dΩ =

_

Ω

[uu + u,x u,x +u,y u,y ] dΩ < ∞. (7.29)

7.1 Scalar product 507

The space Hm(Ω) can also be seen as the completion of C∞(Ω) in the norm

||.||m, and the space Hm

0 (Ω) ⊂ Hm(Ω) is the completion of C∞

0 (Ω) (= the

functions in C∞(Ω) which vanish near the boundary).

On H2(Ω) the scalar product of two functions is defined as

(u, v)H2 =

_

Ω

u(2) • v(2) dΩ =

_

Ω

[uv + u,x v,x +u,y v,y

+u,xx v,xx +u,xy v,xy +u,yx v,yx +u,yy v,yy ] dΩ (7.30)

and the norm is

||u||2 =

_

(u, u)H2 . (7.31)

The extension of these concepts to other spaces Hm(Ω) is obvious.

An expression such as

|u|2 :=

__

Γ

(u,2

xx +u,2

xy +u,2

yx +u,2

yy ) dΩ

_1/2

(7.32)

would be called a semi-norm, because |u|2 = 0 with u = a+bx+c y does not

imply that u = 0.

In abstract terms the FE displacement field uh is the solution of the variational

problem

a(uh, v) = p(v) for all v ∈ Vh ⊂ V , (7.33)

where V is a Hilbert space usually endowed with a Sobolev norm ||.||m, and

p(v) is a continuous linear functional.

An important property of the strain energy product is that it establishes

an equivalent norm on V ,

c1 ||u||m ≤

_

a(u,u) ≤ c2 ||u||m (7.34)

where c1 and c2 are independent of u. Formally this so-called energy norm

||u||E :=

_

a(u,u) = (S,E)1/2 =

__

Ω

S •E dΩ

_1/2

(7.35)

is a only a semi-norm. To actually be a norm on V , the space V must not

allow rigid-body motions (that is, enough supports must be provided), because

otherwise the energy norm cannot separate the elements of V . This property

guarantees that if the norm of u − ˆu is zero, then u = ˆu:

||u − ˆu||E = 0 ⇒ u = ˆu . (7.36)

In this book the same letter p is used for the loads that constitute the load case

p and the load case p itself. In an abstract sense, any load case p constitutes

a functional p(ϕi) on Vh,

508 7 Theoretical details

p(ϕi) :=

_

Ω

p ϕi dΩ = (p, ϕi) (7.37)

where it is understood that the functional may contain additional terms, as

in

p(ϕi) :=

_

Ω

pϕi dΩ +

_

Γ

tϕi ds +P ϕi(x) , (7.38)

if edge loads, t, and point loads, P, are also present but the simplest form is

(7.37).

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