7.10 The dual space

Back

Recall Sobolev’s Embedding Theorem, see p. 46, which states that:

If Ω is a bounded domain in Rn with a smooth boundary, and if 2m > n,

then

Hi+m(Ω) ⊂ Ci(￣Ω) (7.300)

and there exist constants ci < ∞ such that for all u ∈ Hi+m(Ω)

||u||

Ci( ЇΩ)

≤ ci ||u||

Hi+m(Ω) . (7.301)

This seems to be an abstract theorem with no immediate consequences

for structural mechanics—besides of course clarifying our ideas about point

loads—(see Sect. 1.14, p. 44). But we wish to comment on some interesting

consequences of this theorem.

Recall that the norm ||u||

Ci( ЇΩ) of a function is the maximum absolute value

of u and its derivatives up to order i on ￣Ω. Hence if two deflection surfaces

7.10 The dual space 553

Fig. 7.16. If two solutions are close in the sense that a(u − ˆu, u − ˆu) _ 1, the

maximum stresses must not be the same

w1 and w2 of a Kirchhoff plate are close in the sense of the Sobolev space

H2(Ω)—if their strain energy is about the same—the maximum deflections

of the two surfaces must also be nearly the same:

||w1 − w2||2   1 ⇒ maxw1 ∼ maxw2 . (7.302)

This follows from (7.301)—using a somewhat symbolic notation in the last

step,

max |w1 − w2| = ||w1 − w2||

C0( ЇΩ)

≤ c0 ||w1 − w2||2   c0 · 1 , (7.303)

if we assume that the constant c0 is not too pessimistic, i.e., too large.

For displacement fields u = [ux, uy]T of plates—which we typically associate

with the space H1(Ω) = H1(Ω) × H1(Ω)—this is not necessarily true

||u1 − u2||1   1 ⇒ max |u1| ∼ max |u2| (7.304)

because the inequality 2m = 2· 1 > 2 = n is not true.

To study the consequences of this theorem more systematically, we need to

introduce the concept of the dual space of a Sobolev space Hm(Ω). The dual

space is defined as the set of all continuous linear functionals p(.) on Hm(Ω)

as for example

p(w) :=

_

Ω

pwdΩ. (7.305)

In the following the focus is on the Sobolev space H2(Ω), endowed with the

norm

||w||2 :=

__

Ω

(w2 + w,2

x +w,2

y +w,2

xx +w,2

xy +w,2

yx +w,2

yy ) dΩ

_1/2

, (7.306)

554 7 Theoretical details

which is the energy space of Kirchhoff plates.

For a very special reason we focus initially on the subspace H2

0 (Ω) ⊂

H2(Ω) of this space. All the functions w ∈ H2(Ω) with vanishing deflection

and slope on the boundary, w = ∂w/∂n = 0, constitute this subspace H2

0 (Ω).

If a plate is clamped, the deflections w lie in H2

0 (Ω). The nice feature of H2

0 (Ω)

is, that its dual can be identified with the Sobolev space H−2(Ω)3 [202].

To understand this “negative” space, recall that the regularity of the functions

in Hm increases with the index m. The opposite is true with regard to

the negative spaces H−m: the more negative, the worse the regularity. Such

functions “borrow” their regularity from their brethren in Hm.

Take for example the Green’s function G0(y, x) of a taut rope (prestressed

with a force H = 1) with its triangular shape. This function has no second

derivative at the source point y, the foot of the point load, so that the integral

_ l

0

G

__

0 (y, x) δw(x) dx ( )__ = d2

dx2 (7.307)

makes no sense. The point load δ0 = G__

0 does not lie in L2(0, l) = H0(0, l),

because the integral of G__

0 squared does not exist:

_ l

0

[G

__

0 (y, x)]2 dx = ∞. (7.308)

Hence the point load must lie in a weaker space, in some negative Sobolev

space H−m(0, l), namely H−2(0, l).

To understand this choice, note that if the virtual displacement δw lies in

H2(0, l), then integration by parts can be applied twice, and because G0 and

δw have zero boundary values (the rope is fixed at its ends), the result is

_ l

0

G

__

0 (y, x) δw(x) dx = [G

_

0 δw]l

0

−

_ l

0

G

_

0(y, x) δw

_(x)dy

= [G

_

0 δw − G0 δw

_]l

0 +

_ l

0

G0(y, x) δw

__(x) dx

=

_ l

0

G0(y, x) δw

__(x) dx (7.309)

i.e., the work done by the point load acting through δw can be expressed in

terms of the work done by the distributed load δw__ acting through G0.

The point load δ0 = G__

0 is called the generalized second derivative of G0

and because this technique can always be applied if δw ∈ H2(0, l) it is said

that the point load δ0 lies in H−2(0, l).

From an engineering point of view, the concept of generalized derivatives

is an application of Betti’s theorem W1,2 = W2,1. “If W1,2 seems to make no

sense or cannot be calculated, then try W2,1!”.

3 We spare the reader a definition of negative Sobolev spaces, because the essence

of such spaces hopefully will become clear in the following discussion.

7.10 The dual space 555

Fig. 7.17. In the unit ball B1 = {w | ||w||E = 1} the normalized exact solution

w/||w||E gets “the most mileage” out of p, i.e. the virtual work exceeds that of any

other normalized virtual deflection ˆ w/|| ˆ w||E.

Continuous functionals

If a functional p is continuous, there exists a constant c such that

|p( ˆ w)| ≤ c || ˆ w||E . (7.310)

The lowest bound c—divide the equation by || ˆ w||E—is defined as the norm

of the functional p

||p ||−E := sup

ˆ w∈V

ˆ w_=0

|p( ˆ w)|

|| ˆ w||E

= sup

ˆ w∈V

|| ˆ w||

E=1

|p( ˆ w)| . (7.311)

If w is the solution of the load case p, then

|p( ˆ w)|

|| ˆ w||E

=

|a(w, ˆ w)|

|| ˆ w||E

≤

||w||E || ˆ w||E

|| ˆ w||E

= ||w||E (7.312)

i.e., ||w||E is an upper bound and because of

|p(w)|

||w||E

=

|a(w,w)|

||w||E

= ||w||E (7.313)

it is also the lowest upper bound. Hence the norm ||p ||−E of a load case p is

just the norm of the solution

||p ||−E = ||w||E =

_ l

0

M2

EI

dx (in a beam) . (7.314)

556 7 Theoretical details

This means that the exact solution w is that deflection in V which gets “the

most mileage” out of p in the sense of (7.311); see Fig. 7.17.

It seems intuitively clear that any surface load or line load p represents a

continuous functional

|p(δw)| < c||δw||2 . (7.315)

(We switch again to Sobolev norms).

But does this also hold true for point loads acting on a Kirchhoff plate?

The answer is yes. The Dirac delta δ0—a point load of magnitude P = 1—

belongs to H−2(Ω). The reason is that the functions w ∈ H2(Ω) lie also in

C(Ω) and because the embedding H2(Ω) ⊂ C(Ω) is—according to Sobolev’s

Embedding Theorem continuous, (m − i > n/2 or 2 − 0 > 1)—it follows

max

x∈Ω

|w(x)| ≤ c ||w||2 (7.316)

where the constant c does not depend on w. Thus any function w ∈ H2(Ω) is

guaranteed to have a bounded value w(x) at every point x ∈ Ω, and therefore

an expression such as

δ0(w) :=

_

Ω

δ0(y − x)w(y) dΩy = w(x) (7.317)

makes sense, and is a continuous functional on H2(Ω)

|δ0(w)| = |w(x)| ≤ c ||w||2 . (7.318)

That is if ||w||2 → 0 then also |δ0(w)| → 0. So if the strain energy of a slab

is zero (||w||E = a(w,w)1/2 and ||w||2 are equivalent norms) then w ≡ 0 and

no single point is allowed to break ranks while in a plate (2-D elasticity) this

is possible: the influence function for the point support is zero—the plate

does not move, ||u||E = 0—but one single point leaves the plate and travels

downward by one unit length, see Fig. 1.73 p. 103. Hence the conclusion is

• A slab with finite strain energy, w ∈ H2(Ω), is smooth, i.e., the deflection is

continuous—no sudden jumps—and the maximum value of w is bounded.

But it is not guaranteed that all functions w ∈ H2(Ω) have a well-defined

slope at all the points x ∈ Ω, because the embedding of H2(Ω) into C1(Ω)

is not continuous, because the inequality 2−1 > 1 is not true. Hence a single

moment M = 1 (Dirac delta δ1) is not a continuous functional on H2(Ω).

But the embedding of H3(Ω) into C1(Ω) is continuous, so δ1 ∈ H−3(Ω),

and δ2 lies in H−4 and δ3 lies in H−5:

H

δ3

−5 H

δ2

−4 H

δ1

−3 H

δ0

−2 H

−1 H0 = L2(Ω) H1 H

C0

2 H

C1

3 H

C2

4 H

C3

5

7.10 The dual space 557

Fig. 7.18. The deflection curve w caused by the moment M lies in H2(0, l) but not

in H3(0, l), otherwise the bending moment would have to be continuous, because

H3(0, l) ⊂ C2(0, l), 3 − 2 > 1/2

From a theoretical point of view the only load cases p that are admissible

lie in the dual of the energy space Hm(Ω), i.e., there must exist a bound c

such that

|p(w)| ≤ c ||w||m . (7.319)

Because the constant c is just the norm of the solution of the load case p,

i.e., c = ||w||m and because the norm ||w||m and the energy norm ||w||E =

a(w,w)1/2 are equivalent, the constant c is proportional to ||w||E.

Unbounded point functionals

Normal structural loads do lie in the dual space. The solution can be approximated

by minimizing the distance in the energy. This even holds true—at

least for the classical point loads P = 1 and M = 1—in beams; see Fig.

7.18. But in higher dimensions point loads (point functionals, Dirac deltas)

are critical.

558 7 Theoretical details

To prescribe the displacement u1(x) at a particular point x of a plate

makes no sense, because the embedding of the energy space H1(Ω) :=

H1(Ω) × H1(Ω) of the displacement fields u = [ux, uy]T into C0(Ω) :=

C0(Ω) × C0(Ω) is not continuous.

The displacement field u(x) = [ln(ln(1/r)), 0]T for example has finite energy,

in any circular domain Ω1−ε = {(r, ϕ)| 0 ≤ r ≤ 1 − ε, 0 ≤ ϕ ≤ 2 π} but

the horizontal displacement ux = ln(ln(1/r)) is infinite at r = 0.

Hence the Dirac delta δ0 does not lie in the dual of H1(Ω), because there

is no constant c such that for all u ∈ H1(Ω)

|(δx0

,u)| = |ux(x)| ≤ c ||u||1 . (7.320)

Rather we are lead to conclude that c = ∞. This even makes sense, because

a point load will generate a stress field with infinite strain energy and so

c = ||u||1

∼=

a(u,u)1/2 = ∞.

Even more critical are the stresses (the first derivatives), because a result

such as

|(δxx

1 ,u)| = |σxx(x)| ≤ c1 ||u||1 (7.321)

would require that the embedding of the energy space H1(Ω) into C1(Ω)

is continuous—which is not true in 2-D and 3-D elasticity. Hence the point

functional δxx

1 which extracts the stress σxx at a particular point (post processing!)

is not a continuous functional on the energy space H1(Ω). If we pick

an arbitrary point x ∈ Ω, there is no global bound on the stress, say, σxx(x)

at this point, i.e, which is a bound for the stress σxx(x) of all displacement

fields u in H1(Ω) in the sense of (7.321). A displacement field u can have a

bounded strain energy, a(u,u) < ∞, (the norms ||u||1 and ||u||E are equivalent)

but the stresses may become infinite at some points inside Ω. This is no

contradiction.

• Hence, if we calculate the stress at a point, we apply an unbounded point

functional, even though we think we only evaluate the polynomial function

which represents the stress distribution.

Also note that if two displacement fields have nearly zero distance in the

metric of the Sobolev space H1(Ω), ||u −ˆu||1   1, it is not guaranteed that

the maximum stresses are about the same; see Fig. 7.16, p. 553.

Riesz’ representation theorem

We have mentioned Riesz’ representation theorem before, but it deserves more

than a place in a footnote because this theorem is central to Green’s functions

and finite elements.

Extracting information from a structure means—in an abstract sense—to

apply a functional J(u) to the solution

7.10 The dual space 559

J(u) = u(x) J(u) = σxx(x) J(u) =

_

Ω

u dΩ etc. (7.322)

According to Riesz’ representation theorem for each linear, bounded functional

J() there is an element z ∈ V such that

a(z, u) = J(u) . (7.323)

The function z is of course the (generalized) Green’s function G.

In structural mechanics most functionals are unbounded

J(u) = u(x) J(u) = σ(x) J(u) = u,x (x) . . . (7.324)

that is it cannot be guaranteed that the functional is less than the energy of

u times a global constant (not depending on the single u)

|J(u)| ≤ c ||u||E (7.325)

(a displacement field u can be infinite at one point x, that is J(u) = ux(x) =

∞, but the energy is finite, ||u||E < ∞) and also the strain energy of the

Green’s functions Gi (= z) is infinite

a(Gi,Gi) = ||Gi||2

E = ∞ (7.326)

so that—theoretically at least—Riesz’ representation theorem is not applicable.

But we know that if we replace point values, i.e. point functionals

J(u) = u(x), by average values

u(x) → ￣u(x) =

1

|Ωε|

_

Ωε

u dΩ (7.327)

then the functionals J() are bounded and the corresponding generalized

Green’s function ||Gi||E < ∞ have finite energy. So we may assume that

Riesz theorem is “very nearly” applicable to our problems.

Now to each mesh belongs a test and trial space Vh ⊂ V and an abstract

operator P which maps each functional J() onto a functional4 Jh()

P : J() ∈ V

_ ⇒ Jh() ∈ V

_

h (7.328)

such that (see Fig. 1.45 p. 67)

J(v) = Jh(v) foreachv ∈ Vh . (7.329)

To the mapping J() → Jh() corresponds a mapping

V " G → Gh ∈ Vh (7.330)

4 V

_

and V

_

h are the duals of V and Vh that is the set of all functionals defined on

V and Vh resp.

560 7 Theoretical details

Fig. 7.19. The element size h

of the Riesz element.

The interesting aspect of this operator P is that it allows to characterize

FE solutions via the functionals Jh(): namely in FE analysis we choose uh ∈ Vh

in such a way that for all J()

J(uh) = a(G, uh) = a(Gh, uh) = (p,Gh) = Jh(u) (7.331)

or, to keep it short,

J(uh) = Jh(u) . (7.332)

Surprisingly this statement is equivalent to

a(uh, v) = (p, v) v ∈ Vh . (7.333)

For a proof of (7.329) and (7.332) see (1.228) p. 69.

Note that all this happens automatically. A mesh is a space Vh, is an

operator P, is an assemblage of Green’s functions Ghi

and all this before even

one single load case has been solved on this mesh.

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