7.12 Important equations and inequalities

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For the convenience of the reader we repeat some definitions: in a load case p

loads are applied, for example,

− EAu

__(x) = p(x) 0< x < l u(0) = 0, N(l) = P (7.374)

while in a load case δ displacements are prescribed

− EAu

__(x) = 0 0 < x < l u(0) = 0, u(l) = δ . (7.375)

V denotes the trial or test space, that is the set of all possible virtual displacements

that are compatible with the support conditions of the structure.

S is the solution space. In a load case p the two spaces coincide, S = V ,

while in a load case δ it is S = uδ ⊕ V where uδ is a function with the

properties uδ(0) = 0, uδ(l) = δ, see Sect. 1.7, p. 22. The symbol ⊕ means,

that any function u in S is the sum of uδ plus a function v ∈ V .

Vh ⊂ V is the space spanned by the nodal unit displacements ϕi. That is

the generic element of Vh is vh =

_

i vi ϕi(x).

Sh is the composition of a function uhδ

plus the space Vh

Sh = uhδ

⊕ Vh (7.376)

where uhδ

is a displacement field, that satisfies the homogeneous geometric

boundary conditions such as u(0) = 0 exactly and the inhomogeneous geometric

boundary conditions such as u(l) = δ either exactly (Sh ⊂ S) or

approximately (Sh        ⊂ S).

In a load case p the FE solution lies in Vh and in a load case δ it lies in

Sh.

7.12 Important equations and inequalities 569

In a load case p the FE solution is an expansion in terms of the nodal unit

displacements

uh =

_

i

ui ϕi(x) (7.377)

and in a load case δ a function uhδ

is added

uh = uhδ

(x) +

_

i

ui ϕi(x) = uhδ

(x) + uh

V (x) (7.378)

to satisfy the (partially) inhomogeneous geometric boundary conditions

uhδ

(0) = 0 uhδ

(l) = δ . (7.379)

The exact solution u satisfies

a(u, v) = p(v) v ∈ V , (7.380)

and the FE solution satisfies

a(uh, vh) = p(vh) vh ∈ Vh . (7.381)

In a load case δ the load p is zero and u = uδ + uV , where uV ∈ V , so that

(7.380) is equivalent to

a(uV , v) = −a(uδ, v) v ∈ V , (7.382)

and in the same sense in the finite element case, see (7.378),

a(uh

V , vh) = −a(uhδ

, vh) vh ∈ Vh . (7.383)

Note that in FE analysis we replace the right-hand side a(uhδ

, ϕi) by the virtual

work done by the negative7 fixed end forces resulting from uhδ

, i.e.,

a(uhδ

, ϕi) = phδ(ϕi) = fi . (7.384)

An important role in FE analysis plays Green’s first identity

G(u, ˆu) =

_ l

0

−EAu

__ ˆudx + [N ˆu]l

0

−

_ l

0

EAu

_ ˆu

_

dx = 0 (7.385)

which allows to switch at will between external and internal virtual work

G(uh, vh) = ph(vh) − a(uh, vh) = 0, (7.386)

where

7 negative because of actio instead of reactio

570 7 Theoretical details

ph(vh) : =

_ l

0

−EAu

__

h vh dx + [Nh, vh]l

0 (7.387)

a(uh, vh) : =

_ l

0

EAu

_

h v

_

h dx . (7.388)

Note that if u is the exact solution and ˆu ∈ V then

G(u, ˆu) = δΠ(u, ˆu) =

_

d

dε

Π(u + ε ˆu)

_

ε=0

(7.389)

is the first variation of the potential energy.

Boundary conditions as u(0) = 0 or u(l) = δ are essential boundary conditions

and boundary conditions as N(0) = −P or N(l) = P are natural

boundary conditions. In a problem such as

− EAu

__(x) = p(x) 0< x < l, u(0) = 0 N(l) = P (7.390)

the essential boundary condition enters the definition of the trial and solution

space V

V = {u ∈ H1(0, l) | u(0) = 0} (7.391)

while the natural boundary condition enters the definition of the virtual external

work p(ˆu) in the associated variational formulation

a(u, ˆu) =

_ l

0

EAu

_ ˆu

_

dx =

_ l

0

p ˆudx + P · ˆu(l) =: p(ˆu) ˆu ∈ V . (7.392)

It is called a natural boundary condition, because the variational solution

satisfies N(l) = P.

The boundary integrals in Green’s identities are L2 products (or simply

products in 1-D problems) of conjugate boundary terms as [N ˆu]l

0 = N(l) ·

ˆu(l) − N(0) · ˆu(0). A boundary value problem is well posed if of two such

conjugate boundary terms one is prescribed and the other is unknown.

Under regular conditions the strain energy product constitutes a norm on

V

||u||E :=

_

a(u, u) (7.393)

which is equivalent to the Sobolev norm, i.e., there exist constants c1 and c2

such that

c1 ||u||m ≤ ||u||E ≤ c2 ||u||m , (7.394)

and the strain energy product is a continuous bilinear form on V × V

|a(u, v)| ≤ c3 ||u||m ||v||m , (7.395)

7.12 Important equations and inequalities 571

and the virtual work is continuous on V

p(v) ≤ c ||v||m . (7.396)

If u is the solution of the load case p then G(u, u) = 0 implies

a(u, u) = p(u) (7.397)

and because of the definition (7.393) and (7.314) we have

||u||E =

_

a(u, u) = p(u)

||u||E

= sup

v∈V

v_=0

p(v)

||v||E

=: ||p||−E . (7.398)

The strain energy product of the FE solution uh and a test function vh ∈ Vh

can be identified with the work done by the forces in a load case ph acting

through vh

a(uh, vh) = ph(vh) (7.399)

where ph(vh) is obtained from a(uh, vh) by integration by parts, that is

ph(vh) = G(uh, vh) − a(uh, vh) vh ∈ Vh (7.400)

where G(uh, vh) is Green’s first identity.

The FE method

uh ∈ Vh : a(uh, vh) = p(vh) vh ∈ Vh (7.401)

is therefore equivalent to

ph(vh) = p(vh) vh ∈ Vh Equivalence theorem . (7.402)

Let u be the exact solution, uh the FE solution, and e = u − uh the error;

• On V

a(e, v) = p(v) − a(uh, v) = p(v) − ph(v) v ∈ V . (7.403)

• Hence in particular

p(u) − p(uh) = p(e) = a(e, u) = p(u) − ph(u) (7.404)

and as well

p(uh) = ph(u) symmetry (7.405)

which is of course Betti’s theorem.

572 7 Theoretical details

• Let e = G0 − Gh0

then follows

u(x) − uh(x) = a(G0 − Gh0

, u) = (δ0, u) − (δh

0 , u) , (7.406)

i.e., the work done by the exact solution u acting through the jump terms

and element residuals of the approximate Dirac delta δh

0 = {j, r} has the

same value as the error.

• For test functions vh ∈ Vh ⊂ V the right-hand side in (7.403) is zero

because p(vh) = ph(vh), and hence on Vh the error is orthogonal to the

test functions

a(e, vh) = 0 vh ∈ Vh Galerkin orthogonality . (7.407)

• The residual forces p − ph are orthogonal to the test functions

a(e, vh) = p(vh) − ph(vh) = 0 vh ∈ Vh. (7.408)

• If p = 0 as in stability problems, the FE load ph is orthogonal to each test

function

ph(vh) = 0 vh ∈ Vh. (7.409)

• The FE load ph is orthogonal to the displacement error e

a(uh, e) = ph(e) = 0. (7.410)

• The unit load cases pi are orthogonal to the error e

a(ϕi, e) = pi(e) = 0. (7.411)

• The FE solution uh attains the smallest possible value for the strain energy

product of the error e on Vh, i.e., the FE solution has the shortest distance

in the energy metric from the true solution

a(e, e) ≤ a(u − vh, u − vh) vh ∈ Vh , (7.412)

because for any wh ∈ Vh

a(e + wh, e + wh) = a(e, e)

_ _ _

>0

+2 a(e,wh)

_ _ _

=0

+a(wh, wh)

_ _ _

>0

(7.413)

and choosing wh = uh − vh gives the result. C´ea’s lemma is based on

(7.412), the equivalence (7.394) and the continuity (7.395)

c1 ||e||2

m

≤ a(e, e) = a(e, u − vh) + a(e, vh − uh)

= a(e, u − vh) ≤ c3 ||e||m ||u − vh||m (7.414)

and therefore

||e||m = ||u − uh||m ≤ c3

c1

inf

vh∈Vh

||u − vh||m . (7.415)

7.12 Important equations and inequalities 573

• In a load case p the internal energy of the FE solution is less than the

internal energy of the exact solution (we drop the factor 1/2 on both

sides)

a(uh, uh) ≤ a(u, u) in a load case p , (7.416)

because

0 < a(u, u) = a(uh + e, uh + e)

= a(uh, uh) + 2 a(e, uh)

_ _ _

=0

+a(e, e)

_ _ _

>0

. (7.417)

• In a load case δ where Π(u) = 1/2 a(u, u) the internal energy of the FE

solution exceeds the internal energy of the exact solution

a(u, u) ≤ a(uh, uh) in a load case δ . (7.418)

The proof was given in Sect. 1.7 on p. 16.

• In a load case p holds

a(u, u) = a(uh, uh) + a(e, e) ’Pythagoras c2 = a2 + b2 ’ (7.419)

or ’the energy of the error is the error in the energy’

a(e, e) = a(u, u) − a(uh, uh) , (7.420)

because

a(u, u) = a(uh − e, uh − e) = a(uh, uh) − 2 a(e, uh)

_ _ _

=0

+a(e, e) . (7.421)

In a load case δ the equation a(e, uh) = 0 is not true, because uh =

uhδ

+ uh

V

            ∈ Vh, while a(e, vh) =0 is true if vh ∈ Vh.

• The potential energy of the FE solution exceeds the potential energy of

the exact solution

Π(u) ≤ Π(uh) , (7.422)

because

Π(uh) = Π(u − e) =

1

2 a(u, u) − a(u, e) +

1

2 a(e, e) − p(u) + p(e)

= Π(u)−a(u, e) + p(e)

_ _ _

G(u,e)=0

+

1

2 a(e, e)

_ _ _

>0

. (7.423)

• The virtual external work done by p acting through uh is less than the

work done acting through u

p(uh) = a(uh, uh) < a(u, u) = p(u) . (7.424)

574 7 Theoretical details

• The fact that the residual forces are orthogonal to the ϕi

p(ϕi) − ph(ϕi) = 0 ϕi ∈ Vh (7.425)

does not suffice to guarantee, that the FE solution uh interpolates the

exact solution at the nodes. This would be true, if the residual forces

were orthogonal to the Green’s functions G0[xi] = G0(y, xi) of the nodal

displacements ui = u(xi)

p(G0[xi]) − ph(G0[xi])

?

= 0. (7.426)

But the nodal Green’s functions G0[xi] do not lie in Vh

G0(y, xi)          =

_

j

uj(xi) ϕj(y) . (7.427)

The exception are 1-D problems where—in standard situations—the inclusion

G0(y, xi) ∈ Vh is true. The reason is, that (i) the Green’s functions

G0(y, xi) are expansions in terms of homogeneous solutions and (ii) the

element shape functions ϕei

form a complete set of linearly independent

homogeneous solutions of the governing equation.

• Let Gh0

the FE approximation of G0, then holds

uh(x) =

_

Ω

G0(y, x) ph(y) dΩy =

_

Ω

Gh0

(y, x) p(y) dΩy

=

_

Ω

u(y) δh(y − x) dΩy (7.428)

where δh is the approximate Dirac delta, i.e., that assemblage of external

loads that attempts to imitate the action of the true Dirac delta or simply

the “right-hand side” of Gh0

.

This means that the FE solution can be written in two ways

uh(x) =

⎧⎪⎨

⎪⎩

_

i

ϕi(x) ui = (δh

0 , u)

_

Ω

Gh0(y, x) p(y) dΩy = (Gh0

, p)

(7.429)

so that

uh(x) =

_

i

ϕi(x) ui =

_

Ω

Gh0

(y, x) p(y) dΩy

=

_

Ω

_

i

ϕi(y) uGj

(x) p(y) dΩy =

_

i

uGi

(x) fi

= uT

G f = uT

GKu = uT

GKT u = fT

G u (7.430)

or in short (see Fig. 1.52 p. 77)

7.12 Important equations and inequalities 575

uh(x) = uT

G f = fT

G u. (7.431)

Note that this holds true for any quantity, σh(x) = . . ., Vh(x) = . . . if the

nodal vectors uG and fG respectively are replaced by the corresponding

vectors of the Green’s function Gi.

• Betti’s theorem and the principle of virtual work, δWe = δWi, imply

Betti  __ _

u(x) = p(G0[x])

_ _ _

δWe

= a(G0[x], u)

_ _ _

δWi

(7.432)

and therefore as well

u(x) − uh(x) = p(G0[x]) − ph(G0[x]) = a(G0[x], u − uh) . (7.433)

• The Galerkin orthogonality implies that

a(G0[x] − Gh0

[x], vh) = 0 vh ∈ Vh (7.434)

and

a(Gh0

[x], u − uh) = 0 because Gh0

[x] ∈ Vh . (7.435)

• Which proves that on Vh the kernel Gh0

is a perfect replacement for the

kernel G0

vh(x) = a(Gh0

[x], vh) = (δh

0 , vh) vh ∈ Vh . (7.436)

Which is also true for higher kernels Ghi

as well.

• All this jointly implies that, see also (1.451) p. 158,

u(x) − uh(x) = p(G0[x]) − ph(G0[x])

= p(G0[x] − Gh0

[x]) − ph(G0[x] − Gh0

[x])

= a(G0[x] − Gh0

[x], u − uh) , (7.437)

and this allows an estimate such as

|u(x) − uh(x)| ≤ ||G0[x] − Gh0

[x]||E ||u − uh||E , (7.438)

where

||u||E := a(u, u)1/2 (7.439)

is the energy norm.

• If the Green’s function G0 = g0 + uR is split into a fundamental solution

g0 and a regular part uR and the FE approximation Gh0

= g0 +uh

R as well,

the previous estimate can be replaced by

|u(x) − uh(x)| ≤ ||uR[x] − uh

R[x]||E ||u − uh||E . (7.440)

576 7 Theoretical details

• If the exact solution u lies in Vh, i.e., if u = uh the error of any approximate

Green’s function Ghi

is orthogonal to the load case p (the right-hand side

of u), see Sect. 1.17, p. 57,

p(Gi) − p(Ghi

) = a(Gi − Ghi

, u) = a(Gi − Ghi

, uh) = 0. (7.441)

This means, for example, that even though the Green’s function G1 for

the stresses does not lie in Vh, the stresses are exact σ = σh = p(Gh1

).

• The turnstile character of the symmetric strain energy

p(ˆu) ← a(u, ˆu) → ˆp(u) (7.442)

plays a very important role in FE analysis. Betti’s theorem rests on this

character

W1,2 = p(ˆu) = a(u, ˆu) = ˆp(u) = W2,1 (7.443)

and also Tottenham’s equation is an application of Betti’s theorem but

with a special twist, namely that (7.443) remains true if u and ˆu are replaced

by the FE solutions uh and ˆuh of the load cases p and ˆp respectively

W1,2 = p(ˆuh) = a(u, ˆuh) = a(ˆu, uh) = ˆp(uh) = W2,1 . (7.444)

• Note that the Galerkin orthogonality

a(u − uh, ˆuh) = 0 (7.445)

a(ˆu − ˆuh, uh) = 0 (7.446)

implies

a(u, ˆuh) = a(uh, ˆuh) (7.447)

a(ˆu, uh) = a(ˆuh, uh) (7.448)

which establishes

a(u, ˆuh) = a(ˆu, uh) (7.449)

and therewith proves (7.444).

• The FE solution can be written in two ways

uh(x) =

⎧⎪⎪⎨

⎪⎪⎩

_

i

ϕi(x) ui

_ l

0

Gh0

(y, x) p(y) dy

(7.450)

or (see Fig. 1.52 p. 77)

7.12 Important equations and inequalities 577

uh(x) =

_

i

ϕi(x) ui =

_ l

0

Gh0

(y, x) p(y) dy

=

_ l

0

_

i

ϕi(y) uGj

(x) p(y) dy =

_

i

uGi

(x) fi

= uT

G f = uT

GKu = uT

GKT u = fT

G u. (7.451)

This extends naturally to any quantity as for example

σh(x) =

⎧⎪⎪⎨

⎪⎪⎩

_

i

σ(ϕi(x)) ui = fT

G u

_ l

0

Gh1

(y, x) p(y) dy = uT

G f

(7.452)

where uG and fG are now the nodal displacements and equivalent nodal

forces respectively of the Dirac delta δ1.

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