2.1 General Equations

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The basic equations for the finite element method are the weak formulation of the balance

of momentum, Equation (1.328) and the weak formulation of the balance of energy,

Equation (1.351). For mechanical applications in which the temperature is assumed to be

known, only the balance of momentum is needed in order to determine the displacement

fields:

_

V0

SKLδEKL dV = _

A0t

T

K

(N)δUK dA + _

V0

ρ0f KδUK dV − ρ0

_

V0

D2UK

Dt2 δUK dV.

(2.1)

In the present chapter, primarily linear applications are envisaged. The term “linear” relates

to the material, which is assumed to be linear elastic, and to the strain formulation. Consequently

(see Equation (1.420)),

SKL = [γ KL − βKL(θ)T ] + _KLMN(θ)EMN (2.2)

and

EKL ≈ ˜EKL = 12(UK;L + UL;K) (2.3)

or

EKL ≈ 12

(UK,L + UL,K) (2.4)

for rectangular coordinates. In the rest of the chapter, rectangular coordinates will be

assumed and the covariant differentiation will be replaced by simple differentiation. Now,

Equation (2.1) can be written as

_

V0

δ ˜EKL_KLMN(θ ) ˜EMN dV = _

A0t

T

K

(N)δUK dA + _

V0

ρ0f KδUK dV

+ _

V0

[βKL(θ)T − γ KL]δ ˜EKL dV − ρ0

_

V0

D2UK

Dt2 δUK dV. (2.5)

The Finite Element Method for Three-dimensional Thermomechanical Applications Guido Dhondt

2004 John Wiley & Sons, Ltd ISBN: 0-470-85752-8

64 LINEAR MECHANICAL APPLICATIONS

Equation (2.5) shows that the residual and the thermal stresses can be considered as loads.

Because of the symmetry relations satisfied by _KLMN, βKL and γ KL, substitution of

Equation (2.4) into Equation (2.5) yields

_

V0

UM,N_KLMN(θ )δUK,L dV = _

A0t

T

K

(N)δUK dA + _

V0

ρ0f KδUK dV

+ _

V0

[βKL(θ)T − γ KL]δUK,L dV − ρ0

_

V0

D2UK

Dt2 δUK dV. (2.6)

Now, an assumption is made that can be considered as the quintessence of the finite

element method. The volume V0 is split in smaller volumes called “finite” elements:

V0 =_

e

V0e (2.7)

and the displacement field within each of these volumes is assumed to be a continuous

function of the displacement in discrete points i, called “nodes”:

U(X) =

N

_

i=1

ϕi (X)Ui . (2.8)

The functions ϕi are called shape functions.

In Equation (2.8), the position X is characterized by global coordinates. In practice, it

is advantageous to define local coordinates (ξ, η, ζ) within each element satisfying −1 ≤

ξ, η, ζ ≤ 1 (this applies to brick elements; the range for other types of elements will be

discussed shortly) and to express both the global coordinates and the displacements as a

function of discrete values at selected positions:

U(X) =

N

_

i=1

ϕi(ξ, η, ζ)U(Xαi ) (2.9)

X =

M

_

i=1

ψi(ξ, η, ζ)Xβi . (2.10)

If the discrete positions and the shape functions for X and U are the same, the formulation is

called isoparametric (Zienkiewicz and Taylor 1989). Here, only isoparametric formulations

will be considered. Accordingly,

U(X) =

N

_

i=1

ϕi(ξ, η, ζ)U(Xi ) (2.11)

X =

N

_

i=1

ϕi(ξ, η, ζ)Xi . (2.12)

LINEAR MECHANICAL APPLICATIONS 65

Equation (2.11) reads in component formulation

UK(X) =

N

_

i=1

ϕi(ξ, η, ζ)UiK (2.13)

where UiK is the component K of the displacement in node i. Hence,

UK,L(X) =

N

_

i=1

ϕi,L(ξ, η, ζ)UiK (2.14)

where

ϕi,L(ξ, η, ζ) := ∂ϕi

∂XL

(ξ, η, ζ)

= ∂ϕi

∂ξ

∂ξ

∂XL

+ ∂ϕi

∂η

∂η

∂XL

+ ∂ϕi

∂ζ

∂ζ

∂XL

. (2.15)

The terms ∂ϕi/∂ξ are obtained through direct differentiation, while ∂ξ/∂XL can be determined

by inverting ∂XL/∂ξ :

∂ξ

∂XL

= 1

J ∗ cofactor_∂XL

∂ξ

_ (2.16)

where

J

∗ := det_∂X

∂γ

_

, γ (ξ, η, ζ) (2.17)

is the Jacobian determinant of the transformation X(γ ). The quantities ∂XL

∂ξ are obtained

through direct differentiation of Equation (2.12).

Splitting the integrals in Equation (1.8) across the elements e and using Equation (2.14)

yields

_

e

_

V0e

N

_

i=1

N

_

j=1

ϕj,N_KLMN(θ)ϕi,LUjMδUiK dVe =_

e

_

A0e

N

_

i=1

T

K

(N)ϕiδUiK dAe

+_

e

_

V0e

N

_

i=1

ρ0f KϕiδUiK dVe +_

e

_

V0e

N

_

i=1

[βKL(θ)T − γ KL]ϕi,LδUiK dVe

−_

e

_

V0e

N

_

i=1

N

_

j=1

ρ0ϕiϕj

D2U K

j

Dt2 δUiK dVe (2.18)

66 LINEAR MECHANICAL APPLICATIONS

or, removing everything that is not a function of space from the integrals

_

e

N

_

i=1

N

_

j=1

__

V0e

ϕj,N_KLMN(θ)ϕi,L dVe

_

UjMδUiK

=_

e

N

_

i=1

__

A0e

T

K

(N)ϕi dAe

_

δUiK +_

e

N

_

i=1

__

V0e

ρ0f Kϕi dVe

_

δUiK

+_

e

N

_

i=1

__

V0e

[βKL(θ)T − γ KL]ϕi,L dVe

_

δUiK

−_

e

N

_

i=1

N

_

j=1

__

V0e

ρ0ϕiϕj dVe

_ D2U K

j

Dt2 δUiK. (2.19)

If we define for each element e a vector containing all displacements belonging to the

element

_U_e

=



U11

U12

U13

U21

...

UN1

UN2

UN3



e

(2.20)

one can write for Equation (2.19)

_

e

δ _U_T

e _K_e _U_e

=_

e

δ _U_T

e _F_e

−_

e

δ _U_T

e _M_e

D2

Dt2 _U_e (2.21)

where the components of _K_, _F_ and _M_ satisfy

_K_e(iK)(jM)

= _

V0e

ϕi,L_KLMN(θ)ϕj,N dVe (2.22)

_F_e(iK)

= _

A0e

T

K

(N)ϕi dAe + _

V0e

ρ0f Kϕi dVe + _

V0e

[βKL(θ)T − γ KL]ϕi,L dVe

(2.23)

_M_e(iK)(jM)

= _

V0e

ρ0ϕiϕj dVe. (2.24)

LINEAR MECHANICAL APPLICATIONS 67

where _K_e is the element stiffness matrix, _F_e is the element force vector and _M_e is the

element mass matrix. Notice that _K_e is symmetric because of the symmetry of _KLMN,

cf Equation (1.156). By defining a displacement vector _U_ containing all displacements

of the complete model and a localization matrix _L_e localizing the degrees of freedom of

element e in _U_:

_U_e

= _L_e _U_ (2.25)

one obtains for Equation (2.21)

δ _U_T _

_

e

_L_T

e _K_e _L_e

_

_U_

= δ _U_T _

_

e

_L_T

e _F_e

_

− δ _U_T _

_

e

_L_T

e _M_e _L_e

_ D2

Dt2 _U_ . (2.26)

Since Equation (2.26) must be satisfied for an arbitrary virtual displacement δ _U_, one

finally obtains

_K_ _U_ + _M_

D2

Dt2 _U_ = _F_ (2.27)

where

_K_ :=_

e

_L_T

e _K_e _L_e (2.28)

_M_ :=_

e

_L_T

e _M_e _L_e (2.29)

_F_ :=_

e

_L_T

e _F_e (2.30)

are the global stiffness matrix, global mass matrix and global force vector respectively.

Notice how the introduction of the shape functions transformed the integral equation (2.1)

into a set of linear algebraic equations over space, Equation (2.27). Instead of having to

look for a solution everywhere in space, the unknowns are reduced to the displacements

in a finite number of nodes. Equation (2.27) is the basic finite element equation to be

solved for mechanical problems. It generally results in a system of hundreds of thousands

of equations requiring special solution techniques (Sloan 1989), (Ashcraft et al. 1999).

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