fem: stationary functional approach

FEM: Stationary Functional Approach

Mohammad Tawfik #WikiCourses

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Introduction to the Finite

Element Method

Stationary Functional Approach




Few definitions

Over-simplified versions!




A functional …

• A functional is a “function of functions” that

produces a real/complex number

• In mechanics problems, usually, the

functional used is the total energy

functional which contains the potential

energy, the kinetic energy, and the

externally work done on the system




A functional …

• A functional may be presented in the form:

Domain

nnmn

nmn

dxdxxxfxxfG

xxfxxfI

...,...,,...,,...,

,...,,...,,...,

1111

111




Variation …

• Variation of a functional is the “differentiation” of

the functional with respect to one or more of its

entries (functions)

• Note that the Variation of the functional with

respect to the independent variables is always

equal to zero

Domain

nm

m

m

dxdxfdf

dGf

df

dGf

df

dG

fffI

......

,...,,

12

2

1

1

21




Strain Energy …

• Strain energy is the amount of mechanical

energy stored in a structure, due to the

deflection of the structure.

• An expression for the strain energy may

be given by

Volume

dVU 2

1




Applications




The bar tensile problem

• The total energy of the elastic structure is

given as the difference between the strain

energy and the work done by the

externally applied forces





• An expression for the total energy for a

bar, may be given by the following integral

BarLength

dxxFux

uEA .

2

12





• For equilibrium, the total energy needs to

be at a minimum value, that is to say, its

variation is zero

0.

BarLength

dxxFux

u

x

uEA





• Now, let us perform integration by parts,

we get

• Which indicates that

0.2

2

0

BarLength

l

dxxFux

uuEA

x

uuEA

lxx

l

x

uuEA

x

uuEA

x

uuEA

&00

00





• The other term becomes

• Then, the integrand should be equal to

zero:

0.2

2

BarLength

dxxFux

uuEA

02

2

xF

x

uEAu





• And, since the variation of the

displacement is an arbitrary function, it can

not be equal to zero everywhere which

yields

• This is the original differential equation for

the displacement function of a bar subject

to distributed loading along its axis

02

2

xF

x

uEA




A Conclusion …

• So, using the stationary functional

approach, we could start from the total

energy and go all the way through

obtaining the governing differential

equation!

• However, we are more interested in the

week form … so let’s get back to FEM.




Deriving the finite element

model!




Proposing an approximate

solution …

• We get:

euxNxu

euxNxu




Energy variation relation:

• But the nodal values of the function are

independent of the integration

0 gthElementLen

Tee

xx

Te dxxFNuuNNuEA

00

l

e

xx

Te dxxFNuNNEAu




Element Equation

• Variation is arbitrary, therefore, it can not

be zero; hence:

00

l

e

xx dxxFNuNNEA

l

e

l

xx dxxFNudxNNEA00

ee fuk




Beam Bending Problem





• Obtaining the strain energy expression for

the beam under transverse loading, we

get:

• Giving:

l

dxxFwdx

wdEI

0

2

2

2

.2

1

0.0

2

2

2

2

l

dxxFwdx

wd

dx

wdEI





• Using the approximate solution into the

above expression

• We get:

ewxNxw

00

l

Tee

xxxx

Te dxxFNwwNNwEI





• Using the same procedure as for the bar

example above, we get

ee fwk

l

e

l

xxxx dxxFNfdxNNEIk00

&




In conclusion …

• We were able to derive the differential

equation governing the mechanics of an

elastic body, including the boundary

conditions, starting from the total energy

expression!




In conclusion …

• Using the stationary functional approach,

we could get the same FE model without

having to have the governing differential

equstion!

fem: stationary functional approach

Education

wikicourses http

total energy functional

strain energy strain

bar tensile problem

applications fem

system fem

barlengthdxxfuxuxuea

potential energy