Lecture3

Institute of Structural Engineering Page 1
Method of Finite Elements I
Chapter 3
Variational Formulation &
the Galerkin Method

Today’s Lecture Contents:
• Introduction
• Diﬀerential formulation
• Principle of Virtual Work
• Variational formulations
• Approximative methods
• The Galerkin Approach

The diﬀerential form of physical processes
Physical processes are governed by laws (equations) most probably
expressed in a diﬀerential form:

•  The Laplace equation in two
dimensions:
(e.g. the heat conduction problem)
•  The isotropic slab equation:
•  The axially loaded bar equation:

Focus: The axially loaded bar example.
Consider a bar loaded with constant end load R.

Given: Length L, Section Area A, Young'ʹs modulus E
Find: stresses and deformations.

Assumptions:
The cross-‐‑section of the bar does not change after loading.
The material is linear elastic, isotropic, and homogeneous.
The load is centric.
End-‐‑eﬀects are not of interest to us.
Strength of Materials Approach
Equilibrium equation
( )( )
R
f x R x
A
σ= ⇒ =
x

R

R

L-‐x

f(x)

Constitutive equation (Hooke’s Law)
( )( )
x R
x
E AE
σ
ε = =
Kinematics equation
ε x( )=
δ(x)
x
δ x( )=
Rx
AE

Consider and infinitesimal element of the bar:

Given: Length L, Section Area A, Young'ʹs modulus E
Find: stresses and deformations.

Assumptions:
The cross-‐‑section of the bar does not change after loading.
The material is linear elastic, isotropic, and homogeneous.
The load is centric.
End-‐‑effects are not of interest to us.
The Differential Approach
Equilibrium equation
Constitutive equation (Hooke’s Law)
Eσ ε=
Kinematics equation
du
dx
ε =
Aσ = A(σ + Δσ ) ⇒ A lim
Δx→0
!
Δσ
Δx
= 0 ⇒ A
dσ
dx
= 0
AE
d2
u
dx2
= 0 Strong Form
Boundary Conditions (BC)
u(0) = 0 Essential BC
σ L( )= 0 ⇒
AE
du
dx x=L
= R Natural BC

Deﬁnition
The strong form of a physical process is the well posed
set of the underlying diﬀerential equation with the
accompanying boundary conditions

AE
d2
u
dx2
= 0 Strong Form
AE
du
dx x=L
= R Natural BCx

R

Analytical Solution:
u(x) = uh
= C1
x + C2
& ε(x) =
du(x)
dx
= C1
= const!
v This is a homogeneous 2nd order ODE with known solution:

AE
d2
u
dx2
= 0 Strong Form
AE
du
dx x=L
= R Natural BCx

R


u(x) = uh
= C1
x + C2
u(0)=0
du
dx x=L
=
R
EA
⎯ →⎯⎯⎯
C2
= 0
C1
=
R
EA
AE
d2
u
dx2
= 0 Strong Form
AE
du
dx x=L
= R Natural BC
⇒ u x( )=
Rx
AE
v  Same as in the mechanical approach!
x

R

To fully deﬁne the solution (i.e., to evaluate the values of parameters )
we have to use the given boundary conditions (BC):
C1
,C2

The Strong form – 2D case
A generic expression of the two-‐‑dimensional strong form is:

and a generic expression of the accompanying set of boundary conditions:

: Essential or Dirichlet BCs

: Natural or von Neumann BCs

Disadvantages
The analytical solution in such equations is
i.  In many cases diﬃcult to be evaluated
ii.  In most cases CANNOT be evaluated at all. Why?
•  Complex geometries
•  Complex loading and boundary conditions

su
: supported area with prescribed displacements U
su
sf
: surface with prescribed forces f
sf
f B
: body forces (per unit volume)
U : displacement vector
ε : strain tensor (vector)
σ : stress tensor (vector)
Deriving the Strong form – 3D case

Kinematic Relations
U =
U
V
W
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
, L =
∂
∂x
0 0
0
∂
∂y
0
0 0
∂
∂z
∂
∂y
∂
∂x
0
0
∂
∂z
∂
∂y
∂
∂z
0
∂
∂x
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
, εT
= εxx
εyy
εzz
2εxy
2εyz
2εxz
⎡
⎣⎢
⎤
⎦⎥
ε = LU

2 2 2
2 2
2 2 2
2 2
2 2 2
2 2
1 2 2 2 2
2
2 2 2 2
2
2 2 2 2
2
0 2 0 0
0 0 2 0
0 0 0 2
0 0
0 0
0 0
y x x y
z y y z
z x x z
y z x z x x y
x z z y x y y
x y z x z y z
⎡ ⎤∂ ∂ ∂
−⎢ ⎥∂ ∂ ∂ ∂⎢ ⎥
⎢ ⎥∂ ∂ ∂
−⎢ ⎥
∂ ∂ ∂ ∂⎢ ⎥
⎢ ⎥∂ ∂ ∂
⎢ ⎥−
∂ ∂ ∂ ∂⎢ ⎥=
⎢ ⎥∂ ∂ ∂ ∂
− −⎢ ⎥
∂ ∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥
⎢ ⎥∂ ∂ ∂ ∂
⎢ ⎥− −
∂ ∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥
⎢ ⎥
∂ ∂ ∂ ∂⎢ ⎥− −
⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦
LL1
ε = 0 ⇒
Strain Compatibility – Saint Venant principle

2
2, =
B
T T
xx yy zz xy yz zxτ τ τ τ τ τ⎡ ⎤= ⎣ ⎦
L τ +f = 0
τ L L
0 0 0
on we have where 0 0 0
0 0 0
fs
f
l m n
s m l n
n m l
⎡ ⎤
⎢ ⎥− = ⎢ ⎥
⎢ ⎥⎣ ⎦
Ντ f = 0 N
l, m and n are cosines of the angles between the normal on the
surface and X, Y and Z
Equilibrium Equations
Body loads:
Surface loads:

=τ Cε
C is elasticity matrix and depends on material properties E and ν
(modulus of elasticity and Poisson’s ratio)
1−
=ε C τ
Constitutive Law

Differential Formulation
•  Boundary problem of the linear theory of elasticity: differential
equations and boundary conditions
•  15 unknowns: 6 stress components, 6 strain components and 3
displacement components
•  3 equilibrium equations, 6 relationships between displacements and
strains, material law (6 equations)
•  Together with boundary conditions, the state of stress and
deformation is completely defined
Summary -‐‑ General Form 3D case:

Principle of Virtual Work
•  The principle of virtual displacements: the virtual work of a system of
equilibrium forces vanishes on compatible virtual displacements; the
virtual displacements are taken in the form of variations of the real
displacements
•  Equilibrium is a consequence of vanishing of a virtual work
f f
f
S T ST T B iT i
C
iV V S
dV dV dS= + + ∑∫ ∫ ∫ε τ U f U f U R
Stresses in equilibrium with applied loads
Virtual strains corresponding to virtual displacements
Internal Virtual Work
External Virtual Work

Principle of Complementary Virtual Work
• The principle of virtual forces: virtual work of
equilibrium variations of the stresses and the forces on
the strains and displacements vanishes; the stress ﬁeld
considered is a statically admissible ﬁeld of variation
• Equilibrium is assumed to hold a priori and the
compatibility of deformations is a consequence of
vanishing of a virtual work
• Both principles do not depend on a constitutive law

Variational Formulation
• Based on the principle of stationarity of a functional,
which is usually potential or complementary energy
• Two classes of boundary conditions: essential
(geometric) and natural (force) boundary conditions
• Scalar quantities (energies, potentials) are considered
rather than vector quantities

Principle of Minimum
Total Potential Energy
• Conservation of energy: Work = change in potential,
kinetic and thermal energy
• For elastic problems (linear and non-‐‑linear) a special case
of the Principle of Virtual Work – Principle of minimum
total potential energy can be applied
•  is the stress potential:
/ij ijUτ ε= ∂ ∂
U =
1
2v
∫ εT
Cεdv
W = f BT
U
v
∫ dv + f
S f
s
∫ Uds
U

Principle of Minimum
Total Potential Energy
• Total potential energy is a sum of strain energy and
potential of loads
• This equation, which gives P as a function of deformation
components, together with compatibility relations within
the solid and geometric boundary conditions, deﬁnes the
so called Lagrange functional
• Applying the variation we invoke the stationary condition
of the functional

dP = dU − dW = 0
P = U −W
P

Variational Formulation
• By utilizing the variational formulation, it is possible to
obtain a formulation of the problem, which is of lower
complexity than the original diﬀerential form (strong
form).
• This is also known as the weak form, which however can
also be a_ained by following an alternate path (see
Galerkin formulation).
• For approximate solutions, a larger class of trial functions
than in the diﬀerential formulation can be employed; for
example, the trial functions need not satisfy the natural
boundary conditions because these boundary conditions
are implicitly contained in the functional – this is
extensively used in MFE.

Approximative Methods
Instead of trying to ﬁnd the exact solution of the continuous
system, i.e., of the strong form, try to derive an estimate of what
the solution should be at speciﬁc points within the system.

The procedure of reducing the physical process to its discrete
counterpart is the discretisation process.

Variational Methods
approximation is based on the
minimization of a functional, as those
defined in the earlier slides.
•  Rayleigh-Ritz Method
Weighted Residual Methods
start with an estimate of the the solution and
demand that its weighted average error is
minimized
•  The Galerkin Method
•  The Least Square Method
•  The Collocation Method
•  The Subdomain Method
•  Pseudo-spectral Methods
Approximative Methods

Focus: The axially loaded bar with distributed load example.

u(x) = uh
+ up
= C1
x + C2
−
ax3
6EA
u(0)=0
du
dx x=L
=
R
EA
⎯ →⎯⎯⎯
C2
= 0
C1
=
aL2
2EA
AE
d2
u
dx2
= −ax Strong Form
AE
du
dx x=L
= 0 Natural BC
⇒ u x( )=
aL2
2EA
x −
ax3
6EA
x

To fully deﬁne the solution (i.e., to evaluate the values of parameters )
we have to use the given boundary conditions (BC):
C1
,C2
ax


x

ax

0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2
Length (m)
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2
Length (m)
u x( )=
aL2
2EA
x −
ax3
6EA

and its corresponding strong form:
Given an arbitrary weighting function w
that satisﬁes the essential conditions and
additionally:
If
then,
x

R

AE
d2
u
dx2
= −ax Strong Form
Boundary Conditions (BC)
σ L( )= 0 ⇒
AE
du
dx x=L
= 0 Natural BC

ax

Multiplying the strong form by w and integrating over L:

Integrating equation I by parts the following relation is derived:
x

ax

Elaborating a li_le bit more on the relation:
x
R

ax

Elaborating a li_le bit more on the relation:
why?
Therefore, the weak form of the problem is deﬁned as
Find
such that:
why?
Observe that the weak form involves derivatives of a lesser order than the
original strong form.
x

ax

Weighted Residual Methods start with
an estimate of the solution and
demand that its weighted average
error is minimized:
•  The Galerkin Method
•  The Least Square Method
•  The Collocation Method
•  The Subdomain Method
•  Pseudo-spectral MethodsBoris Grigoryevich Galerkin – (1871-1945)
mathematician/ engineer

Theory – Consider the general case of a
diﬀerential equation:
Example – The axially loaded bar:
Try an approximate solution to the
equation of the following form
where
are test functions (input) and
are unknown quantities that we need
to evaluate. The solution must satisfy
the boundary conditions.
Since
is an approximation, substituting
it in the initial equation will result in
an error:
Choose the following approximation
Demand that the approximation
satisﬁes the essential conditions:
The approximation error in this case is:
The Galerkin Method

Assumption 1: The weighted average error of the approximation should
be zero
The Galerkin Method

be zero
Therefore once again integration by parts leads to
But that’s the Weak
Form!!!!!
The Galerkin Method

be zero
Therefore once again integration by parts leads to
But that’s the Weak
Form!!!!!
Assumption 2: The weight function is approximated using the same
scheme as for the solution
Remember that the
weight function must
also satisfy the BCs
Substituting the approximations for both
and
in the weak form,
The Galerkin Method

The following relation is retrieved:
where:
The Galerkin Method

Performing the integration, the following relations are established:
The Galerkin Method

Or in matrix form:
that’s a linear system of equations:
and that’s of course the exact solution. Why?
The Galerkin Method

Now lets try the following approximation:
The weight function assumes the same form:
Which again needs to satisfy the natural BCs, therefore:
Substituting now into the weak form:
Wasn’t that much easier? But….is it correct?
The Galerkin Method
w2
EAu2
− x ax( )( )dx
0
L
∫ = 0 ⇒ u2
= α
L2
3EA

0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2
Length (m)
Strong Form Galerkin-Cubic order Galerkin-Linear
The Galerkin Method

We saw in the previous example that the Galerkin method is based on the approximation
of the strong form solution using a set of basis functions. These are by deﬁnition
absolutely accurate at the boundaries of the problem. So, why not increase the
boundaries?

Instead of seeking the solution of a single bar we chose to divide it into three
interconnected and not overlapping elements
1
2
3
The Galerkin Method
element: 1
element: 2

1
2
3
element: 1
The Galerkin Method
element: 2

1
2
3
The Galerkin Method

The weak form of a continuous problem was derived in a
systematic way:
This part involves only the
solution approximation
This part only involves the
essential boundary
conditions a.k.a. loading
The Galerkin Method

And then an approximation was defined for the displacement field, for example
Vector of
degrees of
freedom
Shape Function
Matrix
The weak form also involves the first derivative of the approximation
Strain Displacement
Matrix
Strain field
Displacement
field
The Galerkin Method

Therefore if we return to the weak form :
The following FUNDAMENTAL FEM expression is derived
and set:
or even be]er
Why??
The Galerkin Method

EA has to do only with material and cross-‐‑sectional
properties
We call
The Finite Element stiﬀness Matrix
Ιf E is a function of {d}
Ιf [B] is a function of {d}
Material Nonlinearity
Geometrical Nonlinearity
The Galerkin Method

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