FEM 10 Common Errors.ppt

Common Errors in Finite Element Analysis
Idealization error Discretization error
Idealization Error
•Posing the problem
•Establishing boundary conditions
•Stress-strain assumption
•Geometric simplification
•Specifying simplification
•Specifying material behaviour
•Loading assumptions
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 1

Discretization Error
•Imposing boundary conditions
•Displacement assumption
•Poor strain approximation due to element distortion
•Feature representation
•Numerical integration
•Matrix ill-conditioning
•Degradation of accuracy during Gaussian elimination
•Lack of inter-element displacement compatibility
•Slope discontinuity between elements

Idealization Error
One must be able to understand the physical nature of an
analysis problem well enough to conceive a proper
idealization. Engineering assumptions are always
required in the process of idealization.
Example:
•Establishing boundary conditions
•Specifying Material behaviour

Establishing Boundary Conditions
There are innumerable examples of how the specification of
improper boundary conditions can lead to either no results or poor
results.
It is impractical to review every possible manner in which one can
error when establishing boundary conditions. The finite element
analyst must gain a sufficient theoretical understanding of
mechanical idealization principles so that he can understand what
boundary conditions are applicable in any particular case.
Consider a bracket loaded by the weight of a servo motor, as
illustrated in Figure 1, 2

Fig 1. Actual Motor Bracket
Fig. 2 Simplified

Idealization Error – Specifying Material Behaviour
Elasto-Plastic state (Incompressible in plastic region)
Elastomeric material (= 0.5, Constitutive relations undefined)
Discretization Error
Discretization is the process where the idealization, having
an infinite number of DOF’s, is replaced with a model
having finite number of DOF’s. The following are errors
associated with discretization.

No of Points = 8
dof per point = 2
Total Equations = 16

Element Errors
•Imposing essential boundary conditions
•Displacement assumption
•Poor strain approximation due to element distortion
•Feature representation
Global Errors
•Numerical integration
•Matrix ill-conditioning
•Degradation of accuracy during Gaussian elimination
•Lack of inter-element displacement compatibility
•Slope discontinuity between elements

Element Error – Imposing Essential Boundary
Conditions
Rigid body motion is displacement of a body in such a manner
that no strain energy is induced. When one considers finite
elements in three-dimensional space, with nodes having three
translations DOF’s restraining rigid body motion can be more
complicated. Consider the 8-node brick element in three-
dimensional space depicted in Figure 4. What are the
minimum nodal displacement restraints required to prevent
rigid body motion of this element?

Fig 4. 8-node Brick Element in 3-D Space
In 3D space, there exists potential for six rigid body modes
( 3 displacements, (x, y, z) and 3 rotations θx, θy and θz )

Fig 5. Restraining One Node
Restraining one node will allow the body to rigidly rotate.

Fig 6. Nodal Restraint as a Ball and Socket Joint
Translational motion arrested but the body tend to rotate

Fig 7. Inhibiting Rigid Body Motion
Displacement is restrained at 5 and 8. This prevents rotation
about z axis.

Most finite element software programs will issue a warning,
“singular system encountered” or “Non-positive definite
system encountered”. The latter warning refers to the fact
that the strain energy is not greater than zero using the
given boundary conditions, suggesting that the structure is
either unstable, as in the case of structural collapse, or not
properly restrained, such that rigid body modes of
displacement are possible.

Element Error – Displacement Assumption
The h-method of finite element analysis endeavours to
minimize this error by using lower order displacement
assumptions (typically linear or quadratic) then refining the
model using more, smaller elements. Using more smaller
elements to gain increased accuracy is known as h-
convergence.

Element Error – Poor Strain Approximation Due to
Element Distortion
Distorted elements influence the accuracy of the finite
element approximation for strain. For instance, some
bending elements (beams, plated, shells) compute
transverse shear stain. This type of elements may have
difficulty computing shear strain when the element becomes
very thin.

Figure Discretization Error

Global Errors
Global errors are those associated with the assembled finite
element model. Even if each element exactly represents the
displacement within the boundary of a particular element,
the assembled model may not represent the displacement
within the entire structure, due to global errors.
Global Error – Numerical Integration
The use of numerical integration instead of closed-form
integration introduces error.

Global Error – Matrix Ill-Conditioning
The finite elements solution can render poor solutions for
displacement due to round off error. Ill-conditioning errors
typically manifest themselves during the solution phase of
an analysis.
Figure: Structure with an Ill-conditioned Stiffness Matrix

A stepped shaft, characterized by two different cross sections,
is depicted in Figure. Two rod elements is used to model the
structure, with the expanded equilibrium equations for
Element I given as:

































3
2
1
3
2
1
1
1
1
0
0
0
0
1
1
0
1
1
F
F
F
U
U
U
L
A
E
(1)
The equilibrium equations for Elements 2 are:

































3
2
1
3
2
1
2
2
2
1
1
0
1
1
0
0
0
0
F
F
F
U
U
U
L
A
E (2)

Substituting the known variables into the equations above, the
stiffness matrices are:



























0
0
0
0
0100
.
0
0100
.
0
0
0100
.
0
0100
.
0
0
0
0
1
1
1
0
1
1
00
.
5
)
0100
.
0
(
00
.
5
1
K


























00
.
1
00
.
1
0
00
.
1
00
.
1
0
0
0
0
1
1
0
1
1
0
0
0
0
00
.
5
)
00
.
5
(
00
.
1
2
K
(3)
(4)
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error -21

The element stiffness matrices are added together to
represent the global stiffness:



































P
F
U
U
U
00
.
0
00
.
1
00
.
1
00
.
0
00
.
1
01
.
1
0100
.
00
.
0
0100
.
0100
. 1
3
2
1
(1)
Boundary conditions are now imposed upon the global system
of Equations, (1). Since U1 = 0 and F1 is unknown, the 3 X 3
system in (1) is replaced by a 2 X 2 systems:





















P
U
U 00
.
0
00
.
1
00
.
1
00
.
1
01
.
1
3
2
(2)

Using Gaussian elimination, Equations (2) is manipulated to
yield:
P
U
P
U
U
101
00
.
0
00990
.
00
.
0
00
.
1
01
.
1
3
3
2




















 
(3)
Consider a small error in computation of k11 given by
Equation(2)





















P
U
U 00
.
0
00
.
1
00
.
1
00
.
1
02
.
1
3
2
(4)

Using Gaussian elimination to solve the system containing
the small error:
2
3
3
1.02 1.00 0.00
51.0
0.00 .0196
U
U P
U P
  
   
  
   
 
   
 
(5)
It may be surprising to note that a 1% error in one of the
entries of the stiffness matrix is responsible for a 50% error
in displacement.

25
EQUILIBRIUM AND COMPATIBILITY IN THE SOLUTION
1.Equilibrium of nodal forces and moments is satisfied. The
structural equations {F} – [K] {Q}= {0} are nodal equilibrium
equations. Therefore, the solution vector {Q} is such that
nodal forces and moments have a zero resultant at every
node.
2.Compatibility prevails at nodes. Elements connected to one
another have the same displacements at the connection
point. (i.e.) elements are compatible at nodes to the extent of
nodal d.o.f. they share.

26
3. Equilibrium is usually not satisfied across interelement
boundaries. Figure 4.4-1 provides a simple example.
Imagine that the elements are constant-strain triangles
and that node 4 is the only node displaced, as shown.
Then σx2 is the only nonzero stress, and the shaded
differential element is not in equilibrium

27
4. Compatibility may or may not be satisfied across
interlement boundaries. The constant-strain triangle and
the plane bilinear element, compatibility is guaranteed
because element sides remain straight even after the
element is deformed.
Other elements, such as plate elements are
incompatible in rotation about an interlement boundary.
Figure 4.4-2 is another case in point. Stretching of the
right edge causes vertical edges of the right element to
bend, and the interlement gap (shaded) appears. (This
element, called either incompatible or nonconforming)

28
5.Equilibrium is usually not satisfied within elements.
Satisfaction of the differential equations of equilibrium at
every point in an element demands a relation among
element d.o.f. that usually does not result from solution of
the global finite element equations [K]{Q} = {F}.
An important property of any reliable element is that its
displacement field be capable of representing all possible
states of constant strain.

29
6. Compatibility is satisfied within elements. We require only
that the element displacement field be continuous and
single-valued. These properties are automatically
provided by polynomial fields.
CONVERGENCE REQUIREMENTS
1.Within each element, the assumed field for φ must contain
a complete polynomial of degree m.
2.Across boundaries between elements, there must be
continuity of φ and its derivatives through order m-1.

30
3. Let the elements be used in a mesh (rather than tested
individually), and let boundary conditions on the mesh be
appropriate to a constant value of any of the mth derivatives
of φ. Then, as the mesh is refined, each element must
come to display that constant value.
For example, if φ = φ(x,y) and П contains first derivatives of
φ, then the lowest order acceptable field has the form
φ = a1 + a2x + a3y in each element, only φ itself need be
continuous across interelement boundaries, and each
element of an appropriately loaded mesh must display a
constant value of φ,x ( or of φ,y for other appropriate loading),
at least as the mesh is refined.

31
This suggests the following criterion for modeling. In a mesh
of low-order elements (such as the bilinear element), the
ratio of stress variation across the element to mean stress
within the element should be small.
It is a simple test that can be performed numerically, so as to
check the validity of an element formulation and its program
implementation. We assume that the element is stable in the
sense described below. Then, if the element passes the
patch test, we have assurance that all convergence criteria
are met.
THE PATCH TEST

32
Procedure. One assembles a small number of elements
into a “patch,” taking care to place at least one node within
the patch, so that the node is shared by two or more
elements, and so that one or more interelement boundaries
exist. Figure 4.6-1 shows an acceptable two-dimensional
patch, built of four-node elements. Boundary nodes of the
patch are loaded by consistently derived nodal loads
appropriate to a state of constant stress.

34
Stability. At the outset we assumed that the element to be
patch-tested is stable. A stable element is one that admits
no zero-energy deformation states when adequately
supported against rigid-body motion. Unstable elements
should be used with caution. They can produce an unstable
mesh, whose displacements are excessive and quite
unrepresentative of the actual structure.

35
“Weak” Patch Test. An element that fails to display constant
stress in a patch of large elements has not necessarily failed
the patch test. If, as the mesh is repeatedly subdivided,
elements come to display the expected state of constant
stress, then the element is said to have passed a “ weak”
patch test, and convergence to correct results is assured.

Convergence
To ensure monotonic convergence of the finite element
solution, both the individual elements and the assemblage
of elements (“the mesh”) must meet certain requirements.
Monotonic Convergence Using the Displacement Based,
h-Method
With mesh refinement, the finite element solution is
expected to convergence, monotonically, to the exact
solution.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 1

Requirements for Monotonic Convergence
(i) Requirements for Each Element’s Displacement Assumption
1. Rigid Body Representation assumption must be able to
account for all rigid body displacement modes of the
element.
2. Uniform Strain Representation: Constant strain states for
all strain components specified in the constitutive
equations of a particular idealization must be represented
within the element as the largest dimension of the element
approaches zero.

(ii) Requirements for the Mesh
1. Compatibility Between Elements: The dependent
variable(s), and p-1 derivatives of the dependent variable,
must be continuous at the nodes and across the inter-
element boundaries of adjacent elements.
2. Mesh Refinement: Each successive mesh refinement must
contain all of the previous nodes and elements in their
original location.
3. Uniform Strain Representation: The mesh must be able to
represent uniform strain when boundary conditions that are
consistent with a uniform strain condition are imposed.

Understanding Convergence Requirements
1. Convergence and Rigid Body Representation

In a one-dimensional element with rectangular coordinates, a
polynomial displacement assumption must have a constant
term to ensure that the type of rigid body motion described
above is allowed.
U(X) = a1 +a2X
2. Convergence and Uniform Strain Representation within an
Element
dx
du
E
E xx
xx 
 

2
2
1
)
( a
dx
du
x
a
a
x
u xx 


 

3. Convergence and Inter-Element Compatibility
Continuity must be maintained across element boundaries,
as well as at the nodes.
A. Incompatibility due to Elements Not Properly Connected:

Figure shows that Nodes 2 and 3 are coincident nodes,
meaning that they have the same spatial coordinated but
belong to separate elements.
Element Node a Node b
1 1 2
2 3 4
Connectivity table before node merge

Connectivity table after node merge
Element Node a Node b
1 1 2
2 2 3
When creating finite element models, a node merging
procedure is typically invoked, such that each pair of coincident
nodes is replaced by a single node, and the connectivity table
is updated to reflect the new connectivity.

B. Incompatibility due to Differing Order Displacement
Assumptions:

Consider two elements in Figure 14, one having a linear
displacement assumption and other having a quadratic
assumption. A gap between the elements occurs since the
displacement for the linear element can only be represented
by a straight line while displacement in the other element is
a quadratic function.

In figure that the mid-side node of Element 1 is connected to
corner nodes of Element 2 and 3. Higher order elements must
be matched such that the mid-side node of one element is
connected to the mid-side node of the other.

C. Incompatibility due to Differing Nodal Variables:

Incompatibility also occurs when element having differing types of nodal
DOF are joined. Figure shows a 2-node beam element attached to one
node of a 8-node element, which have translational DOF only, while
structural elements have both translational and rotational DOF’s.
When joining continuum and structural elements, a special constraint must
be imposed upon the structural element’s rotational DOF, the least 8-node
brick element shown in Figure is well restrained. Its nodes do not have
rotational DOF’s, therefore, at the node where the beam is attached, there
exits no DOF from the brick to couple with the rotational DOF of the beam.
As shown in figure, the beam element will experience rigid body rotation.

D. Incompatibility due to incompatible elements:
Type of incompatibility to be mentioned occurs when elements in
mesh are of the “incompatible type”

Consider the two elements in Figure both are identically
formulated, 4 node quadrilateral surface elements designed
with incompatible modes of displacement. Assume that two
nodes of Element 2 are given a displacement of Δy as
depicted in figure (a). With no other displacement, the
elements would appear as shown. However, displacement
for Element 2 would be computed as shown in figure (18).

4. Convergence and the Discretization Process:
Each successive mesh refinement must contain all of the
previous nodes and elements in their original location.

5. Convergence and Uniform Strain Representation within Mesh
The mesh must be able to represent a uniform strain state,
when suitable boundary conditions are imposed. A Patch
Test has been devised to test the uniform strain condition.
Either displacements or loads can be applied, depending
upon what characteristics of the mesh are to be investigated.

Figure illustrates a mesh for a patch test of two-
dimensional elements although, theoretically, substantial
variation in the geometry (somewhat distorted) of the
elements used for a patch test is permitted.

Elementary Beam, Plate, and Shell Elements
Euler-Beronoulli Beams
Beams are slender structural members, with one dimension
significantly greater that the other two.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Beam - 1

Euler-Beronoulli Beam Assumptions
If Euler- Bernoulli (“elementary”) beam theory is to be used,
some restrictions must be imposed to ensure suitable
accuracy. Six items related to the Euler- Beronulli beam
bending assumptions are considered in brief:
1. Slenderness (Transverse shear neglected, L/h ≥ 10,
Orthogonal planes remain plane and orthogonal)
2. Straight, narrow, and uniform beams (Arch structure –
membrane forces.)
3. Small deformations (Second order terms and dropped
from the Green-Lagrange Strain mertic)

4. Bending loads only (Loads that cause twist or cause x- or
y- displacement of neutral axis are ignored.)
5. Linear, isotropic, homogeneous materials response
6. Only normal stress in the x-coordinate direction (σxx)
significant (σxx vary linearly in the z direction and σzz = 0

Kirchhoff Plate Bending Assumptions
Several restrictions must be imposed when using the
Kirchhoff plate bending formulation if suitable accuracy is
to be obtained; six related items are considered briefly:
1. Thinness
2. Flatness and uniformity
3. Small deformation
4. No membrane deformation
5. Linear, isotropic, homogeneous materials
6. σxx, σyy , σxy the only stress components of significance

1.Thin Plates:
If the ratio of the in-plane dimensions to thickness is greater than ten, a
plate may be considered thin; in other words, thin plates are such that
.
10
/
/ 
 t
b
t
l
(i) Transverse Shear Stress Does Not Affect Transverse
Displacement:
Although transverse displacement of a plat loaded normal to its neutral
surface is affected by both normal stress and transverse shear stress, it is
assumed that the shear stress does not have a significant impact on the
magnitude of transverse displacement. It can be shown that the magnitude
of transverse displacement due to shear stress is small in thin plates.

(ii) Orthogonal Planes Remain Plane and Orthogonal:
In a thin plate, any cross sectional plane, originally orthogonal
to the neutral surface, is assumed to remain plane and
orthogonal when the plate is loaded. As with deep beams,
cross sections in thick plates, originally orthogonal and
planar, will be neither under transverse load, due to shearing
effects.

2. Flat and Uniform
Kirchhoff plates are presumed flat, and as such, loads are
not carried by membrane deformation. If initial curvature
were to exist, some of the load could be carried by
membrane action, hence, curved plates (shells) require a
different approach to account for the combination of
simultaneous membrane and bending deformation.

3. Small Deformation
1. The expression for stain actually contains second order
terms, and these terms are truncated if the strains are
presumed small.
2. Like the Euler-Bernoulli beam, plate curvature can be
expressed in terms of second order derivatives if the
square of the slop is negligible.
3. As a plate deflects its transverse stiffness changes.

Why does the transverse stiffness change with deflection? As
a plate deforms into a curved (or a doubly curved) surface,
transverse loads are resisted by both bending and membrane
deformation. The addition of membrane deformation affects the
transverse stiffness of the plate. Consider figure where a ball is
placed on a very thin “plate”.
Notice that as the amount of plate deflection increases, more
load is carried by membrane tension and less by bending.
Shell theory is required to account for structure that carry loads
through simultaneous bending and membrane deformation

4. No Membrane Deformation
Kirchhoff plates do not account for membrane deformation.
This means that at the neutral surface, there can be no
displacement in the x- or y-coordinate directions, which
would suggest the existence of membrane deformation
5. Signification Stress Components are σ xx, σ yy , σ xy
As in shallow beams, it is assumed that transverse shear
stress in thin plates is relatively insignificant when compared
to the normal stress in the longitudinal direction of the beam,
hence:
σyz = σzx = 0 and σzz = 0

Shear Stress Affects the Edge Reaction Forces
In plate bending stress affects transverse displacement. In
addition, shear stress affects reaction forces on restrained
edges of plates and shells.
The restrained edges of shell and plate structures develop
reaction forces. The magnitude of the reaction is dependent
upon the behaviour in the vicinity very near the restrained
edge. This boundary layer effect cannot be captured without
accounting for transverse shear stress. In addition, even in
models that do account for shear effects, the boundary layer
effect is difficult to model without special care.

FEM 10 Common Errors.ppt

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