FEM and it's applications

Numerical Method
Finite Element Method
Boundary Element Method
Finite Difference Method
Finite Volume Method
Meshless Method

What is FEM ?
• Many physical phenomena in engineering and science can be
described in terms of partial differential equations.
• In general, solving these equations by classical analytical methods for
arbitrary shapes is almost impossible.
• The finite element method (FEM) is a numerical approach by which
these partial differential equations can be solved approximately.
• From an engineering standpoint, the FEM is a method for solving
engineering problems such as stress analysis, heat transfer, fluid flow
and electromagnetics by computer simulation.
• Millions of engineers and scientists worldwide use the FEM to predict
the behaviour of structural, mechanical, thermal, electrical and
chemical systems for both design and performance analyses.

• Its popularity can be gleaned by the fact that over $1 billion is spent
annually in the United States on FEM software and computer time.
• A 1991 bibliography (Noor, 1991) lists nearly 400 finite element books
in English and other languages.
• A web search (in 2006) for the phrase ‘finite element’ using the
Google search engine yielded over 14 million pages of results.
• Mackerle (http://ohio.ikp.liu.se/fe) lists 578 finite element books
published between 1967 and 2005.

Basic Approach of FEM
• Consider a plate with a hole as shown in Figure 1.1 for which we wish
to find the temperature distribution.
• It is straightforward to write a heat balance equation for each point in
the plate.

• However, the solution of the resulting partial differential equation for
a complicated geometry, such as an engine block, is impossible by
classical methods like separation of variables.
• Numerical methods such as finite difference methods are also quite
awkward for arbitrary shapes; software developers have not
marketed finite difference programs that can deal with the
complicated geometries that are commonplace in engineering.
• Similarly, stress analysis requires the solution of partial differential
equations that are very difficult to solve by analytical methods except
for very simple shapes, such as rectangles, and engineering problems
seldom have such simple shapes.

• The basic idea of FEM is to divide the body into finite elements, often
just called elements, connected by nodes, and obtain an approximate
solution as shown in Figure 1.1.
• This is called the finite element mesh and the process of making the
mesh is called mesh generation.
• The FEM provides a systematic methodology by which the solution, in
the case of our example,
The temperature field, can be determined by a computer program.

• For linear problems, the solution is determined by solving a system of
linear equations; the number of unknowns (which are the nodal
temperatures) is equal to the number of nodes.
• To obtain a reasonably accurate solution, thousands of nodes are
usually needed, so computers are essential for solving these
equations.
• Generally, the accuracy of the solution improves as the number of
elements (and nodes) increases, but the computer time, and hence
the cost, also increases.
• The finite element program determines the temperature at each node
and the heat flow through each element.
• The results are usually presented as computer visualizations, such as
contour plots, although selected results are often output on
monitors. This information is then used in the engineering design
process.

The same basic approach is used in other types of problems:
• In stress analysis, the field variables are the displacements;
• In chemical systems, the field variables are material concentrations; and
• In electro-magnetics, the potential field.
• The same type of mesh is used to represent the geometry of the structure
or component and to develop the finite element equations, and
• for a linear system, the nodal values are obtained by solving large systems
(from 103 to 106 equations are common today, and in special
applications,109) of linear algebraic equations.

• The preponderance of finite element analyses in engineering design is
today still linear FEM.
• In heat conduction, linearity requires that the conductance be
independent of temperature.
• In stress analysis, linear FEM is applicable only if the material
behaviour is linear elastic and the displacements are small.
• In stress analysis, for most analyses of operational loads, linear
analysis is adequate as it is usually undesirable to have operational
loads that can lead to nonlinear material behaviour or large
displacements.
• For the simulation of extreme loads, such as crash loads and drop
tests of electronic components, nonlinear analysis is required.

Development of FEM
• The FEM was developed in the 1950s in the aerospace industry. The
major players were Boeing and Bell Aerospace (long vanished) in the
United States and Rolls Royce in the United Kingdom.
• M.J.Turner,R.W.Clough, H.C. Martin and L.J. Topp published one of the
first papers that laid out the major ideas in 1956 (Turner et al., 1956)
• It established the procedures of element matrix assembly and
element formulations, but did not use the term ‘finite elements’.
• The second author of this paper, Ray Clough, was a professor at
Berkeley, who was at Boeing for a summer job. Subsequently, he
wrote a paper that first used the term ‘finite elements’, and he was
given much credit as one of the founders of the method.

• This research coincided with the rapid growth of computer power, and
the method quickly became widely used in the nuclear power, defence,
automotive and aeronautics industries.
• Much of the academic community first viewed FEM very skeptically, and
some of the most prestigious journals refused to publish papers on FEM.
• It is interesting that for many years the FEM lacked a theoretical basis,
i.e. there was no mathematical proof that finite element solutions give
the right answer.
• In the late 1960s, the field aroused the interest of many mathematicians,
who showed that for linear problems, finite element solutions converge
to the correct solution of the partial differential equation (provided that
certain aspects of the problem are sufficiently smooth).
• In other words, it has been shown that as the number of elements
increases, the solutions improve and tend in the limit to the exact
solution of the partial differential equations.

• E. Wilson developed one of the first finite element programs that was
widely used. Its dissemination was hastened by the fact that it was
‘freeware’, which was very common in the early 1960s, as the
commercial value of software was not widely recognized at that time.
• The program was limited to two-dimensional stress analysis.
• Then in 1965, NASA funded a project to develop a general-purpose
finite element program by a group in California led by Dick MacNeal.
• This program, which came to be known as NASTRAN, included a large
array of capabilities, such as two- and three-dimensional stress
analyses, beam and shell elements, for analyzing complex structures,
such as airframes, and analysis of vibrations and time-dependent
response to dynamic loads.
• The initial program was put in the public domain, but it had many
bugs.

• Shortly after the completion of the program, Dick MacNeal and Bruce
McCormick started a software firm that fixed most of the bugs and
marketed the program to industry.
• By 1990, the program was the workhorse of most large industrial firms and
the company, MacNeal-Schwendler, was a $100 million company.
• At about the same time, John Swanson developed a finite element program
at Westinghouse Electric Corp. for the analysis of nuclear reactors.
• In 1969, Swanson left Westinghouse to market a program called ANSYS.
• The program had both linear and nonlinear capabilities, and it was soon
widely adopted by many companies.
• In 1996, ANSYS went public, and it now (in 2018) has a capitalization of
$15.116 billion.

• Another nonlinear software package of more recent vintage is LS-
DYNA.
• This program was first developed at Livermore National Laboratory by
John Hallquist.
• ABAQUS was developed by a company called HKS, which was founded
in 1978. The program was initially focused on nonlinear applications,
but gradually linear capabilities were also added.
• The program was widely used by researchers because HKS introduced
gateways to the program, so that users could add new material
models and elements.
• In 2005, the company was sold to Dassault Systemes for $413 million
and it now (in 2018) has a capitalization of $33.960 Billion.

• In many industrial projects, the finite element database becomes a
key component of product development because it is used for a large
number of different analyses, although in many cases, the mesh has
to be tailored for specific applications. The finite element database
interfaces with the CAD database and is often generated from the
CAD database.
• Unfortunately, in today’s environment, the two are substantially
different. Therefore, finite element systems contain translators, which
generate finite element meshes from CAD databases; they can also
generate finite element meshes from digitization's of surface data.
• The need for two databases causes substantial headaches and is one
of the major bottlenecks in computerized analysis today, as often the
two are not compatible.

• The availability of a wide range of analysis capabilities in one program
makes possible analyses of many Complex real-life problems.
• For example, the flow around a car and through the engine
compartment can be obtained by a fluid solver, called computational
fluid dynamics (CFD) solver.
• This enables the designers to predict the drag factor and the lift of the
shape and the flow in the engine compartment.
• The flow in the engine compartment is then used as a basis for heat
transfer calculations on the engine block and radiator.
• These yield temperature distributions, which are combined with the
loads, to obtain a stress analysis of the engine.

• Similarly, in the design of a computer or microdevice, the
temperatures in the components can be determined through a
combination of fluid analysis (for the air flowing around the
components) and heat conduction analysis.
• The resulting temperatures can then be used to determine the
stresses in the components, such as at solder joints, that are crucial
to the life of the component.
• The same finite element model, with some modifications, can be used
to determine the electromagnetic fields in various situations.
• These are of importance for assessing operability when the
component is exposed to various electromagnetic fields.

• In aircraft design, loads from CFD calculations and wind tunnel tests
are used to predict loads on the airframe.
• A finite element model is then used with thousands of load cases,
which include loads in various maneuvers such as banking, landing,
take-off and so on, to determine the stresses in the airframe.
• Almost all of these are linear analyses; only determining the ultimate
load capacity of an airframe requires a nonlinear analysis.
• It is interesting that in the 1980s a famous professor predicted that by
1990 wind tunnels would be used only to store computer output.
• He was wrong on two counts: Printed computer output almost
completely disappeared, but wind tunnels are still needed because
turbulent flow is so difficult to compute that complete reliance on
computer simulation is not feasible.

• Manufacturing processes are also simulated by finite elements. Thus,
the solidification of castings is simulated to ensure good quality of the
product.
• In the design of sheet metal for applications such as cars and washing
machines, the forming process is simulated to insure that the part can
be formed and to check that after spring-back (when the part is
released from the die) the part still conforms to specifications.
• Thus, FEA has led to tremendous reductions in design cycle time, and
effective use of this tool is crucial to remaining competitive in many
industries.

A question that may occur to you is:
Why has this tremendous change taken place?

• As can be seen from Figure 1.2, the
increase in computational power has been
linear on a log scale, indicating a geometric
progression in speed.
• This geometric progression was first
publicized by Moore, a founder of Intel, in
the 1990s. He noticed that the number of
transistors that could be packed on a chip,
and hence the speed of computers,
doubled every 18 months.
• This came to be known as Moore’s law, and
remarkably, it still holds.

General Description of The Finite Element Method
• In the finite element method, the actual continuum or body of matter, such as a solid,
liquid, or gas, is represented as an assemblage of subdivisions called finite elements.
• These elements are considered to be interconnected at specified joints called nodes or
nodal points.
• The nodes usually lie on the element boundaries where adjacent elements are
considered to be connected.
• Since the actual variation of the field variable (e.g., displacement, stress, temperature,
pressure, or velocity) inside the continuum is not known, we assume that the variation of
the field variable inside a finite element can be approximated by a simple function.
• These approximating functions (also called interpolation models) are defined in terms of
the values of the field variables at the nodes.
• When field equations (like equilibrium equations) for the whole continuum are written,
the new unknowns will be the nodal values of the field variable.
• By solving the field equations, which are generally in the form of matrix equations, the
nodal values of the field variable will be known.
• Once these are known, the approximating functions define the field variable throughout
the assemblage of elements.

• The solution of a general continuum problem by the finite element
method always follows an orderly step-by-step process.
• With reference to static structural problems, the step-by-step
procedure can be stated as follows:
Step (i): Discretization of the structure
• The first step in the finite element method is to divide the structure
or solution region into subdivisions or elements.
• Hence, the structure is to be modelled with suitable finite elements.
• The number, type, size, and arrangement of the elements are to be
decided.

Step (ii): Selection of a proper interpolation or displacement model
• Since the displacement solution of a complex structure under any
specified load conditions cannot be predicted exactly, we assume
some suitable solution within an element to approximate the
unknown solution.
• The assumed solution must be simple from a computational
standpoint, but it should satisfy certain convergence requirements.
• In general, the solution or the interpolation model is taken in the
form of a polynomial.

Step (iii): Derivation of element stiffness matrices and load vectors
• From the assumed displacement model, the stiffness matrix [K(e)] and
the load vector
• of element e are to be derived by using either equilibrium
conditions or a suitable variational principle.
Step (iv): Assemblage of element equations to obtain the overall
equilibrium equations
• Since the structure is composed of several finite elements, the
individual element stiffness matrices and load vectors are to be
assembled in a suitable manner and the overall equilibrium equations
have to be formulated as
where [K] is the assembled stiffness matrix, is the vector of nodal
displacements, and
is the vector of nodal forces for the complete structure.

Step (v): Solution for the unknown nodal displacements
• The overall equilibrium equations have to be modified to account for
the boundary conditions of the problem.
• After the incorporation of the boundary conditions, the equilibrium
equations can be expressed as
• For linear problems, the vector can be solved very easily. However,
for nonlinear problems,
• the solution has to be obtained in a sequence of steps, with each step
involving the modification of the stiffness matrix [K] and/or the load
vector .

Step (vi): Computation of element strains and stresses
• From the known nodal displacements (I), if required, the element
strains and stresses can be computed by using the necessary
equations of solid or structural mechanics.
• The terminology used in the previous six steps has to be modified if
we want to extend the concept to other fields.
• For example, we have to use the term continuum or domain in place
of structure, field variable in place of displacement, characteristic
matrix in place of stiffness matrix, and element resultants in place of
element strains.

Fundamental Concepts (1)
Many engineering phenomena can be expressed by
“governing equations” and “boundary conditions”
Elastic problems
L()  f  0Thermal problems
Fluid flow
Electrostatics
etc. B()  g  0
Boundary Conditions
Governing Equation
(Differential equation)

Example: Vertical machining center
Geometry is
very complex!Elastic deformation
Thermal behavior
etc.
A set of simultaneous
algebraic equationsFEMGoverning
Equation: L()  f  0
B()  g  0 [K]{u} {F}Boundary
Conditions: Approximate!
You know all the equations, but
you cannot solve it by hand

Fundamental
[K]{u} {F}
Concepts (3)
{u}  [K]1
{F}
Property Action
Behavior
Unknown
Property [K] Behavior {u} Action {F}
Elastic stiffness displacement force
Thermal conductivity temperature heat source
Fluid viscosity
velocity
body force
Electrostatic dialectri permittivity electric potential charge

It is very difficult to make the algebraic equations for the entire domain
Divide the domain into a number of small, simple elements
A field quantity is interpolated by a polynomial over an element
Adjacent elements share the DOF at connecting nodes
Finite element: Small piece of structure

Solve the equations, obtaining unknown variabless at nodes.
{u}  [K]1
{F}[K]{u} {F}

Concepts - Summary
- FEM uses the concept of piecewise polynomial interpolation.
- By connecting elements together, the field quantity becomes interpolated
over the entire structure in piecewise fashion.
- A set of simultaneous algebraic equations at nodes.
Kx = F
[K]{u} {F} K: Stiffness matrix
x: Displacement
F: Load
K
Property Action
Behavior
x
F

Typical FEA Procedure by
Commercial Software
User Build a FE model
Computer Conduct numerical analysis
See resultsUser Postprocess
Process
Preprocess

Preprocess (1)
[1] Select analysis type - Structural Static Analysis
- Modal Analysis
- Transient Dynamic Analysis
- Buckling Analysis
- Contact
- Steady-state Thermal Analysis
- Transient Thermal Analysis
Linear Truss2-D[2] Select element type
BeamQuadratic3-D
Shell
Plate
Solid
E, , , , "[3] Material properties

Preprocess (2)
[4] Make nodes
[5] Build elements
connectivity
by assigning
[6] Apply boundary
and loads
conditions

Process and Postprocess
[7] Process
- Solve the boundary value problem
[8] Postprocess
- See the results Displacement
Stress
Strain
Natural frequency
Temperature
Time history

Responsibility of the user
Fancy, colorful contours can200 mm
be produced by any model,
good or bad!!
1 ms pressure pulse
BC: Hinged supports
Load: Pressure pulse
Unknown: Lateral mid point
displacement in the time domain
Results obtained from ten reputable
FEM codes and by users regarded as
expert.*
* R. D. Cook, Finite Element Modeling for Stress Analysis, John
Wiley & Sons, 1995 Displacement(mm)
Time (ms)

Errors Inherent in FEM Formulation
Approximated
Domain
- Geometry is simplified. domain
FEM
- Field quantity is assumed to be a polynomial over an element. (which is not true)
True deformation
Quadratic element Cubic elementLinear element
FEM
- Use very simple integration techniques (Gauss Quadrature)
f(x)
 1     1 1
1
Area: f (x) dx  f f 3


3 
   
x
-1 1

Errors Inherent in Computing
- The computer carries only a finite number of digits.
2 1.41421356,   3.14159265e.g.)
- Numerical Difficulties
e.g.) Very large stiffness difference
k1 »k2 , k2 
O
P P
[(k1  k2 )  k2 ]u2  P u2  
k2 O

Mistakes by Users
- Elements are of the wrong type
e.g) Shell elements are used where solid elements are needed
- Distorted elements
- Supports are insufficient to prevent all rigid-body motions
- Inconsistent units (e.g. E=200 GPa, Force = 100 lbs)
- Too large stiffness differences  Numerical difficulties

Types of Finite Elements
1-D (Line) Element
(Spring, truss, beam, pipe, etc.)
2-D (Plane) Element
(Membrane, plate, shell, etc.)
3-D (Solid) Element
(3-D fields - temperature, displacement, stress, flow velocity)

Typical 3-D Solid Elements
Tetrahedron:
linear (4 nodes) quadratic (10 nodes)
Hexahedron (brick):
Penta:
Avoid using the linear (4-node) tetrahedron element in 3-D
stress analysis (Inaccurate! However, it is OK for static
deformation or vibration analysis).

Substructures (Superelements)
Substructuring is a process of analyzing a large structure as
a collection of (natural) components. The FE models for these
components are called substructures or superelements (SE).
Physical Meaning:
A finite element model of a portion of structure.
Mathematical Meaning:
Boundary matrices which are load and stiffness matrices
reduced (condensed) from the interior points to the exterior or
boundary points.

Advantages of Using Substructures/Superelements:
 Large problems (which will otherwise exceed your
computer capabilities)
Less CPU time per run once the superelements have
been processed (i.e., matrices have been saved)
Components may be modeled by different groups
Partial redesign requires only partial reanalysis (reduced
cost)
Efficient for problems with local nonlinearities (such as
confined plastic deformations) which can be placed in
one superelement (residual structure)
Exact for static stress analysis





Disadvantages:
 Increased overhead for file management
 Matrix condensation for dynamic problems introduce
new approximations
 ...

I . Spring Element
One Spring Element
x
i j
fi ui uj fjk
Two nodes:
Nodal displacements:
Nodal forces:
Spring constant (stiffness):
Spring force-displacement relationship:
i, j
ui, uj (in, m, mm)
fi, fj (lb, Newton)
k (lb/in, N/m, N/mm)
F  k with   u j  ui
Linear
NonlinearF
k

k  F /  (> 0) is the force needed to produce a unit stretch.

Consider the equilibrium of forces for the spring.
we have
At node i,
fi 
and at node j,
F  k (u j  ui )  kui  ku
j
f j  F  k (u j  ui )  kui  ku j
In matrix form,
 k ui  fik  
  

 k  u fk  j   j 
or,
ku  f
where
k = (element) stiffness matrix
u = (element nodal) displacement vector
f = (element nodal) force vector
Note that k is symmetric. Is k singular or nonsingular? That is,
can we solve the equation? If not, why?

Spring System
x
k2k1
1
u1, F1
2
u2, F2
3
u3, F3
For element 1,
k1  k1 u1  1
 f1 
  

 k  1
k u f 1 1  2   2 
element 2,
 k2 u2  2
k2  f1 
u   f 2

 k k  3   2 2 2
m
where fi is the (internal) force acting on
m (i = 1, 2).
local node i of element
Assemble the stiffness matrix for the whole system:
Consider the equilibrium of forces at node 1,
F1  f1
at node 2,
1
1 2
F2  f2 
and node 3,
F3  f2
f1
2

That is,
F1  k1u1  k1u2
F2  k1u1  (k1  k2 )u2  k2 u3
F3  k2u2  k2 u3
In matrix form,
k1  k1 0
 k2
u1 
 
 F1 
 

  k2 u2   F2    

 0  k2 k2 u3  F3

or
KU  F
K is the stiffness matrix (structure matrix) for the spring system.
An alternative way of assembling the whole stiffness matrix:
“Enlarging” the stiffness matrices for elements 1 and 2, we
have
1
k1  k1 0u1   f1 
   1


 k1

0
0

k1 0u2    f2 
   0 0u3  0 
 0
k2
0 u1  0
   2 
0
0
 k2 u2    f1 
 

2  k2 k2 u3   f2

0 
 k1 k1  k
0


Adding the two matrix equations (superposition), we have
1
k1  k1 0 u1   f1 
 

2  1
 k1 k1  k2  k2 u2    f2  f1 
 

2
 0  k2 k2 u3  f2 
This is the same equation we derived by using the force
equilibrium concept.
Boundary and load conditions:
Assuming,
we have

u1  0 and F2  F3  P
k1  k1 0  0  F1 
  
 k1 k1  k2  k2 u2    P 
  

 0
which reduces to
 k2 k2 u3   P

k1  k2  k2 u2   P
  

  k2 k2 u3  
P

and
F1 
Unknowns are
k1u2
u2 
 U  and the reaction force F (if desired).1
u3 

Solving the equations, we obtain the displacements
u2  2 P / k1 
 


u3  2 P / k1  P / k2 
and the reaction force
F1  2P
Checking the Results
 Deformed shape of the structure
 Balance of the external forces
 Order of magnitudes of the numbers

II. Bar Element
Consider a uniform prismatic bar:
ujui
fi fji j
L
A
E
u  u( x)
  (x)
  (x)
length
cross-sectional area
elastic modulus
displacement
strain
stress
Strain-displacement relation:
du
  (1)
dx
Stress-strain relation:
  E (2)
x A,E
L

Stiffness Matrix --- Direct Method
Assuming that the displacement u is varying linearly along
the axis of the bar, i.e.,
u(x)  1 
x  u 
x
u (3) 
 L i j
L
we have
u  u 
L
j i
   ( = elongation) (4)
L
E
L
  E  (5)
We also have
F
A
  (F = force in bar) (6)
Thus, (5) and (6) lead to
EA F   k (7)
L
EA
L
where k  is the stiffness of the bar.
The bar is acting like a spring in this case and we conclude
that element stiffness matrix is

EA EA 


k  k   L
EA
Lk     EA k k   
 L L 
or
1  1EA 
k  (8) 
 1 1L 
This can be verified by considering the equilibrium of the forces
at the two nodes.
Element equilibrium equation is
1  1ui  fi EA 
 (9)  


u j   f j

 1 1L
Degree of Freedom (dof)
Number of components of the displacement vector at a
node.
For 1-D bar element: one dof at each node.
Physical Meaning of the Coefficients in k
The jth column of k (here j = 1 or 2) represents the forces
applied to the bar to maintain a deformed shape with unit
displacement at node j and zero displacement at the other node.

Equation Solving
Direct Methods (Gauss Elimination):
 Solution time proportional to NB2
(N is the dimension of
the matrix, B the bandwidth)
 Suitable for small to medium problems, or slender
structures (small bandwidth)
 Easy to handle multiple load cases
Iterative Methods:
 Solution time is unknown beforehand
 Reduced storage requirement
 Suitable for large problems, or bulky structures (large
bandwidth, converge faster)
 Need solving again for different load cases

Stress Calculation
The stress in an element is determined by the following
relation,
x 
 
 x 
 
 y 
 
 E  y  EBd (39)
 
 xy   xy 
where B is the strain-nodal displacement matrix and d is the
nodal displacement vector which is known for each element
once the global FE equation has been solved.
Stresses can be evaluated at any point inside the element
(such as the center) or at the nodes. Contour plots are usually
used in FEA software packages (during post-process) for users
to visually inspect the stress results.
The von Mises Stress:
The von Mises stress is the effective or equivalent stress for
2-D and 3-D stress analysis. For a ductile material, the stress
level is considered to be safe, if
e  Y
where e
material.
is the von Mises stress and Y the yield stress of the
This is a generalization of the 1-D (experimental)
result to 2-D and 3-D situations.

The von Mises stress is defined by
1 2 2 2
e  (  )  (  )  (  ) (40)1 2 2 3 3 1
2
in which 1 ,2 and3 are the three principle stresses at the
considered point in a structure.
For 2-D problems, the two principle stresses in the plane
are determined by
2
    P x y x y 2
1    xy  



2 2

(41)
2
 P x y x y 2
2    xy  
 2 2
Thus, we can also express the von Mises stress in terms of
the stress components in the xy coordinate system. For plane
stress conditions, we have,
e  (x  y )  3(x y xy )2 2
(42)
Averaged Stresses:
Stresses are usually averaged at nodes in FEA software
packages to provide more accurate stress values. This option
should be turned off at nodes between two materials or other
geometry discontinuity locations where stress discontinuity does
exist.

FEA Stress Plot (Q8, 493 elements)

Discussions
1) Know the behaviors of each type of elements:
T3 and Q4:
T6 and Q8:
linear displacement, constant strain and stress;
quadratic displacement, linear strain and stress.
2) Choose the right type of elements for a given problem:
When in doubt, use higher order elements or a finer mesh.
3) Avoid elements with large aspect ratios and corner angles:
Aspect ratio = Lmax Lmin/
where Lmax and Lmin are the largest and smallest characteristic
lengths of an element, respectively.
Elements with Bad Shapes
Elements with Nice Shapes

4) Connect the elements properly:
Don’t leave unintended gaps or free elements in FE models.
A C
DB
Improper connections (gaps along AB and CD)

Nature of Finite Element Solutions
 FE Model – A mathematical model of the real structure,
based on many approximations.
 Real Structure -- Infinite number of nodes
(physical points or particles), thus infinite number
of DOF’s.
 FE Model – finite number of nodes, thus finite number
of DOF’s.
Ö Displacement field is controlled (or constrained) by the
values at a limited number of nodes.
Recall that on an element :
4
u   N u
 1
Stiffening Effect:
 FE Model is stiffer than the real structure.
 In general, displacement results are smaller
in magnitudes than the exact values.

Hence, FEM solution of displacement provides a lower
bound of the exact solution.
 (Displacement)
Exact Solution
FEM Solutions
No. of DOF’s
The FEM solution approaches the exact solution from
below.
This is true for displacement based FEA!

Numerical Error
Error Mistakes in FEM (modeling or solution).
Type of Errors:
 Modeling Error (beam, plate … theories)
 Discretization Error (finite, piecewise …)
 Numerical Error ( in solving FE equations)
Example (numerical error):
u1 u2
P
x2 k21 k1
FE Equations:
k  k u P  1  


1 1
  


 k k  k u  

01 1 2 2
and Det K  k1 k 2 .
The system will be singular if k2 is small compared with k1.

2 1
ku2
u12
k  k
u1
2 1
k
u2

k1
u u12
k  k1 2
u1
 Large difference in stiffness of different parts in FE
model may cause ill-conditioning in FE equations.
Hence giving results with large errors.
 Ill-conditioned system of equations can lead to large
changes in solution with small changes in input
(right hand side vector).
P/k1
k2 >> k1 (two line apart):
Ö System well conditioned.
u  u 
P
1
P/k1
k2 << k1 (two lines close):
Ö System ill-conditioned.
u  k1
1 2
u  u 
P
1

Convergence of FE Solutions
The selection of the approximation functions in each element will be made
to satisfy the necessary conditions to ensure the convergence of the
method, i.e., that the approximate solution will eventually get closer and
closer to the exact solution as we refine the mesh, i.e. as we divide the
domain into more and more elements. The two necessary conditions for a
finite element method to converge are the following:

As the mesh in an FE model is “refined” repeatedly, the FE solution will converge
to the exact solution of the mathematical model of the problem (the model based on
bar, beam, plane stress/strain, plate, shell, or 3-D elasticity theories or assumptions).
Type of Refinements:
h-refinement: reduce the size of the element (“h” refers to the
typical size of the elements);
Increase the order of the polynomials on an
element (linear to quadratic, etc.; “h” refers to
the highest order in a polynomial);
re-arrange the nodes in the mesh;
Combination of the h- and p-refinements
(better results!).
p-refinement:
r-refinement:
hp-refinement:

Adaptivity (h-, p-, and hp-Methods)
 Future of FE applications
 Automatic refinement of FE meshes until converged
results are obtained
 User’s responsibility reduced: only need to generate a
good initial mesh
Error Indicators:
Define,
 --- element by element stress field (discontinuous),
*
--- averaged or smooth stress (continuous),
*
E =  - --- the error stress field.
Compute strain energy,
M
U  U ,
1
σT
E1
σdV ;U  
Vi
i i
2i1
1M
 U *
, σ*T
E1
σ*
dV ;U *
U *
 i
2i
i1 Vi
1M
U  U σT
E1
σ, U  
Vi
dV ;E E i E i E E
2i1
where M is the total number of elements, Vi is the volume of the
element i.

One error indicator --- the relative energy error:
1/ 2
 U 
  (0   1)E
  .
U  U E 
The indicator  is computed after each FE solution.
of the FE model continues until, say
  0.05.
=> converged FE solution.
Refinement

Advantages of the FEM
Can readily handle very complex geometry:
- The heart and power of the FEM
Can handle a wide
- Solid mechanics
- Fluids
variety of engineering problems
- Dynamics - Heat problems
- Electrostatic problems
Can handle complex restraints
- Indeterminate structures can be solved.
Can handle complex loading
- Nodal load (point loads)
- Element load (pressure, thermal, inertial forces)
- Time or frequency dependent loading
15

Disadvantages of the FEM
A general closed-form solution, which would permit
to examine system response to changes in various
parameters, is not produced.
one
The FEM obtains only "approximate" solutions.
The FEM has "inherent" errors.
Mistakes by users can be fatal.
16

Applications of Finite Element Method

Examples
Crash Analysis for a Car (from LS-DYNA3D)

Modeling of gear coupling
Can Drop Test

Crack Growth Analysis
In this pressure vessel example, FEA software allows for the prediction of crack
growth along arbitrary paths that do not correspond to element boundaries.

FEM and it's applications

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