The document discusses various methods of analysis in engineering, particularly focusing on finite element methods (FEM) used for solving complex problems in fields such as metal forming and structural analysis. It outlines the historical development of FEM, its applications, and different types of solvers used, including implicit and explicit methods. Additionally, it covers the assumptions made in these analyses and the non-linearities that may be encountered during simulations.

1. Contact: abhishek.hukkerikar@rwth-aachen.de

2. WHAT IS ANALYSIS?
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"In 1 million to 10 million years they
might be able to make a plane that
would fly."
-The New York Times, 1903
Imagination
ModelApplication
200 meters384,400 kilometers

3. METHODS OF ANALYSES
Mode of
analyses
Analytical &
numerical methods
Elementary theory
Slip Line
field theory
Finite
Element
Method
(FEM)
Finite Difference
method (FDM)
Upper &
Lower Bound
method
Empirical
Methods
Similarity theory
Visio plastic
method
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Image courtesy: Lecture notes, Fundamental of solving methods,
Prof Dr. –Ing. G. Hirt

4. ELEMENTARY PLASTICITY THEORY
Approach:
Establishment of kinetics relative to
the process.
Establishment of differential
equations suiting the process &
the simplification.
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5. UPPER & LOWER BOUNDARY METHOD
Method approximates the values of deforming forces to be
higher or lower than actual forces.
“Any estimate of the collapse load of a structure made by
equating the rate of the energy dissipation internally to the
rate at which external forces do work, in some assumed
pattern of the deformation will be greater than or equal to
the correct load”
-W.F. Hosford & R.M Caddell
Assumptions of the method:
 Material being deformed is isotropic & homogenous
 There is no effect of work hardening
 No friction exists between work piece & tool interface.
 Plane strain conditions assumed.
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William Hosford

6. due to
frictionless
upsetting
due to
Symmetr
y
Symmetr
y plane
Slip
line on
the
edge
SLIP-LINE THEORY
Here flow pattern from point to point while
deformation is considered & analyzed.
Slip line refers to the planes of maximum shear
stress which are inclined at 45o to the principle
planes.
Assumptions of the method:
 Material being deformed is isotropic &
homogenous
 There is no effect of work hardening & strain rate.
 No friction exists between work piece & tool
interface.
 Plane strain conditions assumed.
 Effect of temperature , strain rate, & time
neglected.
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Image courtesy: Lecture notes, Fundamental of solving methods,
Prof Dr. –Ing. G. Hirt

7. COMPARISON
FEM
• Material flow analysis &
local states of stress &
strain described.
• Various boundary
conditions can be applied.
• Multi-axial stress in
consideration
Analytical methods
• Only Global analysis is done.
• Material homogeneity is assumed.
• 2-deminsional conditions.
• Temperature effects neglected.
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8. FINITE ELEMENT ANALYSIS
A brief history
 Concept was developed by the works of Richard
Courant & Alexander Hrennikoff (early 40’s).
 Idea was originated to solve complex problems
of civil engineering & structural analysis.
 Idea was promoted by Boeing to compute sweep
of airplane wings (mid 50’s).
 M.J Turner & Ray W. Clough articles established
the applications of FEA (mid 50’s).
 Idea was also used to compute roof of Munich
Olympic stadium (late 60’s) 8
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Richard Courant
R.W.Clough

9. AREAS OF APPLICATIONS
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Engineering
• Fluid mechanics
• Thermodynamics
• Metal Forming etc
Biological Sciences
• Botany
• Zoology
• Archeological Anthropology
• Paleontology
General application
• Geology
• Astrophysics

10. ENGINEERING APPLICATIONS
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C-blade Forging &
Manufacturing
Department of electronics &
Telecommunications, University of
Naples,Italy
Hochschule Regensburg,
Biomechanik
Numerical assessment of static & seismic behavior of
the Basilica of Santa Maria all’Impruneta (Italy)
Department of Atomic & Solid state
physics University of Cornell
Lehrstuhl Numerische Mathematik,
Ruprecht-Karls-Universität, Heidelberg

11. HIERARCHY OF FEM
Physical Problem
Establish Finite element
model of the physical problem
Solve the problem
Interpret the result 11
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12. SPACE INCREMENTATION
Finite Elements:
 Every model is sub-divided into
finite elements. Their junction
points are called as nodes.
 Model assumes that forces act at
nodes & stresses & strain exist at
the finite element.
 Reliability of FEA depends on
number of finite elements.
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13. STIFFNESS MATRIX
 
FuK
L
AE






















2
1
2
1
F
F
u
u
11
11
  
)( 121 uu
L
AE
F 







































3
2
1
3
2
1
110
121
011
F
F
F
u
u
u
L
AE
13
Example: Beams protruding from fixed surface
u
u
2
u1
F1 F2
Stiffness Coeffecient Force
)( 212 uu
L
AE
F 


Displacement
Stiffness Matrix

14. SPACE INCREMENTATION
Element types:
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Image courtesy: Lecture notes, Fundamental of solving methods,
Prof Dr. –Ing. G. Hirt

15. SPACE INCREMENTATION
Meshing
Network of nodes is called a mesh.
There are 2 broad mesh-generation
methods.
 Unstructured( Formed
automatically) A
 Structured (Formed by grid based
sub-dividing of geometry) B 15
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16. SPACE INCREMENTATION
Meshing:
 Accuracy of results always
depends on the assumptions.
 Fine mesh is considered where
there are stress & strain
gradients.
 A coarse mesh is used in the
areas of reasonably constant
stress or areas of interest.
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17. PROTOCOLS
 Gaps are not permitted
during meshing.
 Nodes are numbered
sequentially.
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18. SPACE INCREMENTATION
Approaches:
LAGRANGE’S approach
 Mesh is bound to the material
 Mesh will be distorted with
increasing deformation.
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Courtesy: FHWA. USA

19. SPACE INCREMENTATION
EULER’S Approach
 Mesh is fixed & not bound
to the material.
 Material flows through
fixed mesh.
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20. SPACE INCREMENTATION
Remeshing
Why is it necessary?
 Formation of unacceptable shapes due to large local deformations.
 High relative motion between die surface & deforming material.
 Large displacement causes computational problems.
 Difficulties encountered in incorporating die boundary shapes with
increase in relative displacement.
To overcome above difficulties, periodic redefining of mesh is necessary 21
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21. SPACE INCREMENTATION
Remeshing comprises of following
steps:
1. Assignment of new mesh
system to work piece
2. Transfer of information (strain,
strain rate, & temperature)
from the old to the new mesh
through interpolation.
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Image courtesy: emerald.com

22. SOLVERS
For simulation of metal forming, following 2 solutions
are used:
 Implicit method ( Stable, iterative, high
computational effort)
 Explicit method (conditionally stable, no iteration,
less computational effort)
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23. IMPLICIT SOLVERS
 Studies reveal that this solver is
useful in smaller & 2D problems.
 Each time step or increment has
to be treated as unconditionally
stable process.
 Large time steps lead to larger
iterations & process do not
converge.
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(Newton Raphson method )Non-
linear analysis of reinforced
concrete beam

24. IMPLICIT SOLVERS
In the implicit approach a solution to the set of finite
element equations involves iteration until a convergence
criterion is satisfied for each increment.
 Here computation is divided into several calculation time
steps.
 At the end of each time step(increment) the equilibrium
between internal & external load must be reached.
 Else iteration continues.
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25. EXPLICIT SOLVERS
 The finite element equations in the explicit approach are
reformulated as being dynamic.
 In this form they can be solved directly to determine the
solution at the end of the increment, without iteration.
 Two methods are followed for time step calculations.
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26. EXPLICIT SOLVERS
Here largest allowable time step for a stable solution depends on:
 Highest Eigen frequency occurring (ωmax )in the system
 Corresponding damping (ξ)
∆tm ≤ (2/ωmax)* ((1+ξ2)0.5-ξ)
Sonic frequency & smallest element Le are estimated as follows:
∆ t ≤Le /C with C=(E/ρ)0.5
To compensate the disadvantage of extremely small time step, will
be reduced through increasing the density or shortening the process
time.
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27. COMPUTATIONAL TIME REQUIRED FOR
EXPLICIT/IMPLICIT METHODS
implicit: Complexity ~ number of freedom degrees x wave front
explicit: Complexity ~ number of freedom degrees
Complexity
Implicit
Explicit
Model-size
(Calculation time)
Efficiency
Statics Structural
dynamics
Highly
dynamic
Implicit Explicit
Image courtesy: Lecture notes, Fundamental of solving methods,
Prof Dr. –Ing. G. Hirt

28. NON-LINEARITIES IN FEM
Following Non-linearities are encountered during the
simulations.
 Geometrical Non-linearity
 Material Non-linearity
 Contact variance (Change of boundary conditions)
 Friction
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29. GEOMETRICAL NON-LINEARITY
In practical cases it is not
uncommon to encounter
strain of magnitude 2 or more
due to :
 Large elongation
 Large rotation
 Portions of rigid body
movements 30
Abhishek&Jitendra
Courtesy: MRF tyres, India

30. GEOMETRICAL NON LINEARITY
Hydroforming
Operation Tools
Upper part
Lower partTube
In consideration of
geometrical nonlinearity
Geometrical nonlinearity
neglected
Image courtesy: Lecture notes, Fundamental of solving methods,
Prof Dr. –Ing. G. Hirt

31. MATERIAL NON-LINEARITY
Occurs when:
 Transition of elastic to plastic
phase
 Depends on ρ,θ, λ, CP
Note: This non-linearity is
important when considering
thermal effects z.B hot forming or
for calculation of temperature
increase during forming process.
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Courtesy: COMSOL, USA

32. Material-nonlinearity (flow curve)
MATERIAL NON-LINEARITY DURING TENSILE TEST
Image courtesy: Lecture notes, Fundamental of solving methods,
Prof Dr. –Ing. G. Hirt
kf = kf (v, v)
(strain hardened!)

considered Not considered
kf = 100 N/mm2 = const.Initial mesh

33. CONTACT NON-LINEARITY
Changing contact changes:
a.) Mechanical Boundaries
b.) Thermal Boundaries.
Types of contacts in metal forming
1.) Contacts with rigid tools
2.) Contacts with deforming tools
3.) Self contact
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Courtesy: ICS, Switzerland

34. FRICTION
Friction is non-linear. Friction leads to asymmetrical
equation system. This increases the calculation
complexity.
Categorization:
1. τ<µσN -- Sticking friction
2. τ=µσN – Slide friction
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35. WHY IS FEM ADVANTAGEOUS OVER OTHER
SOLVING METHODS
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without friction
with friction

36. Upsetting
without friction with friction
Comparision with & without
friction during upsetting
Image courtesy: Lecture notes, Fundamental of solving methods,
Prof Dr. –Ing. G. Hirt

37. VOTE OF THANKS & REFERENCES
Sincerely indebted to:
Prof Dr. –Ing. G. Hirt, Head of the department, IBF, RWTH Aachen
Dipl.-Ing. Simon Seuren, IBF, RWTH Aachen
Institute of Metal Forming, RWTH Aahcen
References:
 Fundamentals of solving methods in metal forming by Prof.Dr.- Ing.G.Hirt
 Metal forming & finite element method –Atlan,Oh, Kobayashi
 Manufacturing process III –A.C.Niranjan
 Comparison of the implicit and explicit finite element methods
using crystal plasticity- F.J. Harewood , P.E. McHugh
Web resources:
 National program on technology enhanced learning, Dr. R. Krishnakumar, IIT madras.
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38. THANK YOU FOR YOUR PATIENCE & KIND ATTENTION!
FOR FURTHER DETAILS, QUERIES, AND SUGGESTIONS,
CONTACT US ON:
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Abhishek&Jitendra
abhishek.hukkerikar@rwth-aachen.de