# Abstract: step in any simulation. In practice of

Abstract:
Structural analysis is an essential tool for
design engineers. Mesh generation is the basic step in any simulation. In
practice of finite-element stress analysis, the engineer first needs to know if
key stresses are converging, and second if they have converged to a reasonable
level of accuracy. In order to achieve results that are reliable when using the
finite element method one has to use an acceptable mesh with respect to the
shape and size of the elements. Mesh quality and Mesh density are linked with
the solution accuracy. This paper presents a mesh convergence study on the hub
which is most challenging part in the automotive clutch disc assembly.

Finite Element Analysis (FEA) is widely used in analysis since it has many
advantages compared to analytical method. Also Finite Element Analysis (FEA)
problems like that cannot solve by analytical methods. FEA is used for
reduction of the cost as well as faster designing of any component. It gives
the two-way solution “if any change is design is made then what?” Finite
Element Analysis (FEA) is numerical technique to find the approximate solution
of Partial Differential Equations (PDE) due to impact of Force, Vibration and
Heat etc.

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Finite Element Analysis (FEA) is extensively used in many
departments one of which is automobile industry. In automobile industry mainly
Finite Element Analysis (FEA) is used in Power Transmission systems, the main
objective of Power Transmission system is to transfer efficient amount of power
to the wheels from engine, thus researchers are concentrating on optimal design
of the Power Transmission system. Hub is used as Power Transmitting part
present in the clutch disc assembly. Frictional pads which actually transmits
the Power from the flywheel to hub in the clutch plate and from there to output
shaft. Hub also acts as a torque absorption unit, it absorbs the torque at time
of engagement and disengagement of clutch disc with the help of springs located
in drive plate.

The aim of this study is to perform a Mesh Convergence
study on hub. This paper tries to achieve acceptance of the specific solution
by considering the Mesh Discretization error. “If I set everything right, load,
Boundary conditions, material properties, element size then how do I know that
I arrived with fine mesh to capture the results?”

Mesh Discretization Error: Displacement is the unknown variable in any Finite Element
Analysis (FEA), displacement is calculated at every node in the model. Every
element in the Finite Element Analysis (FEA) has its own shape function
associated with the same. The stress are obtained by first derivative of
displacement field, that can be obtained from equation 1,

(1)

The stress values from the ANSYS are average of stresses
from all elements attached to that node. This introduces the error in magnitude
of the stress value which is referred as the mesh discretization error.
Phenomenon of mesh discretization error is shown in Fig 1. As mesh is coarse
difference is more between stress values of the adjacent elements.

?x

?x, ?y, ?xy = 171.25, 81.25, 25

Fig
1 Mesh Discretization Error

Nodal stress and Elemental stress value become close if the
mesh density increased as well as the solution accuracy increases but at the
same time required for the simulation also increases, but at certain point
there is no change in stress value even after the refinement in mesh. For this
reason analyst should required to check the discretization errors in FE
solution.

ANSYS Error Estimation:

1)
Percentage Error
Energy Norm (SEPC): SEPC is rough estimate of
the stress error over the entire set of selected elements. (Obtained from PRERR
command in ANSYS APDL)

2)
Element Stress
Deviations (SDSG): SDSG is measure amount
between elemental and nodal stress. The difference between averaged and
unaveraged stress gives an idea about mesh density. (Obtained from PRESOL and
PRNSOL command in ANSYS APDL)

3)
Element Energy Error
(SERR): SERR is the energy associated with the
stress mismatches all the nodes of the element. (Obtained from PRESOL and
PRNSOL command in ANSYS APDL)

4)
Maximum and Minimum
Stress Bounds (SMXB, SMNB): The stress bounds
are used to determine the effect of mesh discretization error on the maximum
stress. (Obtained from stress plots PLNSOL command in ANSYS APDL)

FE solution will converge towards the exact solution with
increasing number of elements or with increasing order of polynomial in the
element. If the following requirements are fulfilled, then the solution is
converged.

1.
The element must be
able to represent constant strain which is possible where small elements are
used. Small elements have strains are close to constant over the element.

2.
In order to avoid the
gaps, adjutant elements must be compatible i.e. the connecting nodes have the
same possibility to move as concerned element nodes.

Finite Element Accuracy Criteria:

Four types used as a study of mesh discretization error
analysis, this calculated Global as well as Local mesh discretization errors.

1)
Type 1A (SEPC_Model):
The error norm of the entire model must be less than 15%.

2)
Type 1B (SEPC_Local):
3 layers of elements are selected as local area at high stress, the error norm
of the local area of high stress must be less than 10%.

3)
Type 2 (Coefficient of variation): In local area of high stress, the average stress value of
local elements have coefficient of variation stress less than 7%. COV (Coefficient
of variation) is calculated as ratio of nodal stress to the mean stress value.

4)
Type 3 (%Error): The
difference between the stress component and stress considering the
discretization in local area of high stress must be less than 7%.

(2)

(Calculated by
ANSYS APDL)

SMX = max stress value in
selected set of element.

5)
Type 4A (RMS_Model):
This type uses the absolute value of SDSG (maximum variation of nodal
component) to Seqv (Von-mises stress) at that element and RMS(Root mean
square) is calculated from that values and it should be less than 15%.

6)
Type 4B (RMS_Local): (SDSG/
Seqv) RMS value of local area having high stress should be less
than 10%.

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