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.

Introduction: Nowadays,

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.

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%.