The study focuses on investigation of mesh convergence test for two-dimensional disk rim of high-pressure turbine using energy norm as an alternate approach. Numerical methods solve the real-time complex problems, represented by partial differential equations using discretization of time and space variables. The criterion for finite element method to obtain a converged solution to the problem and arrive at accurate results from simulations is optimized input data. However, all discrete methods introduce discretization error into the solution. The increase in the number of nodes, i.e mesh density reduces the error induced leading to the converged solution. The mesh density is increased in iterations, the trend related to convergence of solution, and error induced are observed. The study used CATIA® and ANSYS® for modelling and analysis, and the results obtained showed the effective use of energy norm technique being used as criterion for evaluation and testing to obtain a converged solution. The normalized error percentage was observed to be in the range of 6 to 8% for converged solution.

1. http://www.iaeme.com/IJMET/index.asp 765 editor@iaeme.com
International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 01, January 2019, pp. 765-775, Article ID: IJMET_10_01_078
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=1
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication Scopus Indexed
MESH CONVERGENCE TEST FOR FINITE
ELEMENT METHOD ON HIGH PRESSURE GAS
TURBINE DISK RIM USING ENERGY NORM:
AN ALTERNATE APPROACH
Nithesh Naik, Prajwal Shenoy*, Nithin Nayak, Swetank Awasthi and Rashmi Samant
Department of Mechanical and Manufacturing Engineering,
Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, Udupi,
Karnataka, India - 576104
Department of Mechatronics Engineering,
Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, Udupi,
Karnataka, India - 576104
ABSTRACT
The study focuses on investigation of mesh convergence test for two-dimensional
disk rim of high-pressure turbine using energy norm as an alternate approach.
Numerical methods solve the real-time complex problems, represented by partial
differential equations using discretization of time and space variables. The criterion for
finite element method to obtain a converged solution to the problem and arrive at
accurate results from simulations is optimized input data. However, all discrete methods
introduce discretization error into the solution. The increase in the number of nodes, i.e
mesh density reduces the error induced leading to the converged solution. The mesh
density is increased in iterations, the trend related to convergence of solution, and error
induced are observed. The study used CATIA® and ANSYS® for modelling and
analysis, and the results obtained showed the effective use of energy norm technique
being used as criterion for evaluation and testing to obtain a converged solution. The
normalized error percentage was observed to be in the range of 6 to 8% for converged
solution.
Keywords: Gas Turbine, Aircraft, Mesh convergence, Mesh discretization, Energy
norm, FEM.
Cite this Article: Nithesh Naik, Prajwal Shenoy*, Nithin Nayak, Swetank Awasthi and
Rashmi Samant, Mesh Convergence Test for Finite Element Method on High Pressure
Gas Turbine Disk Rim Using Energy Norm: An Alternate Approach, International
Journal of Mechanical Engineering and Technology, 10(01), 2019, pp. 765-775.
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=1
1. INTRODUCTION

2. Nithesh Naik, Prajwal Shenoy*, Nithin Nayak, Swetank Awasthi and Rashmi Samant
http://www.iaeme.com/IJMET/index.asp 766 editor@iaeme.com
A gas turbine is an internal combustion engine which is mainly used in commercial aircrafts.
The high-pressure compressor and high-pressure turbine assembly plays a key role in the entire
process and the stress analysis of these rotating parts is of prime importance as failure of these
components will result in catastrophic failures [1-3].
Structural stress analysis using finite element methods has been extensively carried out in
domains like aerospace for solving complex engineering problems. The success of finite
element method lies in how well the finite element model simulates the actual physical system.
Since finite element model approximates the actual physical system which is a continuum,
errors are inherent in the FE solution [4-8]. Validity of the FE solution having a discrete finite
element mesh is checked by performing mesh convergence study until the solution converges
to a stable value with an acceptable level of error. The mesh convergence test is very much
significant in analysis of aircraft components, which involves complicated geometry, and non-
linearity for which theoretical solution is not readily available and computational time and cost
is of high priority [9-10].
The study aims to develop an efficient process using ANSYS® tools which will help us to
test the convergence of the FE solution using energy error norm and establish a criterion in
terms of percentage error in energy to test the convergence of the solution. The mesh
convergence test conducted on feature and loading conditions encountered such as load slot and
lock slot in a typical gas turbine engine of commercial aircraft engine.
2. METHODOLOGY
2.1. Modelling
The high-pressure turbine (HPT) module is an integral part of an aircraft engine, which uses
the energy from the combustion gases to run the high-pressure compressor and accessory
gearbox. The HPT module consists of HPT rotor and HPT stator [11]. The HPT rotor usually
has one or two stages. The turbine blades are mounted along the disk in the axial direction on
the disk periphery. The dovetail portion consists of two or three tangs wherein the blade
platform will meet the disk. The turbine disk of dovetail shape as shown in Figure 1 is mainly
subjected to centrifugal loads and loads transferred through blades, which results in peaking of
stress at the min neck region.
Figure 1 Turbine Disk rim half sector model geometry

3. Mesh Convergence Test for Finite Element Method on High Pressure Gas Turbine Disk Rim
Using Energy Norm: An Alternate Approach
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2.2. Boundary Conditions
The model developed using CATIA® with nominal dimensions and reflective symmetry half
sector model is considered. The material properties considered for analysis having Young’s
Modulus – 3e7 Psi, Poisson Ratio – 0.3, Density – 7e-4lb/inch3
The element type selected is 8-node SOLID 273. Since the model is reflective symmetric,
the displacements along θ direction are constrained by rotating the nodes of symmetry planes
in cylindrical coordinate system. The speed of 1500 rad/sec about the engine axis and a constant
pressure of 50000 units on the pressure face to simulate blade loads. The axial displacement is
constrained at the bore to constrain axial movement as shown in Figure 2.
Figure 2 FE model with applied load and boundary conditions
2.3. Mesh Refinement
The critical regions of the turbine disk rim, minimum neck radius and slot bottom region are
the peak stress concentration regions identified for mesh convergence study [12], as shown in
Figure 3. The model is meshed by controlling the element size in the minimum neck, slot bottom
region. The iterations carried out by varying the mesh density in the region of interest in
successive iterations, as shown in Table 1.
Figure 3 FE model indicating critical locations identified for mesh convergence study

4. Nithesh Naik, Prajwal Shenoy*, Nithin Nayak, Swetank Awasthi and Rashmi Samant
http://www.iaeme.com/IJMET/index.asp 768 editor@iaeme.com
Table 1 Elements at critical locations in turbine disk rim
Number of Elements
Iteration Along Width
Minimum
Neck 1
Minimum
Neck 2
Minimum
Neck 3
Slot
Bottom
Along the
Depth
Width (W) Along the Edge (E) Depth (D)
1 30 3 3 4 6 2
2 40 6 6 8 9 3
3 60 12 12 12 12 4
4 80 18 18 16 20 5
5 100 24 24 24 30 6
Each critical location analyzed for five iterations is shown in Figure 4. The peak stress
values and percentage error in energy for all the iterations and the stress values at which the
solution converges graphically is also established.
Figure 4 Five Iterations of elements at critical locations in turbine disk rim
3. RESULTS AND DISCUSSIONS
The variation of stresses and error along the region of interest wherein stresses generally peak
are also studied. The nodal and surface stress gradients for the peak stress element along with
the corresponding error in energy are noted for all the iterations. Finally, the graphical
convergence is established by plotting the peak stress values as a function of element size and
the normalized percentage error in energy corresponding to the converged peak stress element
is also highlighted.

5. Mesh Convergence Test for Finite Element Method on High Pressure Gas Turbine Disk Rim
Using Energy Norm: An Alternate Approach
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Figure 5 Stress plots indicating the location of peak stress for entire disk rim for iteration 4
The peak stress location along with the peak nodal and surface stress value is shown in
Figure 5. The convergence was observed after 4 iterations and the surface stress was found to
be 205818 units for the converged solution. The corresponding normalized percentage error in
energy was found to be 8.6% as shown in Figure 6.
Figure 6 Normalized error plot for turbine disk rim iteration 4

6. Nithesh Naik, Prajwal Shenoy*, Nithin Nayak, Swetank Awasthi and Rashmi Samant
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Figure 7 Stress plots indicating the location of peak equivalent stress and stress value for each critical
location for the converged solution (Iteration 4)
The peak equivalent nodal and surface stress for each critical location along with the
corresponding normalized percentage error in energy is obtained for all the iterations to observe
the variation of percentage normalized error in energy with respect to stress values for each of
the critical feature. Peak surface equivalent stress plot for the converged solution in case of each
critical location is shown in Figure 7.
3.1. Variation of Surface Stress and Normalized Percentage Energy Error
Convergence was observed after 4 iterations as seen in Figure 8 and Figure 9 and stress
gradients in the peak stress region was around 5000 units and the stress gradients were quite
high in the slope region. The peak equivalent stress values, stress gradients and percentage
normalized error in energy for peak stress elements is tabulated for min neck-1, min neck-2,
min neck-3 and slot bottom region for all the iterations in Table 2, 3, 4 and 5 respectively.

7. Mesh Convergence Test for Finite Element Method on High Pressure Gas Turbine Disk Rim
Using Energy Norm: An Alternate Approach
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Figure 8 Variation of surface equivalent stress along min neck and slot bottom region
Figure 9 Variation of normalized percentage energy error along the min neck and slot bottom region
Table 2 Stress and error values for min neck- 1 region for all iterations
Element size
(mil)
Gradient (103
)
Peak equivalent
stress (103
)
Percentage
error in energy
Iteration WxExD N-N N-S S-S Surface Nodal Normalized
1 53x42x15 30.8 16.5 36.5 78.8 65.6 17%
2 40x22x10 22.0 20.4 35.0 89.1 84.5 20%
3 27x11x7 13.0 5.7 9.9 91.8 89.1 12%
4 20x7x6 13.4 3.3 9.4 91.7 89.9 9%
5 16x5x5 1.9 3.3 3.0 91.8 91.0 7%

8. Nithesh Naik, Prajwal Shenoy*, Nithin Nayak, Swetank Awasthi and Rashmi Samant
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Table 3 Stress and error values for min neck- 2 region for all iterations
Element size
(mil)
Gradient (103
)
Peak equivalent
stress (103
)
Percentage
error in energy
Iteration WxExD N-N N-S S-S Surface Nodal Normalized
1 53x42x15 33.9 21.1 42.0 95.8 77.9 15%
2 40x22x10 25.3 19.3 24.6 106.6 96.9 18%
3 27x11x7 5.7 6.1 6.4 108.3 103.9 11%
4 20x7x6 3.0 4.4 3.8 108.1 106.3 8%
5 16x5x5 2.7 2.6 1.6 108.5 106.6 6%
Table 4 Stress and error values for min neck- 3 region for all iterations
Element size
(mil)
Gradient (103
)
Peak equivalent
stress (103
)
Percentage
error in energy
Iteration WxExD N-N N-S S-S Surface Nodal Normalized
1 53x42x15 21.0 43.8 48.6 192.3 169.6 21%
2 40x22x10 11.2 23.9 29.2 199.4 186.8 15%
3 27x11x7 13.6 12.5 9.9 205.9 193.9 10%
4 20x7x6 4.6 10.6 11.6 205.8 198.5 8%
5 16x5x5 2.7 6.9 3.6 207.0 201.2 6%
Table 5 Stress and error values for slot bottom for all iterations
Element size
(mil)
Gradient (103
)
Peak equivalent
stress (103
)
Percentage
error in energy
Iteration WxExD N-N N-S S-S Surface Nodal Normalized
1 53x42x15 7.9 6.8 8.2 176.9 171.4 12%
2 40x22x10 4.8 3.6 3.1 176.6 174.6 9%
3 27x11x7 2.1 2.5 1.8 176.4 175.1 7%
4 20x7x6 0.8 1.3 0.6 176.6 175.6 4%
5 16x5x5 0.2 1.0 0.4 176.8 175.9 3%

9. Mesh Convergence Test for Finite Element Method on High Pressure Gas Turbine Disk Rim
Using Energy Norm: An Alternate Approach
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3.2. Mesh Convergence
The asymptotic stress convergence of nodal and surface equivalent stress along with the
variation of normalized percentage energy error for each critical location is shown in the Figure
10, 11 and 12. The results indicate that the region of peak stress concentration was in minimum
neck 3 region of the disk rim. Accordingly, the rate of convergence of the surface stress was
faster than the nodal stress.
Figure 10 Mesh convergence plot showing asymptotic stress convergence along with variation of
percentage energy error for min neck-1 region
Figure 11 Mesh convergence plot showing asymptotic stress convergence along with variation of
percentage energy error for min neck-2 region

10. Nithesh Naik, Prajwal Shenoy*, Nithin Nayak, Swetank Awasthi and Rashmi Samant
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Figure 12 Mesh convergence plot showing asymptotic stress convergence along with variation of
percentage energy error for min neck-3 region
Normalized percentage error in energy varied in accordance to the stress values i.e. the
maximum value in the element having peak stress and the value decreased with decrease in the
stress values. However, case the normalized error in energy in the region having high stress
gradients is well below that of element having peak stress.
4. CONCLUSIONS
The study shows that the peak stress value increases with the decrease in min neck radius due
to stress concentration. It is observed that the surface stress converges at a faster rate than the
nodal stress. Also, the stress value converges at a slower rate with decrease in min neck radius.
The normalized error in energy in the region having high stress gradients is well below that of
element having peak stress. The solution converged after 4 iterations the corresponding
normalized percentage error in energy for the element having peak stress was in range of 6% to
8%. Thus, the percentage normalized error in energy was found to independent of magnitude
of peak stress value. Based on the results obtained we can conclude that the energy error norm
can be implemented as an alternate criterion to test the convergence of the FE solution.
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11. Mesh Convergence Test for Finite Element Method on High Pressure Gas Turbine Disk Rim
Using Energy Norm: An Alternate Approach
http://www.iaeme.com/IJMET/index.asp 775 editor@iaeme.com
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