A STUDY ON ENERGY NORM AS AN ALTERNATE APPROACH TO MESH CONVERGENCE ON CIRCUMFERENTIAL DISK DOVETAIL OF AIRCRAFT ENGINE COMPRESSOR

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1 International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 13, December 2018, pp , Article ID: IJMET_09_13_131 Available online at ISSN Print: and ISSN Online: IAEME Publication Scopus Indexed A STUDY ON ENERGY NORM AS AN ALTERNATE APPROACH TO MESH CONVERGENCE ON CIRCUMFERENTIAL DISK DOVETAIL OF AIRCRAFT ENGINE COMPRESSOR Nithesh Naik Department of Mechanical and Manufacturing Engineering Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, Udupi Ritesh Bhat Department of Mechanical and Manufacturing Engineering Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, Udupi Prajwal Shenoy* Department of Mechatronics Engineering Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, Udupi George Varghese Department of Mechanical and Manufacturing Engineering Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, Udupi Suhas Kowshik CS Department of Mechanical and Manufacturing Engineering Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, Udupi, Karnataka, India * prajwal.shenoy@manipal.edu ABSTRACT Finite element methods solve complicated computational problems, which requires optimized input data to obtain a converged solution to the problem and arrive at accurate results from simulations. However, the calculations obtained by interpolations of derivatives at the nodes result in mesh discretization error. It is observed that the increase in the mesh density the reduces the error, Hence, in the current study, finite element analysis of the complicated models related to gas turbines is performed with the mesh density being increased iteratively, to observe the trend of convergence of the solution along with the error associated. In the current study, dovetail shaped axisymmetric model present in aircraft engine compressor disks for editor@iaeme.com

2 Nithesh Naik, Ritesh Bhat, Prajwal Shenoy, George Varghese, Suhas Kowshik CS mounting the compressor blades along the disk periphery is modelled using software CATIA and analysis is carried out using ANSYS. The results showed the response of peak stress values in correspondence to energy error showing that the energy norm error can be used as criteria for testing the convergence of the solution. The study shows that normalized percentage error in energy is in the range of 6% to 8% for the converged solution. Key words: High-pressure turbine, Compressor, Mesh convergence, Energy norm, FEM. Cite this Article: Nithesh Naik, Ritesh Bhat, Prajwal Shenoy, George Varghese, Suhas Kowshik CS, A Study on Energy Norm as an Alternate Approach to Mesh Convergence on Circumferential Disk Dovetail of Aircraft Engine Compressor, International Journal of Mechanical Engineering and Technology 9(13), 2018, pp INTRODUCTION 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 [9-11]. 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 [1-3]. Since finite element model is an approximation of the actual physical system which is a continuum, errors are inherent in the FE solution. 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 [4-7]. The aircraft components having complicated geometry and involves nonlinear computations needs higher computational time and cost is of higher priority. Thus, FEM based analysis involving such computations require mesh convergence test for the analysis to obtain converged and accurate solution The study aims to develop an efficient process using ANSYS tools, 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 [8]. The mesh convergence test was conducted on feature and loading conditions encountered such as load slot and lock slot present in circumferential disk dovetail in a typical gas turbine engine of commercial aircraft engine. 2. METHODOLOGY 2.1. Modelling Circumferential dovetail is a feature present in aircraft engine compressor disks for mounting the compressor blades along the disk periphery. Dovetail region is subjected to different types of loads such as centrifugal loading, temperature and gas loads. Figure 1. shows the 2D axisymmetric model with reflective symmetry is created which captures the entire 3D dovetail geometry, as the stresses are constant in hoop direction editor@iaeme.com

3 A Study on Energy Norm as an Alternate Approach to Mesh Convergence on Circumferential Disk Dovetail of Aircraft Engine Compressor Figure 1. 2D disk dovetail model geometry Disk dovetail geometry is generated for 4 different minimum neck radius in mil (where 1 mil = one thousandth of an inch) i.e. 150 mil, 100 mil, 60 mil and 30 mil respectively. The material properties of the component is: Young s Modulus 3e7 psi, Poisson s ratio 0.3, Density 7e-4 lb. /in 3 [6] The geometry was modelled using CATIA modelling software. The analysis is carried by importing the geometry to ANSYS. Nodes on the symmetry plane are constrained in Y direction as shown in figure since the model is having reflective symmetry. Uniform pressure of units is applied on the pressure face to simulate loads due to blade pressure. Angular velocity of 2000 rad/sec is applied about the engine axis as shown in Figure 2. Figure 2. FE model with applied load and boundary conditions with zoomed in view of min neck region 2.2. Mesh Refinement The stresses are observed to be at peak at minimum neck region of the circumferential dovetail and therefore this region is meshed using mapped mesh with successive mesh refinement to capture the stresses accurately [5-6]. Multiple iterations are carried out for each min neck radius by gradually decreasing the element size along the radius as shown in Figure 3. Element type used is 2D PLANE 42 with axisymmetric option [7] editor@iaeme.com

4 Nithesh Naik, Ritesh Bhat, Prajwal Shenoy, George Varghese, Suhas Kowshik CS Figure 3 Mesh refinement with each iteration for min neck radius of 150 mil 3. RESULTS AND DISCUSSIONS The variation of stresses and error along the region of interest wherein stresses (Figure 4) generally peak are shown. 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. Figure 4. Stress plots indicating the location of peak stress for min neck radius of 150mil with 50x4 elements at the min neck region (Iteration 5) The region of peak stress is as shown in Figure 5. Similar plots were obtained for disk dovetail models with min neck radius of 100mil, 60mil and 30mil. The variation of surface stress and normalized percentage energy error along the min neck radius is shown in Figure editor@iaeme.com

5 A Study on Energy Norm as an Alternate Approach to Mesh Convergence on Circumferential Disk Dovetail of Aircraft Engine Compressor Figure 5. Normalized Error plot for disk dovetail with min neck radius of 150 mil with 50x4 elements at the min neck region Figure 6. Variation of surface equivalent stress along min neck radius 1(50 mil) load slot for all iterations Figure 7. Variation of normalized percentage energy error along min neck radius 1(50 mil) load slot for all iterations

6 Nithesh Naik, Ritesh Bhat, Prajwal Shenoy, George Varghese, Suhas Kowshik CS It can be observed from Figure 7 that the surface stresses converge after 4 iterations corresponding to stress value of units for min neck radius of 150mil. As observed in case of load slot model we find high stress gradients in the slope region for the converged solution. Similar charts were obtained for 100, 60 and 30 mil min neck radius. We observe that the normalized percentage error in energy decreases with each iteration and is found to be around 6% for the converged iteration (25x3 elements).similar charts were obtained in case of disk dovetail model with 100, 60 and 30 mil min neck radius. The Table 1 shows the stress and error values for all iterations for 150mil and 100mil min neck radius with the varied number of elements. Similarly, the Stress and error values for all iterations for 60mil and 30mil min neck radius are listed in Table 2. No. of Elements Table 1 Stress and error values for all iterations for 150mil and 100mil min neck radius 150 mil min neck radius 100 mil min neck radius Surface Nodal % normalized N_S Surface Nodal % normalized stress stress error gradient stress stress error 3x % % x % % x % % x % % x % % 5.9 Table 2 Stress and error values for all iterations for 60mil and 30mil min neck radius N_S gradient 60 mil min neck radius 30 mil min neck radius No. of Elements Surface stress Nodal stress % normalized error N_S gradient Surface stress Nodal stress % normalized error N_S gradient 3x % % x % % x % % x % % Mesh convergence The asymptotic convergence of nodal and surface equivalent stress along with the variation of normalized percentage energy error for each slot radius is shown in the figure 8, 9, 10 and 11. The plot of mesh convergence test showing variation of percentage energy error for stress convergence of 150, 100, 60 and 30 mil min neck radius are observed. Figure 8 Plot of mesh convergence test showing variation of percentage energy error for stress convergence of 150 mil min neck radius editor@iaeme.com

7 A Study on Energy Norm as an Alternate Approach to Mesh Convergence on Circumferential Disk Dovetail of Aircraft Engine Compressor Figure 9 Plot of mesh convergence test showing variation of percentage energy error for stress convergence of 100 mil min neck radius Figure 10 Plot of mesh convergence test showing variation of percentage energy error for stress convergence of 60 mil min neck radius Figure 11 Plot of mesh convergence test showing variation of percentage energy error for stress convergence of 30 mil min neck radius As observed, the peak stress value increases with the decrease in min neck radius due to stress concentration. The surface stress converges at a faster rate than the nodal stress. The stress value converges at a slower rate with decrease in min neck radius editor@iaeme.com

8 Nithesh Naik, Ritesh Bhat, Prajwal Shenoy, George Varghese, Suhas Kowshik CS Figure 12 Normalized percentage energy error overlay plots for all min neck radii with increasing mesh density The normalized error in energy in the region having high stress gradients is well below that of element having peak stress. The normalized percentage plots as shown in Figure 12 confers that the solution converged after 4 iterations in all the cases and as observed in the previous case the corresponding normalized percentage error in energy for the element having peak stress was found to be around 6% to 8%. The percentage-normalized error in energy is independent of magnitude of peak stress value. 4. CONCLUSIONS The study shows that the peak stress value increases with the decrease in min neck radius due to stress concentration. The rate of convergence of the surface stress is faster 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 is independent of magnitude of peak stress value. The results depicts and concludes that the suitability of energy norm as an alternate technique in FE based analysis and utilization as a criterion to test the mesh convergence. REFERENCES [1] Ainsworth, M. and Oden, J.T. A posteriori error estimation in finite element analysis. Computer Methods in Applied Mechanics and Engineering. 1997; Vol.142, pp.1-88, Elsevier. [2] Prudhomme, S, Oden.J.T, Westermann.T, Bass.J and Botkin.M.E. Practical methods for a posteriori error estimation in engineering applications. International Journal for Numerical Methods in Engineering. 2003; No.56, pp [3] Shaofan Li, Xiaohu Liu and Anurag Gupta. Smart element method I. The Zienkiewicz- Zhu feedback. International Journal for Numerical Methods in Engineering. 2005; Vol.62: [4] J.Z.Zhu, O.C.Zienkiewicz. Adaptive Techniques in the finite element method- Communications in Applied Numerical Methods. 1988; Vol. 4, pp editor@iaeme.com

9 A Study on Energy Norm as an Alternate Approach to Mesh Convergence on Circumferential Disk Dovetail of Aircraft Engine Compressor [5] Antonio Huerta, Antonio Rodriguez-Ferran and Pedro Diez. Error Estimation and adaptively for nonlinear FE analysis, International Journal for Applied Mathematics and Computer Science. 2002; Vol. 12, No. 1, pp [6] Hyung-Seok Oh and R.C.Batra. Application of Zienkiewicz-Zhu s error estimate with super convergent patch recovery to hierarchical p-refinement. Finite Elements in Analysis and Design. 1998; Vol. 31, pp , Elsevier. [7] Pointer J. Understanding Accuracy and Discretization Error in an FEA Model. ANSYS 7.1 Conference, Woodward Governor Company [8] Shah, Chandresh. Mesh Discretization Error and Criteria for accuracy of Finite Element Solutions. ANSYS Users Conference, Pittsburgh, PA, [9] Thamir K. Ibrahim, M.M. Rahman, M.K. Mohammed and Firdaus Basrawi. Statistical analysis and optimum performance of the gas turbine power plant. International Journal of Automotive and Mechanical Engineering. 2016; 13: [10] Ibrahim TK, Rahman M. Effects of isentropic efficiencies on the performance of combined cycle power plants. International Journal of Automotive & Mechanical Engineering. 2015; 12: [11] Ibrahim TK, Rahman M. Effect of the isentropic efficiency and enhancing strategies on the performance of gas turbine. Journal of Mechanical Engineering and Sciences. 2013; 4: [12] Naik Nithesh, George Varghese, Suhas Kowshik CS. Investigations on Mesh Discretization Error in Fem Based Structural Analysis using Ansys. International Journal of Advanced Computational Engineering and Networking. 2014; 2: editor@iaeme.com