# Topology Optimization of Modified Piston, Bridge Structures and Subway Tunnel Problem

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2 [6] H. Zhang et al. presented an element based search scheme to identify load surfaces. The load surfaces are formed by the connection of the real boundary of elements and the pressures are transferred directly to corresponding element nodes. In the present work the subway tunnel problem as discussed in H. Zhang et al. is modified. Here in modified subway tunnel problem the pressure load is applied on upper, right and left side of design domain to mimic the affect of pressure exerted by soil and rocks on practical subway tunnel structures. A point load is applied on the upper side to show the affect of a concentrated mass of soil and rock at any point. Here the position of point load is not specified as it is assumed that the concentrared mass of soil and rock can be present anywhere above subway tunnel structure. [7] Ekrem Buyukkaya, Muhammet Cerit presented thermal analyses on a conventional (uncoated) diesel piston, made of aluminum silicon alloy and steel. Secondly, thermal analyses are performed on pistons, coated with MgO ZrO2 material by means of using a commercial code, namely ANSYS. For modified bridge structures and subway tunnel problem the material taken is steel and for piston the material is an Aluminium - Silicon alloy (AlSi). For steel and AlSi alloy properties are taken from [7]. [8] Dheeraj Gunwant, Anadi Misra compared and validated the result of ANSYS based Optimality criterion with the results obtained by Element Exchange Method. The mathematical approach of Optimality Criteria used in this in this work is taken from [8]. 2.METHOLOGY 2.1.The Optimality Criterion Approach Compliance = V fu dv + S tu ds +.(1) Where, u = Displacement field f = Distributed body force (gravity load etc.) Fi = Point load on ith node ui = ith degree of freedom t = Traction force S = Surface area of the continuum V = Volume of the continuum The Lagrangian for the optimization problem is defined as: L(x j) = u T Ku + Λ ( jv j - V o ) + λ 1 ( Ku F ) + 2 j + (x min x j) + 3 j ( x j - 1 ).....(2) Where Λ, λ 1, λ 2 and λ 3 are Lagrange multipliers for the various constraints. The optimality condition is given by: =0 where j = 1,2,3..n.. (3) Now compliance, C = u T Ku. (4) Differentiating eq. (1) w. r. t. x j, the optimality condition can be written as: B j = - = 1..(5) The Compliance sensitivity can be evaluated as using equation: = - p(x j) p-1 u j T k j u j...(6) Based on these expressions, the design variables are updated as follows: x j new = max (x min - m), if x j B j n ( x min, x min - m) =x jb jn, if max(x min - m)<x j B jn <min (1, x j +m) =min (1, x j +m), if min(1, x j +m) x j B jn (7) Where, m is called the move limit and represents the maximum allowable change in a single OC iteration. Also, n is a numerical damping coefficient, and is usually taken to be ½. The Lagrange multiplier for the volume constraint Λ is determined at OC iteration using a bisection algorithm. x j is the value of the density variable at each iteration step. u j is the field at each iteration step determined from the equilibrium equations Ojective Function and Constraint Equations The objective function in topology optimization is minimization of structural compliance with a constraint on the material volume. The objective function is given by equation : The force vector is given by C(x)= u T Ku = Subject to Min C (x) = F T u (8) F= K(x)u (9) j T K j(x j)u j..(10) jv j V o.(11) 0 < x min x j 1, where j = 1, 2, 3 n 2015, IRJET.NET- All Rights Reserved Page 1870

3 3. SPECIMEN GEOMETRY AND BOUNDARY CONDITIONS 3.1 Four Point Load Bridge Structure Figure 1 shows design domain for first modified bridge structure (Four Point Load Bridge Structure). The length and height of bridge is 8m and 4m. The left and right side of bridge structure is fully constrained that is their degree of freedom in all directions is zero. It is subjected to four point loads. Two point loads are present at the middle point of upper and lower side while other two are applied at end points of upper side as shown by red arrows in the figure. The material taken is steel having Young s modulus of 200 GPa and Poisson s ratio of.3. The volume fraction taken is.4. The magnitude of applied load is 20000N. 3.3 Piston Subjected to Pressure and Point Loads Figure 3 shows design domain for piston structure. The length and height of piston is 80 mm. The left and right sides of the domain are constrained in the x-direction, representing the cylinder walls, and the center of the bottom edge is fully constrained. The pressure applied at the upper side is 2 MPa and a point load of N is applied at middle and end points of upper side. Here red arrows at the middle and end points of upper side shows applied point load (20000 N) and other two arrows present at some distance from end points shows pressure load (2 MPa). The material taken is AlSi alloy having Young s modulus of 90 GPa and Poisson s ratio of.3. The volume fraction taken is.4. Fig -1: Design domain of four point load bridge structure 3.2 Three Point Load Bridge Structure Figure 2 shows design domain for second modified bridge structure (Three Point Load Bridge Structure). The length and height of bridge is 8m and 4m. The lower half portion of left and right side is fully constrained. Two point loads are applied at end points of upper side and the third one at the middle point of upper side as shown by red arrows in the figure. The material is steel having Young s modulus of 200 GPa and Poisson s ratio of.3. The volume fraction taken is.4. The magnitude of applied load is 20000N. Fig -3: Design domain of piston structure 3.4 Modified Subway Tunnel Problem Figure 4 shows design domain for modified subway tunnel problem. The length and height of are 8m and 4m. The design domain is subjected to pressure load of 2 MPa on left, right and upper sides as shown by red arrows acting above blue region. It is also acted upon by a point load of N on upper side as shown by red arrow acting in the blue region. The material is steel having Young s modulus of 200 GPa and Poisson s ratio of.3. The volume fraction taken is.4. Fig -2: Design domain of three point load bridge structure Fig -4: Design domain of modified subway tunnel problem 2015, IRJET.NET- All Rights Reserved Page 1871

4 4. RESULTS AND DISCUSSION 4.1 Four Point Load Bridge Structure Figure 5 shows the optimized image for four point load bridge structure. Here red region shows solid material and white region shows void. Vertical blue lines represent constrained sides and red arrows shows applied point load. Fig -6: Deformed and undeformed shape for optimized topology Fig -5: Optimized topology for four point load bridge structure Table -1: Maximum s in X any Y directions Max. positive X- Max. negative X-.815 X 10-4 mm.865 X 10-4 mm In fig. 6 the black lines shows undeformed shape and blue region deformed shape. The gap between black linings and blue region shows extent of deformation. As evident from fig. there is considerable amount of deformation at the point of application of load and at the constrained positions there is no deformation. Table -3: Compliance and iteration values Compliance initial value (1 st iteration)(x) Compliance final value (y) No. of iterations N-m.0448 N-m Max. negative Y-.126 X 10-2 mm % reduction of compliance w.r.t initial value (x-y)/y 100) % Table 1 shows that there is a relatively large in negative Y direction as compared to X direction and there is no in positive Y direction. Table 3 shows that there is a % decrement in compliance. Table -2: Maximum von Mises, X and Y stress components Max. X tensile stress N/m 2 Max. X compressive stress N/m 2 Max. Y tensile stress N/m 2 Max. Y compressive stress N/m 2 Max. von Mises stress N/m 2 Fig -7: Compliance vs iteration plot for four point load bridge structure 2015, IRJET.NET- All Rights Reserved Page 1872

5 Fig. 7 shows that compliance decreases sharply from 1 st iteration to 7 th iteration and from 8 th iteration onwards having a relatively flat slope. 4.2 Three Point Load Bridge Structure Figure 8 shows the optimized image for three point load bridge structure. Here a butterfly type structure is obtained as optimized topology. Fig -9: Deformed and undeformed shape for optimized topology As evident from fig. 9 there is considerable amount of deformation at the point of application of load and at the constrained positions there is no deformation. Table -6: Compliance and iteration values Fig -8: Optimized topology for three point load bridge structure Table -4: Maximum s in X any Y directions Max. positive X-.722 X 10-3 mm Compliance initial value (1 st iteration)(x) Compliance final value (y) No. of iterations N-m N-m Max. negative X-.768 X 10-3 mm % reduction of compliance w.r.t initial value (x-y)/y 100) % Max. negative Y-.151 X 10-2 mm Here also there is a relatively large in negative Y direction as compared to X direction and there is no in positive Y direction. Table 6 shows that there is a % decrement in compliance but here the number of iterations for convergence are very large as compared to previous example. Table-5: Maximum von Mises, X and Y stress components Max. X tensile stress N/m 2 Max. X compressive stress N/m 2 Max. Y tensile stress N/m 2 Max. Y compressive stress 1.1 X 10 6 N/m 2 Max. von Mises stress 1.53 X 10 6 N/m 2 Fig -10: Compliance vs iteration plot for three point load bridge structure Compliance has a sharp slope from 1 st iteration to 4 th iteration, then from 4 th to 9 th iteration it form a small 2015, IRJET.NET- All Rights Reserved Page 1873

6 parabolic like curve. From 10 th to 99 th (last iteration) the slope is almost horizontal. 4.3 Piston Subjected to Pressure and Point Loads Figure 11 shows the optimized image for piston. Here the shape obtained is different from that mentioned in [5]. In [5] the branches in the optimized structure are more widespread and thin whereas the optimized piston shown below has thick branches and they are not as widespread as in [5]. Fig -12: Deformed and undeformed shape for optimized topology Here the deformation is maximum for the points where pressure load and point loads are acting simultaneously i.e. at middle and end points of upper side and at constrained positions deformation is zero i.e. at fixed sides and middle constrained point. Fig -11: Optimized topology for piston Table -7: Maximum s in X any Y directions Table -9: Compliance and iteration values Compliance initial value (1 st iteration)(x) X 10 7 N-mm Max. positive X- Max. negative X- Max. negative Y mm mm mm Compliance final value (y) No. of iterations 15 % reduction of compliance w.r.t initial value (x-y)/y 100) X 10 6 N-mm 79.57% Here the maximum in negative Y direction is very as compared to X direction s. Table 9 shows that there is a % decrement in compliance and the convergence is obtained in very less number of iterations. Table-8: Maximum von Mises, X and Y stress components Max. X tensile stress N/mm 2 Max. X compressive stress N/mm 2 Max. Y tensile stress N/mm 2 Max. Y compressive stress N/mm 2 Max. von Mises stress N/mm 2 Fig -13: Compliance vs iteration plot for piston 2015, IRJET.NET- All Rights Reserved Page 1874

7 The graph falls sharply from 1 st iteration to 4 th iteration and afterwards has an almost horizontal slope. 4.4 Modified Subway Tunnel Problem For modified subway tunnel problem an arch type structure resembling a subway tunnel is obtained as an optimal topology of the given design domain. Fig -15: Deformed and undeformed shape for optimized topology Fig -14: Optimized topology for modified subway tunnel problem At the point of application of load and the lines where pressure is applied the deformation is very large as compared to regions around constrained points. Table -12: Compliance and iteration values Table -10: Maximum s in X any Y directions Max. positive X-.23 mm Compliance initial value (1 st iteration)(x) Compliance final value (y) N-m N-m Max. negative X- Max. negative Y-.234 mm.476 mm No. of iterations 13 % reduction of compliance w.r.t initial value (x-y)/y 100) 70.25% The s shown in table 10 are nearly same in magnitude. Table 12 shows that there is a % decrement in compliance and the convergence is obtained in 13 iterations. Table-11: Maximum von Mises, X and Y stress components Max. X tensile stress 4.8 X 10 7 N/m 2 Max. X compressive stress 2.98 X 10 7 N/m 2 Max. Y tensile stress 5.53 X 10 7 N/m 2 Max. Y compressive stress 2.03 X 10 8 N/m 2 Max. von Mises stress 2.55 X 10 8 N/m 2 Fig -16: Compliance vs iteration plot for modified subway tunnel The compliance has a sharp descend from 1 st iteration to 2 nd iteration. From 2 nd iteration to 4 th iteration there is a decrease in slope. From 4 th to 5 th iteration there is further 2015, IRJET.NET- All Rights Reserved Page 1875

8 decrement in slope and from 5 th iteration to last iteration (13 th ) the slope is relatively flat. 5. CONCLUSIONS This work gives an insight into the application of Optimality Criteria Method to bridge problems, piston and subway tunnel problem under different loading and boundary conditions. For both bridge structures considered truss type topologies are obtained which suggests the best suited arrangement of material for a bridge subjected to prescribed loading and boundary conditions. In regard of piston problem the optimal topology needs further refinement by subsequent shape and size optimization procedures. The optimal topology obtained in case of modified subway tunnel problem resembles the actual subway tunnel structure. The subway tunnel in this work is only an illustrative example regarding applicability of OCM. In fact for actual subway tunnel many factors like elasticity foundation, seepage and gravity load due to soil and rocks are to be considered. This work also demonstrates the capability of OCM in topology optimization problems subjected to designdependent pressure loads. REFERENCES [1] Bruggi, Matteo, and Paolo Venini. "Topology optimization of incompressible media using mixed finite elements." Computer methods in applied mechanics and engineering 196, no. 33 (2007): [2] Guo, Xu, Weisheng Zhang, and Li Zhang. "Robust structural topology optimization considering boundary uncertainties." Computer Methods in Applied Mechanics and Engineering 253 (2013): [3] Sigmund, Ole, and Peter Michael Clausen. "Topology optimization using a mixed formulation: an alternative way to solve pressure load problems."computer Methods in Applied Mechanics and Engineering 196, no. 13 (2007): [4] Bruggi, Matteo, and Carlo Cinquini. "An alternative truly-mixed formulation to solve pressure load problems in topology optimization." Computer Methods in Applied Mechanics and Engineering 198, no. 17 (2009): [5] Lee, Edmund, and Joaquim RRA Martins. "Structural topology optimization with design-dependent pressure loads." Computer Methods in Applied Mechanics and Engineering 233 (2012): [6] Zhang, Hui, Xiong Zhang, and Shutian Liu. "A new boundary search scheme for topology optimization of continuum structures with design-dependent loads."structural and Multidisciplinary Optimization 37, no. 2 (2008): [7] Buyukkaya, Ekrem, and Muhammet Cerit. "Thermal analysis of a ceramic coating diesel engine piston using 3-D finite element method." Surface and Coatings Technology 202, no. 2 (2007): [8] Gunwant, Dheeraj, and Anadi Misra. "Topology optimization of continuum structures using optimality of criterion approach in Ansys." Int. J. Adv. Eng. Technol 5, no. 1 (2012): BIOGRAPHIES Akash Dhiman obtained his bachelor s degree (B.Tech.) in Mechanical Engineering from College of Engineering Roorkee (Roorkee), Uttarakhand, in the year 2012 and M. Tech. in Design and Production Engineering from G. B. Pant University of Agriculture and Technology, Pantnagar, Uttarakhand in the year His area of interest is topology optimization. Anadi Misra obtained his Bachelor s, Master s and doctoral degrees in Mechanical Engineering from G. B. Pant University of Agriculture and Technology, Pantnagar, Uttarakhand, with a specialization in Design and Production Engineering. He has a total research and teaching experience of 28 years. He is currently working as professor in the Mechanical Engineering department of College of Technology, G. B. Pant University of Agriculture and Technology, Pantnagar and has a vast experience of guiding M. Tech. and Ph. D. students. 2015, IRJET.NET- All Rights Reserved Page 1876

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