Document Type : Original Article
Authors
- Mahmoud َAlfouneh ^{} ^{1}
- Behrouz Keshtegar ^{2}
^{1} Mechanical Engineering Department, University of Zabol, Zabol, Iran
^{2} Civil Engineering Department, University of Zabol, Zabol, Iran
Abstract
Since deterministic topology optimization (TO) does not consider the uncertainties in the structure, including materials, loading and geometric dimensions, it may provide the optimal designs with the lowest state of reliability and safety. To
solve this problem, reliability-based topology optimization (RBTO) is used, which is actually a combination of TO methods with reliability-based design methods (RBDO) based on a mathematical framework and process. In this article, by considering four TO methods including moving iso-surface threshold (MIST), SIMP, evolutionary structural optimization-extended finite element (XFEM-ESO) and level-set (LS), and considering a constraint or objective function, an optimal volume fraction (Vf) is obtained for TO by the bisection method. Then, with the aid of mean volume fraction, TO is performed and its optimized results are applied by an advanced reliability analysis method, i.e. accelerated dynamical mean value (ADMV), taking into account the uncertainties and their standard deviations to extract the most probable probability point (MPP). Having the MPP and the constraint, the bisection algorithm is used again and the optimized volume fraction for the RBTO model, and thereby the optimal layout for the RBTO solution, is achieved. Several examples are presented to validate and highlight the optimization capability of the RBTO method using a structural model and the mentioned TO methods, and the results are compared together. Based on the results, it is shown that the combination of RBDO and TO approach is able to result in powerful, stable, safe, and reliable structures completely different from the TO results.
Keywords
- Deterministic topology optimization
- Reliability-based topology optimization
- Bisection method
- MIST method
- Level-set method
- SIMP
- XFEM-ESO method
Main Subjects
- Bendsøe, M. P. and Kikuchi, N., “Generating Optimal Topologies in Structural Design Using a Homogenization Method”, Computer Methods in Applied Mechanics and Engineering, 1988. 71(2): pp. 197-224.
- Pereira, R. L., Lopes, H. N., and Pavanello, R., “Topology Optimization of Acoustic Systems with A Multiconstrained BESO Approach”, Finite Elements in Analysis and Design, 2022. 201: pp. 103701.
- Chen, L., Lu, C., Lian, H., Zhao, w., Li, S., Chen, H., and SPA Bordas, “Acoustic Topology Optimization of Sound Absorbing Materials Directly from Subdivision Surfaces with Isogeometric Boundary Element Methods”, Computer Methods in Applied Mechanics and Engineering, 2020. 362: pp. 112806.
- He, M., Zhang, X., Dos Santos Fernandez, L., Molter, A., Xia, L., and Shi, T., “Multi-Material Topology Optimization of Piezoelectric Composite Structures for Energy Harvesting”, Composite Structures, 202:265.1 p. 113783.
- He, M., He, M., Zhang, X., and Xia, L., “Topology Optimization of Piezoelectric Energy Harvesters for Enhanced Open-Circuit Voltage Subjected to Harmonic Excitations”, Materials, 2022. 15, DOI: 10.3390/ma15134423.
- Eschenauer, H.A. and Olhoff, N., “Topology Optimization of Continuum Structures: A Review”, Mech. Rev., 2001. 54(4): pp. 331-390.
- Van Dijk, N.P., Maute, K., Langelaar, M., Van Keulen, F., “Level-Set Methods for Structural Topology Optimization: A Review”, Structural and Multidisciplinary Optimization, 2013. 48: pp. 437-472.
- Zuo, W. and Saitou, K., “Multi-Material Topology Optimization Using Ordered SIMP Interpolation”, Structural and Multidisciplinary Optimization, 2017. 55: pp. 477-491.
- Yang, B., Cheng, C., Wang, X., Meng, Z., and Homayouni-Amlashi, A., “Reliability-Based Topology Optimization of Piezoelectric Smart Structures with Voltage Uncertainty”, Journal of Intelligent Material Systems and Structures, 2022. 33(15): pp. 1975-1989.
- Zheng, B., Chang, C. J., and Gea, H. C., “Topology Optimization of Energy Harvesting Devices Using Piezoelectric Materials”, Structural and Multidisciplinary Optimization, 2009. 38: pp. 17-23.
- Kharmanda, G., Olhoff, N., Mohamed, A., and Lemaire, M., “Reliability-Based Topology Optimization”, Structural and Multidisciplinary Optimization, 2004. 26: p 295-307.
- Maute, K. and Frangopol, D. M., “Reliability-Based Design of MEMS Mechanisms by Topology Optimization”, Computers & Structures, 2003. 81(8-11): pp. 813-824.
- Cho, K. H., Park, J. Y., Im, M. G., and Han, S. Y., “Reliability-Based Topology Optimization of Electro-Thermal-Compliant Mechanisms with A New Material Mixing Method”, International Journal of Precision Engineering and Manufacturing, 2012. 13: pp. 693-699.
- Mohammadzadeh, H. and Abolbashari, M. H., “Reliability Based Topology Optimization for Maximizing Stiffness and Frequency Simultaneously”, Modares Mechanical Engineering, 2017. 17(4): pp. 111-116.
- Habashneh, M. and Movahedi Rad, M., “Reliability Based Geometrically Nonlinear Bi-Directional Evolutionary Structural Optimization of Elasto-Plastic Material”, Sci Rep, 2022. 12(1): pp. 5989.
- Zheng, J., Yuan, L., Jiang, C., and Zhang, Z., “An Efficient Decoupled Reliability-Based Topology Optimization Method Based on A Performance Shift Strategy”, Journal of Mechanical Design, 2023. 145.(6)
- Zhang, X. and Ouyang, G., “A Level Set Method for Reliability-Based Topology Optimization of Compliant Mechanisms”, Science in China Series E: Technological Sciences, 2008. 51(4): pp. 443-455.
- Huang, X. and Xie, M., “Evolutionary Topology Optimization of Continuum Structures: Methods and Applications”, John Wiley & Sons, 2010.
- Huang, X. and Xie, Y. M., “A Further Review of ESO Type Methods for Topology Optimization”, Structural and Multidisciplinary Optimization, 2010. 41: pp. 671-683.
- Wang, M.Y., Wang, X., and Guo, D., “A Level Set Method for Structural Topology Optimization”, Computer Methods in Applied Mechanics and Engineering, 2003. 192(1)-p 227-246.
- Challis, V. J., “A Discrete Level-Set Topology Optimization Code Written in Matlab”, Structural and Multidisciplinary Optimization, 2010. 41: pp. 453-464.
- Bendsøe, M. P., “Optimal Shape Design as A Material Distribution Problem”, Structural Optimization, 1989. 1: pp. 193-202.
- Zhou, M. and Rozvany, G. I., “The COC Algorithm, Part II: Topological, Geometrical and Generalized Shape Optimization”, Computer Methods in Applied Mechanics and Engineering, 1991. 89(1-3): pp. 309-336.
- Tong, L. and Lin, J., “Structural Topology Optimization with Implicit Design Variable—Optimality and Algorithm”, Finite Elements in Analysis and Design, 2011. 47(8): pp. 922-932.
- Alfouneh, M. and Tong, L., “Maximizing Modal Damping in Layered Structures Via Multi-Objective Topology Optimization”, Engineering Structures, 2017. 132: pp. 637-647.
- Alfouneh, M. and Tong, L., “Damping Design of Flexible Structures with Graded Materials Under Harmonic Loading”, Journal of Vibration and Acoustics, 2018. 140 (5).
- Keshtegar, and Alfouneh, M., “SVR-TO-APMA: Hybrid Efficient Modelling and Topology Framework for Stable Topology Optimization with Accelerated Performance Measure Approach”, Computer Methods in Applied Mechanics and Engineering, 2023. 404: pp. 115762.
- Alfouneh, and Keshtegar, B., “STO-DAMV: Sequential Topology Optimization and Dynamical Accelerated Mean Value for Reliability-Based Topology Optimization of Continuous Structures”, Computer Methods in Applied Mechanics and Engineering, 2023. 417: pp. 116429.
- Abdi, M., “Evolutionary Topology Optimization of Continuum Structures Using X-FEM and Isovalues of Structural Performance”, 2015, University of Nottingham.
- Querin, O. M., Steven, G.P., and Xie, Y.M., “Evolutionary Structural Optimisation (ESO) Using A Bidirectional Algorithm”, Engineering Computations, 1998. 15(8): pp. 1031-1048.
- Teimouri, M. and Asgari, M., “Multi-Objective BESO Topology Optimization for Stiffness and Frequency of Continuum Structures”, Eng. Mech, 2019. 72(2): pp. 181-190.
- Xia, Q., Shi, , and Xia, L., “Topology Optimization for Heat Conduction by Combining Level Set Method and BESO Method”, International Journal of Heat and Mass Transfer, 2018. 127: pp. 200-209.
- Richardson, C.L., Hegemann, , Sifakis, E., Hellrung, J., and Teran, J. M., “An XFEM Method for Modeling Geometrically Elaborate Crack Propagation in Brittle Materials”, International Journal for Numerical Methods in Engineering, 2011. 88(10): pp. 1042-1065.
- Riccio, A., Caruso, U., Raimondo, A., and Sellitto, A., “Robustness of XFEM Method for the Simulation of Cracks Propagation in Fracture Mechanics Problems”, American Journal of Engineering and Applied Sciences, 2016. 9(3): pp. 599-610.
- Bendsoe, M. P. and Sigmund, O., “Topology Optimization: Theory, Methods, and Applications”, 2013: Springer Science & Business Media.
- Bendsøe, M.P. and Sigmund, O., “Material Interpolation Schemes in Topology Optimization”, Archive of Applied mechanics, 1999. 69: pp. 635-654.
- Zhang, W., Zhong, W., and Guo, X., “An Explicit Length Scale Control Approach in SIMP-Based Topology Optimization”, Computer Methods in Applied Mechanics and Engineering, 2014. 282: pp. 71-86.
- Marck, G., Nemer, M., Harion, J. L., Russeil, S., and Bougeard, D., “Topology Optimization Using the SIMP Method for Multiobjective Conductive Problems”, Numerical Heat Transfer, Part B: Fundamentals, 2012. 61(6): pp. 439-470.
- Brackett, D., Ashcroft, I., and Hague, R., “Topology Optimization for Additive Manufacturing in 2011 International Solid Freeform Fabrication Symposium”, 2011, University of Texas at Austin.
- Sigmund, O., “A 99 Line Topology Optimization Code Written in Matlab”, Structural and Multidisciplinary Optimization, 2001. 21: pp. 120-127.
- Luo, Q. and Tong, L., “Optimal Designs for Vibrating Structures Using a Moving Isosurface Threshold Method with Experimental Study”, Journal of Vibration and Acoustics, 2015. 137(6): pp. 061005.
- Vasista, S. and Tong, L., “Design and Testing of Pressurized Cellular Planar Morphing Structures”, AIAA journal, 2012. 50(6): pp. 1328-1338.
- Chen, W., Tong, L., and Liu, S., “Concurrent Topology Design of Structure and Material Using a Two-Scale Topology Optimization”, Computers & Structures, 178: pp. 119-128.
- Lu, Y. and Tong, L., “Concurrent Multiscale Topology Optimization of Metamaterials for Mechanical Cloak”, Computer Methods in Applied Mechanics and Engineering, 409: pp. 115966.
- Gordon, G. and Tibshirani, R., “Karush-Kuhn-Tucker Conditions”, Optimization, 2012. 10(725/36): pp. 725.
- Luo, Q. and Tong, L., “Structural Topology Optimization for Maximum Linear Buckling Loads by Using a Moving Iso-Surface Threshold Method”, Structural and Multidisciplinary Optimization, 52(1): pp. 71-90.
- Luo, Q. and Tong, L., “Optimal Designs for Vibrating Structures Using a Moving Isosurface Threshold Method with Experimental Study”, Journal of Vibration and Acoustics, 2015. 137(6): pp. 1-22.
- Wang, S. and Wang, M. Y., “Radial Basis Functions and Level Set Method for Structural Topology Optimization”, International Journal for Numerical Methods in Engineering, 65(12): pp. 2060-2090.
- Yoon, G. H. and Kim, Y. Y., “Element Connectivity Parameterization for Topology Optimization of Geometrically Nonlinear Structures”, International Journal of Solids and Structures, 42(7): pp. 1983-2009.
- Wei, P., Li, Z., Li, X., and Wang, M. Y.,“An 88-Line MATLAB Code for The Parameterized Level Set Method Based Topology Optimization Using Radial Basis Functions”, Structural and Multidisciplinary Optimization, 2018. 58: pp. 831-849.
- Van Dijk, N., Yoon, G., Van Keulen, F., and Langelaar, M., “A Level-Set Based Topology Optimization Using the Element Connectivity Parameterization Method”, Structural and Multidisciplinary Optimization, 2010. 42: 269-282.
- Allaire, G., Jouve, F., and Toader, A. M., “Structural Optimization Using Sensitivity Analysis and A Level-Set Method”, Journal of computational physics, 2004. 194(1): pp. 363-393.
- Keshtegar, B. and Chakraborty, S., “Dynamical Accelerated Performance Measure Approach for Efficient Reliability-Based Design Optimization with Highly Nonlinear Probabilistic Constraints”, Reliability Engineering & System Safety, 178: pp. 69-83.
- Mousavi, S.M., Mostafavi, E. S., Jaafari, A., Jaafari, A., and Hosseinpour, F., “Using Measured Daily Meteorological Parameters to Predict Daily Solar Radiation”, Measurement, 76: pp. 148-155.
- Meng, Z., Li, G., Wang, B. P., and Hao, P., “A Hybrid Chaos Control Approach of The Performance Measure Functions for Reliability-Based Design Optimization”, Computers & Structures, 146: pp. 32-43.
- Keshtegar, B., Hao, P., and Meng, Z., “A Self-Adaptive Modified Chaos Control Method for Reliability-Based Design Optimization”, Structural and Multidisciplinary Optimization, 55: pp. 63-75.