Topology optimization with multiple materials, multiple constraints, and multiple load cases
Zhang, Xiaojia Shelly
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Topology optimization is a practical tool that allows for improved structural designs. This thesis focuses on developing both theoretical foundations and computational algorithms for topology optimization to effectively and efficiently handle many materials, many constraints, and many load cases. Most work in topology optimization is restricted to linear material with limited constraint settings for multiple materials. To address these issues, we propose a general multi-material topology optimization formulation with material nonlinearity. This formulation handles an arbitrary number of materials with flexible properties, features freely specified material layers, and includes a generalized volume constraint setting. To efficiently handle such arbitrary constraints, we derive an update scheme, named ZPR, that performs robust updates of design variables associated with each constraint independently. The derivation is based on the separable feature of the dual problem of the convex approximated primal subproblem with respect to the Lagrange multipliers, and thus the update of design variables in each constraint only depends on the corresponding Lagrange multiplier. This thesis also presents an efficient filtering scheme, with reduced-order modeling, and demonstrates its application to 2D and 3D topology optimization of truss networks. The proposed filtering scheme extracts valid structures, yields the displacement field without artificial stiffness, and improve convergence, leading to drastically improved computational performance. To obtain designs under many load cases, we present a randomized approach that efficiently optimizes structures under hundreds of load cases. This approach only uses 5 or 6 stochastic sample load cases, instead of hundreds, to obtain similar optimized designs (for both continuum and truss approaches). Through examples using Ogden-based, bilinear, and linear materials, we demonstrate that proposed topology optimization frameworks with the new multi-material formulation, update scheme, and discrete filtering lead to a design tool that not only finds the optimal topology but also selects the proper type and amount of material with drastically reduced computational cost.
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