Nonlinear structural design using multiscale topology optimization

topology optimization;nonlinear materials;computational homogenization;macroscopic\roverall response;adjoint sensitivity analysis

Nakshatrala, Praveen Babu

关键词: topology optimization;    nonlinear materials;    computational homogenization;    macroscopic\roverall response;    adjoint sensitivity analysis;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/46735/Praveen%20Babu_Nakshatrala.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

This dissertation presents developments in nonlinear multiscale topology optimization. Our contributionconsists of three parts: a multiscale design framework for nonlinear elastostatic problems;extension to nonlinear elastodynamics; and an elastoplastic design methodology for transientdynamics. We systematically develop adjoint sensitivity analyses and incorporate them intogradient-based optimization update algorithms.In the multiscale elastostatic formulation, we design composite material microstructures bycombining topology optimization, computational homogenization and parallel programming intoa multilevel design framework. The design problem is posed as a multilevel topology optimizationproblem: a macroscopic problem that optimizes the constituent volume fraction field and themicroscopic problems, at each macroscopic material point that optimize the unit cell morphologies.Homogenization theory is used to compute a homogenized macroscale response withoutfully resolving the high frequency oscillations corresponding to the microscale. For compositestructures exhibiting nonlinear response, closed form expressions relating composite microstructuredesign parameters to their homogenized properties do not exist in general. Hence, we resortto computational homogenization to compute the macroscopic effective properties. The presenceof nonlinearities and the iterative nature of the design process makes the problem computationallychallenging to work with. We resolve this through the use of Message Passing Interface andutilize this framework to design structures for maximum stiffness.We extend the above framework to nonlinear multiscale elastodynamics where we designmaterialmicrostructures to achieve effective energy propagation in structures subjected to impactloading. The design process is formulated under the assumption that the primary wave of interesthas much longer wavelength compared to the microstructural length scale. Under suchan assumption, static homogenization theory holds and is utilized to compute the macroscopic effective properties.Finally, we further extend themultiscale transient elastodynamic formulation to design elastoplasticmaterialsystems for impact mitigation. Using themultiscale elastodynamic computationalframework, we replace the unit cell computations with the local constitutive evolution equationsof small deformation elastoplasticity. Furthermore, to extend the adjoint sensitivity formulation,we account for the history dependence of internal state variables. Using these sensitivities andtopology optimization, we distribute two elastoplastic materials within a design domain to effectivelydissipate the energy in an impact problem.