Nonlinear structural design using multiscale topology optimization

This dissertation presents developments in nonlinear multiscale topology optimization. Our contribution consists of three parts: a multiscale design framework for nonlinear elastostatic problems; extension to nonlinear elastodynamics; and an elastoplastic design methodology for transient dynamics. We systematically develop adjoint sensitivity analyses and incorporate them into gradient-based optimization update algorithms. In the multiscale elastostatic formulation, we design composite material microstructures by combining topology optimization, computational homogenization and parallel programming into a multilevel design framework. The design problem is posed as a multilevel topology optimization problem: a macroscopic problem that optimizes the constituent volume fraction field and the microscopic problems, at each macroscopic material point that optimize the unit cell morphologies. Homogenization theory is used to compute a homogenized macroscale response without fully resolving the high frequency oscillations corresponding to the microscale. For composite structures exhibiting nonlinear response, closed form expressions relating composite microstructure design parameters to their homogenized properties do not exist in general. Hence, we resort to computational homogenization to compute the macroscopic effective properties. The presence of nonlinearities and the iterative nature of the design process makes the problem computationally challenging to work with. We resolve this through the use of Message Passing Interface and utilize this framework to design structures for maximum stiffness. We extend the above framework to nonlinear multiscale elastodynamics where we designmaterial microstructures to achieve effective energy propagation in structures subjected to impact loading. The design process is formulated under the assumption that the primary wave of interest has much longer wavelength compared to the microstructural length scale. Under such an assumption, static homogenization theory holds and is utilized to compute the macroscopic effective properties. Finally, we further extend themultiscale transient elastodynamic formulation to design elastoplasticmaterial systems for impact mitigation. Using themultiscale elastodynamic computational framework, we replace the unit cell computations with the local constitutive evolution equations of small deformation elastoplasticity. Furthermore, to extend the adjoint sensitivity formulation, we account for the history dependence of internal state variables. Using these sensitivities and topology optimization, we distribute two elastoplastic materials within a design domain to effectively dissipate the energy in an impact problem.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mechanical Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2014

Author dc:creator

• Nakshatrala, Praveen Babu

Contributors dc:contributor

• Tortorelli, Daniel A.
• Duarte, C. Armando
• Geubelle, Philippe H.
• Lambros, John

• topology optimization
• nonlinear materials
• computational homogenization
• macroscopic overall response
• adjoint sensitivity analysis

Statement dc:rights

• Copyright 2013 Praveen Babu Nakshatrala

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/46735

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/46735