With the development of Additive Manufacturing (AM) technology, architected materials are attracting increased attention owing to their ability to manipulate material behavior. Through AM technology, many unconventional multiscale and multiphysical effects can be realized. Topology optimization (TO) presents a systematic mathematical approach to the design of optimal materials and structures.
Fig. 1: Simultaneous structural and material design approach 
To further extend the design space, we developed a simultaneous material and structural optimization approach, in which the macroscale design domain can be divided by any number of subregions as the practical requirements or the designer's insight, and the microstructures are assumed to be periodic in each subregion (Fig. 1). In this sense, both the macroscale structure and all the unit cells can be designed to improve the structural performance. In addition, by linearization technique, both the macroscale structure and unit cells can be optimized in parallel to save considerable computational efforts in our level set topology optimization framework .
Nevertheless, homogenization-based TO has a salient limitation as it does not consider how the unit cells are interconnected within the material. Since the unit cells are often disconnected within the structure (Fig. 2), the architecture of the material becomes implausible in reality. To overcome such issue, we introduced a connectivity index approach to optimally design the multiscale structures with multiple well-connected microstructures (Fig. 3) . Currently, this approach is adopted to study coupled multiphysics design problems where the structure interacts with its surroundings, e.g., mechanical-thermal systems, through which the quantitative benefit and potential introduced by multi-scale architecture can be understood clearly.
Fig. 2: Multiscale optimized design of the L beam example with connectivity issue of microstructures (a) macroscale structure and (b) microstructures
Fig. 3: Optimized L beam design through connectivity index approach (a) macrostructure and (b) microstructures 
Stress optimization is one of the key factors for structural design in a wide range of engineering problems. Designs such as the one from Fig. 2 present stress concentrations that might lead to structural failure in practice.
(a) Design domain
(b) Von Mises stress field
Fig. 2: Benchmark L-bracket example.
It is acknowledged that considering stress within a topology optimization framework is hard and partially unresolved. Notably, Le et al. (2010) presented a practical solution for stress-constrained design in the context of density-based topology optimization. However, stress constraints are still hard to achieve with other methods, such as level sets. Our level set method has been capable to address stress-based shape and topology optimization.
Fig. 3: Stress-based optimal designs for the L-bracket.
Our approach uses a handful of numerical ingredients to deal with the challenges of stress-based design, namely, singularity problem, the local nature of the stress and the highly nonlinear stress behavior. The application of a p-norm aggregation function and other techniques allows the method to address stress-constraints and multiple load and stress cases.
Fig. 4: Multiple load and stress cases.
Future work will consider the application of our stress-based level set method to different problems, e.g., microstructural stresses, stress control, etc. We gratefully acknowledge Prof. Julian Norato, from the University of Connecticut, for the collaboration in this work; and, also, the Engineering and Physical Sciences Research Council (EPSRC) from the United Kingdom for the financial support in this research.
A coupling between unsteady aerodynamics and structural vibration gives rise to the flutter phenomenon: steady or growing oscillations which can lead to catastrophic structural failure.
(a) Aeroelastic damping curves for a scale NASA CRM wing
(b) Aeroelastic mode shape at the flutter point (zero damping)
Fig. 5: Aeroelastic stability information is contained in the vibrational damping parameters: Negative damping implies stability and vice-versa. We calculate structural and aerodynamic behaviour via the finite element and doublet-lattice methods, linking them via surface splines.
While aircraft optimization, with an emphasis on avoiding aeroelastic instability, has been studied since the 1960's, few have utilised a topology optimization approach. We are employing the level set method to represent and optimize components of wings, such as the skin thickness distribution.
(a) Example level set field
(b) Resulting skin thickness distribution
Fig. 6: Wing structures are represented by the level set method:Skin thickness is defined at the finite element nodes, and is mapped via the level set function such that binary topologies are obtained.
Wing weight is typically the measure to be optimized, with constraints placed on the eigenvalues of the flutter equation; such ensures stability under the flight conditions of interest. The level set optimization results in aeroelastically-tailored structures, which leverage the known aeroelastic phenomena such as mass-balancing and bend-twist coupling. Our approach has proven capable of significantly reducing weight while maintaining flutter and divergence speed.
(a) NASA CRM wing planform
(b) Delta wing planform
Fig. 7: Optimal wing structures produced by the level set method. For each, the flutter speed is equal to or great than the reference design.
Future works will include extension to three-dimensional wing structures, and inclusion of the aerodynamic planform into the optimization process. We gratefully acknowledge the support of the European Office of Aerospace Research and Development, Air Force Office of Scientific Research, and NASA Langley for this research.
Topology optimization, which determines the optimum shape and layout for given objective function and constraints, is considered to be capable of giving out optimum configuration out of large design spaces and hence has a high potential when it is combined with different regimes of physics (multiphysics) and scales (multiscale). In the application context, however, the potential is not fully exploited partially due to the challenges come out of non-technical barriers. First, there is a limited number of systematic object-oriented approaches that reduce the recurring tasks of coding. It is a limiting factor considering the wide background of the researchers who are interested in the structural optimization. Moreover, the numerical treatments are also in need in addition to the building blocks of the topology optimization, say getting consistent sensitivities, by which the overall implementation involves manual, repetitive tasks which are prone to user-induced errors. In this respect, the robust framework on which topology optimization is running is highly required.
OpenMDAO, an open-source computing platform for Multidisciplinary Design Optimization (MDO) created and maintained by NASA Glenn, is employed in this regards. Written in Python with Numpy and Scipy numerical libraries, the optimization technique can be delivered to the wider users regardless of their computing architectures (e.g., operating system) or technical background. The automatic visualization module applied to OpenMDAO also helps users to understand the method more clearly. Also, equipped with state-or-arts numerical recipes (e.g., MAUD), a burden of implementation of the topology optimization is even reduced further.
Fig. 8: A hierarchical structure within OpenMDAO.
In this regards, we implemented both SIMP (Solid isotropic material with penalization) and Level-set based topology optimizations within OpenMDAO, which are two typical techniques of topology optimizations. Even though the basis of the language is written in the context of MDO, commonalities found within general optimization enables such integration. XDSM diagrams visualize a flow of the data and decomposition of the techniques..
Fig. 9: (a) XDSM diagram of SIMP (b) XDSM diagram of level-set topology optimization, composed of Hamilton-Jacobi solver and sequential linear problem (SLP) solver.
Although their update schemes are different, both SIMP and level-set topology optimization shares finite element analysis and sensitivities; a reusability of the modules are thus guaranteed. Specialized C++ modules are thereby written to obtain finite element analysis and level-set solution advection and exposed to Python using Cython library, which seamlessly links between two different language architecture. Note that such flexibility of the language is another benefit of Python-based API, as does in openMDAO. For demonstration, typical compliance minimization benchmark of the cantilever beam is solved using both methods.
Fig. 10: Numerical examples of compliance minimization of the cantilever beam. (a) SIMP without filter (b) Level-set method.
Further works include adding auxiliary components for optimization and expansion of the finite element library. The completed works would be distributed to the public, by which we hope this highly reconfigurable modules of topology optimization help people who have interest in design optimization. We gratefully acknowledge NASA Glenn for the collaboration in this work.