Large-scale 3D topology optimization is often limited by memory demands as geometric, connectivity, and neighborhood data grow with problem size. We address this bottleneck using a sparse OpenVDB-based RKPM framework that replaces explicit Voronoi-cell and neighbor storage with local, on-the-fly stencil operations.
Computational fluid structure interactions (FSI) pose significant numerical challenges due to the inherently non-linear and time-dependent nature of coupled multiphysics. An additional difficulty is the discretization of the fluid-structure interface to accurately capture the complicated dynamic motion and deformation of the fluid-structure interface and to impose coupling conditions. The complexity further increases when considering topology optimization of such FSI problems, since the interface not only deforms under the action of the fluid, but its shape also changes throughout the optimization procedure.
1. Modified Immersed Finite Element Method
To address the main challenges associated with topology optimization of FSI problems, the level set topology optimization is combined with the modified immersed finite element method (mIFEM) developed by Wang and Zhang (2013). The mIFEMthat leads to a formulation with separate solid and fluid domains. The fluid can see the solid through an artificial fluid region overlapping with the solid whereas the solid sees the fluid as a traction on the FSI interface. A useful feature of this formulation is that no assumption is made on the discretization methods used to solve the governing equations of each domain. Here, for the Eulerian fluid domain the finite element method (FEM) is used whereas the Lagrangian solid domain is analyzed using the reproducing kernel particle method (RKPM).
2. Compliance Minimization Under FSI
Most topology optimization works on FSI consider the leaflet example which is shown in Fig. 2. The setup consists of a rectangular beam with fixed support on the bottom which is placed inside a fluid field.
The optimized designs for two Reyonlds numbers, 0.01 and 100, are shown in Fig.3. It can be observed that the two solutions have several differences, especially on the FSI interface. The boundary is more rounded for the low Re but it becomes more flat, nearly bended inwards, for the high Re.
Upon examining the pressure fields in for the two cases in Fig. 4, it can be seen that pressure around the FSI interface is positive for the low Re acting inwards towards the structure. Spherical shapes are ideal for such hydrostatic loads which may explain the tendency of the boundary to become more rounded. On the other hand, the pressure at the FSI interface is negative for the high Re case, which may cause the boundary to go in the opposite direction.
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In large-scale level set topology optimization, the operations involved in morphing the geometry are slow. Moreover, simulating the physical characteristics, such as stress and stiffness is computationally expensive. This prohibitively expensive computational issues are addressed using a VDB, a tree data structure to represent the geometry. Moreover, we also implement the finite element analysis using parallel computing software PETSc, which significantly speeds up the topology optimization process.
1. VDB Data Structure
VDB is a sparse and hierarchical tree data structure (like octrees) that efficiently represents geometries, resulting in fast convergence characteristics. In the first example, we show a geomertrical optimization of a helix-torus structure obtained using a grid resolution of 2000 x 800 x 2000 = 3.2 billion voxels.
Fig. 1: Geometrical optimization of a helix-torus structure.
2. Finite Element Analysis using PETSc
The torsion ball problem is to design a lightweight structure which is stiff under a torsion load. The structure is clamped under torsion load. The stiffness of the ball is characterized using 6 million finite elements. The compliance of the structure is minimized for a volume constraint of 3%, and the optimization converges in 3 hours.
Fig. 2: Optimized lightweight torsion ball that is stiff under torsion.
The torsion ball problem obtained using a higher resolution mesh (20 million elements) is shown below for a volume constraint of 1%.
The third problem is to design a lightweight bridge structure that is stiff under a pressure load on top with its corners pinned. Due to symmetry, only one half of the domain is modelled. In Figure 5a, the evolution of the bridge obtained using 1 milion elements is shown. In Figure 5b, the optimum bridge topology obtained using 33 million elements is shown.
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The level set method topology optimization for 2D and 3D convective heat transfer problems is presented here.
1. Heat Dissipation Problem
First of all, we show the the optimized heat dissipation structure which is a lightweight design that dissipates heat being produced everywhere.The evolution of the topoology is shown in Fig. 1a obtained using 1 million elements. The optimized structure shown in Fig. 1b is obtained using 32 million elements, showing branches emanating from the heat sink and drawing heat from everywhere in the designdomain.
2. Heat Flow Problem
Stokes flow and Darcy potential flow, which are linear flow models, are used to simulate the flow using the finite element method. The resulting velocity field is used in a convection-diffusion model to simulate the heat transfer using the finite element method. A linear combination of the pressure drop and the average temperature is considered as the objective function, which is minimized subject to a volume constraint. The optimized topologies obtained for different problem cases for Stokes flow and Darcy potential flow are presented.
The schematic of second problem domain is shown in Fig. 2. The boundary conditions of the flow entering through the centerline on left side at 
and exiting through the centerline on the right hand side is shown in Fig. 2. The domain is subject constant heating of . The boundary conditions are at the fluid entry region and at the fluid exit region. The FE mesh is discretized into elements. Due to symmetry, only the top half of FE domain is modeled. The objective is to minimize average temperature pressure drop subject to a volume constraint of 50%.
The topologies obtained for Darcy flow for different values of k are shown in Fig. 3. It can be seen that the number of channels increase as k increases.
The velocity and temperature distribution is shown in Fig. 4. For k = 0.4, most of the flow goes through the center and results in high temperatures. For k = 0.7, the flow is distributed more uniformly, resulting in lower temperatures.
The topologies obtained for Stokes flow for different values of k are shown in Fig. 5. For lower values of k, the width of the channels going through the center is high. On the other hand, for higher values of k, the width of the channels going around the center are high.
Similar to the Darcy flow case, the velocity and temperature distribution is shown in Fig. 6. For k = 0.4, most of the flow goes through the center and results in high temperatures. For k = 0.7, the flow is distributed more uniformly, resulting in lower temperatures.
Now we are trying to solve 3D problem. The schematic of 3D problem domain is shown in Fig. 7, which is under constant heating of , with flow entering the domain through the center line on the left side with , and exiting through the center on the right side is shown in Fig. 7. The FE mesh is discretized into = 0.52 million elements. Due to symmetry, only a quarter of FE domain is modeled. The objective is to minimize average temperature pressure drop subject to a volume constraint of 15%.
The optimized topology obtained using Darcy potential flow is shown in Fig. 8 with k = 0.7 and k = 0.4.
The optimized topology obtained using Stokes flow is shown in Fig. 9 with k = 0.7 and k = 0.4.
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.
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.
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.
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.
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Thermoelasticity is a type of elastic behavior where temperature change is coupled with stress state within the structure. The phenomenon is widely considered in aerospace engineering, where the temperature change is always involved during its operation and fabrication. Often, these thermally induced behaviors have an adverse effect on the structure, for example, an unintended warping. To design that alleviates such a negative effect, topology optimization considering thermoelasticity has been widely investigated and proven to design structures that are optimal when mechanical and thermal loads are imposed simultaneously. However, there are only a few works of literature that considering a nonlinear thermoelastic case, which refers to the case where either mechanical or thermal loads induce a large deformation of the structure.
In this research, we propose level-set based topology optimization of thermoelastic nonlinear structure, which is essential when structure experiences a large deformation.
Fig. 1. Benchmark bi-clamped beam. Mechanical point load and thermal expansion are simultaneously imposed.
1. Mechanical Nonlinearity
Contrary to the linear case, the optimal design layouts change with respect to the sign and magnitude of when structural nonlinearity is considered during the optimization. Figure 2 demonstrates such material layouts, for prescribed mechanical loads only.
Fig. 2. Optimized material layouts for prescribed mechanical loads .
Fig. 3. Comparison of load-carrying capacity between design layouts obtained by the topology optimization considering linear (dotted line) and the nonlinear (solid) structural analysis.
For a range of deformation of the nonlinear optimum structures gradually changes in proportion with the mechanical loading. A large deformation is found to be salient when since the maximum displacement is in the order of height of the structure. In the linear case, however, tip displacements are computed only for a load range of , and shows a displacement of substantially higher magnitude throughout the range: when the load is higher than 0.0016, a structural equilibrium is not attainable for a standard Newton-Raphson method. It is therefore concluded that the linear design has much less structural load capacity due to the presence of slender, compressed members populated in the layout.
2. Thermoelastic Design
We extend the previous discussion on the nonlinear elastic design to nonlinear thermoelasticity. The effect of uniform temperature change is examined herein, by fixing the mechanical load while gradually increasing within the range of . Such a range roughly coincides with the amount of thermal expansion found in high-temperature operation conditions.
Fig. 4. Optimized layouts considering thermoelastic loads, obtained each by employing linear and nonlinear thermoelastic analysis.
For each layout, thermoelastic load that corresponds to one assumed during the optimization is employed for the analysis. The resulting displacements are shown in Fig. 5. Deflection of the linear thermoelastic optimums is plotted against changing and marked by dotted lines and compared with that obtained for the nonlinear design optima. Since the mechanical nonlinearity is not significant in the assumed load case , the displacements of the linear and nonlinear coincides near . As one may expect, the deflections of the design layouts concerning linear thermoelasticity envelopes these of the nonlinear elasticity, which denotes the relative optimality of the designs of nonlinear layouts. However, as shown in where V-shape layouts are optimum solutions in the linear optimization, a strong deviation between the linear and nonlinear results are shown. The V-shape structure buckles therein, which is inevitable when only the linear thermoelasticity is considered as such structural instability is not taken into account in the linear optimization. As a result, it is concluded that even in the case where both the mechanical and thermal loads are small, often nonlinear thermoelasticity are necessarily considered. This finding again emphasizes the needs of nonlinear consideration in thermoelastic structure design.
Fig. 5. Comparison of the optimal compliances with respect to changing .
Through the investigation, the incorporation of the nonlinearity is found to be significant in both mechanically-driven and thermally-driven nonlinear regimes, again demonstrate the importance of incorporating the nonlinearity to the thermoelastic design.
The authors acknowledge the support from DARPA (Award number HR0011-16-2-0032). H. Alicia Kim also acknowledges the support of the Engineering and Physical Sciences Research Council (grant number EP/M002322/2).
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 [1]](/sites/default/files/m2do/images/all/fig_1.png)
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 [2].
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) [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.
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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.
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.
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.
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.
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