Full-scale 3D structural topology optimization using adaptive mesh refinement based on the level-set method
Graphical abstract
Introduction
As it is well known, topology optimization (TO) methods have a high degree of flexibility to find an optimal structure within a given design domain. Therefore, TO methods have been widely used in structural design, for example, lightweight designs in aerospace industry [1], aesthetics in civil engineering [2], high heat flux dissipation in thermal management [[3], [4], [5], [6]], compliant mechanism design [7,8], etc. In addition, the rapid development of advanced manufacturing technology allows us to build complex high-performance products based on the TO approach in an easier way. During the past several decades, a number of typical shape/topology optimization methods have been proposed, such as the homogenization based approach [9], the density-based approach [[10], [11], [12]], the level-set-based approach [13,14], the evolutionary structural optimization approach (ESO) [15,16], and the moving morphable components (MMC) approach [17].
Among these methods, the level-set-based method is able to track boundary movements accurately. The basic idea is to incorporate an implicit representation of boundarys, and the level-set function is updated by solving the level-set equation, commonly known as a Hamilton–Jacobi equation [18,19]. The idea of applying the level-set method to structural optimization was proposed in the early work of Wang et al. [13] and Allaire et al. [14]. Soon after that, several level-set updating strategies had been proposed to overcome the shortcomings in the early level-set methods due to the inability of nucleating new holes during the iterations. Allaire et al. [20] adopted the bubble method [21] to the level-set-based shape optimization using topological derivatives. In this work, new holes can be generated during the optimization process. Wang et al. [22] presented an extended level-set method based on one of their previous works [13]. In this extended method [22], an extended velocity which has a non-zero value in the material domain is introduced and there is no need to reinitialize the level-set function to maintain the property of a signed distance function, hence, allowing the introduction of new holes in a material domain. Yamada et al. [23] proposed a level-set-based TO method where the level-set is updated using the RDE. This method allows not only shape but also topological changes, and allows the geometrical complexity of the optimal configuration to be specified.
In addition, large-scale (or high resolution) shape/topology optimization has attracted an increasing scientific interest because it can discover novel designs that cannot be achieved by using coarse meshes [24]. For example, for a compliance minimization problem, the optimal structure is a plate- or shell-like structure with various thickness rather than a truss-like structure. In other words, a coarse grid will limit the solution space of the optimization problem [25]. Such effect due to the use of a coarse grid is named as artificial length-scale effect. To overcome such a limitation, a high resolution structural TO formulation should be considered. To this end, parallel computing should be used to handle large computations.
Early parallel computing research in TO was mainly found in the density-based method. Borrvall and Petersson [26] proposed a CPU parallel strategy using the regularized intermediate density control method and the method of moving asymptotes (MMA) is used as the update scheme. Aage et al. [27] proposed a fully distributed TO framework using the Portable and Extendable Toolkit for Scientific Computing (PETSc). In their consequential work, a full-scale aircraft wing with more than one billion 3D elements was optimized on a cluster with 8000 processors [24]. In this work, MMA is adopted as the update scheme. Liu et al. [28] presented a narrow-band TO framework on a sparsely populated grid. Their technique can accommodate computational domains with over one billion grid voxels on a single shared-memory multiprocessors platform, allowing to generate optimal structures with both feature-rich shapes and exceptional mechanical performance. Their aircraft wing design also resemble the one from Aage et al. [24]. Wu et al. [29] presented a scalable algorithm running on a graphics processing unit (GPU) which can efficiently handle models with several millions of elements. They presented a variety of optimized structures with different shapes such as interior structures within closed surfaces, exposed support structures, and surface models. They then printed out the design structures. Martinez-Frutos et al. [30] also accelerated TO computation using GPU, and the maximum number of mesh elements is up to 2,097,125 elements.
Very recently, the utilization of parallel computation in level-set-based shape/topology optimization algorithms is in rapid development. Liu et al. [31] suggested a fully parallel parameterized level-set method with compactly supported radial basis functions (CSRBFs) based on both uniform and non-uniform structured meshes. In this work, a numerical example with at most 7 million elements was conducted. They also reached the same conclusion as Sigmund et al. [25] with regard to the artificial length-scale effect. That is, for a compliance minimization problem, the optimal structure is a plate- or shell-like structure with various thickness rather than a truss-like structure. Kambamptati et al. [32] presented a large-scale level-set TO method, in which the spatially adaptive and temporally dynamic volumetric dynamic B+ (VDB) tree data structure is adopted to address the shortcomings arising from the challenges in both slow convergence and high memory consumption. In this work, a numerical example with at most 34 million elements was computed. Several research efforts have been made to achieve large-scale 3D TO in different physical problems such as laminar incompressible flow [33,34], unsteady thermal-fluid problem [35], etc.
On the other hand, the recent development of mesh adaption techniques offers shape/topology optimization a possibility to achieve a better quality of boundaries by imposing mesh refinement around the solid/void material interface while coarser mesh elements remain in the rest of the domains. Furthermore, mesh adaption can be used to reduce the computational cost during the iterative process, especially when the maximum allowed volume fraction is relatively small, because after several iterations, a huge percentage of the computational domain becomes void. As a result, although the void domains contribute little to the overall functional performance, it still costs a huge percentage of the overall computational resources which can be regarded as a waste. Therefore, mesh adaption enjoys the advantage not only of high resolution of boundary, but also of cheaper computational cost.
The idea of adaptive grid design combining numerical grid-generation methods and adaptive finite element methods (FEM) was proposed in the early work of Kikuchi et al. [36]. Since this pioneering work, several successful implementations of adaptive mesh-based shape/topology optimization have been reported. Persson used mesh adaption (or mesh evolution) in between shape optimization iterations and showcased this in his thesis [37]. Micheletti et al. [38] proposed a density-based TO driven by anisotropic mesh adaption based on a recovery-based posteriori error estimator, which allows elimination of the checkerboard phenomenon often seen in density-based TO method. Yaji et al. [39] constructed a shape optimization method based on the convected level-set method, in which the mesh adaption is applied to the level-set to concentrate fine meshes in an area near the level-set interface. Kim et al. [40] presented a simple FreeFEM code to represent high-resolution boundaries of the optimal shape using the RDE-based TO method and adaptive mesh refinement, in which a fixed mesh is used to get a rough topology at first, and an adapted mesh is used to reach the final design with refined boundaries.
The benefits of mesh adaption become more evident in full-scale 3D cases. Nana et al. [41] implemented a fully-automated adaptation strategy for density-based TO method on 3D unstructured tetrahedral meshes, in which the density gradient is used to locate the solid–void interface and h-adaptation is applied for a better definition of this interface. Baiges et al. [42] proposed a fully parallelized level-set-based TO framework using a sparse grid stochastic collocation method, to calculate the statistical metrics of the topology optimization under uncertainty formulation, and a parallel adaptive mesh refinement method available in RefficientLib, to efficiently solve each of the stochastic collocation nodes. From the results, we can find that the optimal solution obtained with a deterministic design sometimes cannot converge to a symmetric design due to the lack of symmetry of the initial mesh.
Unlike mesh adaption strategies presented in above-mentioned works, Allaire, Dapogny, and Frey [43] independently proposed a mesh evolution algorithm based on the level-set method for shape/topology optimization, in which the level-set function is moving on an unstructured mesh, and the surface corresponding to the zero level-set is remeshed. Feppon et al. applied this mesh revolution strategy in several multi-physics optimization problems in his thesis [44] such as Lamé system, thermal conduction problem, and thermal fluid–structure coupling problem [45,46] for both 2D and 3D cases. In these works, a proposed initial guess needs to be discretized by means of a mesh which is deformed to a better shape. And the design results sometimes may vary with the initial guess.
As mentioned above, notable achievements have been made by applying those TO methodologies. However, challenging issues still exist: (1) in fixed mesh-based full-scale 3D TO algorithms, high resolution designs can be achieved using very fine meshes, which in turn, requires high computational cost, i.e., CPU cores and memory usage. (2) Adaptive mesh refinement is a powerful tool to achieve a high quality of boundaries. However, to date, only a few research works have extended the mesh adaption technique from 2D to 3D cases. (3) In the traditional level-set-based TO methods, the optimal solution usually depends on the initial guess. Additionally, mesh dependency issues are often observed in TO designs. To date, only a few research work reported effective strategies to mitigate initial guess/mesh dependency simultaneously. (4) Though the generative designs have been introduced in the real world industry for several years, pre/post-processing always requires a considerable manual effort which is very time consuming. More importantly, the process of threshold and smoothing inevitably introduces the loss of accuracy.
In this paper, we address the above mentioned challenges that impede an efficient full-scale 3D TO design. The main contributions of this paper can be summed up as follows:
- (1)
the RDE-based level-set TO method is formulated to solve the compliance minimization problem. Topological derivative acts as the source term in the RDE. Together with mesh refinement around the zero-level-set, the present method can mitigate the dependency to initial guess/mesh resolution to some extent. This feature is in contrast to what is generally seen in most of the traditional level-set-based shape optimization methods;
- (2)
by properly combining parallel computing and mesh adaption in the proposed TO method, one can efficiently achieve a high-resolution and clear boundary by solving a smaller problem than with a fixed-mesh framework. This feature is in contrast to what is generally required by most large-scale 3D TO frameworks;
- (3)
the boundary representation (B-Reps) is sampled with the volumetric representation of body-fitted tetrahedral meshes based on the level-set. The 3D printing data is directly converted from obtained generative meshes. We are free of most of the post-processing steps, i.e., thresholding and smoothing.
The remainder of this paper is organized as follows. Section 2 formulates the mathematical model of the level-set-based TO where the level-set function is updated using the RDE. Section 3 illustrates our parallel strategy, mesh adaption technique and optimization flow chart. In Section 4, several typical numerical examples are conducted and the corresponding runtime analyses are provided to validate the effectiveness of the proposed TO method and of the parallel framework/mesh adaption. Section 5 presents several 3D-printed prototypes to further support these remarkable features. Lastly, the conclusion is documented in Section 6.
Section snippets
Governing equations
Considering a domain under a surface traction, we intend to find the optimal topological configuration which can minimize the compliance (or maximize the stiffness) under a volume constraint. We assume the surface traction to be applied on a portion of boundary ∂Ω. The governing equation underlying the displacement field u can be formulated by the following linear elasticity equation:where σ(u) = 2μe(u) + λTr(e(u))Id is the stress
Distributed computing
As already briefly explained, in this work, FreeFEM [51,52] is used for the discretization of PDEs while PETSc [[53], [54], [55], [56]] is used for the linear algebra backend.
FreeFEM is a domain-specific language (DSL), internally written in C++. It makes extensive use of operator overloading and templates, something known as generic programming. Some other numerical libraries are integrated inside the DSL, such as PETSc and Mmg. They can be called from within FreeFEM. For parallelism, it
Numerical investigations
In this section, five design examples are studied to examine the effectiveness, stability and extensibility of the proposed TO algorithm. All numerical experiments are performed on the same workstation with at most 80 processes, and with a memory size of 256 GB. The CPU is an Intel(R) Xeon(R) E5-2698 V4, using the GNU compiler, OpenMPI, and the same linear solver parameters.
As already explained in feature 2, the optimal solution has a more complex configuration with a smaller value of τ in eq.
Manufacture of the prototype
Now, we translate the design results obtained in Section 4 to prototypes using 3D printing technology. The benefits of the body-fitted adaptive mesh can be further demonstrated in this section.
Conclusions
In this paper, we have presented an RDE-based level-set TO algorithm, in which we integrate mesh adaption in a parallel computing framework to solve full-scale 3D problems. The main idea is to utilize the level-set-based mesh evolution to achieve an essentially clear and high resolution solid–void material boundary, and at the same time to pursue a cheaper computational cost by coarsening the mesh distributed in the void domain. To this end, first, the compliance minimization TO problem has
Credit author statement
Hao Li: Conceptualization, Software, Validation, Writing – original draft. Takayuki Yamada: Methodology, Software, Supervision, Writing-Review. Pierre Jolivet: Software, Validation, Writing- Review & Editing. Kozo Furuta: Methodology, Validation, Writing-Review. Tsuguo Kondoh: Methodology, Supervision, Writing-Review. Kazuhiro Izui: Supervision, Writing-Review. Shinji Nishiwaki: Conceptualization, Supervision, Writing-Review, Funding acquisition.
Declaration of competing interest
The authors declare that they have no competing financial interests for this paper.
Acknowledgement
The authors wish to thank to two anonymous reviewers for their helpful comments on earlier drafts of the manuscript. The authors would like to thank the developers of the Mmg platform for their helpful advice with regards to the numerical implementation of mesh adaption. Fruitful discussions with Prof. Atsushi Suzuki from Cybermedia Center, Osaka University, and Dr. Minghao Yu from Dalian University of Technology are also gratefully acknowledged.
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