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2D design domain and level-set model. a Level-set function and zero level-set. b Design domain and boundary
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This paper proposes a new level set-based topology optimization (TO) method using a parallel strategy of Graphics Processing Units (GPUs) and the isogeometric analysis (IGA). The strategy consists of parallel implementations for initial design domain, IGA, sensitivity analysis and design variable update, and the key issues in the parallel implement...
Citations
... However, this method has three limitations due to the large design space, which is based on density. First, high-resolution topology optimization still requires expensive computing resources and considerable computing time, although various excellent algorithms have been proposed, such as the multi-CPU framework [16][17][18] , GPU computation [19][20][21][22] , and adaptive mesh refinement 23,24 . Second, since the optimization problem is highly nonlinear and nonconvex, the optimized metamaterials are closely related to the initialization. ...
... The reference structures (absent voids and cloaks) consist of 40 × 40 × 40 periodically tessellated base cells selected from the metamaterial dataset. We design an invisibility cloak for such a situation by excising a cylinder with a radius of 20 3 and subsequently filling the outer circular region (comprising a cylinder with an outer radius of 40 3 ) with material to design the invisibility cloak. Our surrounding structure is an isotropic structure identified from the dataset to emulate materials with isotropic properties, such as soil. ...
Mechanical metamaterial is a synthetic material that can possess extraordinary physical characteristics, such as abnormal elasticity, stiffness, and stability, by carefully designing its internal structure. To make metamaterials contain delicate local structures with unique mechanical properties, it is a potential method to represent them through high-resolution voxels. However, it brings a substantial computational burden. To this end, this paper proposes a fast inverse design method, whose core is an advanced deep generative AI algorithm, to generate voxel-based mechanical metamaterials. Specifically, we use the self-conditioned diffusion model, capable of generating a microstructure with a resolution of 128 3 to approach the specified homogenized tensor matrix in just 3 seconds. Accordingly, this rapid reverse design tool facilitates the exploration of extreme metamaterials, the sequence interpolation in metamaterials, and the generation of diverse microstructures for multi-scale design. This flexible and adaptive generative tool is of great value in structural engineering or other mechanical systems and can stimulate more subsequent research.
... Numerical examples show that the proposed topology optimization method can improve the electromechanical coupling coefficient. Since then, there have been many other works combining LSM and IGA (Xia et al. 2017;Nguyen et al. 2020;Gao et al. 2022;Jiang et al. 2022). Hou et al. (2017) first proposed an MMC-based explicit ITO, which using explicit design parameters to represent the geometry of structural components. ...
In this paper, a new isogeometric topology optimization (ITO) method based on the moving iso-surface threshold (MIST) method is proposed, and the corresponding MATLAB code is provided. The same nonuniform rational B-splines (NURBS) basis functions are used to construct a geometrical model and evaluate the objective function for minimal compliance problems. In MIST-based ITO, the physical response function is calculated by using the same NURBS basis functions as the geometry model. First, the physical response function values of control points are calculated by using the NURBS basis function and the physical response function values of the Gauss points. Second, the physical response function values of the knots (the element nodes) are obtained by fitting the control points using NURBS basis functions. Finally, the physical response surface is formed by connecting its nodal values. The structure topology is iteratively updated by using an iso-surface with an appropriate threshold to cut the physical response surface. Compared to traditional MIST, MIST-based ITO can improve the computational accuracy and computational efficiency of high-order elements. Several numerical examples demonstrate the effectiveness of the proposed method, verifying the validity of isogeometric topology optimization MATLAB codes in implementing MIST_based_ITO, which is provided in Online Appendix 1.
... Based on such a parallel framework, Aage et al. [24] proposed a powerful computational morphogenesis tool, which realized an optimized design of airplane wing with more than one-billion voxels. In addition to the CPU-based ones, several GPU-based parallel strategies were proposed to accelerate the optimization process [25][26][27]. ...
Structural topology optimization has undergone tremendous developments in the past three decades. Making the most of high-performance computing resources contributes to broadening the application of topology optimization in large-scale design problems. In this paper, a subdomain-based parallel strategy is proposed for general three-dimensional topology optimization. The optimization process is significantly accelerated through subdomain division, matrix calculation, and hard-kill algorithm. This strategy is integrated into an efficient and compact Python code, which is valid for design space with an arbitrary shape. The complete code, given in Appendix 1, can be easily extended to tackle different kinds of optimization problems. Four compliance minimization problems are taken as examples to demonstrate the efficiency of the proposed strategy. This work has potential applications in areas such as mechanical engineering, advanced manufacturing, and architectural design.
... Mahdavi et al. [38] used a master-slave programming paradigm to accomplish parallel topology optimisation with 2D compliance minimisation on a coarse-grain, multiple instructions, multiple data, and sharedmemory architecture. Several studies implemented parallel computing for the SIMP method [39,40], the LSM [41,42], and the ESO method [43] on GPU. ...
The bi-directional evolutionary structural optimisation (BESO) has attracted much interest in recent decades. However, the high computational cost of the topology optimisation method hinders its applications in large-scale industrial designs. In this study, a parallel BESO method is developed to solve high-resolution topology optimisation problems. An open-source computing platform, FEniCS, is used to parallelise the finite element analysis (FEA) and optimisation steps. Significant improvements in efficiency have been made to the FEA and the filtering process. An iterative solver, a reanalysis approach and a hard-kill option in
BESO have been developed to reduce the computational cost of the FEA. An isotropic filter scheme is used to eliminate the time-consuming elemental adjacency search process. The efficiency and effectiveness of the developed method are demonstrated by a series of numerical examples in both 2D and 3D. It is shown that the parallel BESO can efficiently solve problems with more than 100 million tetrahedron elements on a 14-core CPU server. This work holds great potential for high-resolution design problems in engineering and architecture.
Keywords: Topology optimisation, Bi-directional evolutionary structural optimisation, FEniCS, Parallel computing, High-resolution
... Reducing the grain size in the assembly-free GPU implementation is another way to boost the computational speed [17]. GPU acceleration has also been used in [18] for level set method (LSM) and isogeometric analysis (IGA) based topology optimization. ...
We propose an efficient implementation of a new hybrid topology optimization algorithm based on multigrid approach that combines the parallelization strategy of CPU using OpenMP and heavily multithreading capabilities of modern Graphics Processing Units (GPU). In addition to that, significant computational efficiency in memory requirement has been achieved using homogenization strategy. The algorithm has been integrated with versatile computing platform of MATLAB for ease of use and customization. The bottlenecking repetitive solution of the state equation has been solved using an optimized geometric multigrid approach along with CUDA parallelization enabling an order of magnitude faster in computational time than current state of the art implementations. The main novelty lies in the efficient implementation wherein on the fly computation of auxiliary matrices in the multigrid scheme and modification in interpolation schemes using homogenization strategy removes memory limitation of GPUs. Memory hierarchy of GPU has also been exploited for further optimized implementations. All these enable solution of structures involving hundred millions of three dimensional brick elements to be accomplished in a standard desktop computer or a workstation. Performance of the proposed algorithm is illustrated using several examples including design dependent loads. Results obtained indicate the excellent performance and scalability of the proposed approach.
... It is only in recent years that some work has been done to improve computing efficiency, and it is still in its infancy. Reference [40] proposed a new level set-based topology optimization (TO) method using a parallel strategy of Graphics Processing Units (GPUs) and IGA. Reference [41] proposed and investigate new robust preconditioners for spacetime IGA of parabolic evolution problems. ...
In this paper, a new computation scheme based on parallelization is proposed for Isogeometric analysis. The parallel computing is introduced to the whole progress of Isogeometric analysis. Firstly, with the help of the “tensor-product” and “iso-parametric” feature, all the Gaussian integral points in particular element can be mapped to a global matrix using a transformation matrix that varies from element. Then the derivatives of Gauss integral points are computed in parallel, the results of which can be stored in a global matrix. And a middle layer is constructed to assemble the final stiffness matrices in parallel. The numerical example results show that: the method presented in the paper can reduce calculation time and improve the use rate of computing resources.
... Le et al. [41] , Duvigneau [42] and Zhang et al. [43] applied IGA method to heat transfer problems. Wang et al. [44][45][46][47] applied the isogeometric analysis technique in the field of structural topology optimization and prove its efficiency. ...
... plr sin φdφ (45) where F t represents the friction resistance, F t 1 and F t 2 respectively represent the friction resistance caused by shear flow and pressure flow. The calculation of dynamic characteristics mainly involves the oil film stiffness and oil film damping of the lubrication zone, both of which contain direct terms and cross terms respectively. ...
This paper attempts to solve the Reynolds equation of the lubrication problem in hybrid bearing using high-order isogeometric analysis method. The analysis model consists of two parts: the static characteristic calculation part composed of oil film static pressure distribution, oil film thickness and oil film bearing capacity; the dynamic characteristic calculation part including damping coefficient and stiffness coefficient. A mapping strategy based on NURBS basis function is used to realize the application of imposed Dirichlet condition inside the analysis domain of hybrid bearing lubrication characteristics. The analysis of hybrid bearing in fast reactor with complex dynamic and static pressure oil cavity structure shows that: (1) In high-order isogeometric calculation framework, imposed boundary conditions are applied accurately, and dynamic and static characteristics calculation results are given; (2) Compared with the finite element method, when using high-order isogeometric analysis method to solve the viscous fluid motion, the solution requires fewer degrees of freedom to achieve grid independence, and the computational efficiency is also improved in terms of time.
... In addition, truncated hierarchical B-spline (Xie et al. 2020) and T-splines (Zhao et al. 2020) are applied to the ITO method to enhance its applicability to arbitrarily shaped design domains. A level-set based ITO method that integrates the NURBS-based IGA method and the parameterized level set method was developed by (Wang and Benson 2016a), which was extended to arbitrary geometrical constraint (Wang and Benson 2016b) and GPU parallel strategy (Xia et al. 2017). In recent years, level-set based ITO method has been extended for various design problems concerning flexoelectric material (Ghasemi et al. 2017), stress (Jahangiry and Tavakkoli 2017), heat conduction (Jahangiry and Jahangiri 2019), fundamental eigenfrequency (Xu et al. 2019) and graded microstructures design (Yu et al. 2020). ...
This work proposed an evolutionary topology optimization using isogeometric analysis for continuum structures with smoothed boundary representation. Specifically, continuum structure is analyzed and optimized in fixed background IGA mesh. The design domain of continuum structure is a spatially continuous design variable field with discrete topology design variable. Moreover, the corresponding sensitivity field is constructed by the same NURBS formalism as design and analysis. Hence, the structural topology with smoothed boundary is iteratively updated by intersecting the sensitivity field with an appropriate threshold. Moreover, the post-processing phase involving the CAD reconstruction of the optimized geometry is trivial for 2D problems and it needs a few extra operations for 3D cases. Several 2D and 3D numerical examples including the fully CAD compatible and editable post-processing procedure are performed to demonstrate the effectiveness and efficiency of the proposed method, as well as its potential in engineering application.
... There are generally three main ways of improving the computational efficiency of topology optimization. One way is to make full use of computational resources and use hardware acceleration, such as GPU acceleration on unstructured meshes (Zegard and Paulino 2013), GPU parallel strategy for PLSM (Xia et al. 2017), and the fully parallel level set method . Another way is to find the shortcomings of the existing algorithm, and speed up by improving the algorithm, such as the extended level set method (Wang et al. 2007) to avoid the initialization of H-J function in traditional LSM, accelerated fixed-point formulation (Li et al. 2020) for compliance minimization problem, and accelerated optimization by means of triple acceleration method (Liao et al. 2019), deep learning method (Kallioras et al. 2020) or discrete cosine transform , etc. ...
This article aims to present a novel topological design approach, which is inspired by the famous density method and parametric level set method, to control the structural complexity in the final optimized design and to improve computational efficiency in structural topology optimization. In the proposed approach, the combination of radial basis function and the SIMP formula is introduced to describe the distribution of the fictitious density field in the design domain. By changing the radius and distribution of radial function, the structural complexity can be controlled. Meanwhile, it is found that the proposed method can naturally avoid checkerboard design. In order to improve the computational efficiency affected by the number of design variables, we propose to redefine the support points so that the number of support points is much smaller than that of the observation points of the radial function. It follows that the number of design variables can be reduced to a great extent. Several numerical examples are tested to show the feasibility and effectiveness of the presented method.
... Here, we do not perform any computation on GPU and instead stick to multi-processor CPU. However, we mention that GPU based topology optimization has been applied to density based methods [124], ESO (evolutionary structural optimization) [123] and to level-set based method [187]. ...
The main contribution of this thesis is the theoretical and the numerical study shape and topology optimization for nonlinear phenomena, like contact, plasticity, and fracture using the level-set method. One application of the contact boundary condition for an idealized-bolt model is also proposed.The governing equations of the three physics dealt with in this thesis: contact, plasticity and damage, are theoretically not shape-differentiable. In each case, we construct an approximation by penalization, regularization or a combination of the two. The approximations for contact and plasticity are shown to be well-posed and to admit solutions that converge to the exact solution. For each physics, the shape sensitivity analysis is performed on the approximate model and the resultant adjoint problem is shown to be well-posed under technical assumptions. The shape optimization is implemented numerically using a level-set method with body-fitted remeshing, which captures the boundary of the shapes while allowing for topology changes. Numerical results are presented in 2D and 3D. We also discuss high-performance computing for linear elasticity and for fracture model, and present a few 3D results.
