Large-scale Topology Optimization Research Articles

Concurrent Island scanning pattern and large-scale topology optimization method for laser powder bed fusion processed parts

Concurrent Island scanning pattern and large-scale topology optimization method for laser powder bed fusion processed parts

A fast dimension reduction framework for large-scale topology optimization of grid-layout offshore wind farm collector systems

The construction of large-scale offshore wind farms faces the difficulty of solving the optimization problem of collector systems. Due to its NP-hard feature, this problem cannot be solved straightforwardly. This paper proposes an efficient and precise solution framework suitable for large-scale collector system optimization and raises the main influencing factors of time complexity. First, according to a heuristic idea, an arc selection algorithm is established to tighten the feasible region. Second, an optimization framework based on the greedy algorithm and mixed integer quadratic programming (MIQP) is established with excellent efficiency and accuracy. The overall problem is decomposed into the master problem of topology optimization and the sub-problem of cable selection. A bi-level model that adopts both greedy algorithm and global optimization is proposed, iterating through the master problem and the sub-problem by transferring parameters and updating constraints. Third, a large number of examples are analyzed, and the advantages of the proposed algorithms and model are comprehensively verified. In the end, time complexity analysis is carried out, and the empirical formula of solution time is obtained by fitting. Based on the proposed method, large-scale problems can be solved much faster than traditional methods and reach a balance of both speed and quality.

Inertial projected gradient method for large-scale topology optimization

We present an inertial projected gradient method for solving large-scale topology optimization problems. We consider the compliance minimization problem, the heat conduction problem and the compliant mechanism problem of continua. We use the projected gradient method to efficiently treat the linear constraints of these problems. Also, inertial techniques are used to accelerate the convergence of the method. We consider an adaptive step size policy to further reduce the computational cost. The proposed method has a global convergence property. By numerical experiments, we show that the proposed method converges fast to a point satisfying the first-order optimality condition with high accuracy compared with the existing methods. The proposed method has a low computational cost per iteration, and is thus effective in a large-scale problem.

An adaptive and scalable artificial neural network-based model-order-reduction method for large-scale topology optimization designs

An adaptive and scalable artificial neural network-based model-order-reduction method for large-scale topology optimization designs

Multi-functional topology optimization of Victoria cruziana veins.

The growth and development of biological tissues and organs strongly depend on the requirements of their multiple functions. Plant veins yield efficient nutrient transport and withstand various external loads. Victoria cruziana, a tropical species of the Nymphaeaceae family of water lilies, has evolved a network of three-dimensional and rugged veins, which yields a superior load-bearing capacity. However, it remains elusive how biological and mechanical factors affect their unique vein layout. In this paper, we propose a multi-functional and large-scale topology optimization method to investigate the morphomechanics of Victoria cruziana veins, which optimizes both the structural stiffness and nutrient transport efficiency. Our results suggest that increasing the branching order of radial veins improves the efficiency of nutrient delivery, and the gradient variation of circumferential vein sizes significantly contributes to the stiffness of the leaf. In the present method, we also consider the optimization of the wall thickness and the maximum layout distance of circumferential veins. Furthermore, biomimetic leaves are fabricated by using the three-dimensional printing technique to verify our theoretical findings. This work not only gains insights into the morphomechanics of Victoria cruziana veins, but also helps the design of, for example, rib-reinforced shells, slabs and dome skeletons.

A partition and microstructure based method applicable to large-scale topology optimization

Topology optimization (TO) of large-scale structures is a computationally demanding process that challenges its widespread adoption in industrial design. We present a new partition based TO framework applicable to the design of large-scale problems and apply it to mechanical stiffness optimization problems. The method employs, first, a data-driven physical partitioning of strain energy contours to partition the design domain, and then a partition based TO. In contrast to conventional topology optimization in which the number of design variables is typically equal to the number of elements in the discretized domain, the proposed method assigns density design variables to each spatial partition leading to significant computational cost reduction and convergence acceleration. The constitutive matrix required for finite element analysis is iteratively determined according to the Hashin-Shtirkman upper bounds based on partition densities. Once the optimized partition densities are achieved, the manufacturable binary structure is realized by mapping a set of generated high-performance isotropic microstructure cells onto partition elements. To validate the capability, effectiveness, and efficiency of the proposed method, several numerical examples are provided. The optimized structures using macrostructural analysis exhibit comparable performance to the conventional SIMP method for small-size problems. Besides, the TO results for the large-scale problems suggest significant computational cost efficiency.

Accelerating Large-scale Topology Optimization: State-of-the-Art and Challenges

Large-scale structural topology optimization has always suffered from prohibitively high computational costs that have till date hindered its widespread use in industrial design. The first and major contributor to this problem is the cost of solving the Finite Element equations during each iteration of the optimization loop. This is compounded by the frequently very fine 3D models needed to accurately simulate mechanical or multi-physical performance. The second issue stems from the requirement to embed the high-fidelity simulation within the iterative design procedure in order to obtain the optimal design. The prohibitive number of calculations needed as a result of both these issues, is often beyond the capacities of existing industrial computers and software. To alleviate these issues, the last decade has opened promising pathways into accelerating the topology optimization procedure for large-scale industrial sized problems, using a variety of techniques, including re-analysis, multi-grid solvers, model reduction, machine learning and high-performance computing, and their combinations. This paper attempts to give a comprehensive review of the research activities in all of these areas, so as to give the engineer both an understanding as well as a critical appreciation for each of these developments.

Discrete variable topology optimization for compliant mechanism design via Sequential Approximate Integer Programming with Trust Region (SAIP-TR)

The discrete variable topology optimization method based on Sequential Approximate Integer Programming (SAIP) and Canonical relaxation algorithm demonstrates its potential to solve large-scale topology optimization problem with 0–1 optimum designs. However, currently, this discrete variable method mainly applies to the minimum compliance problem. The compliant mechanism design is another widely studied topic with distinguishing features. First, the objective function for the compliant mechanism design is non-monotonic with the material usage. Second, since de facto hinges always occur, the minimum length scale control is indispensable for manufacturability. These two issues are well studied in the SIMP approach but bring great challenges when topology optimization problems are formulated in the frame of discrete variables. The present paper generalizes this discrete variable method for the compliant mechanism design problems with minimum length scale control. Firstly, the sequential approximate integer programming with trust region (SAIP-TR) framework is proposed to directly restrict the variation of discrete design variables. Different from the continuous variable optimization, the non-linear trust region constraint can be formulated as a linear constraint under the SAIP framework. By using a merit function, two different trust region adjustment strategies that can self-adaptively adjust the precision of the sub-problems from SAIP-TR are explored. Secondly, a geometric constraint to control the minimum length scale for the material phase and void phase in the framework of discrete design variables is proposed, which suppresses de facto hinges and reduces stress concentration in optimum design. The related issue of feasibility of sub-problem is discussed. By combining the SAIP-TR framework with the geometric constraint, some different hinge-free compliant mechanism designs are successfully obtained.

Three-dimensional adaptive mesh refinement in stress-constrained topology optimization

Structural optimization software that can produce high-resolution designs optimized for arbitrary cost and constraint functions is essential to solve real-world engineering problems. Such requirements are not easily met due to the large-scale simulations and software engineering they entail. In this paper, we present a large-scale topology optimization framework with adaptive mesh refinement (AMR) applied to stress-constrained problems. AMR allows us to save computational resources by refining regions of the domain to increase the design resolution and simulation accuracy, leaving void regions coarse. We discuss the challenges necessary to resolve such large-scale problems with AMR, namely, the need for a regularization method that works across different mesh resolutions in a parallel environment and efficient iterative solvers. Furthermore, the optimization algorithm needs to be implemented with the same discretization that is used to represent the design field. To show the efficacy and versatility of our framework, we minimize the mass of a three-dimensional L-bracket subject to a maximum stress constraint and maximize the efficiency of a three-dimensional compliant mechanism subject to a maximum stress constraint.

Consistent boundary conditions for PDE filter regularization in topology optimization

Design variables in density-based topology optimization are typically regularized using filtering techniques. In many cases, such as stress optimization, where details at the boundaries are crucially important, the filtering in the vicinity of the design domain boundary needs special attention. One well-known technique, often referred to as “padding,” is to extend the design domain with extra layers of elements to mitigate artificial boundary effects. We discuss an alternative to the padding procedure in the context of PDE filtering. To motivate this augmented PDE filter, we make use of the potential form of the PDE filter which allows us to add penalty terms with a clear physical interpretation. The major advantages of the proposed augmentation compared with the conventional padding is the simplicity of the implementation and the possibility to tune the boundary properties using a scalar parameter. Analytical results in 1D and numerical results in 2D and 3D confirm the suitability of this approach for large-scale topology optimization.

Hybrid algorithms for handling the numerical noise in topology optimization

This paper presents new hybrid methods for the identification of optimal topologies by combining the teaching–learning based optimization (TLBO) and the method of moving asymptotes (MMA). The topology optimization problem is parameterizing with a low dimensional explicit method called moving morphable components (MMC), to make the use of evolutionary algorithms more efficient. Gradient-based solvers have good performance in solving large-scale topology optimization problems. However, in unconventional cases same as crashworthiness design in which there is numerical noise in the gradient information, the uses of these algorithms are unsuitable. The standard evolutionary algorithms can solve such problems since they don’t need gradient information. However, they have a high computational cost. This paper is based upon the idea of combining metaheuristics with mathematical programming to handle the probable noises and have faster convergence speed. Due to the ease of computations, the compliance minimization problem is considered as the case study and the artificial noise is added in gradient information.

Towards solving large-scale topology optimization problems with buckling constraints at the cost of linear analyses

This work presents a multilevel approach to large-scale topology optimization accounting for linearized buckling criteria. The method relies on the use of preconditioned iterative solvers for all the systems involved in the linear buckling and sensitivity analyses and on the approximation of buckling modes from a coarse discretization. The strategy shows three main benefits: first, the computational cost for the eigenvalue analyses is drastically cut. Second, artifacts due to local stress concentrations are alleviated when computing modes on the coarse scale. Third, the ability to select a reduced set of important global modes and filter out less important local ones. As a result, designs with improved buckling resistance can be generated with a computational cost little more than that of a corresponding compliance minimization problem solved for multiple loading cases. Examples of 2D and 3D structures discretized by up to some millions of degrees of freedom are solved in Matlab to show the effectiveness of the proposed method. Finally, a post-processing procedure is suggested in order to reinforce the optimized design against local buckling.

On-the-fly model reduction for large-scale structural topology optimization using principal components analysis

Despite a solid theoretical foundation and straightforward application to structural design problems, 3D topology optimization still suffers from a prohibitively high computational effort that hinders its widespread use in industrial design. One major contributor to this problem is the cost of solving the finite element equations during each iteration of the optimization loop. To alleviate this cost in large-scale topology optimization, the authors propose a projection-based reduced-order modeling approach using proper orthogonal decomposition for the construction of a reduced basis for the FE solution during the optimization, using a small number of previously obtained and stored solutions. This basis is then adaptively enriched and updated on-the-fly according to an error residual, until convergence of the main optimization loop. The method of moving asymptotes is used for the optimization. The techniques are validated using established 3D benchmark problems. The numerical results demonstrate the advantages and the improved performance of our proposed approach.

A method using successive iteration of analysis and design for large-scale topology optimization considering eigenfrequencies

Repeatedly solving the generalized eigenvalue problems by far dominates the computational cost in large-scale topology optimization involving natural frequency constraints. This study proposes a method for dynamic topology optimization problems considering natural frequencies using successively executed iterations for the structural analysis and design. By using the Rayleigh quotients as approximations of the natural frequencies and achieving sequential approximation of the eigenpairs through inverse iteration-like procedures to improve the eigenvectors along with the topological evolution of the structure, the method avoids solving the time-consuming eigenvalue problem in each design iteration. This makes the method particularly suitable for large-scale frequency-constrained topology optimization problems. The convergence property of the method is analyzed under the assumption of sufficiently small design changes between two successive design iterations. Numerical examples regarding frequency and frequency gap constraints show that this method is able to realize concurrent convergence of the eigenvalue analysis and design optimization, and is more efficient than the conventional double-loop approach.

On using a zero lower bound on the physical density in material distribution topology optimization

The current paper studies the possibility of allowing a zero lower bound on the physical density in material distribution based topology optimization. We limit our attention to the standard test problem of minimizing the compliance of a linearly elastic structure subject to a constant forcing. First order tensor product Finite Elements discretize the problem. An elementwise constant material indicator function defines the discretized, elementwise constant, physical density by using filtering and penalization. To alleviate the ill-conditioning of the stiffness matrix, due to the variation of the elementwise constant physical density, we precondition the system. We provide a specific spectral analysis for large matrix sizes for the one-dimensional problem with Dirichlet–Neumann conditions in detail, even if most of the mathematical tools apply also in a d-dimensional setting, d≥2. It is easy to find an elementwise constant material indicator function so that the resulting preconditioned system matrix is singular when allowing the vanishing physical densities. However, for a large class of material indicator functions, the corresponding preconditioned system matrix has a condition number of the same order as the system matrix for the case when the physical density is one in all elements. Finally, we critically report and illustrate results from numerical experiments: as a conclusion, it is indeed possible to solve large-scale topology optimization problems, allowing a vanishing physical density, without encountering ill-conditioned system matrices during the optimization.

3D topology optimization for cost and time minimization in additive manufacturing

As the frontier of modern-day engineering challenges pushes forward, the integration of multiple strategies to reduce manufacturing cost and increase component performance has engineers turning to tools such as topology optimization (TO) and additive manufacturing (AM). Recent focus on these topics has led to the bridging of the gap between these two tools and the making of their integration in the conventional design cycle as seamless as possible. This paper expands upon existing mathematical constructs by providing an algorithm to minimize the cost and time associated with additively manufactured parts within a three-dimensional topology optimization framework. The formulation has been constructed in such a manner to accommodate large-scale topology optimization problems, including a filtering scheme requiring minimal storage of additional mesh information and an iterative finite element analysis solver. A rigorous trade-off analysis is conducted to determine the optimal contribution of additive manufacturing factors to minimize build time. A perimeter method-inspired approach for optimization of the surface area is explored, suggesting benefits for AM-specific process mechanics. Multiple academic example problems included in this work illustrate the applicability of this approach to three-dimensional geometries; physical models of these example problems created via fused filament fabrication serve to validate the numerical results obtained herein.

Large-scale topology optimization incorporating local-in-time adjoint-based method for unsteady thermal-fluid problem

This note briefly reports on the applicability of the local-in-time (LT) adjoint-based method to a large-scale topology optimization problem with unsteady thermal-fluid. The basic idea of the LT method is to divide a time-dependent optimization problem into reasonable subproblems to reduce memory cost. We demonstrate that the proposed method solves the large-scale topology optimization problem by incorporating the LT method, the lattice Boltzmann method, and parallel computing.

Optimal duct layout for HVAC using topology optimization

Automated design procedure for duct layout has potential for further improving system efficiency. Currently, duct layout is often based on the discretion of the designer and might miss possible areas for energy improvement or meeting system targets, for example, minimal pressure drop, temperature uniformity, and contaminant removal, just to mention a few. Since the initial duct layout design can be considered as a topology problem, topology optimization methods have great potential to advance green building efficiency and design. Topology optimization has been successfully applied to various physics problems, such as structural, heat conduction, and fluidics, yet its application for the built environment, HVAC&R and its subsystems, remains highly unexplored. In the current article, the applicability of topology optimization for determining optimal duct layout is presented. A survey of successfully applied fluid topology optimization cases is presented in the article with outlooks of its applicability to initial duct design. Also, some initial investigations are compared to an existing design. New interpretations are given for intermediate porous flow results in the presented examples. The computational cost of performing large-scale topology optimization is currently one of the biggest limitations of its applicability but with the continuous improvements in computing technology it is indeed a possibility.

A Large-scale Topology Optimization Method for Three-dimensional Transient Thermal-fluid Flows

A Large-scale Topology Optimization Method for Three-dimensional Transient Thermal-fluid Flows

流体問題を対象とした大規模トポロジー最適化

The aim of this research is to construct a large-scale topology optimization method using the lattice Boltzmann method (LBM) in fluid flow problems. Since the LBM is formulated with a linear equation rather than the nonlinear Navier-Stokes equation, its algorithm can be easily constructed. Therefore, the explicit scheme of the LBM makes it especially suitable when implementing large-scale parallel computations. The design sensitivities are derived based on the adjoint variable method, in which the optimization problem is formulated by taking into account the transient property of lattice Boltzmann equation. As a result, the adjoint equation can be derived as a simple time evolution equation, called adjoint lattice Boltzmann equation, which is also suitable when implementing large-scale parallel computations. Additionally, our optimization algorithm is based on the one-shot method, in order to greatly reduce the numerical cost of sensitivity calculations. The efficacy of our proposed method is tested with a pressure drop minimization problem and a heat sink design problem.