Topology Optimization Problem Research Articles

Application of thermal energy storage technology in power grid topology

In order to solve the problem of grid topology optimization, the author proposes the application of renewable energy and energy storage technology in the grid topology. The author first defines the grid graph data model, then designs a grid topology analysis framework, and finally realizes several grid topology analysis applications on this basis. The experimental results show that graph database can better support the concurrent analysis of large-scale users, and the average time required for analysis is significantly less and the advantages are greater. When the number of users reached 200, it took 0.07 seconds for graph database and 0.13 seconds for relational database. In conclusion, the power grid topology analysis method based on renewable energy and energy storage technology can greatly improve the performance and meet the needs of practical scheduling.

Explicit Isogeometric Topology Optimization Method with Suitably Graded Truncated Hierarchical B-Spline

This work puts forward an explicit isogeometric topology optimization (ITO) method using moving morphable components (MMC), which takes the suitably graded truncated hierarchical B-Spline based isogeometric analysis as the solver of physical unknown (SGTHB-ITO-MMC). By applying properly basis graded constraints to the hierarchical mesh of truncated hierarchical B-splines (THB), the convergence and robustness of the SGTHB-ITOMMC are simultaneously improved and the tiny holes occurred in optimized structure are eliminated, due to the improved accuracy around the explicit structural boundaries. Moreover, an efficient computational method is developed for the topological description functions (TDF) of MMC under the admissible hierarchical mesh, which consists of reducing the dimensionality strategy for design space and the locally computing strategy for hierarchical mesh. We apply the above SGTHB-ITO-MMC with improved efficiency to a series of 2D and 3D compliance design problems. The numerical results show that the proposed SGTHB-ITO-MMC method outperforms the traditional THB-ITO-MMC method in terms of convergence rate and efficiency. Therefore, the proposed SGTHB-ITO-MMC is an effective way of solving topology optimization (TO) problems.

A Modified Bi-Directional Evolutionary Structural Optimization Procedure with Variable Evolutionary Volume Ratio Applied to Multi-Objective Topology Optimization Problem

Natural frequency and dynamic stiffness under transient loading are two key performances for structural design related to automotive, aviation and construction industries. This article aims to tackle the multi-objective topological optimization problem considering dynamic stiffness and natural frequency using modified version of bi-directional evolutionary structural optimization (BESO). The conventional BESO is provided with constant evolutionary volume ratio (EVR), whereas low EVR greatly retards the optimization process and high EVR improperly removes the efficient elements. To address the issue, the modified BESO with variable EVR is introduced. To compromise the natural frequency and the dynamic stiffness, a weighting scheme of sensitivity numbers is employed to form the Pareto solution space. Several numerical examples demonstrate that the optimal solutions obtained from the modified BESO method have good agreement with those from the classic BESO method. Most importantly, the dynamic removal strategy with the variable EVR sharply springs up the optimization process. Therefore, it is concluded that the modified BESO method with variable EVR can solve structural design problems using multi-objective optimization.

A bionic topology optimization method with an additional displacement constraint

<abstract><p>Displacement is an important measure of stiffness, and its constraint must be considered in many real engineering designs. However, traditional volume-constrained compliance minimization methods for load-bearing structures do not deal with displacements of practical importance directly. Based on this situation, the paper extends an improved bionic topology optimization method to solve the topology optimization problem with an additional displacement constraint. The updates of density design variables are based on an improved bone remodeling algorithm rather than gradient information employed by traditional methods. An explicit relationship between the threshold in the bone remodeling algorithm and target node displacement is constructed to satisfy displacement constraint. As a result, one will obtain a topology with an optimal cost-weighted sum of stiffness and mass while the target node displacement does not exceed its predefined limit. 2D and 3D examples are given to demonstrate the effectiveness of the proposed method.</p></abstract>

Topological optimization algorithm for mechanical-electrical coupling of periodic composite materials

<abstract><p>In this paper, a topology optimization algorithm for the mechanical-electrical coupling problem of periodic composite materials is studied. Firstly, the homogenization problem of the mechanical-electrical coupling topology optimization problem of periodic composite materials is established by the multi-scale asymptotic expansion method. Secondly, the topology optimization algorithm for the mechanical-electrical coupling problem of periodic composite materials is constructed by finite element method, solid isotropic material with penalisation method and homogenization method. Finally, numerical results show that the proposed algorithm is effective to calculate the optimal structure of the periodic composite cantilever beam under the influence of the mechanical-electrical coupling.</p></abstract>

Phase field topology optimisation for 4D printing

This work concerns a structural topology optimisation problem for 4D printing based on the phase field approach. The concept of 4D printing as a targeted evolution of 3D printed structures can be realised in a two-step process. One first fabricates a 3D object with multi-material active composites and apply external loads in the programming stage. Then, a change in an environmental stimulus and the removal of loads cause the object to deform in the programmed stage. The dynamic transition between the original and deformed shapes is achieved with appropriate applications of the stimulus. The mathematical interest is to find an optimal distribution for the materials such that the 3D printed object achieves a targeted configuration in the programmed stage as best as possible. Casting the problem as a PDE-constrained minimisation problem, we consider a vector-valued order parameter representing the volume fractions of the different materials in the composite as a control variable. We prove the existence of optimal designs and formulate first order necessary conditions for minimisers. Moreover, by suitable asymptotic techniques, we relate our approach to a sharp interface description. Finally, the theoretical results are validated by several numerical simulations both in two and three space dimensions.

A Simple and Efficient Structural Topology Optimization Implementation Using Open-Source Software for All Steps of the Algorithm: Modeling, Sensitivity Analysis and Optimization

This work analyzes the implementation of a continuous method of structural topology optimization (STO) using open-source software for all stages of the topology optimization problem: modeling, sensitivity analysis and optimization. Its implementation involves three main components: numerical analysis using the Finite Element Method (FEM), sensitivity analysis using an Adjoint method and an optimization solver. In order to allow the automated numerical solution of Partial Differential Equations (PDEs) and perform a sensitivity analysis, FEniCS and Dolfin Adjoint software are used as tools, which are open-source code. For the optimization process, Ipopt (Interior Point OPTimizer) is used, which is a software package for nonlinear optimization scale designed to find (local) solutions of mathematical optimization problems. The topological optimization method used is based on the SIMP-Solid Isotropic Material with Penalization interpolation. The considered problem is the minimization of compliance/maximization of stiffness, considering the examples of recurrent structures in the literature in 2D and 3D. A density filtering algorithm based on Helmholtz formulation is used. The complete code involves 51 lines of programming and is presented and commented in detail in this article.

Topology optimization under microscale uncertainty using stochastic gradients

This paper considers the design of structures made of engineered materials, accounting for uncertainty in material properties. We present a topology optimization approach that optimizes the structural shape and topology at the macroscale assuming design-independent uncertain microstructures. The structural geometry at the macroscale is described by an explicit level set approach, and the macroscopic structural response is predicted by the eXtended Finite Element Method (XFEM). We describe the microscopic layout by either an analytic geometric model with uncertain parameters or a level-cut from a Gaussian random field. The macroscale properties of the microstructured material are predicted by homogenization. Considering the large number of possible microscale configurations, one of the main challenges of solving such topology optimization problems is the computational cost of estimating the statistical moments of the cost and constraint functions and their gradients with respect to the design variables. Methods for predicting these moments, such as Monte Carlo sampling, and Taylor series and polynomial chaos expansions often require a large number of random samples resulting in an impractical computation. To reduce this cost, we propose an approach wherein, at every design iteration, we only use a small number of microstructure configurations to generate an independent, stochastic approximation of the gradients. These gradients are then used either with a gradient descent algorithm, namely Adaptive Moment (Adam), or the globally convergent method of moving asymptotes (GCMMA). Three numerical examples from structural mechanics are used to show that the proposed approach provides a computationally efficient way for macroscale topology optimization in the presence of microstructural uncertainty and enables the designers to consider a new class of problems that are out of reach today with conventional tools.

Computational Acceleration of Topology Optimization Using Deep Learning

Topology optimization is a computationally expensive process, especially when complicated designs are studied, and this is mainly due to its finite element analysis and iterative solvers incorporated into the algorithm. In the current work, we investigated the application of deep learning methods to computationally accelerate topology optimization. We tested and comparatively analyzed three types of improved neural network models using three different structured datasets and achieved satisfactory results that allowed for the generation of topology optimized structures in 2D and 3D domains. The results of the studies show that the improved Res-U-Net and U-Net are reliable and effective methods among deep learning approaches for the computational acceleration of topology optimization problems. Moreover, based on the results, it is evaluated that Res-U-Net gives better results than U-Net for higher iterations. We also showed that the proposed CNN method is highly accurate and required much less training time compared to existing methods.

The Boussinesq System with Nonsmooth Boundary Conditions: Existence, Relaxation, and Topology Optimization

In this paper, we tackle a topology optimization problem which consists in finding the optimal shape of a solid located inside a fluid that minimizes a given cost function. The motion of the fluid is modeled thanks to the Boussinesq system which involves the unsteady Navier-Stokes equation coupled to a heat equation. In order to cover several models presented in the literature, we choose a non-smooth formulation for the outlet boundary conditions and an optimization parameter of bounded variations. This paper aims at proving existence of solutions to the resulting equations, along with the study of a relaxation scheme of the non-smooth conditions. A second part covers the topology optimization problem itself for which we proved the existence of optimal solutions and provides the definition of first order necessary optimality conditions.

Chaotic marine predators algorithm for global optimization of real-world engineering problems

A novel metaheuristic called Chaotic Marine Predators Algorithm (CMPA) is proposed and investigated for the optimization of engineering problems. CMPA integrates the exploration merits of the recently proposed Marine Predators Algorithm (MPA) with the chaotic maps exploitation capabilities. Several chaotic maps were applied in the proposed CMPA to govern MPA parameters that eventually led to controlled exploration and exploitation of search. This study makes an initial attempt to explore and employ CMPA in decoding complex and challenging design and manufacturing problems. For performance evaluation of the proposed algorithm, CEC 2020 numerical problems having different dimensions and five widely adopted constrained design problems were solved. For all problems, both qualitative and qualitative results are examined and discussed. Moreover, two case studies of multi-pass turning were examined by the proposed CMPA algorithm to optimize the cutting operation with a minimum cost of production per unit objective. Furthermore, the suggested CMPA algorithm has been investigated for solving a real-world structural topology optimization problem. Statistical analysis is performed, and the results of CMPA are compared with twelve distinguished algorithms. Outcomes of the proposed variant algorithm on the benchmarks demonstrate its significantly improved performance relative to other optimizers including a variant of MPA and two state-of-the-art IEEE CEC competitions winners algorithms. Findings from the manufacturing process exhibit CMPA proficiency in solving arduous real-world design problems.

Parallel BESO framework for solving high-resolution topology optimisation problems

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. A PDE-based 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-cores CPU server. This work holds great potential for high-resolution design problems in engineering and architecture.

Topological optimization of constructive solid geometry of lightweight structures

The work is devoted to solving the problem of topological optimization in the design of lightweight structures. The advantages and disadvantages of popular topological optimization methods are analyzed. One of the characteristic problems of all methods is the dependence of their results on grid partitioning. Besides, there is a necessity to recast the geometry model of the structure. To solve this problem, it is proposed to perform the formation of cavities based on the ESO method directly at the constructive solid geometry from the shape elements. The main stages of the proposed approach are described, the use of clustering to simplify the shape of the surfaces of the formed cavities is justified.

Evolutionary structural optimization of vulcanization molds

A method of evolutionary topology optimization of vulcanization molds according to the criteria of the maximum temperature deviation in the volume of the rubber mixture from the set value and the mechanical compliance of the structure is proposed. To take into account the transient of the mold heating process, the condition of specific heat absorption in the volume of the processed rubber mixture is proposed. The problem of topology optimization of a mold for the manufacture of a rubber diaphragm has been solved. The analysis of the received results from the point of view of efficiency of the decision and manufacturability of a design is carried out. An engineering interpretation of the results of topology optimization has been carried out.

Addressing topology optimization with overhang constraints for structures subjected to self-weight loads

This paper investigates the topology optimization of structures subjected to self-weight loads with self-supporting constraints for additive manufacturing. The integration of topology optimization procedures and additive manufacturing techniques can make the most of their advantages, and there is significant interest today in integrating both approaches. Imposing overhang constraints in topology optimization has been addressed, but primarily for classical topology optimization problems with fixed external loads, not design-dependent loads. This work combines an effective numerical procedure for contour evaluation with a modified version of the power-law model for low densities to eliminate the problems that arise when self-weight loads are considered. The overhang edge detection is based on the Smallest Univalue Segment Assimilating Nucleus (SUSAN) method, and a variable mask size technique is used to avoid eventual dripping problems. The proposed constraint function evaluates the overhang globally and allows control of the formation of unsupported contours for maximum stiffness design problems when self-weight loads are present. Several numerical experiments demonstrate the proposed method's effectiveness and robustness.

A framework for plasticity‐based topology optimization of continuum structures

AbstractIn this paper, a framework is proposed for topology optimization of continuum structures considering plasticity. The method merges the rigid‐plastic analysis and the density‐based topology optimization. To obtain a clean black‐and‐white design, the density in the objective function is penalized using an exponential function. The solution of the final plasticity‐based topology optimization problem exhibits as a sequence of second‐order cone programming (SOCP) problems that can be resolved efficiently using the advanced primal‐dual interior point method. Compared to the conventional stress‐constrained topology optimization techniques, the developed method accounts for plasticity and the finite element analysis of structures does not need to be carried out separately. Furthermore, the proposed method requires no relaxation techniques for imposing local stress‐constrain and possesses good computational efficiency for large‐scale problems.

Design and performance enhancement of thermal-fluid system based on topology optimization

The present work is based on the Finite Element Method as the discretization technique coupled to a density approach, an alternative interpolation function in this paper. A new dimensionless objective function combining minimum energy consumption and maximum thermal performance of topology optimization is proposed, and the performance of the approach and the optimized structure in this paper are verified. The result shows: that the alternative interpolation function can effectively solve the checkerboard and gray cell problems in topology optimization; the new objective function can reduce the vibration problem in the calculation process caused by nonlinearity; in the same conjugate heat transfer systems, the objective function value obtained by the Finite Element Method is 2.24% higher than that of Finite Volume Method and 4.26% higher than that of the Lattice Boltzmann Method; under high Reynolds number, topology structure shows superior comprehensive performance, which is increased by 19.5% -65.2%, and energy consumption per heat transfer can be reduced by up to 38.85%.

Economic topology optimization of District Heating Networks using a pipe penalization approach

In the presented study, a pipe penalization approach for the economic topology optimization of District Heating Networks is proposed, drawing inspiration from density-based topology optimization. For District Heating Networks, the upfront investment is a crucial factor for the rollout of this technology. Today, the pipe routing is usually designed relying on a linearization of the underlying heat transport problem. This study proposes to solve the optimal pipe routing problem as a non-linear topology optimization problem, drawing inspiration from density-based topology optimization. The optimization problem is formulated around a non-linear heat transport model and minimizes a detailed net present value representation of the heating network cost. By relaxing the combinatorial problem of pipe placement, this approach remains scalable for large-scale applications. In a design study on a realistic medium-sized network with 160 houses, a strong influence of economic parameters on the optimal network topology was observed. For this case, the optimization algorithm converges to a discrete network topology and near-discrete pipe design in about 10 min by using the proposed intermediate pipe penalization strategy. The optimal discrete network design found by the algorithm showed to outperform simple rounding post-processing steps by up to 2.8% of their respective net present value.

A Topology Optimization Method Based on the Edge-Based Smoothed Finite Element Method

In this paper, we combined the edge-based smoothed finite element method (ES-FEM) with topology optimization. The edge-based gradient smoothing operation was introduced to overcome the accuracy-loss of the classical finite element method raised by the coarse mesh and “overly stiff” phenomenon. By employing the ES-FEM, design variables can be related to the smoothed edge, thus more design variables can be adaptively obtained without additional remeshing. Two classical topology optimization problems were considered, namely compliance minimization and stress-constrained topology optimization. We presented several numerical examples, among which the compliance minimization examples illustrated the potential of the proposed method, and the advantages of applying such a numerical method in topology optimization were demonstrated through the stress-constrained topology optimization.

Multiscale topology optimization of electromagnetic metamaterials using a high-contrast homogenization method

This study proposes a multiscale topology optimization method for electromagnetic metamaterials using a level set-based topology optimization method that incorporates a high-contrast homogenization method. The high-contrast homogenization method can express wave propagation behavior in metamaterials for various frequencies. It can also capture unusual properties caused by local resonances, which cannot be estimated by conventional homogenization approaches. We formulated multiscale topology optimization problems where objective functions are defined by the macroscopic wave propagation behavior, and microstructures forming a metamaterial are set as design variables. Sensitivity analysis was conducted based on the concepts of shape and topological derivatives. As numerical examples, we offer optimized designs of metamaterials composed of multiple unit cell structures working as a demultiplexer based on negative permeability. The mechanism of the obtained metamaterials is discussed based on homogenized coefficients.