Fixed Design Domain Research Articles

Shape and topology optimization method with generalized topological derivatives

This paper introduces a novel method for shape and topology optimization based on a generalized approach for evaluating topological derivatives, which are essential for the integration of shape and topology optimization. Traditionally, evaluating these derivatives presents significant mathematical challenges due to the discontinuity introduced by the insertion of a hole within the domain of interest. To overcome this issue, the study employs Helmholtz-type partial differential equations (PDEs) to construct a filtered objective functional. This approach ensures differentiability across the material and void phases and continuity over the fixed design domain while maintaining the same evaluation value as the original objective functional. By considering differentiability, continuity conditions, and the relationship between shape and topological derivatives during asymptotic analysis, generalized topological derivatives are obtained through established mathematical procedures. These topological derivatives exhibit a direct correlation with the PDE solutions and demonstrate satisfactory smoothness, thereby facilitating refined shapes in optimization strategies. Furthermore, an effective shape update algorithm is proposed, which directly integrates topological derivatives into structural optimization problems, simplifying their implementation and improving efficiency. Finally, the efficacy of the proposed methodology is demonstrated through its application to various optimal design problems, including stiffness maximization, compliant mechanisms, and eigenfrequency maximization. Verification results further highlight its potential to enhance existing methods for addressing more practical and complex optimization challenges.

Phase-field methods for spectral shape and topology optimization

We optimize a selection of eigenvalues of the Laplace operator with Dirichlet or Neumann boundary conditions by adjusting the shape of the domain on which the eigenvalue problem is considered. Here, a phase-field function is used to represent the shapes over which we minimize. The idea behind this method is to modify the Laplace operator by introducing phase-field dependent coefficients in order to extend the eigenvalue problem on a fixed design domain containing all admissible shapes. The resulting shape and topology optimization problem can then be formulated as an optimal control problem with PDE constraints in which the phase-field function acts as the control. For this optimal control problem, we establish first-order necessary optimality conditions and we rigorously derive its sharp interface limit. Eventually, we present and discuss several numerical simulations for our optimization problem.

Structural topology optimization with an adaptive design domain

Topology optimization has rapidly developed as a powerful tool of structural design in multiple disciplines. Conventional topology optimization techniques usually optimize the material layout within a predefined, fixed design domain. Here, we propose a subdomain-based method that performs topology optimization in an adaptive design domain (ADD). A subdomain-based parallel processing strategy that can vastly improve the computational efficiency is implemented. In the ADD method, the loading and boundary conditions can be easily changed in concert with the evolution of the design space. Through the automatic, flexible, and intelligent adaptation of the design space, this method is capable of generating diverse high-performance designs with distinctly different topologies. Five representative examples are provided to demonstrate the effectiveness of this method. The results show that, compared with conventional approaches, the ADD method can improve the structural performance substantially by simultaneously optimizing the layout of material and the extent of the design space. This work might help broaden the applications of structural topology optimization.

Cellular structure design based on free material optimization under connectivity control

Cellular structures exhibit exceptional multifunction performance at relatively low densities and have wide industrial applications, but their effective design still remains challenging. The study aims to find a manufacturable cellular structure of valid geometry under certain volume usage budget, given a fixed design domain under external loadings. It is achieved via first introducing the concept of free material optimization (FMO), which provides an ultimately best structure among all possible elastic continua. After this, two novel control strategies are developed in combination with inverse homogenization to ultimately produce a cellular structure of valid geometric connectivity. It mainly includes approaches of material space reduction via hierarchical clustering, and a novel physics-based connectivity control to tailor the geometric connectivity between neighboring microstructures. Performance of the approach is tested on various 2D examples, in comparison with structures generated via classical (multiscale) topology optimization approaches.

Topology optimization of coated structure using moving morphable sandwich bars

An explicit topology optimization method for coated structures is proposed by using moving morphable sandwich bars (MMSBs). An MMSB acting as a composite bar has two different material parts with an inner part of the base material and an exterior layer of the coating material. The geometries of an MMSB are mapped onto two different density fields using a fixed grid and the same shortest distance function. The densities of an element are determined depending on whether the element lies inside or outside the boundaries of the inner bars/the hollowed bars. By treating the coordinates of the ends and thicknesses of the base and coating layers of a sandwich bar as design variables, the sandwich bar can move, morph, change its thicknesses, and overlap with others in a fixed design domain to form optimal shapes of the base and coating structures. The minimum thickness of the base structure as well as the coating layer is independently and precisely controlled in an explicit way by simply adjusting the lower bounds of the thickness variables of the sandwich bars. The uniform thickness control of the coating layer can be obtained without any filtering techniques or additional constraints. The proposed method is also extended for topology optimization of the coated structures with two coating layers. The numerical examples of minimizing structural compliance show that the proposed method works effectively and produces good results with smooth and fast convergence rate.

Research of Topology Optimization of Nodal Density Based on Bilinear Interpolation

With the purpose to overcome the numerical instabilities and to generate more distinct structural layouts in the topology optimization, by using bilinear interpolation function, a topology optimization model of density interpolation based on nodal density is established, smooth density field is constructed. This method can ensure that the density field in the fixed design domain owns C0 continuity, and checkerboard patterns are naturally avoided in the nature of mathematics. After adding the sensitivity filtering, the optimal structures are smoother and have lesser details, which is helpful for manufacturing. Two numerical examples show that not only checkerboard pattern can be solved by the proposed method, but also the middle density nodal can be suppressed effectively.

レベルセット法を用いた熱構造連成問題に対するトポロジー最適化(機械力学,計測,自動制御)

In structural designs considering thermal loading, maximization of temperature diffusivity as well as structural stiffness are important and indispensable goals that optimal solutions to design problems need to achieve to reduce operating temperatures and extend product durability. In this paper, a new topology optimization method, based on the level set method and Finite Element Method and including design-dependent effects, is constructed for coupled thermal and structural problems, and multi-objective optimization problems having generic heat transfer boundaries in a fixed design domain are formulated. First, a topology optimization method for coupled thermal and structural problems that uses a level set method incorporating fictitious interface energy is briefly explained. Next, a new objective function that can take into account both temperature diffusivity and stiffness maximization is formulated, based on the concept of total potential energy maximization for the thermal and structural problems. An optimization algorithm that uses the Finite Element Method when solving the equilibrium equation and updating the level set function is then constructed. Finally, several numerical examples are presented to confirm the usefulness of the proposed method.

レベルセット法に基づく熱拡散最大化問題のトポロジー最適化(機械力学,計測,自動制御)

In structural designs considering thermal loading, to control thermal stress and minimize decreases in material strength at high temperatures, it is important to maximize the thermal diffusivity of structures, in addition to the usual maximization of stiffness that optimal designs achieve. This paper presents a new level set-based topology optimization method for thermal problems with generic heat transfer boundaries in a fixed design domain that includes design-dependent effects, using level set boundary expressions and the Finite Element Method. First, a topology optimization method using a level set model incorporating fictitious interface energy is briefly discussed. Next, an optimization problem is formulated using the concept of total potential energy to address the design of mechanical structures that aim to minimize the mean temperature of the structure under thermal loading. An optimization algorithm that uses the Finite Element Method when solving the equilibrium equation and updating the level set function is then constructed. Finally, several numerical examples are provided to confirm the utility of the proposed optimization method.

Topology Optimization for Heat Convection Problems Including Design-Dependent Effects

In this paper, a topology optimization method is constructed for thermal problems with generic heat transfer boundaries in a fixed design domain that include design-dependent effects. First, the topology optimization method for thermal problems is briefly explained using a homogenization method for the relaxation of the design domain, where a continuous material distribution is assumed, to suppress numerical instabilities and checkerboards. Next, a method is developed for handling heat transfer boundaries between material and void regions that appear in the fixed design domain and move during the optimization process, using the Heaviside function as a function of node-based material density to extract the boundary of the target structure being optimized so that the heat transfer boundary conditions can be set. Shape dependencies concerning heat transfer coefficients are also considered in the topology optimization scheme. The optimization problem is formulated using the concept of total potential energy and an optimization algorithm is constructed using the Finite Element Method and Sequential Linear Programming. Finally, several numerical examples are presented in order to confirm the usefulness of the proposed method.

Structural Design of Piezoelectric Actuator Considering Polarization Direction and Continuous Approximation of Material Distribution

In the design of piezoelectric actuator the concept of compliant mechanism combined with piezoelectric materials has been used to magnify either geometric or mechanical advantage. The polarization of piezoelectric materials is considered to improve actuation since the piezoelectric polarization has influences on the performance of the actuator. The topology design of compliant mechanism can be formulated as an optimization problem of material distribution in a fixed design domain and continuous approximation of material distribution(CAMD) method has demonstrated its effectiveness to prevent the numerical instabilities in topology optimization. The optimization problem is formulated to maximize the mean transduction ratio subject to the total volume constraints and solved using a sequential linear programming algorithm. The performance improvement of Moonie actuator design confirms an effect of polarization direction and CAMD.

분극방향과 재료분포의 연속적 근사방법을 고려한 압전형 액추에이터의 구조설계

In this paper, the polarization of piezoelectric materials is considered to improve actuation since the piezoelectric polarization has influences on the performance of the actuator. The topology design of compliant mechanism can be formulated as an optimization problem of material distribution in a fixed design domain and continuous approximation of material distribution (CAMD) method has demonstrated its effectiveness to prevent the numerical instabilities in topology optimization. The optimization problem is formulated to maximize the mean transduction ratio subject to the total volume constraints and solved using a sequential linear programming algorithm. The effect of CAMD and the performance improvement of actuator are confirmed through Moonie actuator and PZT suspension design.

Continuous approximation of material distribution for topology optimization

AbstractIn this paper, we propose a checkerboard‐free topology optimization method without introducing any additional constraint parameter. This aim is accomplished by the introduction of finite element approximation for continuous material distribution in a fixed design domain. That is, the continuous distribution of microstructures, or equivalently design variables, is realized in the whole design domain in the context of the homogenization design method (HDM), by the discretization with finite element interpolations. By virtue of this continuous FE approximation of design variables, discontinuous distribution like checkerboard patterns disappear without any filtering schemes. We call this proposed method the method of continuous approximation of material distribution (CAMD) to emphasize the continuity imposed on the ‘material field’. Two representative numerical examples are presented to demonstrate the capability and the efficiency of the proposed approach against some classes of numerical instabilities. Copyright © 2004 John Wiley & Sons, Ltd.

Topology Optimization Method by Continuous Approximation of Material Distribution

We propose a new method for topology optimization, which is free from numerical instabilities such as checkerboard patterns and mesh-dependency, without introducing any additional constraint parameters. This aim is accomplished by the introduction of finite element approximation for continuous material distribution in a fixed design domain. That is, the continuous distribution of microstructures, or equivalently design variables, is realized in the whole design domain in the context of the homogenization design method (HDM), by the discretization with finite element interpolations. By virtue of this continuous FE approximation of design variables, discontinuous distribution like checkerboard patterns disappears without any filtering schemes. We call this technique the method of continuous approximation of material distribution (CAMD) to emphasize the continuity imposed on the "material field". Two representative numerical examples are presented to demonstrate the capability and the efficiency of the proposed approach against the numerical instabilities.