Topology optimization of nonlinear structures under displacement loading

Stress-based topology optimization and nonlinear structural topology optimization is gaining increasing attention in order to make topology optimization more realistic. Thus, this paper extends current concepts of topology optimization to the design of structures made of nonlinear materials. An extended bi-directional evolutionary structural optimization (BESO) method for stress minimization topology optimization of material nonlinear structures is proposed in this work. BESO method based on discrete variables can effectively avoid the well-known singularity problem in density-based methods with low-density elements. The maximum von Mises stress is approximated by the p-norm global stress. The sensitivity information for designing variable updates is derived in detail by adjoint method. As for the highly nonlinear stress behavior, the updated scheme takes advantages from two filters respectively of the sensitivity and topological variables to improve convergence. Moreover, the filtered sensitivity numbers are combined with their historical sensitivity information to further stabilize the optimization process. The effectiveness of the proposed method is demonstrated by several 2D benchmark design problems.

Topology optimization with meshless density variable approximations and BESO method

Topology optimization of material nonlinear continuum structures under stress constraints

During the last 20 years, structural optimization has become one of the most important topics of engineering applications. Design optimization of structure has been an interesting area of research in the field of engineering design for its ability to short the design cycle and to enhance product quality. Significant research activity has occurred in the area of structural optimization in the last decade. Especially for topology optimization of structure, many new theoretical, algorithmic, and computational contributions have resulted by researchers and engineers. Topology optimization is a powerful tool for global and multi-scale design of macrostructures, microstructures, and the cell of prescribed composite materials. The population based evolutionary algorithms have emerged as powerful mechanism for finding optimum solutions of complex optimization problems in engineering during the last two decades. Evolutionary computation is the study of computational systems which use ideas and get inspiration from natural evolution and adaptation [1]. The thinking has wide application in various engineering fields, such as computer science, artificial intelligence, operations research. Genetic algorithm is another kind of bio-inspired optimization method and it is playing an increasingly important role in studies of complex adaptive systems. Its application ranges from adaptive agents in economic theory to the use of machine learning techniques in the design of complex devices and structures, such as aircraft turbines and integrated circuits [2]. Optimization of structures can be classified into three categories: sizing, shaping, and topology optimization. In the topology optimization, it is concerned with the structure members and connectivity between members. In general, it is easily represented by discrete variables rather than by those used for continuous optimization problems. Topology optimization is the most difficult and complex among three categories and it is special useful in developing innovative conceptual designs. Structural optimization, in particular the topology optimization, has been identified as one of the most challenging tasks in structural design. Various techniques and approaches have been established during the last two decades. Topology optimization usually referred to as layout optimization or general shape optimization [3]. It lets engineers get the optimal topology of structure or new configurations during product design phase, as they are implementing the design of the size

Bi-directional evolutionary topology optimization of continuum structures subjected to inertial loads

This study deals with comparing the stiffness design of geometrically nonlinear structures and linear structures using topology optimization. Bi-directional Evolutionary Structures Optimization (BESO) is used for the design process. The geometrically nonlinear behavior of the structures is analyzed using a total Lagrangian finite element formulation and the equilibrium is achieved by Newton-Raphson iterative scheme. The topology optimization of linear and nonlinear modeling is performed. The sensitivity of the objective function is found with the adjoint method and the optimization problem is solved using BESO’s update method. Objective function of complementary work is evaluated. A special technique, the continuation method, is applied to eliminate the instability of nonlinear structure optimization. ANSYS APDL is also used to do FEA of optimal topology to verify the effectiveness of geometrically nonlinear modelling. The results show that differences in stiffness of structures optimized using linear and nonlinear modelling are generally small but it can be large in some cases, especially structure highly involving buckling behaviour.

Stress-based multi-material structural topology optimization considering graded interfaces

Запропоновано метод оптимізації неперервних функції багатьох змінних, який втілює підходи методів рою часток та моделювання штучної імунної системи. Проведено дослідження з метою визначення оптимальних налаштувань алгоритму, що реалізує запропонований метод. Показано, що запропонований метод може бути застосований як складова інформаційної технології супроводу підтримки управлінських рішень та розв’язання задач оптимізації технологічних процесів у металургійному виробництві. Метою роботи є розробка оригінального гібридного методу глобальної оптимізації на основі методу штучної імунної системи та методу рою частинок, а також формулювання рекомендацій щодо налаштування його параметрів. Запропонований метод має прискорити прийняття рішень при керуванні технологічними процесами у металургійному виробництві та підвищити їх точність. Методологія забезпечення рішення полягає у поєднанні основних пошукових операторів еволюційних методів оптимізації на основі рою часток і моделювання штучної імунної системи людини. Використовуються традиційні для ройового пошуку кроки з пошуку рішення та обміну інформацією про знайдені локальні оптимуми. Такий пошук доповнюється принципом змагальності, запозиченим у методі штучної імунної системи, для чого популяція поділяється на менші рої або пошукові команди. Також запропоновані кроки контролю різноманіття в рої та радіусу розсіювання часток. Останнє дозволяє вирішувати задачі як безумовної, так і умовної оптимізації у неперервному просторі високої розмірності. Наукова новизна отриманих у роботі результатів полягає в формулюванні нового еволюційного методу оптимізації в неперервному просторі на основі ройового інтелекту, який на відміну від відомих раніше ройових методів використовує принцип змагання підгруп рою та оператор стиснення популяції, властивих методу моделювання штучних імунних систем. Емпірично встановленні параметри методу, що визначають його ефективність. Висновки. Застосування запропонованого еволюційного методу пошукової оптимізації до мінімізації тестових функцій у неперервному просторі розмірністю до 20 показало його ефективність у порівнянні з відомими раніше. В подальшому актуальним бачиться застосування викладеного алгоритму для розв’язання задач оптимізації технологічних процесів у металургійному виробництві. Зокрема, даний алгоритм вбачається ефективним для задач шихтування, оптимізації використання феросплавів у ливарному виробництві, а також при прогнозуванні механічних властивостей готової продукції.

SummaryA novel density‐based topology optimization framework for plastic energy absorbing structural designs with maximum damage constraint is proposed. This framework enables topologies to absorb large amount of energy via plastic work before failure occurs. To account for the plasticity and damage during the energy absorption, a coupled elastoplastic ductile damage model is incorporated with topology optimization. Appropriate material interpolation schemes are proposed to relax the damage in the low‐density regions while still ensuring the convergence of Newton‐Raphson solution process in the nonlinear finite element analyses. An effective method for obtaining path‐dependent sensitivities of the plastic work and maximum damage via adjoint method is presented, and the sensitivities are verified by the central difference method. The effectiveness of the proposed methodology is demonstrated through a series of numerical examples. Copyright © 2017 John Wiley & Sons, Ltd.

Design optimization of beam structures is a significant topic as beams are an efficient load-carrying component in engineering applications. Most of the earlier researches concentrate on the structural optimization of beams with invariant cross-section topology, and design optimization of periodic beam structures, which cross-section topology varies along the axial direction, has not been extensively investigated. This work presents a novel approach based on the relaxed Saint-Venant solution to conduct microstructural topology optimization of periodic beam structures for minimum structural compliance. One benefit of adopting Saint-Venant solution based compliance formulation is that the strain energy induced by transverse shear loading is incorporated in the structural compliance, which is not reflected in that based on first-order homogenization of the asymptotic homogenization (AH) theory, so that a more reasonable objective function is adopted for the optimization problem. In addition, a material connectivity constraint, which is constructed by restraining the ratio of the strain energy calculated from relaxed Saint-Venant solution to that obtained from first-order homogenization of the AH theory, is further proposed to prevent material separation or to strengthen material connection through the thickness direction. The detailed sensitivity analysis of the objective function and the constraints are carried out with the adjoint method, and several numerical examples are given to show the validity of the relaxed Saint-Venant solution based compliance formulation and the effectiveness of the proposed material connectivity constraint.

This paper presents structural topology optimization applied to the coupled magneto-structural problem. The design goals are to minimize mechanical compliance and to maximize total magnetic force. To calculate the compliance and magnetic force, coupled magneto-structural analysis is performed using the finite-element method. From the solution of a magnetostatic analysis, distribution of the magnetic body force is obtained using the virtual air gap scheme. The structural analysis from the calculated force distribution then provides the mechanical compliance. The design sensitivity analysis for the optimization is performed using the adjoint method, and the derived sensitivity is verified by comparing it with the finite difference sensitivity. The optimization problem is formulated and solved using the sequential linear programming (SLP) method. To show the validity of the proposed analysis and optimization approach, design of the solenoid actuator is presented.

This article proposes a bacterial-chemotaxis-based bio-inspired algorithm (BCBTOA) to solve structural topology optimization problems in two and three-dimensional continuous, which describe the chemotactic strategy as a simulated process for removing material from a structure. A chemotaxis-based regularization numerical scheme was employed for common checkerboards and mesh dependence issues. The algorithm minimizes compliance and maximizes stiffness, so it is applied to various configurations of beams to show its efficiency, performance, and versatility. Then, the algorithm is evaluated to determine its effectiveness and computational performance. Results are compared with the Sequential Element Rejection and Admission (SERA) method and the Soft Bidirectional Evolutionary Structural Optimization (Soft BESO) method using several performance metrics. The results showed that BCBTOA possessed a better computational performance on some benchmark problems and showed layouts with entirely clear gray areas.

In order to overcome the difficulties caused by singular optima, in the present paper, a new method for the solutions of structural topology optimization problems is proposed. The distinctive feature of this method is that instead of solving the original optimization problem directly, we turn to seeking the solutions of a sequence of approximated problems which are formulated by relaxing the constraints of the original problem to some extent. The approximated problem can be solved efficiently by employing the algorithms developed for sizing optimization problems because its solution is not singular. It can also be proved that when the relaxation parameter is tending to zero, the solution of the approximated problem will converge to the solution of the original problem uniformly. Numerical examples illustrate the effectiveness and validity of the present approach. Results are also compared with those obtained by traditional methods.

This article proposes an efficient parameterized level set method (PLSM) for structural topology optimization design with minimum compliance as the objective function and volume fraction as the constraint condition. In the stage of establishing the optimization model, the level set function (LSF) is interpolated using compactly-supported radial basis functions (CS-RBFs) to transform the Hamilton-Jacobi partial differential equation (PDE) into ordinary differential equations (ODEs), thereby making the evolution of the LSF more convenient and efficient, and ensuring the smoothness of the optimization result boundary. The method of moving asymptotes (MMA) is used for solving the established optimization model, and meanwhile the shape sensitivity constraint factor is added to improve computational efficiency. During the evolution of the LSF, an approximate re-initialization scheme is employed to prevent the gradient of the LSF boundary from being too large or too small, thereby improving the numerical stability and the convergence speed of structural topology optimization process. Furthermore, the proposed method is also extensible and applicable to topology optimization of multi-material structures. The feasibility and effectiveness of this method have been verified through several typical numerical examples involving topology optimization of single-material structures and multi-material structures within the framework of minimum compliance design. HIGHLIGHTS An efficient parameterized level set method using the approximate re-initialization scheme is proposed for structural topology optimization. The approximate re-initialization scheme is employed when solving the parameterized level set function via the MMA algorithm. This scheme can prevents the gradient of the level set function boundary from being too large or too small, making the level set function update more stable and accelerating the convergence speed of structural topology optimization. The effectiveness and feasibility of the proposed method are demonstrated through examples of single-material and multi-material structural topology optimization.

An optimization method of adjoint assimilation based on the cumulative gradient: An application for correction of the bottom friction coefficient in the tidal wave model