Lightweight design of bus frames from multi-material topology optimization to cross-sectional size optimization

A comprehensive solution for bus frame design is proposed to bridge multi-material topology optimization and cross-sectional size optimization. Three types of variables (material, topology and size) and two types of constraints (static stiffness and frequencies) are considered to promote this practical design. For multi-material topology optimization, an ordered solid isotropic material with penalization interpolation is used to transform the multi-material selection problem into a pure topology optimization problem, without introducing new design variables. Then, based on the previously optimal topology result, cross-sectional sizes of the bus frame are optimized to further seek the least mass. Sequential linear programming is preferred to solve the two structural optimization problems. Finally, an engineering example verifies the effectiveness of the presented method, which bridges the gap between topology optimization and size optimization, and achieves a more lightweight bus frame than traditional single-material topology optimization.     

Multi-material topology optimization for the transient heat conduction problem using a sequential quadratic programming algorithm

Transient heat conduction analysis involves extensive computational cost. It becomes more serious for multi-material topology optimization, in which many design variables are involved and hundreds of iterations are usually required for convergence. This article aims to provide an efficient quadratic approximation for multi-material topology optimization of transient heat conduction problems. Reciprocal-type variables, instead of relative densities, are introduced as design variables. The sequential quadratic programming approach with explicit Hessians can be utilized as the optimizer for the computationally demanding optimization problem, by setting up a sequence of quadratic programs, in which the thermal compliance and weight can be explicitly approximated by the first and second order Taylor series expansion in terms of design variables. Numerical examples show clearly that the present approach can achieve better performance in terms of computational efficiency and iteration number than the solid isotropic material with penalization method solved by the commonly used method of moving asymptotes. In addition, a more lightweight design can be achieved by using multi-phase materials for the transient heat conductive problem, which demonstrates the necessity for multi-material topology optimization.     

Conceptual design of reinforced concrete structures using topology optimization with elastoplastic material modeling

Design of reinforced concrete structures is governed by the nonlinear behavior of concrete and by its different strengths in tension and compression. The purpose of this article is to present a computational procedure for optimal conceptual design of reinforced concrete structures on the basis of topology optimization with elastoplastic material modeling. Concrete and steel are both considered as elastoplastic materials, including the appropriate yield criteria and post‐yielding response. The same approach can be applied also for topology optimization of other material compositions where nonlinear response must be considered. Optimized distribution of materials is achieved by introducing interpolation rules for both elastic and plastic material properties. Several numerical examples illustrate the capability and potential of the proposed procedure. Copyright © 2012 John Wiley & Sons, Ltd.     

Multi-material plastic part design via the level set shape and topology optimization method

This work presents a new multi-material level set topology optimization method which is developed especially for designing plastic parts. Instead of representing the structure using multiple level set functions, this new method employs only one level set function to describe the material/void interface. The injection moulding filling simulation is used to determine the material/material interfaces. Because plastic parts are targeted, domain-specific knowledge is carefully investigated to improve the optimization algorithm. Both homogeneous and heterogeneous fibre-reinforced plastics are considered as potential material phases. For the latter, one extra design freedom, fibre orientation distribution, is introduced. Instead of generating incremental interior voids, which complicates the mould design and part ejection, shape-fixed interior voids could be predefined inside the design domain for functional or assembly purposes. This is represented by an additional level set function. A few numerical examples are studied to demonstrate the effectiveness of the proposed method.     

Simultaneous single-loop multimaterial and multijoint topology optimization

As the aerospace and automotive industries continue to strive for efficient lightweight structures, topology optimization (TO) has become an important tool in this design process. However, one ever-present criticism of TO, and especially of multimaterial (MM) optimization, is that neither method can produce structures that are practical to manufacture. Optimal joint design is one of the main requirements for manufacturability. This article proposes a new density-based methodology for performing simultaneous MMTO and multijoint TO. This algorithm can simultaneously determine the optimum selection and placement of structural materials, as well as the optimum selection and placement of joints at material interfaces. In order to achieve this, a new solid isotropic material with penalization-based interpolation scheme is proposed. A process for identifying dissimilar material interfaces based on spatial gradients is also discussed. The capabilities of the algorithm are demonstrated using four case studies. Through these case studies, the coupling between the optimal structural material design and the optimal joint design is investigated. Total joint cost is considered as both an objective and a constraint in the optimization problem statement. Using the biobjective problem statement, the tradeoff between total joint cost and structural compliance is explored. Finally, a method for enforcing tooling accessibility constraints in joint design is presented.     

Hierarchical optimization for topology design of multi-material compliant mechanisms

Multi-material topology optimization enables potential design possibilities in the multiphysics and structural designing fields. In this article, a bi-level hierarchical optimization method is introduced to address the multi-material design of compliant mechanisms. The hierarchical optimization develops decomposition approaches allowing the original complex multi-material optimization problem to be reduced to set of low-order single-material optimization sub-problems. The solution of the complex multi-material problem is found as a vector of the single-material sub-problems solutions. All the local sub-problems are solved with the solid isotropic material with penalization method independently, and a stiffness spreading technique is worked out to coordinate components of the global solution of the original problem. Several numerical examples are presented to demonstrate the validity of this method.     

Compliant mechanism design with non-linear materials using topology optimization

In this paper, compliant mechanism design with non-linear materials using topology optimization is presented. A general displacement functional with non-linear material model is used in the topology optimization formulation. Sensitivity analysis of this displacement functional is derived from the adjoint method. Optimal compliant mechanism examples for maximizing the mechanical advantage are presented and the effect of non-linear material on the optimal design are considered.     

Topology optimization of multi-material for the heat conduction problem based on the level set method

This article presents a numerical approach of topology optimization with multiple materials for the heat conduction problem. The multiphase level set model is used to implicitly describe the geometric boundaries of material regions with different conductivities. The model of multi-material representation has no emergence of the intermediate density. The optimization objective is to construct the optimal heat conductive paths which improve the efficiency of heat transfer. The dissipation of thermal transport potential capacity is taken as the objective function. The sensitivity analysis is implemented by the adjoint variable method, which is the foundation of constructing the velocity field of the level set equation. The optimal result is gradually realized by the evolution of multi-material boundaries, and the topological changes are naturally handled during the optimization process. Finally, the numerical examples are presented to demonstrate the feasibility and validity of the proposed method for topology optimization of the heat conduction problem.     

An efficient approach for reliability-based topology optimization

This article presents an efficient approach for reliability-based topology optimization (RBTO) in which the computational effort involved in solving the RBTO problem is equivalent to that of solving a deterministic topology optimization (DTO) problem. The methodology presented is built upon the bidirectional evolutionary structural optimization (BESO) method used for solving the deterministic optimization problem. The proposed method is suitable for linear elastic problems with independent and normally distributed loads, subjected to deflection and reliability constraints. The linear relationship between the deflection and stiffness matrices along with the principle of superposition are exploited to handle reliability constraints to develop an efficient algorithm for solving RBTO problems. Four example problems with various random variables and single or multiple applied loads are presented to demonstrate the applicability of the proposed approach in solving RBTO problems. The major contribution of this article comes from the improved efficiency of the proposed algorithm when measured in terms of the computational effort involved in the finite element analysis runs required to compute the optimum solution. For the examples presented with a single applied load, it is shown that the CPU time required in computing the optimum solution for the RBTO problem is 15–30% less than the time required to solve the DTO problems. The improved computational efficiency allows for incorporation of reliability considerations in topology optimization without an increase in the computational time needed to solve the DTO problem.     

Achieving minimum length scale in topology optimization using nodal design variables and projection functions

A methodology for imposing a minimum length scale on structural members in discretized topology optimization problems is described. Nodal variables are implemented as the design variables and are projected onto element space to determine the element volume fractions that traditionally define topology. The projection is made via mesh independent functions that are based upon the minimum length scale. A simple linear projection scheme and a non‐linear scheme using a regularized Heaviside step function to achieve nearly 0–1 solutions are examined. The new approach is demonstrated on the minimum compliance problem and the popular SIMP method is used to penalize the stiffness of intermediate volume fraction elements. Solutions are shown to meet user‐defined length scale criterion without additional constraints, penalty functions or sensitivity filters. No instances of mesh dependence or checkerboard patterns have been observed. Copyright © 2004 John Wiley & Sons, Ltd.     

Evolutionary topology optimization using design space adjustment based on fixed grid

Design space optimization for topology based on fixed grid is proposed and its superiority to conventional topology optimization is shown. In the conventional topology optimization, the design domain is fixed. It is, however, desirable to make the design domain evolve into a better one during optimization process by increasing or decreasing the number of design pixels or variables, which we call design space optimization. A breakthrough in obtaining sensitivities when design space expands has been made recently with necessary mathematical background, but due to coupling effect and others, sensitivity results have not been satisfactory. Three innovative implementations are developed in this paper. Firstly, the proper characteristics of artificial material are defined. The second one is to decouple neighbouring elements for exact design space sensitivities. The previous design space optimization has been tedious because only one layer can be added. So, the third innovation is a new expansion strategy with multi‐layers based on design space sensitivities. As a result, the proposed evolutionary method can get an optimum much faster than ever before especially for large‐scale problems. It is also conjectured that this gives higher probability of getting the global optimum, as confirmed by numerical examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Synthesis of shape and topology of multi-material structures with a phase-field method

In this paper, we present a phase-field method to the problem of shape and topology synthesis of structures with three materials. A single phase model is developed based on the classical phase-transition theory in the fields of mechanics and material sciences. The multi-material synthesis is formulated as a continuous optimization problem within a fixed reference domain. As a single parameter, the phase-field model represents regions made of any of the three distinct material phases and the interface between the regions. The Van der Waals–Cahn-Hilliard theory is applied to define a dynamic process of phase transition. The Γ-convergence theory is used for an approximate numerical solution to this free-discontinuity problem without any explicit tracking of the interface. Within this variational framework, we show that the phase-transition theory leads to a well-posed problem formulation with the effects of “domain regularization” and “region segmentation” incorporated naturally. The proposed phase-field method is illustrated with several 2D examples that have been extensively used in the recent literature of topology optimization, especially in the homogenization based methods. It is further suggested that such a phase-field approach may represent a promising alternative to the widely-used homogenization models for the design of heterogeneous materials and solids, with a possible extension to a general model of multiple material phases.     

Reducing dimensionality in topology optimization using adaptive design variable fields

Topology optimization methodologies typically use the same discretization for the design variable and analysis meshes. Analysis accuracy and expense are thus directly tied to design dimensionality and optimization expense. This paper proposes leveraging properties of the Heaviside projection method (HPM) to separate the design variable field from the analysis mesh in continuum topology optimization. HPM projects independent design variables onto element space over a prescribed length scale. A single design variable therefore influences several elements, creating a redundancy within the design that can be exploited to reduce the number of independent design variables without significantly restricting the design space. The algorithm begins with sparse design variable fields and adapts these fields as the optimization progresses. The technique is demonstrated on minimum compliance (maximum stiffness) problems solved using continuous optimization and genetic algorithms. For the former, the proposed algorithm typically identifies solutions having objective functions within 1% of those found using full design variable fields. Computational savings are minor to moderate for the minimum compliance formulation with a single constraint, and are substantial for formulations having many local constraints. When using genetic algorithms, solutions are consistently obtained on mesh resolutions that were previously considered intractable. Copyright © 2009 John Wiley & Sons, Ltd.     

A multi-objective approach for multi-material topology and shape optimization

An automated multi-material approach that integrates multi-objective Topology Optimization (TO) and multi-objective shape optimization is presented. A new ant colony optimization algorithm is presented and applied to solving the TO problem, estimating a trade-off set of initial topologies or distributions of material. The solutions found usually present irregular boundaries, which are not desirable in applications. Thus, shape parameterization of the internal boundaries of the design region, and subsequent shape optimization, is performed to improve the quality of the estimated Pareto-optimal solutions. The selection of solutions for shape optimization is done by using the PROMETHEE II decision-making method. The parameterization process involves identifying the boundaries of different materials and describing these boundaries by non-uniform rational B-spline curves. The proposed approach is applied to the optimization of a C-core magnetic actuator, with two objectives: the maximization of the attractive force on the armature and the minimization of the volume of permanent magnet material.     

Eigenvalue topology optimization via efficient multilevel solution of the frequency response

The article presents an efficient solution method for structural topology optimization aimed at maximizing the fundamental frequency of vibration. Nowadays, this is still a challenging problem mainly because of the high computational cost required by spectral analyses. The proposed method relies on replacing the eigenvalue problem with a frequency response one, which can be tuned and efficiently solved by a multilevel procedure. Connections of the method with multigrid eigenvalue solvers are discussed in details. Several applications demonstrating more than 90% savings of the computational time are presented as well.     

Multimaterial multijoint topology optimization

In this paper, a methodology that solves multimaterial topology optimization problems while also optimizing the quantity and type of joints between dissimilar materials is proposed. Multimaterial topology optimization has become a popular design optimization technique since the enhanced design freedom typically leads to superior solutions; however, the conventional assumption that all elements are perfectly fused together as a single piece limits the usefulness of the approach since the mutual dependency between optimal multimaterial geometry and optimal joint design is not properly accounted for. The proposed methodology uses an effective decomposition approach to both determine the optimal topology of a structure using multiple materials and the optimal joint design using multiple joint types. By decomposing the problem into two smaller subproblems, gradient‐based optimization techniques can be used and large models that cannot be solved with nongradient approaches can be solved. Moreover, since the joining interfaces are interpreted directly from multimaterial topology optimization results, the shape of the joining interfaces and the quantity of joints connecting dissimilar materials do not need to be defined a priori. Three numerical examples, which demonstrate how the methodology optimizes the geometry of a multimaterial structure for both compliance and cost of joining, are presented.     

Density gradient-based adaptive refinement of analysis mesh for efficient multiresolution topology optimization

In topology optimization, the finite-element analysis of the problem is generally the most computationally demanding task of the solution process. In order to improve the efficiency of this phase, in this article we propose to represent regions with zero density gradient by a coarser analysis mesh. The design is instead represented in a uniform mesh. We motivate the density gradient-based adaptive refinement by discussing the topological meaning of the density gradient and how it can help avoid loss of information during projections or interpolations between design and analysis meshes. We also study the adaptiveness of the mesh and its ability to detect the topology change of the design. An a posteriori error analysis is performed as well. Furthermore, we provide theoretical and numerical considerations on the reduction of the number of degrees of freedoms of the adaptive analysis mesh with respect to the uniform case. This translates into a faster solution of the analysis, as we show numerically. Finally, we solve several test problems, including large 3D problems that we solve in parallel on computer cluster, demonstrating the applicability of our procedure in large scale computing and with iterative solvers.     

随机参数平面连续体结构的频率拓扑优化设计

摘 要 研究几何和物理参数均为随机变量的平面连续体结构在结构基频约束下的拓扑优化设计问题。以结构总质量均值极小化为目标函数,以结构的形状拓扑信息为设计变量,以结构基频概率可靠性指标为约束条件,构建了随机结构拓扑优化设计数学模型。利用代数综合法,导出了随机参数结构动力响应的均值和均方差的计算表达式。采用渐进结构优化的求解策略与方法,通过两个算例验证了文中模型及求解方法的合理性和可行性。     

Geometrical design of thermoelectric generators based on topology optimization

This paper discusses an application of the topology optimization method for the design of thermoelectric generators. The proposed methodology provides the optimized geometry in accordance with various arbitrary conditions such as the types of materials, the volume of materials, and the temperature and shape of the installation position. By considering the coupled equations of state for the thermoelectric problem, we introduce an analytical model subject to these equations, which mimics the closed circuit composed of thermoelectric materials, electrodes, and a resistor. The total electric power applied to the resistor and the conversion efficiency are formulated as objective functions to be optimized. The proposed optimization method for thermoelectric generators is implemented as a geometrical optimization method using the solid isotropic material with penalization method used in topology optimizations. Simple relationships are formulated between the density function of the solid isotropic material with penalization method and the physical properties of the thermoelectric material. A sensitivity analysis for the objective functions is formulated with respect to the density function and the adjoint equations required for calculating it. Depending on the sensitivity, the density function is updated using the method of moving asymptotes. Finally, numerical examples are provided to demonstrate the validity of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.