Structural and Multidisciplinary Optimization, volume 68, issue 2, publication number 25

Discrete variable topology optimization using multi-cut formulation and adaptive trust regions

Publication type: Journal Article
Publication date: 2025-02-11
scimago Q1
SJR: 1.181
CiteScore: 7.6
Impact factor: 3.6
ISSN: 1615147X, 16151488
Abstract
We present a new framework for efficiently solving general topology optimization (TO) problems that find an optimal material distribution within a design space to maximize the performance of a part or structure while satisfying design constraints. These problems can involve convex or non-convex objective functions and may include multiple candidate materials. The framework is designed to greatly enhance computational efficiency, primarily by diminishing optimization iteration counts and thereby reducing the frequency of solving associated state equilibrium partial differential equations (PDEs) (e.g., through the finite element method (FEM)). It maintains binary design variables and addresses the large-scale mixed integer nonlinear programming (MINLP) problem that arises from discretizing the design space and PDEs. The core of this framework is the integration of the generalized Benders’ decomposition and adaptive trust regions. Specifically, by formulating the master sub-problem (decomposed from the MINLP) as a multi-cut optimization problem and enabling the estimation of the upper and lower bounds of the original objective function, significant acceleration in solution convergence is achieved. The trust region radius adapts based on a merit function. To mitigate ill-conditioning due to extreme parameter values, we further introduce a parameter relaxation scheme where two parameters are relaxed in stages at different paces, improving both solution quality and efficiency. Numerical tests validate the framework’s superior performance, including minimum compliance and compliant mechanism problems in single-material and multi-material designs. We compare our results with those of other methods and demonstrate significant reductions in optimization iterations (and therefore the number of FEM analyses required) by about one order of magnitude, while maintaining comparable optimal objective function values and material layouts. As the design variables and constraints increase, the framework maintains consistent solution quality and efficiency, underscoring its good scalability. We anticipate this framework will be especially advantageous for TO applications involving substantial design variables and constraints and requiring significant computational resources for FEM analyses (or PDE solving).
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Elsevier
Multi-material structural discrete variable topology optimization with minimum length scale control under mass constraint
Liu H., Wang C., Zhang Y., Liang Y.
Computer Methods in Applied Mechanics and Engineering scimago Q1 wos Q1 , 
2024-02-01,  citations by CoLab: 11 ,  Abstract  
Topology optimization of multi-material structure has a larger design space compared with single material optimization, and it also requires efficient material selection methods to provide references for designers. The multi-material additive manufacturing ensures the manufacturability of topology design of multi-material structures, but also puts forward new manufacturability requirements, such as the minimum size of multi-material and the connection interface. This paper focuses on the development of a topology optimization algorithm for multi-material structures under the total mass constraint based on discrete variables. Recursive multiphase materials interpolation (RMMI) scheme is utilized to establish the relationship between discrete design variables and element elastic modulus and density, and sequential approximate integer programming (SAIP) and canonical relaxation algorithms are constructed to update the discrete design variables. It is shown that this method can obtain multi-material topology design with clear interface display, and has the ability of free material coexistence and degradation, which indicates that the discrete variables have more advantages than the continuous variables in the RMMI scheme. Based on this, the material selection under mass constraints is discussed in depth, and it is pointed out that the traditional determination of material coexistence or degradation in design based on specific stiffness has limitations. Thus, the present work expands the necessary conditions of material coexistence. Then, a minimum size control method of multi-material design based on post-processing is proposed to solve the minimum size geometry constraint problem by SAIP. Numerical examples show that this method can not only precisely control the minimum size of each material phase (including void phase), but also resolve the problem of interface and boundary diffusion of the multi-material phase, and significantly improve the manufacturability of multi-material design. Finally, it is applied to optimization problems with complex design domain and irregular meshes which are closer to engineering design.
Elsevier
Stress-based topology optimization approach using binary variables and geometry trimming
Kiyono C.Y., Picelli R., Sivapuram R., De Leon D.M., Silva E.C.
Finite Elements in Analysis and Design scimago Q1 wos Q1 , 
2023-12-01,  citations by CoLab: 4 ,  Abstract  
In this paper a new approach to handle stress-based topology optimization problems by using the Topology Optimization of Binary Structures method is presented. The design update is carried out with binary values (0 or 1) and a boundary identification scheme is employed to smooth the structural contours to avoid artificial stress concentrations that can occur because of the jagged nature of the topology optimization process. Because of the boundary identification, re-meshing is necessary at each iteration. To minimize the discontinuity of the moving domain through the iterations, we define two domains. The first one is the extended domain (called topology domain) which is fixed, meshed only in the beginning of the optimization process. It is where the design variables are defined, and the mass constraint and its sensitivity are calculated. The second one (called analysis domain) is the structure with the boundary already identified, where the finite element analysis is carried out and the objective function and its sensitivity are calculated. The objective function sensitivity must be interpolated to the optimization domain only where the design variables indicate solid regions. A spatial filtering technique is applied to avoid numerical instabilities and to extrapolate to void regions. Numerical examples are presented to demonstrate the methodology efficiency.
Institute of Electrical and Electronics Engineers (IEEE)
Quantum Topology Optimization via Quantum Annealing
Ye Z., Qian X., Pan W.
IEEE Transactions on Quantum Engineering scimago Q1 Open Access , 
2023-04-11,  citations by CoLab: 13 , 
Elsevier
Three-field floating projection topology optimization of continuum structures
Huang X., Li W.
Computer Methods in Applied Mechanics and Engineering scimago Q1 wos Q1 , 
2022-09-01,  citations by CoLab: 34 ,  Abstract  
Topology optimization using the variable substitution among three fields can achieve a design with desired solid and/or void features. This paper proposes a three-field floating projection topology optimization (FPTO) method using the linear material interpolation. The implicit floating projection constraint is used as an engine for generating a 0/1 solution at the design field. The substitution filtering and projection schemes enhance the length scale and solid/void features to accelerate the formation of structural topology in the physical field. Meanwhile, the three-field FPTO method can be extended to robust formulation, which obtains the eroded, intermediate, and dilated designs with the same topology. The most distinct feature of the FPTO method lies in the adoption of the linear material interpolation scheme , which makes many topology optimization problems straightforward. As an example, the proposed three-field FPTO algorithm is further applied to the design of shell-infill structures using the linear multi-material interpolation scheme. The distribution of the shell material is generated through a simple filtering scheme, and the shell thickness is accurately controlled by the filter radius . Numerical examples are presented to demonstrate the effectiveness and advantage of the proposed three-field FPTO method. • A three-field floating projection topology optimization method is proposed. • The algorithm can use the linear material interpolation scheme. • The algorithm is extended straightforwardly for robust formulation. • A simple approach for the topology optimization of shell-infill structures is proposed.
Springer Nature
Sensitivity analysis of discrete variable topology optimization
Sun K., Liang Y., Cheng G.
Structural and Multidisciplinary Optimization scimago Q1 wos Q1 , 
2022-08-05,  citations by CoLab: 14 ,  Abstract  
This paper studies sensitivity analysis for discrete variable topology optimization. Minimum compliance of plane stress structures is considered. The element thickness is the design variable and is named as element density, whose value is 0 or 1. According to the concerned element density and its surrounding density distribution, all the design elements are classified into three types: white interface elements, black elements, and white isolated elements. Their sensitivities are studied by shape sensitivity analysis, topological and configuration derivative, respectively. The white interface element sensitivity obtained by shape sensitivity justifies the sensitivity filter. Based on theoretical deduction and inspired by the analytical, topological derivative formula, the black element sensitivity for inserting the square hole that is consistent with the background finite element mesh is a linear combination of three quadratic forms of stress components. The combination coefficients are dependent on material constants and irrelevant to the stress and strain state, which can be determined by parameter fitting through special load conditions. The white isolated element sensitivity can also be determined by parameter fitting inspired by the configuration derivative. The obtained formula resolves the paradox of the white isolated element sensitivity. The present can further solidify the theoretical foundation for the discrete variable topology optimization methods via Sequential Approximate Integer Programming (SAIP).
Elsevier
Extended level set method: A multiphase representation with perfect symmetric property, and its application to multi-material topology optimization
Noda M., Noguchi Y., Yamada T.
Computer Methods in Applied Mechanics and Engineering scimago Q1 wos Q1 , 
2022-04-01,  citations by CoLab: 24 ,  Abstract  
This paper provides an extended level set (X-LS) based topology optimiza- tion method for multi material design. In the proposed method, each zero level set of a level set function {\phi}ij represents the boundary between materials i and j. Each increase or decrease of {\phi}ij corresponds to a material change between the two materials. This approach reduces the dependence of the initial configuration in the optimization calculation and simplifies the sensitivity analysis. First, the topology optimization problem is formulated in the X-LS representation. Next, the reaction-diffusion equation that updates the level set function is introduced, and an optimization algorithm that solves the equilibrium equations and the reaction-diffusion equation using the fi- nite element method is constructed. Finally, the validity and utility of the proposed topology optimization method are confirmed using two- and three- dimensional numerical examples.
Springer Nature
Topology optimization of turbulent fluid flow via the TOBS method and a geometry trimming procedure
Picelli R., Moscatelli E., Yamabe P.V., Alonso D.H., Ranjbarzadeh S., dos Santos Gioria R., Meneghini J.R., Silva E.C.
Structural and Multidisciplinary Optimization scimago Q1 wos Q1 , 
2022-01-06,  citations by CoLab: 23 ,  Abstract  
One of the current challenges for topology optimization methods is the consideration of high Reynolds fluid flow analysis, especially including turbulence models. The issues in current pseudo-density-based methods are threefold. The fluid boundaries are unknown during optimization, the convergence to $$\left\{ 0,1\right\}$$ designs might be highly dependent on the tuning of the optimization parameters and it is difficult to specify the maximum value of the inverse permeability to avoid the presence of fluid flowing inside the modeled solid medium. This paper proposes a methodology to tackle these three problems. The Topology Optimization of Binary Structures (TOBS) method and a geometry trimming procedure are employed to create the TOBS-GT method. This method uses a binary $$\left\{ 0,1\right\}$$ design variable, which naturally creates explicit fluid boundaries during optimization and avoids the need for tuning the material model interpolation parameters. The geometry trimming procedure removes the solid regions and create a CAD model with only the fluid analysis domain and smooth walls. Since there is no solid region inside the analysis mesh, the problem of having fluid flowing through a solid region is avoided. The k- $$\varepsilon$$ and k- $$\omega$$ turbulence models are chosen to illustrate that the method may be applied to any turbulence model. The equilibrium equations are solved using the finite element method. The total fluid energy dissipation is minimized considering a fluid volume constraint. Numerical results show that the TOBS-GT method is well-fitted for topology optimization of turbulent fluid flow problems.
Elsevier
A new multi-material topology optimization algorithm and selection of candidate materials
Huang X., Li W.
Computer Methods in Applied Mechanics and Engineering scimago Q1 wos Q1 , 
2021-12-01,  citations by CoLab: 47 ,  Abstract  
With the availability of multi-material additive manufacturing , topology optimization of a multi-material structure and the selection of candidate materials become increasingly important. This paper aims to develop a new multi-material topology optimization algorithm and propose the guidelines for the selection of candidate materials from the database. The multi-material design variables and their inequality relationship are built on volume fractions of multiple materials within each element. The multi-material design variables are relaxed and multiple floating projection constraints simulate their discrete constraints by pushing design variables towards 0 or 1. Meanwhile, their inequality relationship is enforced by their variation limits. The proposed multi-material topology optimization algorithm can be applied to the compliance minimization problem constrained by a single mass or multiple volumes, as demonstrated in numerical examples. This paper mainly focuses on a single mass constraint, and 2D and 3D numerical examples systematically demonstrate that an optimized design under a mass constraint achieves a lower compliance when more materials appear in the final design simultaneously. Furthermore, we establish an approach to predict the inclusion or exclusion of a material from the final design, and propose the conditions for the co-existence of candidate materials, which guide users to select candidate materials from the database. • A new multi-material topology optimization algorithm is proposed. • The algorithm can be directly applied to a single mass constraint or multiple volume constraints. • The results systematically demonstrate the advantage of the multi-material design. • The approach to predict the occurrence of candidate materials in the final design is developed. • The approach can guide the users to select materials for multi-material additive manufacturing.
Elsevier
On the design of multimaterial structural topologies using Integer Programming
Sivapuram R., Picelli R., Yoon G.H., Yi B.
Computer Methods in Applied Mechanics and Engineering scimago Q1 wos Q1 , 
2021-10-01,  citations by CoLab: 13 ,  Abstract  
This paper presents a framework for the discrete design of optimal multimaterial structural topologies using integer design variables and mathematical programming . The structural optimization problems: compliance minimization subject to mass constraint, and mass minimization subject to compliance constraint are used to design the multimaterial topologies in this work. The extended SIMP interpolation is used to interpolate the different materials available for structural design, and the material phases in each element are represented using binary design variables, one variable per available material. The Topology Optimization of Binary Structure (TOBS) method (Sivapuram and Picelli, 2018) is employed, wherein the nonlinear objective/constraint functions of optimization are sequentially approximated (herein, linearized) to obtain a sequence of Integer Linear Programs (ILPs). A novel truncation error-regulating constraint in terms of the Young’s moduli of the elements is introduced to maintain the sequential approximations valid, by restricting large changes in successive structural topologies. A commercial branch-and-bound solver is used to solve the integer subproblems yielding perfectly binary solutions which guarantee discrete structural topologies with clear material interfaces at each iteration. Adjoint sensitivities are computed to generate the integer subproblems, and the sensitivities are filtered using a conventional mesh-independent sensitivity filter. Few examples show the design of multimaterial structures in the presence of design-dependent loads: hydrostatic pressure loads and self-weight loads. This work also demonstrates through few examples, convergence of optimal multimaterial topologies at inactive constraint values when different type of loadings simultaneously act on the structure. • Integer programming is used in multimaterial structural optimization. • The optimized structures have clear boundaries and clear material interfaces. • A novel truncation error-regulating constraint is proposed for multimaterial problems. • Examples include models with design-dependent pressure and self-weight loads. • Convergence at inactive mass constraint values is demonstrated through few examples.
Springer Nature
Topology optimization of multi-scale structures: a review
Wu J., Sigmund O., Groen J.P.
Structural and Multidisciplinary Optimization scimago Q1 wos Q1 , 
2021-03-08,  citations by CoLab: 353 ,  Abstract  
Multi-scale structures, as found in nature (e.g., bone and bamboo), hold the promise of achieving superior performance while being intrinsically lightweight, robust, and multi-functional. Recent years have seen a rapid development in topology optimization approaches for designing multi-scale structures, but the field actually dates back to the seminal paper by Bendsøe and Kikuchi from 1988 (Computer Methods in Applied Mechanics and Engineering 71(2): pp. 197–224). In this review, we intend to categorize existing approaches, explain the principles of each category, analyze their strengths and applicabilities, and discuss open research questions. The review and associated analyses will hopefully form a basis for future research and development in this exciting field.
Springer Nature
A 101-line MATLAB code for topology optimization using binary variables and integer programming
Picelli R., Sivapuram R., Xie Y.M.
Structural and Multidisciplinary Optimization scimago Q1 wos Q1 , 
2020-09-27,  citations by CoLab: 37 ,  Abstract  
This paper presents a MATLAB code with the implementation of the Topology Optimization of Binary Structures (TOBS) method first published by Sivapuram and Picelli (Finite Elem Anal Des 139: pp. 49–61, 2018). The TOBS is a gradient-based topology optimization method that employs binary design variables and formal mathematical programming. Besides its educational purposes, the 101-line code is provided to show that topology optimization with integer linear programming can be efficiently carried out, contrary to the previous reports in the literature. Compliance minimization subject to a volume constraint is first solved to highlight the main features of the TOBS method. The optimization parameters are discussed. Then, volume minimization subject to a compliance constraint is solved to illustrate that the method can efficiently deal with different types of constraints. Finally, simultaneous volume and displacement constraints are investigated in order to expose the capabilities of the optimizer and to serve as a tutorial of multiple constraints. The 101-line MATLAB code and some simple enhancements are elucidated, keeping only the integer programming solver unmodified so that it can be tested and extended to other numerical examples of interest.
Springer Nature
Discrete variable topology optimization for compliant mechanism design via Sequential Approximate Integer Programming with Trust Region (SAIP-TR)
Liang Y., Sun K., Cheng G.
Structural and Multidisciplinary Optimization scimago Q1 wos Q1 , 
2020-08-27,  citations by CoLab: 27 ,  Abstract  
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.
Elsevier
Smooth topological design of structures using the floating projection
Huang X.
Engineering Structures scimago Q1 wos Q1 , 
2020-04-01,  citations by CoLab: 65 ,  Abstract  
This paper presents an element-based topology optimization using the ersatz material model, which enables us to achieve a smooth design for a wide range of topology optimization problems. The traditional element-based topology formation algorithms based on the solid isotropic material with penalization (SIMP) model seek the pure 0/1 design with a zig-zag boundary, which is less practical for engineering applications. The ersatz material model suits for the simulation of a smooth design under the fixed mesh, but the element-based topology optimization using the ersatz material model does not provide a clear topology. This paper proposes the floating projection constraint which gradually pushes the design variables to a desired 0/1 level for a smooth design. The applications on a series of engineering optimization problems demonstrate the effectiveness of the proposed topology optimization algorithm. Numerical results show that the proposed topology optimization algorithm based on the floating projection and ersatz material model can achieve optimized structures with a smooth boundary for practical applications.
Springer Nature
Topology optimization of fluidic pressure-loaded structures and compliant mechanisms using the Darcy method
Kumar P., Frouws J.S., Langelaar M.
Structural and Multidisciplinary Optimization scimago Q1 wos Q1 , 
2020-01-09,  citations by CoLab: 38 ,  Abstract  
In various applications, design problems involving structures and compliant mechanisms experience fluidic pressure loads. During topology optimization of such design problems, these loads adapt their direction and location with the evolution of the design, which poses various challenges. A new density-based topology optimization approach using Darcy’s law in conjunction with a drainage term is presented to provide a continuous and consistent treatment of design-dependent fluidic pressure loads. The porosity of each finite element and its drainage term are related to its density variable using a Heaviside function, yielding a smooth transition between the solid and void phases. A design-dependent pressure field is established using Darcy’s law and the associated PDE is solved using the finite element method. Further, the obtained pressure field is used to determine the consistent nodal loads. The approach provides a computationally inexpensive evaluation of load sensitivities using the adjoint-variable method. To show the efficacy and robustness of the proposed method, numerical examples related to fluidic pressure-loaded stiff structures and small-deformation compliant mechanisms are solved. For the structures, compliance is minimized, whereas for the mechanisms, a multi-criteria objective is minimized with given resource constraints.
Springer Nature
Further elaborations on topology optimization via sequential integer programming and Canonical relaxation algorithm and 128-line MATLAB code
Liang Y., Cheng G.
Structural and Multidisciplinary Optimization scimago Q1 wos Q1 , 
2019-11-20,  citations by CoLab: 50 ,  Abstract  
This paper provides further elaborations on discrete variable topology optimization via sequential integer programming and Canonical relaxation algorithm. Firstly, discrete variable topology optimization problem for minimum compliance subject to a material volume constraint is formulated and approximated by a sequence of discrete variable sub-programming with the discrete variable sensitivity. The differences between continuous variable sensitivity and discrete variable sensitivity are discussed. Secondly, the Canonical relaxation algorithm designed to solve this sub-programming is presented with a discussion on the move limit strategy. Based on the discussion above, a compact 128-line MATLAB code to implement the new method is included in Appendix 1. As shown by numerical experiments, the 128-line code can maintain black-white solutions during the optimization process. The code can be treated as the foundation for other problems with multiple constraints.

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