Discrete variable topology optimization for compliant mechanism design via Sequential Approximate Integer Programming with Trust Region (SAIP-TR)

Niu B., Liu X., Wallin M., Wadbro E.

In this work, compliant mechanisms are designed by using multi-objective topology optimization, where maximization of the output displacement and minimization of the strain are considered simultaneously. To quantify the strain, we consider typical measures of strain, which are based on the p-norm, and a new class of strain quantifying functions, which are based on the variance of the strain. The topology optimization problem is formulated using the Solid Isotropic Material with Penalization (SIMP) method, and the sensitivities with respect to design changes are derived using the adjoint method. Since nearly void regions may be highly strained, these regions are excluded in the objective function by a projection method. In the numerical examples, compliant grippers and inverters are designed, and the tradeoff between the output displacement and the strain function is investigated. The numerical results show that distributed compliant mechanisms without lumped hinges can be obtained when including the variance of the strain in the objective function.

Zhu B., Zhang X., Zhang H., Liang J., Zang H., Li H., Wang R.

Compliant mechanisms have become an important branch of modern mechanisms. Unlike conventional rigid body mechanisms, compliant mechanisms transform the displacement and force at least partly through the deformation of their structural components, which can offer a great reduction in friction, lubrication and assemblage. Therefore, compliant mechanisms are particularly suitable for applications in microscale/nanoscale manipulation systems. The significant demand of practical applications has also promoted the development of systematic design methods for compliant mechanisms. Several methods have been developed to design compliant mechanisms. In this paper, we focus on the continuum topology optimization methods and present a survey of the state-of-the-art design advances in this research area over the past 20 years. The presented overview can be helpful to those engaged in the topology optimization of compliant mechanisms who desire to be apprised of the field’s recent state and research tendency.

Liang Y., Cheng G.

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.

Liu J.

This paper presents a piecewise length scale control method for level set topology optimization . Different from the existing methods, where a unique lower limit or upper limit was applied to the entire design domain, this new method decomposes the topological design into pieces of strip-like components based on the connectivity condition, and then, the lower or upper limit for length scale control could be piecewise and dynamically defined based on each component’s real-time status (such as position, orientation, or dimension). Specifically, a sub-algorithm of structural skeleton identification and segmentation is developed to decompose the structure and its skeleton. Then, a skeleton segment-based length scale control method is developed to achieve the piecewise length scale control effect. In addition, a special type of length scale constrained topology optimization problem that involves an irregular design domain will be addressed, wherein the complex design domain plus the length scale constraint may make the conventional length scale control methods fail to work. Effectiveness of the proposed method will be proved through a few numerical examples. • Realize the piecewise length scale control for level set topology optimization. • Dynamically and separately define the length scale target based on each component’s real-time status. • Use image processing techniques for structural skeleton identification and segmentation. • Address the length scale constrained topology optimization problem that involves an irregular design domain.

Yang K., Fernandez E., Niu C., Duysinx P., Zhu J., Zhang W.

Spatial gradient information of density field in SIMP (solid isotropic material with penalization) topology optimization is very useful for imposing overhang angle and minimum length (size) manufacturing constraints or achieving shell-infill optimization. However, the computation of density gradient is an approximation since the design space is discretized. There are several operators for this purpose, which arise from the image processing field. This note compares different gradient operators in the context of SIMP topology optimization method and suggests a new computation strategy to improve the accuracy of gradient estimation. We take a case study of spatial gradient-based minimum size constraints. New structural indicator functions are proposed to improve the general applicability of previous gradient-based minimum length constraints. This study is carried out in 2D structure examples to validate the methodology.

Liang Y., Cheng G.

The mathematical essence of structural topology optimization is large-scale nonlinear integer programming . To overcome its huge computational burden, a popular way is to relax the 0–1 variable constraints and transform the integer programming problem into a continuous variable programming problem which can be solved by using gradient based mathematical programming methods. To cope with the variable transformation, the well-known SIMP (Solid Isotropic Material with Penalty) method introduces interpolation schemes for the material properties versus design variables with penalty and achieves great success and popularity. However, there is no doubt that directly tackling the large-scale nonlinear integer programming is very important. This paper solves the structural topology optimization problems with single or multiple constraints by applying the Canonical Dual Theory (CDT) by Gao and Ruan (2010) together with Sequential Approximate Programming approach under the classic structural topology optimization formulations. Following the Sequential Approximate Programming (SAP) frame from the structural optimization, the present paper firstly utilizes sensitivity information to construct the explicit and separable approximate Sequential Quadratic Integer Programming (SQIP) or Sequential Linear Integer Programming (SLIP) subproblems for the topology optimization. And then, the subproblems are solved by applying the Canonical relaxation algorithm based on CDT theory. Their special mathematical structures are exploited to develop analytic solution of Kuhn–Tucker condition of the dual programming. Numerical experiments of two linear and quadratic integer programming problems with random coefficients assert that the Canonical relaxation algorithm can obtain approximate solutions with good properties very efficiently and the dual gap is negligible when the number of design variables increases. Because move limit strategies play a key role in many search algorithms of structural optimization, this paper combines one of two different move limit strategies within the new method. The new method first solves a set of classic topology optimization problems with only material usage constraint, including minimum structural compliance design under constant load, maximum heat transfer efficiency for the heat conduction problem . And then we apply the method to the topology optimization problems with multiple constraints, including minimum structural compliance design under an additional local displacement constraint and minimum structural compliance design under infill constraints. The results of these problems demonstrate that the new method can efficiently solve the discrete variable structural topology optimization problems with multiple nonlinear constraints or many local linear constraints in a unified and systematic way. Beyond that, the new method can achieve integer solutions when combined with the move limit strategy of controlling volume fraction parameter. It can deal with much more design variables than the general branch-and-bound method and makes no use of any sensitivity threshold or heuristic stabilization scheme during the iterative process in comparison with BESO method. Finally, the new method in this paper can be further developed as a general solver for these large-scale discrete variable structural topology optimization problems.

Jansen M.

The goal of this paper is to introduce local length scale control in an explicit level set method for topology optimization. The level set function is parametrized explicitly by filtering a set of nodal optimization variables. The extended finite element method (XFEM) is used to represent the non-conforming material interface on a fixed mesh of the design domain. In this framework, a minimum length scale is imposed by adopting geometric constraints that have been recently proposed for density-based topology optimization with projections filters. Besides providing local length scale control, the advantages of the modified constraints are twofold. First, the constraints provide a computationally inexpensive solution for the instabilities which often appear in level set XFEM topology optimization. Second, utilizing the same geometric constraints in both the density-based topology optimization and the level set optimization enables to perform a more unbiased comparison between both methods. These different features are illustrated in a number of well-known benchmark problems for topology optimization.

Gao J., Song B., Mao Z.

ABSTRACTThis article proposes a novel method for topology optimization with length scale control. In this method, the structural skeleton based on the level set framework is employed. On this basis...

Sivapuram R., Picelli R., Xie Y.M.

In this paper, we use the Topology Optimization of Binary Structures (TOBS) method recently developed by Sivapuram and Picelli (2018) for microstructural optimization. This is the first work in topology optimization addressing various non-volume microstructural constraints with discrete (0/1) design variables. The objective and constraint functions are linearized at each iteration, and the obtained linear problem is solved through Integer Linear Programming (ILP) using sensitivities computed from asymptotic homogenization. A periodic filter is used to make the optimized solutions checkerboard-free and mesh-independent. Volume minimization problems subject to elastic and thermal constraints are considered. The examples consider different sets of constraints, including bulk and shear moduli, square/cubic symmetry, isotropy, thermal conductivity and a combination of them in two and three dimensions. The non-volume constraints are treated explicitly, i.e., without the use of Lagrange multiplier/penalty as used in conventional gradient-based binary topology optimization methods (Huang and Xie, 2010). The resulting microstructures are observed to be convergent in all the examples presented and in agreement with the Hashin–Shtrikman bounds.

Amir O., Lazarov B.S.

A new suite of computational procedures for stress-constrained continuum topology optimization is presented. In contrast to common approaches for imposing stress constraints, herein it is proposed to limit the maximum stress by controlling the length scale of the optimized design. Several procedures are formulated based on the treatment of the filter radius as a design variable. This enables to automatically manipulate the minimum length scale such that stresses are constrained to the allowable value, while the optimization is driven to minimizing compliance under a volume constraint – without any direct constraints on stresses. Numerical experiments are presented that incorporate the following : 1) Global control over the filter radius that leads to a uniform minimum length scale throughout the design; 2) Spatial variation of the filter radius that leads to local manipulation of the minimum length according to stress concentrations; and 3) Combinations of the two above. The optimized designs provide high-quality trade-offs between compliance, stress and volume. From a computational perspective, the proposed procedures are efficient and simple to implement: essentially, stress-constrained topology optimization is posed as a minimum compliance problem with additional treatment of the length scale.

Sivapuram R., Picelli R.

This work proposes an improved method for gradient-based topology optimization in a discrete setting of design variables. The method combines the features of BESO developed by Huang and Xie [1] and the discrete topology optimization method of Svanberg and Werme [2] to improve the effectiveness of binary variable optimization. Herein the objective and constraint functions are sequentially linearized using Taylor's first order approximation, similarly as carried out in [2]. Integer Linear Programming (ILP) is used to compute globally optimal solutions for these linear optimization problems, allowing the method to accommodate any type of constraints explicitly, without the need for any Lagrange multipliers or thresholds for sensitivities (like the modern BESO [1]), or heuristics (like the early ESO/BESO methods [3]). In the linearized problems, the constraint targets are relaxed so as to allow only small changes in topology during an update and to ensure the existence of feasible solutions for the ILP. This process of relaxing the constraints and updating the design variables by using ILP is repeated until convergence. The proposed method does not require any gradual refinement of mesh, unlike in [2] and the sensitivities every iteration are smoothened by using the mesh-independent BESO filter. Few examples of compliance minimization are shown to demonstrate that mathematical programming yields similar results as that of BESO for volume-constrained problems. Some examples of volume minimization subject to a compliance constraint are presented to demonstrate the effectiveness of the method in dealing with a non-volume constraint. Volume minimization with a compliance constraint in the case of design-dependent fluid pressure loading is also presented using the proposed method. An example is presented to show the effectiveness of the method in dealing with displacement constraints. The results signify that the method can be used for topology optimization problems involving non-volume constraints without the use of heuristics, Lagrange multipliers and hierarchical mesh refinement.

Clausen A., Andreassen E.

Most research papers on topology optimization involve filters for regularization. Typically, boundary effects from the filters are ignored. Despite significant drawbacks the inappropriate homogeneous Neumann boundary conditions are used, probably because they are trivial to implement. In this paper we define three requirements that boundary conditions must fulfill in order to eliminate boundary effects. Previously suggested approaches are briefly reviewed in the light of these requirements. A new approach referred to as the “domain extension approach” is suggested. It effectively eliminates boundary effects and results in well performing designs. The approach is intuitive, simple and easy to implement.

Qian X.

Summary We present an approach for controlling the undercut and the minimal overhang angle in density based topology optimization, which are useful for reducing support structures in additive manufacturing. We cast both the undercut control and the minimal overhang angle control that are inherently constraints on the boundary shape into a domain integral of Heaviside projected density gradient. Such a Heaviside projection based integral of density gradient leads to a single constraint for controlling the undercut or controlling the overhang angle in the optimization. It effectively corresponds to a constraint on the projected perimeter that has undercut or has slope smaller than the prescribed overhang angle. In order to prevent trivial solutions of intermediate density to satisfy the density gradient constraints, a constraint on density grayness is also incorporated into the formulations. Numerical results on Messerschmitt–Bolkow–Blohm beams, cantilever beams, and 2D and 3D heat conduction problems demonstrate the proposed formulations are effective in controlling the undercut and the minimal overhang angle in the optimized designs. Copyright © 2016 John Wiley & Sons, Ltd.

Xia L., Xia Q., Huang X., Xie Y.M.

The evolutionary structural optimization (ESO) method developed by Xie and Steven (Comput Struct 49(5):885–896, 162), an important branch of topology optimization, has undergone tremendous development over the past decades. Among all its variants, the convergent and mesh-independent bi-directional evolutionary structural optimization (BESO) method developed by Huang and Xie (Finite Elem Anal Des 43(14):1039–1049, 48) allowing both material removal and addition, has become a widely adopted design methodology for both academic research and engineering applications because of its efficiency and robustness. This paper intends to present a comprehensive review on the development of ESO-type methods, in particular the latest convergent and mesh-independent BESO method is highlighted. Recent applications of the BESO method to the design of advanced structures and materials are summarized. Compact Malab codes using the BESO method for benchmark structural and material microstructural designs are also provided.

Zhang W., Li D., Zhang J., Guo X.

A novel approach, which can control the minimum length scale in topology optimization in a straightforward and explicit way, is proposed in the present paper. This approach is constructed under the so-called Moving Morphable Components (MMC) based solution framework where optimized structural topology can be found by changing the shapes and layout of a set of trapezoid-shaped structural components on a fixed finite element mesh. The eXtended Finite Element Method (XFEM) is employed for structural response analysis and shape sensitivity information is obtained by carrying out numerical integration along the structural boundary. A precision definition of the minimum length scale of a structure is also suggested based on the proposed solution framework. Compared with existing approaches, the proposed approach can achieve minimum length scale control by simply setting lower bounds on a set of geometric design variables. Only a small number of purely geometric constraints imposed on the sizes of intersection regions should be considered explicitly in order to get a complete minimum length scale control. Numerical examples demonstrate the effectiveness of the proposed method.

Ye Z., Pan W.

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).

Liang Y., Cheng G.

Li L., Guo G., Chen X., Chen G., Yang D.

Deng Z., Lei Z., Cheng G., Liang Y.

Lei Z., Liang Y., Cheng G., Yang D., Chen G.

Sun K., Cheng G., Liang Y.

Bacciaglia A., Ceruti A., Liverani A.

AbstractThe topology optimization methodology is widely utilized in industrial engineering for designing lightweight and efficient components. In this framework, considering natural frequencies is crucial for adequately designing components and structures exposed to dynamic loads, as in aerospace or automotive applications. The scientific community has shown the efficiency of Bi-directional Evolutionary Structural Optimization (BESO), showcasing its ability to converge towards optimal solid-void or bi-material solutions for a wide range of frequency optimization problems in continuum structures. However, these methods show limits when the complexity of the domain volume increases; thus, they are well-suited for academic case studies but may fail when dealing with industrial applications that require more complex shapes. The connectivity of the structures resulting from the optimization also plays a fundamental role in choosing the best optimization approach, as some available commercial and open-source codes nowadays return unfeasible sparse structures. An improved voxel-based BESO algorithm has been developed in this work to cope with current limits in lightweight structure optimization. A significant case study has been developed to evaluate the performances of the new methodology and compare it with existing algorithms. In contrast to previous studies, the method we developed guarantees that the final structure respects constraints on the initial design volume and that the structure’s connection is preserved, thus enabling the manufacturing of the component with Additive Manufacturing technologies. The proposed approach can be complemented by smoothing algorithms to obtain a structure with externally appealing surfaces.

Wang X., Wang Z., Ni B.

Quantum computing (QC) is a rapidly growing technology in the field of computation that has garnered significant attention in recent years. This emerging technology has become particularly relevant due to the increasing complexity of optimization problems and their expanding search spaces. As a result, innovative solutions that can surpass the limitations of the current optimization paradigms executed on classic computers are becoming necessary. D-wave, a specialized quantum computer, presents a novel solution for addressing intricate optimization problems with remarkable speed advantages over traditional methods. However, a major hurdle in terms of utilizing the D-wave platform for topology optimization design is the conversion of an optimization problem into formulas that can be comprehended by a quantum annealing machine. This is because the D-wave platform is limited to solving quadratic unconstrained binary optimization problems or Ising model problems, making it necessary to find a way to adapt the task of interest to these specific types of optimization problems. This paper examines the current reality concerning the extremely limited availability of quantum computing resources. We focus on small-scale discrete structural topology optimization problems as a starting point and establish a mapping relationship between quantum bits and the cross-sectional area variables of truss elements. Utilizing this mapping, a quadratic unconstrained binary optimization model is developed with these variables. We propose a nested optimization process with dynamically adjusted cross-sectional areas, which enables the development of a quantum annealing approach for optimizing the topology of discrete variables. Our method is validated through numerical experiments, demonstrating its efficiency.

Zhou J., Wang Y., Chiu L.N., Ghabraie K.

Abstract This paper presents a concurrent topology optimization method for macro and micro phases based on non-penalization smooth-edged material distribution for optimization topology (SEMDOT) method. Although there is existing research on the multiscale design method, grayscale elements are always emerged especially for penalization method for example the solid isotropic material penalization (SIMP) method, also high computational cost are required when large scale of elements are utilized for obtaining high resolution structures. The methodology proposed here aims to apply a new tech called non-penalization SEMDOT method to find the optimum layout on both scales of elements, it is assumed that the macro structure is composed of periodic materials and both element scales are optimized through their linearly interpolated grid points. The effective macroscopic properties are evaluated by the homogenization method. The approach could provide smooth and clear boundaries for multiscale system without grayscale elements or high computational cost. A series of numerical examples are presented to demonstrate the effectiveness and the efficiency of the proposed method.

Yan X., Liang Y., Cheng G., Pan Y., Cai X.

A discrete variable topology optimization method of internally finned ducts in heat exchangers for efficient thermal performance is proposed. Fully developed convective heat transfer (FDCHT) model, which has been extensively employed and well checked in practical thermal engineering applications, is considered here. Under the uniform wall temperature boundary condition, the energy equation of the FDCHT model can be mathematically formulated as a generalized eigenvalue equation, and the total thermal resistance reflecting the thermal performance can be related to the eigenvalue. The well-known eigenvalue optimization formulation and sensitivity analysis is applied to this optimization problem. Significantly, the physical reality, such as precise Nusselt number, is maintained by using the discrete variable method, i.e., Sequential Approximate Integer Programming. Here, only the 0–1 densities denoted to solid and fluid are involved so that blurry intermediate zones and interpolation schemes are avoided. The practical engineering conditions of fixed pumping power and fixed fluid volume flow rate are discussed under the unified framework, respectively. Several designs of internally finned ducts without any blurry zone are obtained. The optimized design conforms to the three-dimensional precise Conjugate Heat Transfer calculation. Numerical results show that contrary to the design method by predefined heat transfer coefficient, the proposed method can automatically obtain the optimal shape and spacing of fins since the spatially varying effect of convective heat transfer has been achieved.

Deng Z., Liang Y., Cheng G.

AbstractFinding optimized structural topology design for maximizing natural frequencies and frequency gaps of continuum structures is crucial for engineering applications. However, two significant numerical issues must be addressed: non‐smoothness caused by multiple frequencies and Artificial Localized Rigid Motion (ALRM) modes due to the violation of the topological constraint related to isolated islands and point‐connections. The above two issues are solved by employing the discrete variable topology optimization method based on Sequential Approximate Integer Programming (SAIP). First, the directional differentiability and multiple frequencies preservation constraints are formulated as the linear integer constraints. And then, the integer programming with these linear integer constraints is established and solved by the discrete variable linear or quadratic programming solver with multi‐constraint Canonical Relaxation Algorithm (CRA). We also prove that the popular Average Modal Frequencies (AMF) strategy, like the Kreisselmeier‐Steinhauser (KS) function, cannot rigorously tackle this non‐smoothness caused by multiple frequencies. Furthermore, to eliminate the ALRM modes and concentrate on the real structural global modes, the burning method is employed to impose the topological constraint of the first Betti number that represents the number of isolated islands and point‐connections. Numerical examples, including 2D and 3D, two‐fold and three‐fold multiple frequencies, natural frequencies and frequency gaps, are presented to show the effectiveness of the proposed method.

Liu H., Wang C., Zhang Y., Liang Y.

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.

Zhou J., Zhao G., Zeng Y., Li G.

A novel moving morphable components (MMC)-based topology optimization method is proposed to solve topology optimization problems of plate structure subjected to out-of-plane load, by introducing a grid structure composed of beam elements to perform finite element analysis with the adaptive pruning of elements (APE) module. The grid structure can describe the mechanical property of the plate structure by using the model with concise expression of stiffness matrix. The APE can reduce the redundant degrees of freedom (DOFs) of the grid structure as an adaptive mesh technology to improve the efficiency of optimization. To validate the proposed method, the benchmark examples are tested in comparison with MMC method using Kirchhoff plate elements, which shows that the proposed method can obtain the consistent optimization results with less calculation cost under specified conditions. In addition, several non-centrosymmetric numerical examples considering model parameter and APE influences are discussed to demonstrate the proposed method’s characteristics in efficiency and stability.

Lei Z., Zhang J., Liang Y., Chen G., Yang D.

In practical applications, some random variables follow multimodal distributions. However, conventional reliability analysis methods in reliability-based topology optimization (RBTO), such as the first order reliability method, often result in considerable computational errors when dealing with problems involving multimodal distributions. Consequently, the corresponding RBTO design is incredible. Moreover, the RBTO problem also faces the challenge of high computational cost. To this end, this paper proposes an efficient two-phase approach for RBTO of continuum structures with multimodal distributions combining sequential approximate integer programming with trust region (SAIP-TR) and direct probability integral method (DPIM). Firstly, DPIM is advanced to address the difficult problems of failure probability estimation and efficient sensitivity analysis under multimodal distributions. Secondly, a reliability-based discrete variable topology optimization framework based on SAIP-TR and DPIM is established, which yields clear topology configurations and facilitates engineering manufacturing. Owing to the merit of SAIP-TR, the original RBTO process is divided into two phases: the first phase performs deterministic topology optimization, and the second phase focuses on RBTO. Moreover, an adaptive selection strategy of representative points, considering structural compliance as performance function, is devised to further enhance computational efficiency. Finally, several examples illustrate high efficiency and accuracy of the proposed approach. The multimodal random variable and the random field are employed separately to describe global and local uncertainties in materials. In contrast, the optimized result considering global material uncertainty is more suitable for additive manufacturing. The proposed approach also presents potential in handling complex RBTO problems with random fields.

Bayat M., Zinovieva O., Ferrari F., Ayas C., Langelaar M., Spangenberg J., Salajeghe R., Poulios K., Mohanty S., Sigmund O., Hattel J.

Additive manufacturing (AM) processes have proven to be a perfect match for topology optimization (TO), as they are able to realize sophisticated geometries in a unique layer-by-layer manner. From a manufacturing viewpoint, however, there is a significant likelihood of process-related defects within complex geometrical features designed by TO. This is because TO seldomly accounts for process constraints and conditions and is typically perceived as a purely geometrical design tool. On the other hand, advanced AM process simulations have shown their potential as reliable tools capable of predicting various process-related conditions and defects hence serving as a second-to-none material design tool for achieving targeted properties. Thus far, these two geometry and material design tools have been traditionally viewed as two entirely separate paradigms, whereas one must conceive them as a holistic computational design tool instead. More specifically, AM process models provide input to physics-based TO, where consequently, not only the designed component will function optimally, but also will have near-to-minimum manufacturing defects. In this regard, we aim at giving a thorough overview of holistic computational design tool concepts applied within AM. The paper is arranged in the following way: first, literature on TO for performance optimization is reviewed and then the most recent developments within physics-based TO techniques related to AM are covered. Process simulations play a pivotal role in the latter type of TO and serve as additional constraints on top of the primary end-user optimization objectives. As a natural consequence of this, a comprehensive and detailed review of non-metallic and metallic additive manufacturing simulations is performed, where the latter is divided into micro-scale and deposition-scale simulations. Material multi-scaling techniques which are central to the process-structure-property relationships, are reviewed next followed by a subsection on process multi-scaling techniques which are reduced-order versions of advanced process models and are incorporable into physics-based TO due to their lower computational requirements. Finally the paper is concluded and suggestions for further research paths discussed.