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This paper presents a computational framework for multimaterial topology optimization under uncertainty. We combine stochastic collocation with design sensitivity analysis to facilitate robust design optimization. The presence of uncertainty is motivated by the induced scatter in the mechanical properties of candidate materials in the additive manu...
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... However, material uncertainty problems typically require many stochastic dimensions, and hence, the required number of tensorial quadrature points increases rapidly. To mitigate this scaling issue, various sample reduction methods have been considered, such as the Smolyak sparse-grid (Rostami et al. 2021(Rostami et al. , 2023, adaptive quadrature schemes (Keshavarzzadeh and James 2019), and dimension reduction techniques (Zheng et al. 2018(Zheng et al. , 2022Jing et al. 2019). Dimension reduction techniques can also be applied to accelerate the Monte Carlo RTO method. ...
This paper presents an efficient computational method for optimal structural design in the presence of uncertain Young’s modulus modeled using discretized random fields. To quantify and propagate the uncertainty, random matrix theory is employed to quantify uncertainty in the context of robust topology optimization (RTO) for the minimization of compliance. Random matrix theory employs statistical inference methods to model the matrix-variate probability distribution of the finite element stiffness matrix. This provides analytical expressions for the mean and the standard deviation of the compliance, a combination of which is minimized in RTO. The novel random matrix theory-based RTO is computationally efficient due to the intrusive nature of the method, and is flexible as its computational performance and robustness remain consistent regardless of the correlation lengths or the variance of the random field, as demonstrated through numerical cases. The random matrix RTO method is applied to several two-dimensional numerical problems where the random fields of the modulus are assigned with ranges of correlation lengths and variances to illustrate the versatility of the method. The performance of random matrix RTO is compared with Monte Carlo RTO and stochastic collocation RTO to explore the efficiency and accuracy of the method.
... In contrast, robust topology optimization (RTO) is concerned with the variability of the structure performance at its mean value, aiming to reduce the sensitivity of the structure performance to uncertainties. Scholars have extensively researched the RTO problem in recent years, mainly focusing on the load (size and direction [Zhao and Wang, 2014;Zheng et al., 2020], the position of action [Wang and Gao, 2019]), material (anisotropy [Jansen et al., 2013;Keshavarzzadeh and James, 2019], stiffness [Asadpoure et al., 2011;Cardoso et al., 2019], and Poisson's ratio [Zheng et al., 2021]), and geometry uncertainty [Guo et al., 2013]. Generally, methods for characterizing uncertain quantities through interval models [Liu et al., 2017;Zhan and Luo, 2019] (including the worstcase [Schevenels et al., 2011;Thore et al., 2017]) or convex models [Bai and Kang, 2021;Hu et al., 2017;Jiang et al., 2015] can be called non-probabilistic RTO. ...
The rapid development of additive manufacturing has made coated structures an innovative configuration with high design flexibility. However, poor forming accuracy and surface roughness during manufacturing will cause uncertainty in surface layer thickness, which results in structure performance deviation and failure to achieve the expected goals. This paper proposes a robust topology optimization method for coated structures considering the surface layer thickness uncertainty to obtain high-quality designs that can resist disturbance by uncertainties. First, an erosion-based approach is used to establish the model of the coated structure surface layer. Second, modeling the surface layer thickness uncertainty applies a random field whose dimensionality of the random fields is reduced by the Expansion Optimal Linear Estimation (EOLE) method. Then, minimizing the weighted sum of the mean and standard deviation of structural compliance is taken as the optimization objective, and robust topology optimization considering uncertainty is established. Finally, estimate the stochastic response by the perturbation technique, then the sensitivity of the objective function with respect to the design variables is derived. Numerical examples show that the structural design obtained with the proposed method has a stronger resistance to uncertainty than the deterministic topology optimization method, proving the method’s effectiveness in this paper.
... On the other hand, it would be possible to develop uncertainty-based optimization approaches to accommodate fabrication-induced imperfections without eliminating the source of uncertain variability [22,[612][613][614]. In this regard, topology optimization considering material and geometry uncertainties has become an active research topic with the advent of various AM technologies [608,[615][616][617][618][619][620][621]. ...
... Robust BESO topology optimization [613] Metal Robust multiphase topology optimization [621] Metal Reliability-based and robust topology optimization [616] Metal Material & geometrical SIMP topology optimization with stochastic collocation methods [617] Metal SIMP-based topology optimization with stochastic collocation methods performance in feature recognition with high efficiency [662][663][664]. ...
Lightweight materials and structures have been extensively studied for a wide range of applications in design and manufacturing of more environment-friendly and more sustainable products, such as less materials and lower energy consumption, while maintaining proper mechanical and energy absorption characteristics. Additive manufacturing (AM) or 3D printing techniques offer more freedom to realize some new designs of novel lightweight materials and structures in an efficient way. However, the rational design for desired mechanical properties of these materials and structures remains a demanding topic. This paper provides a comprehensive review on the recent advances in additively manufactured materials and structures as well as their mechanical properties with an emphasis on energy absorption applications. First, the additive manufacturing techniques used for fabricating various materials and structures are briefly reviewed. Then, a variety of lightweight AM materials and structures are discussed, together with their mechanical properties and energy-absorption characteristics. Next, the AM-induced defects, their impacts on mechanical properties and energy absorption, as well as the methods for minimizing the effects are discussed. After that, numerical modeling approaches for AM materials and structures are outlined. Furthermore, design optimization techniques are reviewed, including parametric optimization, topology optimization, and nondeterministic optimization with fabrication-induced uncertainties. Notably, data-driven and machine learning-based techniques exhibit compelling potential in design for additive manufacturing, process-property relations, and in-situ monitoring. Finally, significant challenges and future directions in this area are highlighted. This review is anticipated to provide a deep understanding of the state-of-the-art additively manufactured materials and structures, aiming to improve the future design for desired mechanical properties and energy absorption.
... However, non-intrusive UQ turns computation unaffordable, while considering the hybrid uncertainties problem; it will cause the dimension curse issue. Some papers relieve or solve this problem through three aspects, (1) reduce the cost of the FEM simulation [64], (2) use sparse grid collocation points [25,63,67] (3)uncertainty dimension reduction [132]. In addition, the intrusive approach, where analytically expressed mean and variance, is explored by literature [66]. ...
Topology optimization is a systemic design that requires simulation and optimization of a system for a single or multiple physics coupling processes. However, it is short of the engineering sense regarding the absence of uncertainties and limitations on applied monophase material. The foundation of this dissertation is to combine homogenization and stochastic processing into topology optimization to formulate a robust multiscale topology optimization approach. Accordingly, this Ph.D. dissertation concerns (1) the multiscale and multiphysics performance of heterogeneous materials/structures embedded with microstructures material, taking into account the uncertainties, (2) for further optimizing the heterogeneous structure at different scales to satisfy target performance. These microstructures may arise from the processing of biological materials, or from dedicated engineered materials, e.g., aerogels, foams, composites, acoustics metamaterials, etc. We parametrize architecture material; study the performances of the microstructure at the macroscopic scale by homogenization method. Then, the homogenization model can be considered a stochastic model with presented uncertainties exhibited in the unit cell. It can be built from a polynomial chaos development. In addition, these parametrized micro geometry features can be mapped into homogenized properties space, which can be utilized as design variables to control the macrostructure performance. Afterward, we combined the topology optimization, homogenization, and uncertainties qualification to (1) design macro topology and micro material distribution to maximum structure stiffness (2) reduce the structure sensitivity to presented uncertainties (e.g., loading and material properties). This proposed general framework has the advance and compatibility ability in solving optimization problems considering the (1) multiple parametrized architectures cells, (2) complex loading problem, (3) hybrid uncertified, etc., with an affordable computation manner.
... The former approach finds optimal design parameters while constraining the performance variations, and the latter approach entails finding the optimal parameters while constraining the failure probability. In both cases, to compute probabilistic quantities such as mean and standard deviation (in robust design optimization) [13][14][15][16][17] and probability of failure (in reliability based design optimization) [18][19][20][21], the performance function needs to be evaluated several times at each design iteration. In this paper we focus on the robust topology optimization problem, which has been the subject of active research in the past two or three decades. ...
... To perform this optimization, the sensitivity of the mechanical advantage is needed. Interested readers are referred to [16] for the details of sensitivity analysis for mechanical advantage. Figure 17 shows the randomization study errors for test data sets which are obtained by randomly initializing the neural net 10 times. ...
We present a low rank approximation approach for topology optimization of parametrized linear elastic structures. The parametrization is considered on loading and stiffness of the structure. The low rank approximation is achieved by identifying a parametric connection among coarse finite element models of the structure (associated with different design iterates) and is used to inform the high fidelity finite element analysis. We build an Artificial Neural Network (ANN) map between low resolution design iterates and their corresponding interpolative coefficients (obtained from low rank approximations) and use this surrogate to perform high resolution parametric topology optimization. We demonstrate our approach on robust topology optimization with compliance constraints/objective functions and develop error bounds for the the parametric compliance computations. We verify these parametric computations with more challenging quantities of interest such as the p-norm of von Mises stress. To conclude, we use our approach on a 3D robust topology optimization and show significant reduction in computational cost via quantitative measures.
... On the other hand, probabilistic approaches make use of the stochastic characterization of the uncertainty of the phenomenon of interest. These methods can either consider the statistical moments of the structural response, leading to robust topology optimization methods [12,15,20,24,26], or they can also include probabilities of failure, leading to risk-averse [11,27] and reliability-based topology optimization methods [21]. In reliability-based topology optimization methods, reliability constraints are imposed to the design. ...
This paper presents a risk-averse approach in the context of fail-safe topology optimization. The main novelty is the minimization of two risk functions quantifying the costs inherent to partial or full collapses, whose occurrence is considered as a source of uncertainty. This provides the designer with the flexibility to explicitly incorporate probabilistic information of occurrence of different structural failures, in contrast to the worst case approach, that penalizes all the damage configurations regardless their probability of occurrence. For the first time in the context of fail-safe topology optimization, a level-set method is employed. The level-set function is updated by means of a reaction–diffusion equation incorporating the topological derivative of the two risk-averse functions considered. Finally, the numerical experiments reveal the capability of the proposed formulations to yield redundant structures less sensitive to inherent losses of stiffness resulting from possible failures, whilst allowing designers to assume an acceptable level of risk. The benefits and drawbacks of the formulations proposed are compared against deterministic and fail-safe worst-case formulations.
... On the other hand, probabilistic approaches make use of the stochastic characterization of the uncertainty of the phenomenon of interest. These methods can consider the statistical moments of the structural response, and therefore be classified as robust topology optimization methods [23,24,25,26,27,28], or they can also include probabilities of failure, leading to risk-averse [29,30] and reliability-based topology optimization methods [31,32,33,34]. Due to the binary nature of failure definition, it is difficult to treat fail-safe requirements within a probabilistic framework [14]. ...
This paper presents a novel probabilistic approach for fail-safe robust topology optimization with the following novelties: (1) the probability for failure to occur at a specified location is considered; (2) the possibility for random failure size is incorporated; (3) a multi-objective problem is pursued encompassing both the expected value of the structural performance and its variance as a robustness criterion. Compared against alternative worst-case-based formulations, the probabilistic framework employed allows designers to assume certain level of risk, avoiding undesirable increments in structural performance due to low probability damage configurations; (4) alternatively to most existing works within fail-safe topology optimization, considering density-based methods, this paper pursues for the first time an optimization technique where the structural boundary is represented implicitly by an iso-level of an optimality criterion field, which is gradually evolved using a bisection method. A key advantage of this technique is that it provides optimized solutions for different volume fractions during the optimization process, allowing to efficiently find a trade-off between structural performance, cost and robustness. Finally, numerical results are included demonstrating the ability of the proposed formulation to provide smooth and clearly defined structural boundaries and to enhance structural robustness with respect to conventional deterministic designs.
... Sigmund and Torquato [10] later used the inverse homogenization method to design multi-material periodic microstructures, achieving materials with extreme thermal expansion coefficients beyond those of periodic microstructures are manufactured on small scales, making manufacturing uncertainty an even more important consideration. One group of approaches in addressing manufacturing uncertainty involves statistical analysis, which has been applied to multi-material topology optimization with uncertainties in material properties [27][28][29][30]. Considering variations in geometry (which is of interest in this research), a probabilistic method using random geometry variations and Monte Carlo simulations has been applied to single-material topology optimization [31], however the number of analyses required in each iteration to compute averages and standard deviations makes the method computationally expensive. ...
To take advantage of multi-material additive manufacturing technology using mixtures of metal alloys, a topology optimization framework is developed to synthesize high-strength spatially periodic metamaterials possessing unique thermoelastic properties. A thermal and mechanical stress analysis formulation based on homogenization theory is developed and is used in a regional scaled aggregation stress constraint method. Since specific load cases are not always known beforehand, a method of worst-case stress minimization is also included to efficiently address load uncertainty. It is shown that the two stress-based techniques lead to thermal expansion properties that are highly sensitive to small changes in material distribution and composition. To resolve this issue, a uniform manufacturing uncertainty method is utilized which considers variations in both geometry and material mixture. Test cases of high stiffness, zero thermal expansion, and negative thermal expansion microstructures are generated, and the stress-based and manufacturing uncertainty methods are applied to demonstrate how the techniques alter the optimized designs. Large reductions in stress are achieved while maintaining robust strength and thermal expansion properties.
In the past three decades, biomedical engineering has emerged as a significant and rapidly growing field across various disciplines. From an engineering perspective, biomaterials, biomechanics, and biofabrication play pivotal roles in interacting with targeted living biological systems for diverse therapeutic purposes. In this context, in silico modelling stands out as an effective and efficient alternative for investigating complex interactive responses in vivo. This paper offers a comprehensive review of the swiftly expanding field of machine learning (ML) techniques, empowering biomedical engineering to develop cutting-edge treatments for addressing healthcare challenges. The review categorically outlines different types of ML algorithms. It proceeds by first assessing their applications in biomaterials, covering such aspects as data mining/processing, digital twins, and data-driven design. Subsequently, ML approaches are scrutinised for the studies on mono-/multi-scale biomechanics and mechanobiology. Finally, the review extends to ML techniques in bioprinting and biomanufacturing, encompassing design optimisation and in situ monitoring. Furthermore, the paper presents typical ML-based applications in implantable devices, including tissue scaffolds, orthopaedic implants, and arterial stents. Finally, the challenges and perspectives are illuminated, providing insights for academia, industry, and biomedical professionals to further develop and apply ML strategies in future studies.
Topology optimization is a powerful tool for structural design, while its computational cost is quite high due to the large number of design variables, especially for multi-material systems. Herein, an incremental interpolation approach with discrete cosine series expansion (DCSE) is established for multi-material topology optimization. A step function with shape coefficients (i.e., ensuring that no extra variables are required as the number of materials increases) and the use of the DCSE together reduces the number of variables (e.g., from 8400 to 120 for the optimization of the clamped-clamped beam with four materials). Remarkably, the proposed approach can effectively bypass the checkerboard problem without using any filter. The enhanced computational efficiency (e.g., a ∼89.2% reduction in computation time from 439.1 s to 47.4 s) of the proposed approach is validated via both 2D and 3D numerical cases.
