Review Special Issues

Uncertain design optimization of automobile structures: A survey


  • Received: 03 December 2022 Revised: 13 December 2022 Accepted: 14 December 2022 Published: 03 January 2023
  • In real life, there are a lot of uncertainties in engineering structure design, and the potential uncertainties will have an important impact on the structural performance responses. Therefore, it is of great significance to consider the uncertainty in the initial stage of structural design to improve product performance. The consensus can be reached that the mechanical structure obtained by the reliability and robustness design optimization method considering uncertainty not only has low failure risk but also has highly stable performance. As a large mechanical system, the uncertainty design optimization of key vehicle structural performances is particularly important. This survey mainly discusses the current situation of the uncertain design optimization framework of automobile structures, and successively summarizes the uncertain design optimization of key automobile structures, uncertainty analysis methods, and multi-objective iterative optimization models. The uncertainty analysis method in the design optimization framework needs to consider the existing limited knowledge and limited test data. The importance of the interval model as a non-probabilistic model in the uncertainty analysis and optimization process is discussed. However, it should be noted that the interval model ignores the actual uncertainty distribution rule, which makes the design scheme still have some limitations. With the further improvement of design requirements, the efficiency, accuracy, and calculation cost of the entire design optimization framework of automobile structures need to be further improved iteratively. This survey will provide useful theoretical guidance for engineers and researchers in the automotive engineering field at the early stage of product development.

    Citation: Xiang Xu, Chuanqiang Huang, Chongchong Li, Gang Zhao, Xiaojie Li, Chao Ma. Uncertain design optimization of automobile structures: A survey[J]. Electronic Research Archive, 2023, 31(3): 1212-1239. doi: 10.3934/era.2023062

    Related Papers:

    [1] DavidWood, Dale K. Hensley, Nicholas Roberts . Enhanced thermal conductance of polymer composites through embeddingaligned carbon nanofibers. AIMS Materials Science, 2016, 3(3): 851-861. doi: 10.3934/matersci.2016.3.851
    [2] G. A. El-Awadi . Review of effective techniques for surface engineering material modification for a variety of applications. AIMS Materials Science, 2023, 10(4): 652-692. doi: 10.3934/matersci.2023037
    [3] Abbas Hodroj, Lionel Teulé-Gay, Michel Lahaye, Jean-Pierre Manaud, Angeline Poulon-Quintin . Nanocrystalline diamond coatings: Effects of time modulation bias enhanced HFCVD parameters. AIMS Materials Science, 2018, 5(3): 519-532. doi: 10.3934/matersci.2018.3.519
    [4] Nikolay A. Voronin . Specialties of deformation and damage of the topocomposite on a ductile substrate during instrumental indentation. AIMS Materials Science, 2020, 7(4): 453-467. doi: 10.3934/matersci.2020.4.453
    [5] Muhamed Shajudheen V P, Saravana Kumar S, Senthil Kumar V, Uma Maheswari A, Sivakumar M, Sreedevi R Mohan . Enhancement of anticorrosion properties of stainless steel 304L using nanostructured ZnO thin films. AIMS Materials Science, 2018, 5(5): 932-944. doi: 10.3934/matersci.2018.5.932
    [6] Daria Wehlage, Robin Böttjer, Timo Grothe, Andrea Ehrmann . Electrospinning water-soluble/insoluble polymer blends. AIMS Materials Science, 2018, 5(2): 190-200. doi: 10.3934/matersci.2018.2.190
    [7] Olayinka Oluwatosin Abegunde, Esther Titilayo Akinlabi, Oluseyi Philip Oladijo, Stephen Akinlabi, Albert Uchenna Ude . Overview of thin film deposition techniques. AIMS Materials Science, 2019, 6(2): 174-199. doi: 10.3934/matersci.2019.2.174
    [8] Wiebke Langgemach, Andreas Baumann, Manuela Ehrhardt, Thomas Preußner, Edda Rädlein . The strength of uncoated and coated ultra-thin flexible glass under cyclic load. AIMS Materials Science, 2024, 11(2): 343-368. doi: 10.3934/matersci.2024019
    [9] Rucheng Zhu, Yota Mabuchi, Riteshkumar Vishwakarma, Balaram Paudel Jaisi, Haibin Li, Masami Naito, Masayoshi Umeno, Tetsuo Soga . Enhancing in-plane uniformity of graphene nanowalls using a rotating platform for solid-state lithium-ion battery. AIMS Materials Science, 2024, 11(4): 760-773. doi: 10.3934/matersci.2024037
    [10] Shuhan Jing, Adnan Younis, Dewei Chu, Sean Li . Resistive Switching Characteristics in Electrochemically Synthesized ZnO Films. AIMS Materials Science, 2015, 2(2): 28-36. doi: 10.3934/matersci.2015.2.28
  • In real life, there are a lot of uncertainties in engineering structure design, and the potential uncertainties will have an important impact on the structural performance responses. Therefore, it is of great significance to consider the uncertainty in the initial stage of structural design to improve product performance. The consensus can be reached that the mechanical structure obtained by the reliability and robustness design optimization method considering uncertainty not only has low failure risk but also has highly stable performance. As a large mechanical system, the uncertainty design optimization of key vehicle structural performances is particularly important. This survey mainly discusses the current situation of the uncertain design optimization framework of automobile structures, and successively summarizes the uncertain design optimization of key automobile structures, uncertainty analysis methods, and multi-objective iterative optimization models. The uncertainty analysis method in the design optimization framework needs to consider the existing limited knowledge and limited test data. The importance of the interval model as a non-probabilistic model in the uncertainty analysis and optimization process is discussed. However, it should be noted that the interval model ignores the actual uncertainty distribution rule, which makes the design scheme still have some limitations. With the further improvement of design requirements, the efficiency, accuracy, and calculation cost of the entire design optimization framework of automobile structures need to be further improved iteratively. This survey will provide useful theoretical guidance for engineers and researchers in the automotive engineering field at the early stage of product development.



    1. Introduction

    It is reported that under some synthesis parameters a silicon nitride coating forms on the sidewalls and at the base of vertically aligned carbon nanofibers (VACNFs) during plasma enhanced chemical vapor deposition (PECVD) synthesis [1,2,3,4,5]. The conical shape of VACNFs, as captured using scanning electron microscopy image in Figure 1, is due to SiNx sheathing layer as shown in transmission electron microscopy images in Figure 2. This coating has proven useful for enhancing mechanical properties and biocompatibility [6]. The increased mechanical properties stem from changing the fiber structure to a more conical shape in comparison to the cylindrical shape of unsheathed carbon-rich VACNFs. Some of the benefits of this coating include the ability of fibers to withst and being pressed into tissue many times without mechanical degradation [3] and sufficient rigidity to undergo many post-processing steps such as spincasting without collapse [3,6,7]. SiNx nano-layers have been shown as promising in photovoltaics as surface passivation [8,9]. In one study, the source of the SiNx sheathing was a thin sacrificial Si layer on the underlying substrate [2]. A different study used drop cast Si microparticles that redeposited onto the VACNFs during synthesis [6]. However, in all of these instances, the only source of silicon was either the bulk substrate or a layer added to the substrate. It was never as a silicon bearing gas, such as silane. VACNFs are conventionally grown from a combination of two gases, a carbon source and a nitrogenous etchant, most typically acetylene and ammonia, respectively. The coating forms during PECVD synthesis through redeposition of any present silicon in combination with nitrogen from ammonia. This SiNx coating is a by-product of a complex interplay of p lasma surface interactions as it is yet to be fully analyzed [10]. The coating can also be fluorescent in the visible spectrum, making it potentially useful for biomarking in cellular interfacing studies, including impalefection, a technique employing VACNFs and other high aspect ratio structures for macromolecular delivery into cells and tissue [11]. Herein is proposed a deposition mechanism for the SiNx coating to the VACNF sidewalls as well as exploration of the photoluminescent properties of the coating.

    Figure 1. SEM image of an array of vertically aligned carbon nanofibers synthesized on Si substrate with SiNx coating. A portion of an uncoated carbon nanofiber is extending from SiNx sheath by about 500 nm at the tip of each fiber.
    Figure 2. TEM image of a VACNF coated with SiNx. The initial stages of the film growth at the freshly form carbon nanofiber surface can be observed near the tip. a) Secondary electron mode showing surface of the nanofiber b) Z-contrast mode showing contrast between carbon core and SiN coating c) transmission mode displaying contrast between CNF and SiNx coating.





    2. Materials and Method

    2.1. VACNF Synthesis

    Vertically aligned carbon nanofibers were synthesized on n-type <100> Si wafers. Dots 2µm in diameter were photo-lithographically patterned onto the wafers. A 50 nm thick nickel catalyst layer was then deposited via electron beam evaporation and liftoff was performed to remove the nickel everywhere except for the previously defined dot pattern. The nanofibers were grown in a custom-built dc-PECVD chamber. The growth parameters for the VACNFs that were used for the fluorescence measurements were 60-sccm C2H2, 100 sccm NH3, 10 Torr, 658 °C, and 3A for 10 min. The fibers that underwent Auger depth profiling were grown at 700 °C with 80 sccm NH3, 40 sccm C2H2, 3 Torr, 350 mA, for 1 hour. In depth explanation of VACNF synthesis can be found elsewhere [12,13,14,15].

    A PECVD deposited SiNx reference film was deposited on a separate p-type <100> Si wafer using an Oxford Instruments Plasmalab 100 PECVD system. The growth parameters were 400 sccm of 5 % SiH4/Ar, 20 sccm of NH3, and 600 sccm of N2 at 650 mT and 350 °C for 17 minutes, resulting in a film 115 nm thick.

    2.2. Characterization

    Characterization of the resultant fibers included optical measurements, imaging, and chemical analysis. Photoluminescence mapping was performed using a Spex Fluorolog 2 at room temperature over an excitation range of 300-500 nm and an emission range of 350-700 nm. Additionally, fluorescence microscopy in a Leica TCS SP2 MP laser scanning confocal system was used to comparatively assess photobleaching of the SiNx coating against transiently expressed fluorescent proteins. EDX was used extensively along with auger electron spectroscopy (AES) depth profiling to determine chemical composition. EDX and SEM imaging was done using a Zeiss Merlin SEM with a Bruker EDX system. AES was performed using a PHI 680 Scanning Auger Nanoprobe.

    3. Results and Discussion

    3.1. Deposition Mechanism

    There are two primary mechanisms as to how the SiNx coating forms; either the silicon is sputtered from the substrate to the sidewalls of the fiber, or hydrogen volatilizes silicon from the substrate creating silane and other compounds in the plasma whereupon it further reacts with nitrogen from the ammonia and is redeposited through CVD processes. It has been shown that it is possible to form SiNx films using rf magnetron sputtering by using a silicon target with argon and ammonia process gases [16]. However, the amount of power supplied to the system is much greater than in a PECVD chamber. Additionally the deposition pressures are usually lower, meaning the particles in the system experience fewer collisions and retain more of their energy before impacting the substrate or target. It seems unlikely that the ions and gas molecules in a PECVD chamber would acquire the energy necessary to physically knock out many Si atoms from the substrate.

    PECVD deposition of silicon nitride is also well established as previously mentioned. However, films deposited via PECVD use a silicon bearing gas, such as silane, as a silicon source, instead of the substrate itself. The silane decomposes and combines with nitrogen (usually from ammonia) on the surface of the substrate.

    The films formed on VACNFs differ from both of these techniques in several ways. The largest difference is the films formed on VACNFs are being formed on three dimensional structures that are continuously growing during deposition instead of on a planar surface that exists before deposition begins. Additionally, for the work presented here, the source of silicon is only from the substrate, as opposed to sputtering and PECVD SiNx films which use a target or a silicon bearing gas respectively. Figure 3 shows the possible mechanisms of deposition. It is important to note that sputtering usually occurs at very low pressures, around 10-5 Pa which correlates to a mean free path of ~104 m. The synthesis of VACNFs however, takes place at hundreds to thous and s of Pa. A growth at 4 Torr has a mean free path of roughly only 100 μm. A significantly lower mean free path indicates that it is likely that any sputtered Si atom would undergo collisions with the other process gases and react with it, leading to a more CVD-like deposition. Alternatively, excited hydrogen molecules could react with the substrate and become subsequently volatilized, further reacting with the process gases and resulting in the SiNx film. It is possible that a combination of these two processes is what is occurring to deposit the film.

    Figure 3. Illustration of possible deposition mechanisms of SiNx coating to VACNF sidewalls (1) Reactive sputtering where ions eject Si atoms from the substrate which go on to react with the gas or (2) Excited hydrogen chemically reacts with the substrate and then volatilizes to further react through CVD processes.

    3.2. Photoluminescence properties of SiN-coated VACNFs

    In the literature there is a large ongoing debate on the origin of the photoluminescence observed in SiNx films. There is a substantial body of work indicating the presence of Si nanoclusters (NCs) in Si-rich SiNx films [17,18,19,20,21,22,23,24,25,26,27]. Many of these studies attribute the photoluminescence of the films to quantum confinement effects (QCE) [23,27,28,29,30]. According to the QCE model, the photoluminescence (PL) peak is inversely proportional to the square of the average size of the Si-NCs, while intensity increases with NC density and improved passivation [17]. Others attribute the PL to the presence of defect-related states, such as nitrogen defects or Si dangling bonds [31]. Another study surmised that the blue, green, and red components of the PL were due to d efects, b and tail recombination, and QCE respectively [32]. SiNx films without any Si-NCs have also been studied, with the PL in those samples being attributed to b and tail recombination [24]. It seems that the Si-NCs do not spontaneously form during co-sputtering of Si and Si3N4 targets or with some PECVD parameters and require annealing of more than 1000 °C to form the nanostructures [24,27]. Yet, it appears that growths that encourage an a-SiNx: H matrix yield in-situ formatting of Si-NCs.

    We observe a strong PL response from SiNx coated VACNFs as shown in Figure 4. There appear to be two emission peaks at 416 nm and 432 nm for both the PECVD SiNx coating and the VACNF coating with an excitation wavelength of 380 nm. Figure 5 shows the PL spectra for 380 nm excitation for both samples. It is possible that this dual peak stems from the presence of a bimodal size distribution of Si-NCs. From Figure 5 it can be observed that the intensity of the sample with VACNFs coated with SiNx ­has nearly twice the intensity of the flat PECVD SiNx film, though this could be due to there simply being a greater amount of material present on the VACNFs, which seems unlikely. Here we will make the argument that this PL response is due to the presence of Si-NCs. First, it is important to show the plausibility of Si-NCs forming. As was previously mentioned, both annealing and high hydrogen content have shown the ability to yield Si-NCs. Since the samples are grown at 700 °C in the presence of C2H2 and NH3, both of these conditions are met. Another key factor is that for the synthesis of Si-NCs, the film must be Si-rich, that is that Si/N ratio is greater than stoichiometric Si3N4. Table 1shows that the coating is in fact Si-rich, lending further credibility to the presence of Si-NCs.

    Figure 4. Fluorescent response from a PECVD SiNx coating (top) and from a SiNx coated VACNF array (bottom).





    Figure 5. PL spectra at 380nm excitation for VACNF SiNx and PECVD SiNx.

    Table 1. Relative atomic concentrations in VACNF SiNx coating.
    Element Atomic %
    Si 41.07
    N 45.75
    C 7.86
    O 5.33
     | Show Table
    DownLoad: CSV

    Figure 6 shows an EDS line scan of a broken VACNF with a SiNx coating. From this line scan it can be seen that many nitrogen troughs are accompanied by silicon peaks, while the inverse is never true. It can be inferred then, that these Si peaks are areas where Si-NCs are present. The lack of nitrogen peaks accompanied by silicon troughs is expected, since nitrogen does not have a crystallographic structure at room temperature. The evidence for Si-NCs is further corroborated by the EDS maps shown in Figure 7. From the EDS maps of the same fiber shown in Figure 6, it can be seen that there are areas where there are bright Si clusters.

    Figure 6. EDS line scan of a broken VACNF coated with SiNx (top) with positionally accurate SEM image of broken VACNF (bottom).
    Figure 7. (a) SEM image of broken VACNFs with SiNx coatings on an aluminum viewing platform. EDS maps were made of this area showing location of (b) Nitrogen (c) Carbon (d) Silicon and (e) Composite of all three.



    Perhaps the most convincing proof of the presence of Si-NCs is shown in Figure 8. Auger electron spectroscopy not only reveals elemental composition, but also bonding state of atoms. Combined with ion milling, AES is a very powerful tool for elemental analysis of samples. From Figure 8 it can be seen that there is a significant amount of Si that is bonded to either O or N, and that after ~1000 μm there is only Si bonded Si, indicating that the substrate has been reached by the probe. It is critical to note that Si bonded Si is in abundance throughout the area probed, well above the substrate, alongside the O- and N- bonded Si. Some of this Si is probably bonded to the carbon that is present to form SiC, since AES only shows bonding states of atoms with significantly different electronegativities. However, there is a much greater amount of “unbonded” Si than C, so even if the full amount of present C is used to form SiC, there is still a large amount of Si that must be bonded to Si. These Si-Si bonds must invariably lead to small Si-NCs.

    Figure 8. AES depth profile showing the composition of the SiNx coating on a VACNF as a function of depth. The inset shows the region that was probed.

    Now that the presence of Si-NCs and strong fluorescence of SiNx coated fibers has been established, we provide example of how this characteristic can have application. VACNF arrays have been reported extensively as a mechanical means for gene delivery into cells and tissue using a method whereby sparse arrays of nanofibers are modified with DNA and pressed into cellular matrices, resulting in widescale cellular penetration and ‘microinjection’ of genetic material; aka ‘impalefection’ [1]. An attractive advantage of this gene delivery technique is the ability of nanofibers to achieve nuclear penetration, as observed using freeze fracture and scanning electron microscopy. Nuclear penetration enables the delivery of transgene cargo directly to the transcriptional control center of the cell, thereby resulting in potentially very rapid transgene expression [33,34]. In previous optical microscopy studies, the nuclear presence of nanofibers has been difficult to characterize due to the lack of optical emission from the nanofibers [11]. Inherently fluorescent fibers thus provide advantage for imaging the presence of nanofibers within cells and tissue and potentially as registry markers for a variety of fluorescent cellular studies, particularly for confocal microscopy where there is value of a registry mark that extends through the depth of the scanned field. Figure 9 shows a vertical slice of a cell undergoing mitosis following nanofiber mediated gene delivery of DNA-constructs encoding GFP-tubulin and H2B-DsRed (monomer). Imaging was performed on a Leica SP1 laser scanning confocal microscope after impaling cells onto a periodic array of DNA-modified nanofibers at a ten micron pitch. In the image, the GFP-tubulin is largely collected within the mitosing-cell’s spindle apparatus and is observed via 488 laser excition and emission collected from 510-520 nm. The condensed chromatin is observed due to the 600-655 nm emission via 458 nm laser excitation of the fluorescently labeled protein H2B-DsRed, whereby H2B is one of the 5 main histone proteins involved in the structure of chromatin in eukaryotic cells. The broad emission from the SiNx-coated VACNFs stems solely from the fluorescent SiNx coating. Since the VACNFs emit over such a broad spectrum, they can be used as positional markers or “registry” marks throughout the visible spectrum. During these studies, the fluorescently labeled proteins were rapidly photobleached by the laser excitation, even though the laser was mechanically set at its lowest available power. However, the nanofiber emission remained stable. Figure 10 shows a plot of the emission intensity across the two emission ranges (same as Figure 9) as a function of time. It is observed that the emission from the fluorescently-labeled proteins, GFP-Tubulin and H2B-DsRed decreases over time, but comparatively the intensity of the emission from VACNFs is unchanged over the time period evaluated. Therefore, in addition to emitting over a broad range, the flu orescence of the VACNFs does not photobleach, wherein the intensity of emission decreases over time due to the presence of intermediate energy levels with long half-lives.

    Figure 9. Fluorescent optical micrographs of a cell undergoing mitosis looking at different emission wavelengths. (a) Emission of 510 nm-520 nm from GFP-tubulin (b) Emission of 600 nm - 655 nm from H2B-DsRed. Arrows indicate location of SiNx coated VACNFs from a top-down view. The VACNFs are not stained or modified with dye in any way. The nanofibers are on a 5 um x 5 um grid as scale markers.
    Figure 10. Fluorescent optical micrographs (top) showing a human osteosarcoma U2OS cell impaled on a SiNx coated VACNF array following impalefection-mediated delivery of genes encoding GFP-tubulin (left) and H2B-dsRed (right). Graphs indicate the change of the fluorescent intensity (I/Imax) of several regions of interest in each image. Laser excitation at 488 nm for the left set, and 545 nm for the right set causes photobleaching of the transgenic fluorescent proteins, but the nanofiber positions remain virtually unchanged over the duration of imaging. The nanofibers are spaced 5 micrometers apart as scale markers.

    4. Conclusion

    Here we have proposed a potential mechanism for deposition of SiNx coating on the sidewalls of VACNFs during PECVD synthesis in addition to exploring the origin of the coating’s fluorescence. It seems most likely that the substrate reacts with the process gases through mechanisms similar to reactive sputtering and CVD to form silane and other silicon bearing compounds before being deposited isotropically as a SiNx coating onto the VACNFs. The case for the presence of Si-NCs is made strong through a combination of the strong fluorescence and elemental analysis of the samples. These broadly-across the whole visible range-luminescent fibers can prove useful as registry markers in fluorescent cellular studies.

    Acknowledgement

    A portion of research was conducted at the Center for Nanophase Materials Sciences, which is sponsored at Oak Ridge National Laboratory by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy.



    [1] Z. Zhang, X. Jia, T. Yang, Y. Gu, W. Wang, L. Chen, Multi-objective optimization of lubricant volume in an ELSD considering thermal effects, Int. J. Therm. Sci., 164 (2021), 106884. https://doi.org/10.1016/j.ijthermalsci.2021.106884 doi: 10.1016/j.ijthermalsci.2021.106884
    [2] C. Yu, J. Liu, J. Zhang, K. Xue, S. Zhang, J. Liao, et al., Design and optimization and experimental verification of a segmented double-helix blade roller for straw returning cultivators, J. Chin. Inst. Eng., 44 (2021), 379–387. https://doi.org/10.1080/02533839.2021.1903342 doi: 10.1080/02533839.2021.1903342
    [3] Y. Cao, J. Yao, J. Li, X. Chen, J. Wu, Optimization of microbial oils production from kitchen garbage by response surface methodology, J. Renew. Sustain. Ener., 5 (2013), 053105. https://doi.org/10.1063/1.4821218 doi: 10.1063/1.4821218
    [4] K. Cao, Z. Li, Y. Gu, L. Zhang, L. Chen, The control design of transverse interconnected electronic control air suspension based on seeker optimization algorithm, P. I. Mech. Eng. D-J. Aut., 235 (2021), 2200–2211. https://doi.org/10.1177/0954407020984667 doi: 10.1177/0954407020984667
    [5] G. Wang, H. Yuan, Stability and critical spinning speed of a flexible liquid-filled rotor in thermal environment with nonlinear variable-temperature, Appl. Math. Model., 95 (2021), 143–158. https://doi.org/10.1016/j.apm.2021.01.056 doi: 10.1016/j.apm.2021.01.056
    [6] G. Wang, H. Yuan, Dynamics and stability analysis of an axially functionally graded hollow rotor partially filled with liquid, Compos. Struct., 266 (2021), 113821. https://doi.org/10.1016/j.compstruct.2021.113821 doi: 10.1016/j.compstruct.2021.113821
    [7] G. Wang, W. Yang, H. Yuan, Dynamics and stability analysis of a flexible liquid-filled rotor in a constant thermal environment, J. Vib. Control, 28 (2021), 2913–2924. https://doi.org/10.1177/10775463211022489 doi: 10.1177/10775463211022489
    [8] D. Zhang, N. Zhang, N. Ye, J. Fang, X. Han, Hybrid learning algorithm of radial basis function networks for reliability analysis, IEEE Trans. Reliab., 70 (2021), 887–900. https://doi.org/10.1109/TR.2020.3001232 doi: 10.1109/TR.2020.3001232
    [9] X. Du, H. Xu, F. Zhu, A data mining method for structure design with uncertainty in design variables, Comput. Struct., 244 (2021), 106457. https://doi.org/10.1016/j.compstruc.2020.106457 doi: 10.1016/j.compstruc.2020.106457
    [10] E. Acar, G. Bayrak, Y. Jung, I. Lee, P. Ramu, S. S. Ravichandran, Modeling, analysis and optimization under uncertainties: A review, Struct. Multidiscip. Optimiz., 64 (2021), 2909–2945. https://doi.org/10.1007/s00158-021-03026-7 doi: 10.1007/s00158-021-03026-7
    [11] M. Zimmermann, S. Königs, C. Niemeyer, J. Fender, C. Zeherbauer, R. Vitale, et al., On the design of large systems subject to uncertainty, J. Eng. Design, 28 (2017), 233–254. https://doi.org/10.1080/09544828.2017.1303664 doi: 10.1080/09544828.2017.1303664
    [12] Y. Noh, K. K. Choi, I. Lee, Reduction of ordering effect in reliability-based design optimizatioin using dimension reduction method, AIAA J. 47 (2009), 994–1004. https://doi.org/10.2514/1.40224
    [13] U. Lee, N. Kang, I. Lee, Selection of optimal target reliability in RBDO through reliability-based design for market systems (RBDMS) and application to electric vehicle design, Struct. Multidiscip. Optimiz., 60 (2019), 949–963. https://doi.org/10.1007/s00158-019-02245-3 doi: 10.1007/s00158-019-02245-3
    [14] K. Wang, P. Wang, X. Chen, L. T. Zhao, Multiobjective optimization design of toll plaza, Math. Probl. Eng., 2020 (2020), 2324894. https://doi.org/10.1155/2020/2324894 doi: 10.1155/2020/2324894
    [15] C. Yu, D. Zhu, Y. Gao, K. Xue, S. Zhang, J. Liao, et al., Optimization and experiment of counter-rotating straw returning cultivator based on discrete element method, J. Adv. Mech. Des. Syst., 14 (2020). https://doi.org/10.1299/jamdsm.2020jamdsm0097
    [16] S. Liu, J. Sun, H. Zhou, F. Wei, M. Lu, M. Lei, Experimental and numerical study on fatigue performance for TIG welding and EB welding of RAFM steel plate, Fusion Eng. Des., 146 (2019), 2663–2666. https://doi.org/10.1016/j.fusengdes.2019.04.076 doi: 10.1016/j.fusengdes.2019.04.076
    [17] J. Mendoza, E. Bismut, D. Straub, J. Köhler, Optimal life-cycle mitigation of fatigue failure risk for structural systems, Reliab. Eng. Syst. Safe., 222 (2022), 108390. https://doi.org/10.1016/j.ress.2022.108390 doi: 10.1016/j.ress.2022.108390
    [18] C. A. Castiglioni, R. Pucinotti, Failure criteria and cumulative damage models for steel components under cyclic loading, J. Constr. Steel Res., 65 (2009), 751–765. https://doi.org/10.1016/j.jcsr.2008.12.007 doi: 10.1016/j.jcsr.2008.12.007
    [19] D. M. Harland, R. D. Lorenz, Space systems failures: Disasters and rescues of satellites, rockets and space probes, in Springer Praxis Books, Springer Praxis, 2005. https://doi.org/10.1007/978-0-387-27961-9
    [20] A. D. Kiureghian, O. Ditlevsen, Aleatory or epistemic? Does it matter?, Struct. Safe., 31 (2009), 105–112. https://doi.org/10.1016/j.strusafe.2008.06.020 doi: 10.1016/j.strusafe.2008.06.020
    [21] M. A. Hariri-Ardebili, F. Pourkamali-Anaraki, Structural uncertainty quantification with partial information, Expert Syst. Appl., 198 (2022), 116736. https://doi.org/10.1016/j.eswa.2022.116736 doi: 10.1016/j.eswa.2022.116736
    [22] K. Bowcutt. A perspective on the future of aerospace vehicle design, in 12th AIAA International Space Planes and Hypersonic Systems and Technologies: American Institute of Aeronautics and Astronautics, 2003.
    [23] J. Fang, Y. Gao, G. Sun, Y. Zhang, Q. Li, Crashworthiness design of foam-filled bitubal structures with uncertainty, Int. J. Non-Lin. Mech., 67 (2014), 120–132. https://doi.org/10.1016/j.ijnonlinmec.2014.08.005 doi: 10.1016/j.ijnonlinmec.2014.08.005
    [24] S. A. Latifi Rostami, A. Kolahdooz, J. Zhang, Robust topology optimization under material and loading uncertainties using an evolutionary structural extended finite element method, Eng. Anal. Bound. Elem., 133 (2021), 61–70. https://doi.org/10.1016/j.enganabound.2021.08.023 doi: 10.1016/j.enganabound.2021.08.023
    [25] X. Wang, Z. Meng, B. Yang, C. Cheng, K. Long, J. Li, Reliability-based design optimization of material orientation and structural topology of fiber-reinforced composite structures under load uncertainty, Compos. Struct., 2022 (2022), 115537. https://doi.org/10.1016/j.compstruct.2022.115537 doi: 10.1016/j.compstruct.2022.115537
    [26] M. E. Riley, R. V. Grandhi, Quantification of model-form and predictive uncertainty for multi-physics simulation, Comput. Struct., 89 (2011), 1206–1213. https://doi.org/10.1016/j.compstruc.2010.10.004 doi: 10.1016/j.compstruc.2010.10.004
    [27] B. Do, M. Ohsaki, M. Yamakawa, Bayesian optimization for robust design of steel frames with joint and individual probabilistic constraints, Eng. Struct., 245 (2021), 112859. https://doi.org/10.1016/j.engstruct.2021.112859 doi: 10.1016/j.engstruct.2021.112859
    [28] Z. Meng, Y. Pang, Y. Pu, X. Wang, New hybrid reliability-based topology optimization method combining fuzzy and probabilistic models for handling epistemic and aleatory uncertainties, Comput. Method. Appl. Mech., 363 (2020), 112886. https://doi.org/10.1016/j.cma.2020.112886 doi: 10.1016/j.cma.2020.112886
    [29] C. Yang, H. Ouyang, A novel load-dependent sensor placement method for model updating based on time-dependent reliability optimization considering multi-source uncertainties, Mech. Syst. Signal Proces., 165 (2022), 108386. https://doi.org/10.1016/j.ymssp.2021.108386 doi: 10.1016/j.ymssp.2021.108386
    [30] G. A. da Silva, E. L. Cardoso, A. T. Beck, Comparison of robust, reliability-based and non-probabilistic topology optimization under uncertain loads and stress constraints, Probabilist. Eng. Mech., 59 (2020), 103039. https://doi.org/10.1016/j.probengmech.2020.103039 doi: 10.1016/j.probengmech.2020.103039
    [31] J. M. King, R. V. Grandhi, T. W. Benanzer, Quantification of epistemic uncertainty in re-usable launch vehicle aero-elastic design, Eng. Optimiz., 44 (2012), 489–504. https://doi.org/10.1080/0305215X.2011.588224 doi: 10.1080/0305215X.2011.588224
    [32] Y. C. Tsao, V. V. Thanh, A multi-objective fuzzy robust optimization approach for designing sustainable and reliable power systems under uncertainty, Appl. Soft Comput., 92 (2020), 106317. https://doi.org/10.1016/j.asoc.2020.106317 doi: 10.1016/j.asoc.2020.106317
    [33] L. Wang, B. Ni, X. Wang, Z. Li, Reliability-based topology optimization for heterogeneous composite structures under interval and convex mixed uncertainties, Appl. Math. Model., 99 (2021), 628–652. https://doi.org/10.1016/j.apm.2021.06.014 doi: 10.1016/j.apm.2021.06.014
    [34] L. A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Set. Syst., 1 (1978), 3–28. https://doi.org/10.1016/0165-0114(78)90029-5 doi: 10.1016/0165-0114(78)90029-5
    [35] H. Lü, K. Yang, X. Huang, H. Yin, W. B. Shangguan, D. Yu, An efficient approach for the design optimization of dual uncertain structures involving fuzzy random variables, Comput. Method. Appl. Mech., 371 (2020), 113331. https://doi.org/10.1016/j.cma.2020.113331 doi: 10.1016/j.cma.2020.113331
    [36] R. E. Moore, Methods and Application of Interval Analysis, SIAM: Philadelphia, USA, 1979. https://doi.org/10.1137/1.9781611970906
    [37] S. P. Gurav, J. F. L. Goosen, F. vanKeulen, Bounded-But-Unknown uncertainty optimization using design sensitivities and parallel computing: Application to MEMS, Comput. Struct., 83 (2005), 1134–1149. https://doi.org/10.1016/j.compstruc.2004.11.021 doi: 10.1016/j.compstruc.2004.11.021
    [38] R. Yuan, D. Ma, H. Zhang, Flow characteristics and grain size distribution of granular gangue mineral by compaction treatment, Adv. Mater. Sci. Eng., 2017 (2017), 2509286. https://doi.org/10.1155/2017/2509286 doi: 10.1155/2017/2509286
    [39] G. Sun, T. Pang, J. Fang, G. Li, Q. Li, Parameterization of criss-cross configurations for multiobjective crashworthiness optimization, Int. J. Mech. Sci., 124125 (2017), 145–157. https://doi.org/10.1016/j.ijmecsci.2017.02.027
    [40] J. Fang, G. Sun, N. Qiu, T. Pang, S. Li, Q. Li, On hierarchical honeycombs under out-of-plane crushing, Int. J. Solids Struct., 135 (2018), 1–13. https://doi.org/10.1016/j.ijsolstr.2017.08.013. doi: 10.1016/j.ijsolstr.2017.08.013
    [41] J. Fu, Q. Liu, K. Liufu, Y. Deng, J. Fang, Q. Li, Design of bionic-bamboo thin-walled structures for energy absorption, Thin-Wall. Struct., 135 (2019), 400–413. https://doi.org/10.1016/j.tws.2018.10.003 doi: 10.1016/j.tws.2018.10.003
    [42] J. Fang, Y. Gao, G. Sun, G. Zheng, Q. Li, Dynamic crashing behavior of new extrudable multi-cell tubes with a functionally graded thickness, Int. J. Mech. Sci., 103 (2015), 63–73. https://doi.org/10.1016/j.ijmecsci.2015.08.029 doi: 10.1016/j.ijmecsci.2015.08.029
    [43] J. Fang, Y. Gao, G. Sun, N. Qiu, Q. Li, On design of multi-cell tubes under axial and oblique impact loads, Thin-Wall. Struct., 95 (2015), 115–126. https://doi.org/10.1016/j.tws.2015.07.002 doi: 10.1016/j.tws.2015.07.002
    [44] J. Fang, G. Sun, N. Qiu, N. H. Kim, Q. Li, On design optimization for structural crashworthiness and its state of the art, Struct. Multidiscip. Optimiz., 55 (2017), 1091–1119. https://doi.org/10.1007/s00158-016-1579-y doi: 10.1007/s00158-016-1579-y
    [45] N. Qiu, Y. Gao, J. Fang, Z. Feng, G. Sun, Q. Li, Crashworthiness analysis and design of multi-cell hexagonal columns under multiple loading cases, Finite Elem. Anal. Des., 104 (2015), 89–101. https://doi.org/10.1016/j.finel.2015.06.004 doi: 10.1016/j.finel.2015.06.004
    [46] S. Kodiyalam, High performance computing for multidisciplinary design optimization and robustness of vehicle structures, in Computational Fluid and Solid Mechanics 2003, Elsevier Science Ltd, (2003), 2305–2307. https://doi.org/10.1016/B978-008044046-0.50566-2
    [47] K. Hamza, K. Saitou, Design Optimization of Vehicle Structures for Crashworthiness Using Equivalent Mechanism Approximations, J. Mech. Des., 127 (2004), 485–492. https://doi.org/10.1115/1.1862680 doi: 10.1115/1.1862680
    [48] X. Gu, G. Sun, G. Li, X. Huang, Y. Li, Q. Li, Multiobjective optimization design for vehicle occupant restraint system under frontal impact, Struct. Multidiscip. Optimiz., 47 (2013), 465–477. https://doi.org/10.1007/s00158-012-0811-7 doi: 10.1007/s00158-012-0811-7
    [49] L. Chen, W. Li, Y. Yang, W. Miao, Evaluation and optimization of vehicle pedal comfort based on biomechanics, Proc. Inst. Mech. Eng. Part D J. Automob. Eng., 234 (2019), 1402–1412. https://doi.org/10.1177/0954407019878355 doi: 10.1177/0954407019878355
    [50] X. L. Zhang, T. Wu, Y. Shao, J. Song, Structure optimization of wheel force transducer based on natural frequency and comprehensive sensitivity, Chin. J. Mech. Eng., 30 (2017), 973–981. https://doi.org/10.1007/s10033-017-0149-6 doi: 10.1007/s10033-017-0149-6
    [51] X. Xu, X. Chen, Z. Liu, Y. Zhang, Y. Xu, J. Fang, et al., A feasible identification method of uncertainty responses for vehicle structures, Struct. Multidiscip. Optimiz., 64 (2021), 3861–3876. https://doi.org/10.1007/s00158-021-03065-0 doi: 10.1007/s00158-021-03065-0
    [52] X. Liu, X. Liu, Z. Zhou, L. Hu, An efficient multi-objective optimization method based on the adaptive approximation model of the radial basis function, Struct. Multidiscip. Optimiz., 63 (2021), 1385–1403. https://doi.org/10.1007/s00158-020-02766-2 doi: 10.1007/s00158-020-02766-2
    [53] J. Fang, Y. Gao, G. Sun, C. Xu, Q. Li, Multiobjective robust design optimization of fatigue life for a truck cab, Reliab. Eng. Syst. Safe., 135 (2015), 1–8. https://doi.org/10.1016/j.ress.2014.10.007 doi: 10.1016/j.ress.2014.10.007
    [54] X. Gu, G. Sun, G. Li, L. Mao, Q. Li, A Comparative study on multiobjective reliable and robust optimization for crashworthiness design of vehicle structure, Struct. Multidiscip. Optimiz., 48 (2013), 669–684. https://doi.org/10.1007/s00158-013-0921-x doi: 10.1007/s00158-013-0921-x
    [55] J. Zhou, F. Lan, J. Chen, F. Lai, Uncertainty optimization design of a vehicle body structure considering random deviations, Automot. Innov., 1 (2018), 342–351. https://doi.org/10.1007/s42154-018-0041-9 doi: 10.1007/s42154-018-0041-9
    [56] J. Zhu, X. Wang, H. Zhang, Y. Li, R. Wang, Z. Qiu, Six sigma robust design optimization for thermal protection system of hypersonic vehicles based on successive response surface method, Chinese J. Aeronaut., 32 (2019), 2095–2108. https://doi.org/10.1016/j.cja.2019.04.009 doi: 10.1016/j.cja.2019.04.009
    [57] X. Wang, L. Shi, A new metamodel method using Gaussian process based bias function for vehicle crashworthiness design, Int. J. Crashworthines., 19 (2014), 311–321. https://doi.org/10.1080/13588265.2014.898932 doi: 10.1080/13588265.2014.898932
    [58] N. Qiu, J. Zhang, F. Yuan, Z. Jin, Y. Zhang, J. Fang, Mechanical performance of triply periodic minimal surface structures with a novel hybrid gradient fabricated by selective laser melting, Eng. Struct., 263 (2022), 114377. https://doi.org/10.1016/j.engstruct.2022.114377 doi: 10.1016/j.engstruct.2022.114377
    [59] L. Chen, P. Ma, J. Tian, X. Liang, Prediction and optimization of lubrication performance for a transfer case based on computational fluid dynamics, Eng. Appl. Comp. Fluid, 13 (2019), 1013–1023. https://doi.org/10.1080/19942060.2019.1663765 doi: 10.1080/19942060.2019.1663765
    [60] X. Xu, Y. Zhang, X. Wang, J. Fang, J. Chen, J. Li, Searching superior crashworthiness performance by constructing variable thickness honeycombs with biomimetic cells, Int. J. Mech. Sci., 235 (2022), 107718. https://doi.org/10.1016/j.ijmecsci.2022.107718 doi: 10.1016/j.ijmecsci.2022.107718
    [61] X. Xu, Y. Zhang, J. Wang, F. Jiang, C. H. Wang, Crashworthiness design of novel hierarchical hexagonal columns, Compos. Struct., 194 (2018), 36–48. https://doi.org/10.1016/j.compstruct.2018.03.099 doi: 10.1016/j.compstruct.2018.03.099
    [62] X. Song, L. Lai, S. Xiao, Y. Tang, M. Song, J. Zhang, et al., Bionic design and multi-objective optimization of thin-walled structures inspired by conchs, Electron. Res. Arch., 31 (2023), 575–598. https://doi.org/10.3934/era.2023028 doi: 10.3934/era.2023028
    [63] J. Fang, Y. Gao, G. Sun, Q. Li, Multiobjective reliability-based optimization for design of a vehicledoor, Finite Elem. Anal. Des., 67 (2013), 13–21. https://doi.org/10.1016/j.finel.2012.11.007 doi: 10.1016/j.finel.2012.11.007
    [64] N. Qiu, Z. Jin, J. Liu, L. Fu, Z. Chen, N. H. Kim, Hybrid multi-objective robust design optimization of a truck cab considering fatigue life, Thin-Wall. Struct., 162 (2021), 107545. https://doi.org/10.1016/j.tws.2021.107545 doi: 10.1016/j.tws.2021.107545
    [65] G. Sun, H. Zhang, J. Fang, G. Li, Q. Li, A new multi-objective discrete robust optimization algorithm for engineering design, Appl. Math. Model., 53 (2018), 602–621. https://doi.org/10.1016/j.apm.2017.08.016 doi: 10.1016/j.apm.2017.08.016
    [66] F. Lei, X. Lv, J. Fang, T. Pang, Q. Li, G. Sun, Injury biomechanics-based nondeterministic optimization of front-end structures for safety in pedestrian-vehicle impact, Thin-Wall. Struct., 167 (2021). https://doi.org/10.1016/j.tws.2021.108087
    [67] H. Lu, D. Yu, Brake squeal reduction of vehicle disc brake system with interval parameters by uncertain optimization, J. Sound Vib., 333 (2014), 7313–7325. https://doi.org/10.1016/j.jsv.2014.08.027 doi: 10.1016/j.jsv.2014.08.027
    [68] J. Wu, Z. Luo, Y. Zhang, N. Zhang, An interval uncertain optimization method for vehicle suspensions using Chebyshev metamodels, Appl. Math. Model., 38 (2014), 3706–3723. https://doi.org/10.1016/j.apm.2014.02.012 doi: 10.1016/j.apm.2014.02.012
    [69] A. Jamali, M. Salehpour, N. Nariman-zadeh, Robust Pareto active suspension design for vehicle vibration model with probabilistic uncertain parameters, Multibody Syst. Dyn., 30 (2013), 265–285. https://doi.org/10.1007/s11044-012-9337-4 doi: 10.1007/s11044-012-9337-4
    [70] X. Gu, W. Wang, L. Xia, P. Jiang, A system optimisation design approach to vehicle structure under frontal impact based on SVR of optimised hybrid kernel function, Int. J. Crashworthines., 26 (2021), 1–15. https://doi.org/10.1080/13588265.2019.1634335 doi: 10.1080/13588265.2019.1634335
    [71] Z. Liu, C. Jiang, Y. Li, Y. Bai, Fatigue life optimization for spot-welded Structures of vehicle body considering uncertainty of welding spots, China Mech. Eng., 26 (2015), 2544–2549. https://doi.org/10.3969/j.issn.1004-132X.2015.18.024 doi: 10.3969/j.issn.1004-132X.2015.18.024
    [72] L. Farkas, D. Moens, S. Donders, D. Vandepitte, Optimisation study of a vehicle bumper subsystem with fuzzy parameters, Mech. Syst. Signal Pr., 32 (2012), 59–68. https://doi.org/10.1016/j.ymssp.2011.11.014 doi: 10.1016/j.ymssp.2011.11.014
    [73] E. Acar, K. Solanki, System reliability based vehicle design for crashworthiness and effects of various uncertainty reduction measures, Struct. Multidiscip. Optimiz., 39 (2009), 311–325. https://doi.org/10.1007/s00158-008-0327-3 doi: 10.1007/s00158-008-0327-3
    [74] Q. Zhao, H. Zhang, Z. Zhu, R. Jiang, L. Yuan, Reliability-based topology optimization for vehicle suspension control arm, Aut. Eng., 40(2018), 313–319. https://doi.org/10.19562/j.chinasae.qcgc.2018.03.011 doi: 10.19562/j.chinasae.qcgc.2018.03.011
    [75] M. Grujicic, G. Arakere, W. C. Bell, H. Marvi, H. V. Yalavarthy, B. Pandurangan, et al., Reliability-based design optimization for durability of ground vehicle suspension system components, J. Mater. Eng. Perform., 19 (2010), 301–313. https://doi.org/10.1007/s11665-009-9482-y doi: 10.1007/s11665-009-9482-y
    [76] M. Rais-Rohani, K. N. Solanki, E. Acar, C. D. Eamon, Shape and sizing optimisation of automotive structures with deterministic and probabilistic design constraints, Int. J. Vehicle Des., 54 (2010), 309–338. https://doi.org/10.1504/IJVD.2010.036839 doi: 10.1504/IJVD.2010.036839
    [77] X. Xu, J. Chen, Z. Lin, Y. Qiao, X. Chen, Y. Zhang, et al., Optimization design for the planetary gear train of an electric vehicle under uncertainties, Actuators, 11 (2022). https://doi.org/10.3390/act11020049
    [78] G. Sun, H. Zhang, R. Wang, X. Lv, Q. Li, Multiobjective reliability-based optimization for crashworthy structures coupled with metal forming process, Struct. Multidiscip. Optimiz., 56 (2017), 1571–1587. https://doi.org/10.1007/s00158-017-1825-y doi: 10.1007/s00158-017-1825-y
    [79] Y. Xu, Y. Gao, C. Wu, J. Fang, Q. Li, Robust topology optimization for multiple fiber-reinforced plastic (FRP) composites under loading uncertainties, Struct. Multidiscip. Optimiz., 59 (2019), 695–711. https://doi.org/10.1007/s00158-018-2175-0 doi: 10.1007/s00158-018-2175-0
    [80] C. van Mierlo, L. Burmberger, M. Daub, F. Duddeck, M. G. R. Faes, D. Moens, Interval methods for lack-of-knowledge uncertainty in crash analysis, Mech. Syst. Signal Proces., 168 (2022). https://doi.org/10.1016/j.ymssp.2021.108574
    [81] C. Lin, F. Gao, Y. Bai, Multiobjective reliability-based design optimisation for front structure of an electric vehicle using hybrid metamodel accuracy improvement strategy-based probabilistic sufficiency factor method, Int. J. Crashworthines., 23 (2018), 290–301. https://doi.org/10.1080/13588265.2017.1317466 doi: 10.1080/13588265.2017.1317466
    [82] F. Y. Li, G. Y. Li, Interval-based uncertain multi-objective optimization design of vehicle crashworthiness, CMC-Comput. Mater. Con., 15 (2010), 199–219. https://doi.org/10.2478/s11533-009-0061-0 doi: 10.2478/s11533-009-0061-0
    [83] J. Li, Y. Fang, Z. Zhan, Y. Jiang, An enhanced surrogate model based vehicle robust design method under materials and manufacturing uncertainties, in ASME International Mechanical Engineering Congress and Exposition, 2016. https://doi.org/10.1115/IMECE2016-67714
    [84] R. J. Yang, L. Gu, C. H. Tho, K. K. Choi, B. Youn, Reliability-based multidisciplinary design optimization of vehicle structures. in 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2002. https://doi.org/10.2514/6.2002-1758
    [85] L. Cao, J. Liu, L. Xie, C. Jiang, R. Bi, Non-probabilistic polygonal convex set model for structural uncertainty quantification, Appl. Math. Model., 89 (2021), 504–518. https://doi.org/10.1016/j.apm.2020.07.025 doi: 10.1016/j.apm.2020.07.025
    [86] X. Meng, J. Liu, L. Cao, Z. Yu, D. Yang, A general frame for uncertainty propagation under multimodally distributed random variables, Comput. Method. Appl. Mech. Eng., 367 (2020). https://doi.org/10.1016/j.cma.2020.113109
    [87] G. Chen, D. Yang, A unified analysis framework of static and dynamic structural reliabilities based on direct probability integral method, Mech. Syst. Signal Proces., 158 (2021). https://doi.org/10.1016/j.ymssp.2021.107783
    [88] X. Y. Long, D. L. Mao, C. Jiang, F. Y. Wei, G. J. Li, Unified uncertainty analysis under probabilistic, evidence, fuzzy and interval uncertainties, Comput. Method. Appl. Mech. Eng., 355 (2019), 1–26. https://doi.org/10.1016/j.cma.2019.05.041
    [89] X. Xu, X. Chen, Z. Liu, J. Yang, Y. Xu, Y. Zhang, et al., Multi-objective reliability-based design optimization for the reducer housing of electric vehicles, Eng. Optimiz., 54 (2022), 1324–1340. https://doi.org/10.1080/0305215X.2021.1923704 doi: 10.1080/0305215X.2021.1923704
    [90] N. Qiu, C. Park, Y. Gao, J. Fang, G. Sun, N. H. Kim, Sensitivity-based parameter calibration and model validation under model error, J. Mech. Des., 140 (2017). https://doi.org/10.1115/1.4038298
    [91] Z. L. Huang, C. Jiang, Y. S. Zhou, Z. Luo, Z. Zhang, An incremental shifting vector approach for reliability-based design optimization, Struct. Multidiscip. Optimiz., 53 (2016), 523–543. https://doi.org/10.1007/s00158-015-1352-7 doi: 10.1007/s00158-015-1352-7
    [92] W. Gao, D. Wu, K. Gao, X. Chen, F. Tin-Loi, Structural reliability analysis with imprecise random and interval fields, Appl. Math. Model., 55 (2018), 49–67. https://doi.org/10.1016/j.apm.2017.10.029 doi: 10.1016/j.apm.2017.10.029
    [93] N. Qiu, Y. Gao, J. Fang, G. Sun, Q. Li, N. H. Kim, Crashworthiness optimization with uncertainty from surrogate model and numerical error, Thin-Wall. Struct., 129 (2018), 457–472. https://doi.org/10.1016/j.tws.2018.05.002 doi: 10.1016/j.tws.2018.05.002
    [94] G. S. Song, M. C. Kim, Theoretical approach for uncertainty quantification in probabilistic safety assessment using sum of lognormal random variables, Nucl. Eng. Technol., 54 (2022), 2084–2093. https://doi.org/10.1016/j.net.2021.12.033 doi: 10.1016/j.net.2021.12.033
    [95] N. A. W. Walton, R. Crowder, N. Satvat, N. R. Brown, V. Sobes, Demonstration of a random sampling approach to uncertainty propagation for generic pebble-bed fluoride-salt-cooled high temperature reactor (gFHR), Nucl. Eng. Des., 395 (2022), 111843. https://doi.org/10.1016/j.nucengdes.2022.111843 doi: 10.1016/j.nucengdes.2022.111843
    [96] X. P. Du, W. Chen, Sequential optimization and reliability assessment method for efficient probabilistic design, J. Mech. Des., 126 (2004), 225–233. https://doi.org/10.1115/1.1649968 doi: 10.1115/1.1649968
    [97] Y. Lei, N. Yang, D. Xia, Probabilistic structural damage detection approaches based on structural dynamic response moments, Smart Struct. Syst., 20 (2017), 207–217. https://doi.org/10.12989/sss.2017.20.2.207 doi: 10.12989/sss.2017.20.2.207
    [98] P. Hoang-Anh, T. Viet-Hung, V. Tien-Chuong, Fuzzy finite element analysis for free vibration response of functionally graded semi-rigid frame structures, Appl. Math. Model., 88 (2020), 852–869. https://doi.org/10.1016/j.apm.2020.07.014 doi: 10.1016/j.apm.2020.07.014
    [99] Z. Chen, Z. Lu, C. Ling, K. Feng, Reliability analysis model of time-dependent multi-mode system under fuzzy uncertainty: Applied to undercarriage structures, Aerosp. Sci. Technol., 120 (2022), 107278. https://doi.org/10.1016/j.ast.2021.107278 doi: 10.1016/j.ast.2021.107278
    [100] P. Wang, H. Zhu, H. Tian, G. Cai, Analytic target cascading with fuzzy uncertainties based on global sensitivity analysis for overall design of launch vehicle powered by hybrid rocket motor, Aerosp. Sci. Technol., 114 (2021), 106680. https://doi.org/10.1016/j.ast.2021.106680 doi: 10.1016/j.ast.2021.106680
    [101] S. Yin, D. Yu, H. Yin, B. Xia, A new evidence-theory-based method for response analysis of acoustic system with epistemic uncertainty by using Jacobi expansion, Comput. Method. Appl. Mech. Eng., 322 (2017), 419–440. https://doi.org/10.1016/j.cma.2017.04.020 doi: 10.1016/j.cma.2017.04.020
    [102] H. Tang, D. Li, J. Li, S. Xue, Epistemic uncertainty quantification in metal fatigue crack growth analysis using evidence theory, Int. J. Fatigue, 99 (2017), 163–174. https://doi.org/10.1016/j.ijfatigue.2017.03.004 doi: 10.1016/j.ijfatigue.2017.03.004
    [103] Z. Zhang, X. X. Ruan, M. F. Duan, C. Jiang, An efficient epistemic uncertainty analysis method using evidence theory, Comput. Method. Appl. Mech. Eng., 339 (2018), 443–466. https://doi.org/10.1016/j.cma.2018.04.033 doi: 10.1016/j.cma.2018.04.033
    [104] X. Tang, K. Yuan, N. Gu, P. Li, R. Peng, An interval quantification-based optimization approach for wind turbine airfoil under uncertainties, Energy, 244 (2022), 122623. https://doi.org/10.1016/j.energy.2021.122623 doi: 10.1016/j.energy.2021.122623
    [105] Z. Niu, H. Zhu, X. Huang, A. Che, S. Fu, S. Meng, et al., Uncertainty quantification method for elastic wave tomography of concrete structure using interval analysis, Measurement, 205 (2022), 112160. https://doi.org/10.1016/j.measurement.2022.112160 doi: 10.1016/j.measurement.2022.112160
    [106] B. Y. Ni, C. Jiang, P. G. Wu, Z. H. Wang, W. Y. Tian, A sequential simulation strategy for response bounds analysis of structures with interval uncertainties, Comput. Struct., 266 (2022), 106785. https://doi.org/10.1016/j.compstruc.2022.106785 doi: 10.1016/j.compstruc.2022.106785
    [107] T. Tang, H. Luo, Y. Song, H. Fang, J. Zhang, Chebyshev inclusion function based interval kinetostatic modeling and parameter sensitivity analysis for Exechon-like parallel kinematic machines with parameter uncertainties, Mech. Mach. Theory, 157 (2021). https://doi.org/10.1016/j.mechmachtheory.2020.104209
    [108] C. Viegas, D. Daney, M. Tavakoli, A. T. de Almeida, Performance analysis and design of parallel kinematic machines using interval analysis, Mech. Mach. Theory, 115 (2017), 218–236. https://doi.org/10.1016/j.mechmachtheory.2017.05.003 doi: 10.1016/j.mechmachtheory.2017.05.003
    [109] M. Ma, L. Wang, Reliability-based topology optimization framework of two-dimensional phononic crystal band-gap structures based on interval series expansion and mapping conversion method, Int. J. Mech. Sci., 196 (2021). https://doi.org/10.1016/j.ijmecsci.2020.106265
    [110] Z. Qiu, X. Li, A new model for the eigenvalue buckling analysis with unknown-but-bounded parameters, Aerosp. Sci. Technol., 113 (2021). https://doi.org/10.1016/j.ast.2021.106634
    [111] B. Xia, H. Lu, D. Yu, C. Jiang, Reliability-based design optimization of structural systems under hybrid probabilistic and interval model, Comput. Struct., 160 (2015), 126–134. https://doi.org/10.1016/j.compstruc.2015.08.009 doi: 10.1016/j.compstruc.2015.08.009
    [112] L. Wang, Z. Chen, G. Yang, Q. Sun, J. Ge, An interval uncertain optimization method using back-propagation neural network differentiation, Comput. Method. Appl. Mech. Eng., 366 (2020). https://doi.org/10.1016/j.cma.2020.113065
    [113] S. H. Chen, L. Ma, G. W. Meng, R. Guo, An efficient method for evaluating the natural frequencies of structures with uncertain-but-bounded parameters, Comput. Struct., 87 (2009), 582–590. https://doi.org/10.1016/j.compstruc.2009.02.009 doi: 10.1016/j.compstruc.2009.02.009
    [114] N. Impollonia, G. Muscolino, Interval analysis of structures with uncertain-but-bounded axial stiffness, Comput. Method. Appl. Mech. Eng., 200 (2011), 1945–1962. https://doi.org/10.1016/j.cma.2010.07.019 doi: 10.1016/j.cma.2010.07.019
    [115] G. Zhao, G. Wen, J. Liu, A novel analysis method for vibration systems under time-varying uncertainties based on interval process model, Probabilistic Eng. Mech., 70 (2022), 103363. https://doi.org/10.1016/j.probengmech.2022.103363 doi: 10.1016/j.probengmech.2022.103363
    [116] L. Wang, J. Liu, C. Yang, D. Wu, A novel interval dynamic reliability computation approach for the risk evaluation of vibration active control systems based on PID controllers, Appl. Math. Model., 92 (2021), 422–446. https://doi.org/10.1016/j.apm.2020.11.007 doi: 10.1016/j.apm.2020.11.007
    [117] W. Gao, D. Wu, C. Song, F. Tin-Loi, X. Li, Hybrid probabilistic interval analysis of bar structures with uncertainty using a mixed perturbation Monte-Carlo method, Finite Elem. Anal. Des., 47 (2011), 643–652. https://doi.org/10.1016/j.finel.2011.01.007 doi: 10.1016/j.finel.2011.01.007
    [118] Z. P. Qiu, X. J. Wang, Solution theorems for the standard eigenvalue problem of structures with uncertain-but-bounded parameters, J. Sound Vib., 282 (2005), 381–399. https://doi.org/10.1016/j.jsv.2004.02.024 doi: 10.1016/j.jsv.2004.02.024
    [119] F. Li, Z. Luo, J. Rong, N. Zhang, Interval multi-objective optimisation of structures using adaptive Kriging approximations, Comput. Struct., 119 (2013), 68–84. https://doi.org/10.1016/j.compstruc.2012.12.028 doi: 10.1016/j.compstruc.2012.12.028
    [120] J. A. Fernandez-Prieto, J. Canada-Bago, M. A. Gadeo-Martos, J. R. Velasco, Optimisation of control parameters for genetic algorithms to test computer networks under realistic traffic loads, Appl. Soft Comput., 12 (2012), 1875–1883. https://doi.org/10.1016/j.asoc.2012.04.018 doi: 10.1016/j.asoc.2012.04.018
    [121] Y. G. Xu, G. R. Li, Z. P. Wu, A novel hybrid genetic algorithm using local optimizer based on heuristic pattern move, Appl. Artif. Intell., 15 (2001), 601–631. https://doi.org/10.1080/088395101750363966 doi: 10.1080/088395101750363966
    [122] C. Jiang, X. Han, G. P. Liu, A sequential nonlinear interval number programming method for uncertain structures, Comput. Meth. Appl. Mech. Eng., 197 (2008), 4250–4265. https://doi.org/10.1016/j.cma.2008.04.027 doi: 10.1016/j.cma.2008.04.027
    [123] J. Cheng, M. Y. Tang, Z. Y. Liu, J. R. Tan, Direct reliability-based design optimization of uncertain structures with interval parameters, J. Zhejiang Univ-Sci. A, 17 (2016), 841–854. https://doi.org/10.1631/jzus.A1600143 doi: 10.1631/jzus.A1600143
    [124] Y. P. Li, G. H. Huang, P. Guo, Z. F. Yang, S. L. Nie, A dual-interval vertex analysis method and its application to environmental decision making under uncertainty, Eur. J. Oper. Res., 200 (2010), 536–550. https://doi.org/10.1016/j.ejor.2009.01.013 doi: 10.1016/j.ejor.2009.01.013
    [125] C. Wang, H. G. Matthies, Non-probabilistic interval process model and method for uncertainty analysis of transient heat transfer problem, Int. J. Therm. Sci., 144 (2019), 147–157. https://doi.org/10.1016/j.ijthermalsci.2019.06.002 doi: 10.1016/j.ijthermalsci.2019.06.002
    [126] S. Nayak, S. Chakraverty, Non-probabilistic approach to investigate uncertain conjugate heat transfer in an imprecisely defined plate, Int. J. Heat Mass Tran., 67 (2013), 445–454. https://doi.org/10.1016/j.ijheatmasstransfer.2013.08.036 doi: 10.1016/j.ijheatmasstransfer.2013.08.036
    [127] Z. Qiu, X. Wang, J. Chen, Exact bounds for the static response set of structures with uncertain-but-bounded parameters, Int. J. Solids Struct., 43 (2006), 6574–6593. https://doi.org/10.1016/j.ijsolstr.2006.01.012 doi: 10.1016/j.ijsolstr.2006.01.012
    [128] Z. Qiu, Y. Xia, H. Yang, The static displacement and the stress analysis of structures with bounded uncertainties using the vertex solution theorem, Comput. Method. Appl. Mech. Eng., 196 (2007), 4965–4984. https://doi.org/10.1016/j.cma.2007.06.022 doi: 10.1016/j.cma.2007.06.022
    [129] Z. P. Qiu, S. H. Chen, I. Elishakoff, Bounds of eigenvalues for structures with an interval description of uncertain-but-non-random parameters, Chaos Solition. Fract., 7 (1996), 425–434. https://doi.org/10.1016/0960-0779(95)00065-8 doi: 10.1016/0960-0779(95)00065-8
    [130] Z. P. Qiu, X. J. Wang, Parameter perturbation method for dynamic responses of structures with uncertain-but-bounded parameters based on interval analysis, Int. J. Solids Struct., 42 (2005), 4958–4970. https://doi.org/10.1016/j.ijsolstr.2005.02.023 doi: 10.1016/j.ijsolstr.2005.02.023
    [131] Z. Qiu, L. Ma, X. Wang, Non-probabilistic interval analysis method for dynamic response analysis of nonlinear systems with uncertainty, J. Sound Vib., 319 (2009), 531–540. https://doi.org/10.1016/j.jsv.2008.06.006 doi: 10.1016/j.jsv.2008.06.006
    [132] D. Meng, T. Xie, P. Wu, C. He, Z. Hu, Z. Lv, An uncertainty-based design optimization strategy with random and interval variables for multidisciplinary engineering systems, Structures, 32 (2021), 997–1004. https://doi.org/10.1016/j.istruc.2021.03.020 doi: 10.1016/j.istruc.2021.03.020
    [133] X. Liu, X. Yu, J. Tong, X. Wang, X. Wang, Mixed uncertainty analysis for dynamic reliability of mechanical structures considering residual strength, Reliab. Eng. Syst. Safe., 209 (2021), 107472. https://doi.org/10.1016/j.ress.2021.107472 doi: 10.1016/j.ress.2021.107472
    [134] J. Wang, Z. Lu, Probabilistic safety model and its efficient solution for structure with random and interval mixed uncertainties, Mech. Mach. Theory, 147 (2020), 103782. https://doi.org/10.1016/j.mechmachtheory.2020.103782 doi: 10.1016/j.mechmachtheory.2020.103782
    [135] D. M. Do, K. Gao, W. Yang, C. Q. Li, Hybrid uncertainty analysis of functionally graded plates via multiple-imprecise-random-field modelling of uncertain material properties, Comput. Method. Appl. Mech. Eng., 368 (2020), 113116. https://doi.org/10.1016/j.cma.2020.113116 doi: 10.1016/j.cma.2020.113116
    [136] J. Wang, Z. Lu, M. Zhong, T. Wang, C. Sun, H. Li, Coupled thermal–structural analysis and multi-objective optimization of a cutting-type energy-absorbing structure for subway vehicles, Thin-Wall. Struct., 141 (2019), 360-373. https://doi.org/10.1016/j.tws.2019.04.026 doi: 10.1016/j.tws.2019.04.026
    [137] M. Luo, Y. Chen, D. Gao, L. Wang, Inversion study of vehicle frontal collision and front bumper collision, Electron. Res. Arch. 31 (2023), 776–792. https://doi.org/10.3934/era.2023039
    [138] W. Li, J. Wang, Z. Du, H. Ma, L. Zhang, L. Duan, Lightweight design method and application of MEWP bracket based on multi-level optimization, Electron. Res. Arch., 30 (2022), 4416–4435. https://doi.org/10.3934/era.2022224 doi: 10.3934/era.2022224
    [139] X. Xu, G. Xu, J. Chen, Z. Liu, X. Chen, Y. Zhang, et al., Multi-objective design optimization using hybrid search algorithms with interval uncertainty for thin-walled structures, Thin-Wall. Struct., 175 (2022), 109218. https://doi.org/10.1016/j.tws.2022.109218 doi: 10.1016/j.tws.2022.109218
    [140] Y. Wu, W. Li, J. Fang, Q. Lan, Multi-objective robust design optimization of fatigue life for a welded box girder, Eng. Optimiz., 50 (2018), 1252–1269. https://doi.org/10.1080/0305215X.2017.1395023 doi: 10.1080/0305215X.2017.1395023
    [141] Y. Wu, L. Sun, P. Yang, J. Fang, W. Li, Energy absorption of additively manufactured functionally bi-graded thickness honeycombs subjected to axial loads, Thin-Wall. Struct., 164 (2021), 107810. https://doi.org/10.1016/j.tws.2021.107810 doi: 10.1016/j.tws.2021.107810
    [142] Y. Su, H. Tang, S. Xue, D. Li, Multi-objective differential evolution for truss design optimization with epistemic uncertainty, Adv. Struct. Eng., 19 (2016), 1403–1419. https://doi.org/10.1177/1369433216643250 doi: 10.1177/1369433216643250
    [143] X. Liu, Q. Fu, N. Ye, L. Yin, The multi-objective reliability-based design optimization for structure based on probability and ellipsoidal convex hybrid model, Struct. Safe., 77 (2019), 48–56. https://doi.org/10.1016/j.strusafe.2018.11.004 doi: 10.1016/j.strusafe.2018.11.004
    [144] T. Vo-Duy, D. Duong-Gia, V. Ho-Huu, T. Nguyen-Thoi, An effective couple method for reliability-based multi-objective optimization of truss structures with static and dynamic constraints, Int. J. Comput. Method., 17 (2020). https://doi.org/10.1142/S0219876219500166
    [145] F. S. Lobato, M. A. da Silva, A. A. Cavalini, V. Steffen, Reliability-based robust multi-objective optimization applied to engineering system design, Eng. Optimiz., 52 (2020), 1–21. https://doi.org/10.1080/0305215X.2019.1577413 doi: 10.1080/0305215X.2019.1577413
    [146] P. Wang, S. Zheng, G. Wu, Multidisciplinary design optimization of vehicle body structure based on collaborative optimization and multi-objective genetic algorithm, Chinese J. Mech. Eng., 47 (2011), 102–108. https://doi.org/10.3901/JME.2011.02.102 doi: 10.3901/JME.2011.02.102
    [147] H. Yin, H. Fang, Y. Xiao, G. Wen, Q. Qing, Multi-objective robust optimization of foam-filled tapered multi-cell thin-walled structures, Struct. Multidiscip. Optimiz., 52 (2015), 1051–1067. https://doi.org/10.1007/s00158-015-1299-8 doi: 10.1007/s00158-015-1299-8
    [148] A. Khakhali, N. Nariman-zadeh, A. Darvizeh, A. Masoumi, B. Notghi, Reliability-based robust multi-objective crashworthiness optimisation of S-shaped box beams with parametric uncertainties, Int. J. Crashworthines., 15 (2010), 443–456. https://doi.org/10.1080/13588261003696458 doi: 10.1080/13588261003696458
    [149] A. U. Ebenuwa, K. F. Tee, Y. Zhang, Fuzzy-based multi-objective design optimization of buried pipelines, Int. J. Uncertain. Fuzz., 29 (2021), 209–229. https://doi.org/10.1142/S0218488521500104 doi: 10.1142/S0218488521500104
    [150] E. Untiedt. A Parametrized Model for Optimization with Mixed Fuzzy and Possibilistic Uncertainty, Springer, Berlin, 2010. https://doi.org/10.1007/978-3-642-13935-2_9
    [151] X. Liu, Z. Zhang, L. Yin, A multi-objective optimization method for uncertain structures based on nonlinear interval number programming method, Mech. Based Des. Struct. Mach., 45 (2017), 25–42. https://doi.org/10.1080/15397734.2016.1141365 doi: 10.1080/15397734.2016.1141365
    [152] T. Xie, Q. Zhou, J. Hu, L. Shu, P. Jiang, A sequential multi-objective robust optimization approach under interval uncertainty based on support vector machines, in 2017 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM), 2017, 2088–2092. https://doi.org/10.1109/IEEM.2017.8290260
    [153] H. Chagraoui, M. Soula, M. Guedri, Robust multi-objective collaborative optimization of complex structures. Adv. Acous. Vib., 5 (2017), 247–258. https://doi.org/10.1007/978-3-319-41459-1_24 doi: 10.1007/978-3-319-41459-1_24
    [154] F. Li, G. Li, G. Sun, Z. Luo, Z. Zhang, Multi-disciplinary optimization for multi-objective uncertainty design of thin walled beams, CMC-Comput. Mater. Con., 19 (2010), 37–56. http://hdl.handle.net/10453/22026
    [155] X. Zhang, W. Sun, J. Huo, X. Ding, Interval multi-objective optimization design based on physical programming, Przeglad Elektrotechiczhy, 88 (2012), 379–381.
    [156] J. Cheng, G. Duan, Z. Liu, X. Li, Y. Feng, X. Chen, Interval multiobjective optimization of structures based on radial basis function, interval analysis, and NSGA-Ⅱ, J. Zhejiang Univ-Sci. A, 15 (2014), 774–788. https://doi.org/10.1631/jzus.A1300311
    [157] X. Feng, J. Wu, Y. Zhang, M. Jiang, Suspension kinematic/compliance uncertain optimization using a chebyshev polynomial approach, SAE Int. J. Mater. Manu., 8 (2015), 257–262. https://doi.org/10.4271/2015-01-0432 doi: 10.4271/2015-01-0432
    [158] X. Li, C. Jiang, X. Han, An uncertainty multi-objective optimization based on interval analysis and its application, China Mech. Eng., 22 (2011), 1100–1106. http://www.cmemo.org.cn/EN/Y2011/V22/I9/1100
    [159] C. Jiang, H. C. Xie, Z. G. Zhang, X. Han, A new interval optimization method considering tolerance design, Eng. Optimiz., 47 (2015), 1637–1650. https://doi.org/10.1080/0305215X.2014.982632 doi: 10.1080/0305215X.2014.982632
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2859) PDF downloads(262) Cited by(3)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog