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22 pages, 10425 KiB  
Article
A High-Precision Inverse Finite Element Method for Shape Sensing and Structural Health Monitoring
by Hongsheng Yan and Jiangpin Tang
Sensors 2024, 24(19), 6338; https://doi.org/10.3390/s24196338 - 30 Sep 2024
Viewed by 699
Abstract
In the contemporary era, the further exploitation of deep-sea resources has led to a significant expansion of the role of ships in numerous domains, such as in oil and gas extraction. However, the harsh marine environments to which ships are frequently subjected can [...] Read more.
In the contemporary era, the further exploitation of deep-sea resources has led to a significant expansion of the role of ships in numerous domains, such as in oil and gas extraction. However, the harsh marine environments to which ships are frequently subjected can result in structural failures. In order to ensure the safety of the crew and the ship, and to reduce the costs associated with such failures, it is imperative to utilise a structural health monitoring (SHM) system to monitor the ship in real time. Displacement reconstruction is one of the main objectives of SHM, and the inverse finite element method (iFEM) is a powerful SHM method for the full-field displacement reconstruction of plate and shell structures. However, existing inverse shell elements applied to curved shell structures with irregular geometry or large curvature may result in element distortion. This paper proposes a high-precision iFEM for curved shell structures that does not alter the displacement mode of the element or increase the mesh and node quantities. In reality, it just modifies the methods of calculation. This method is based on the establishment of a local coordinate system on the Gaussian integration point and the subsequent alteration of the stiffness integration. The results of numerical examples demonstrate that the high-precision iFEM is capable of effectively reducing the displacement difference resulting from inverse finite element method reconstruction. Furthermore, it performs well in practical engineering applications. Full article
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Figure 1

Figure 1
<p>Implementation process of the iQS4 element.</p>
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<p>Local coordinate system of the iQS4 element.</p>
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<p>Geometric representation of a warped iQS4 element.</p>
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<p>Tangent plane local coordinate system.</p>
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<p>Comparison of two methods for establishing a local coordinate system.</p>
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<p>Local coordinate system established at Gaussian integration points.</p>
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<p>Geometric meaning of <math display="inline"><semantics> <mrow> <mi>D</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>L</mi> </mrow> </semantics></math>.</p>
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<p>Implementation process of new iFEM algorithm.</p>
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<p>Geometry and loads of the twisted plate. (<b>a</b>) Geometry of the twisted plate; (<b>b</b>) loads of the twisted plate.</p>
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<p>Meshing methods of the twisted plate. (<b>a</b>) Regular meshing; (<b>b</b>) irregular meshing.</p>
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<p>The maximum <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>U</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </semantics></math> of the twisted plate for different numbers of elements. (<b>a</b>) Regular meshing; (<b>b</b>) irregular meshing.</p>
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<p>The maximum <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>D</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> </mrow> </semantics></math> of the twisted plate for different numbers of elements. (<b>a</b>) Regular meshing; (<b>b</b>) irregular meshing.</p>
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<p>Contour plots of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>U</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </semantics></math> for the twisted plate. (<b>a</b>) FEM; (<b>b</b>) iFEM-n for regular meshing; (<b>c</b>) iQS4 for regular meshing; (<b>d</b>) iFEM-n for irregular meshing; (<b>e</b>) iQS4 for irregular meshing.</p>
Full article ">Figure 13 Cont.
<p>Contour plots of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>U</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </semantics></math> for the twisted plate. (<b>a</b>) FEM; (<b>b</b>) iFEM-n for regular meshing; (<b>c</b>) iQS4 for regular meshing; (<b>d</b>) iFEM-n for irregular meshing; (<b>e</b>) iQS4 for irregular meshing.</p>
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<p>Geometry and loads of the hemispherical shell. (<b>a</b>) Full model; (<b>b</b>) 1/4 model.</p>
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<p>Meshing methods of the hemispherical shell. (<b>a</b>) Regular meshing; (<b>b</b>) irregular meshing.</p>
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<p>The maximum <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>U</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </semantics></math> of the hemispherical shell for different numbers of elements. (<b>a</b>) Regular meshing; (<b>b</b>) irregular meshing.</p>
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<p>The maximum <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>D</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> </mrow> </semantics></math> of the hemispherical shell for different numbers of elements. (<b>a</b>) Regular meshing; (<b>b</b>) irregular meshing.</p>
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<p>Contour plots of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>U</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </semantics></math> for the hemispherical shell. (<b>a</b>) FEM; (<b>b</b>) iFEM-n for regular meshing; (<b>c</b>) iQS4 for regular meshing; (<b>d</b>) iFEM-n for irregular meshing; (<b>e</b>) iQS4 for irregular meshing.</p>
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<p>Geometric model of the containership.</p>
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<p>Internal structure of the containership.</p>
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<p>Loads of the containership.</p>
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<p>Grids of the containership.</p>
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<p>Contour plots of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>U</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> for the containership. (<b>a</b>) FEM; (<b>b</b>) iFEM-H.</p>
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<p>Contour plots of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>U</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> for the containership. (<b>a</b>) FEM; (<b>b</b>) iFEM-H.</p>
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<p>Contour plots of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>U</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> </mrow> </semantics></math> for the containership. (<b>a</b>) FEM; (<b>b</b>) iFEM-H.</p>
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22 pages, 5490 KiB  
Article
Fem-1 Gene of Chinese White Pine Beetle (Dendroctonus armandi): Function and Response to Environmental Treatments
by Jiajin Wang, Songkai Liao, Haoyu Lin, Hongjian Wei, Xinjie Mao, Qi Wang and Hui Chen
Int. J. Mol. Sci. 2024, 25(19), 10349; https://doi.org/10.3390/ijms251910349 - 26 Sep 2024
Viewed by 722
Abstract
Dendroctonus armandi (Tsai and Li) (Coleoptera: Curculionidae: Scolytinae) is regarded as the most destructive forest pest in the Qinling and Bashan Mountains of China. The sex determination of Dendroctonus armandi plays a significant role in the reproduction of its population. In recent years, [...] Read more.
Dendroctonus armandi (Tsai and Li) (Coleoptera: Curculionidae: Scolytinae) is regarded as the most destructive forest pest in the Qinling and Bashan Mountains of China. The sex determination of Dendroctonus armandi plays a significant role in the reproduction of its population. In recent years, the role of the fem-1 gene in sex determination in other insects has been reported. However, the function and expression of the fem-1 gene in Dendroctonus armandi remain uncertain. In this study, three fem-1 genes were cloned and characterized. These were named Dafem-1A, Dafem-1B, and Dafem-1C, respectively. The expression levels of these three Dafem-1 genes vary at different stages of development and between the sexes. In response to different environmental treatments, including temperature, nutrients, terpenoids, and feeding duration, significant differences were observed between the three Dafem-1 genes at different developmental stages and between males and females. Furthermore, injection of double-stranded RNA (dsRNA) targeting the expressions of the Dafem-1A, Dafem-1B, and Dafem-1C genes resulted in increased mortality, deformity, and decreased emergence rates, as well as an imbalance in the sex ratio. Following the interference with Dafem-1A and Dafem-1C, no notable difference was observed in the expression of the Dafem-1B gene. Similarly, after the interference with the Dafem-1B gene, no significant difference was evident in the expression levels of the Dafem-1A and Dafem-1C genes. However, the interference of either the Dafem-1A or Dafem-1C gene results in the downregulation of the other gene. The aforementioned results demonstrate that the Dafem-1A, Dafem-1B, and Dafem-1C genes play a pivotal role in the regulation of life development and sex determination. Furthermore, it can be concluded that external factors such as temperature, nutrition, terpenoids, and feeding have a significant impact on the expression levels of the Dafem-1A, Dafem-1B, and Dafem-1C genes. This provides a crucial theoretical foundation for further elucidating the sex determination mechanism of Dendroctonus armandi. Full article
(This article belongs to the Special Issue Essential Molecules in Life: Regulation, Defense, and Longevity)
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Figure 1
<p>Relative expression of the <span class="html-italic">Dafem-1</span> genes in different developmental stages of <span class="html-italic">D. armandi</span>. (<b>A</b>) <span class="html-italic">Dafem-1</span>A; (<b>B</b>) <span class="html-italic">Dafem-1</span>B; and (<b>C</b>) <span class="html-italic">Dafem-1</span>C. The abbreviations used in the figure are L for Larvae, P for Pupae, and A for Adults. Relative expression levels were normalized to <span class="html-italic">β-actin</span>. All values are presented as mean ± SE (n = 3). Statistical analysis using one-way ANOVA and Tukey’s post hoc test revealed a significant difference at the <span class="html-italic">p</span> &lt; 0.05 level. Capital letters indicate inter-group differences. The asterisk indicates a significant difference between males and females (** <span class="html-italic">p</span> &lt; 0.01, and *** <span class="html-italic">p</span> &lt; 0.001, independent sample <span class="html-italic">t</span>-test). Note: It is not possible to distinguish between larvae and pupae in terms of gender based on their morphology. Consequently, they are referred to as belonging to two sexes.</p>
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<p>Relative expression of the <span class="html-italic">Dafem-1</span> genes in <span class="html-italic">D. armandi</span> under temperature treatments. (<b>A</b>) Larvae <span class="html-italic">Dafem-1</span>A; (<b>B</b>) larvae <span class="html-italic">Dafem-1</span>B; (<b>C</b>) larvae <span class="html-italic">Dafem-1</span>C; (<b>D</b>) pupae <span class="html-italic">Dafem-1</span>A; (<b>E</b>) pupae <span class="html-italic">Dafem-1</span>B; (<b>F</b>) pupae <span class="html-italic">Dafem-1</span>C; (<b>G</b>) adult <span class="html-italic">Dafem-1</span>A; (<b>H</b>) adult <span class="html-italic">Dafem-1</span>B; and (<b>I</b>) adult <span class="html-italic">Dafem-1</span>C. Relative expression levels were normalized to <span class="html-italic">β-actin</span>. All values are presented as mean ± SE (n = 3). Statistical analysis using one-way ANOVA and Tukey’s post hoc test revealed a significant difference at the <span class="html-italic">p</span> &lt; 0.05 level. Capital letters indicate inter-group differences. The asterisk indicates a significant difference between males and females (* <span class="html-italic">p</span> &lt; 0.05, ** <span class="html-italic">p</span> &lt; 0.01, and *** <span class="html-italic">p</span> &lt; 0.001, independent sample <span class="html-italic">t</span>-test).</p>
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<p>Relative expression of the <span class="html-italic">Dafem-1</span> genes in <span class="html-italic">D. armandi</span> under different nutrient treatments. (<b>A</b>) Larvae <span class="html-italic">Dafem-1</span>A; (<b>B</b>) larvae <span class="html-italic">Dafem-1</span>B; (<b>C</b>) larvae <span class="html-italic">Dafem-1</span>C; (<b>D</b>) adult <span class="html-italic">Dafem-1</span>A; (<b>E</b>) adult <span class="html-italic">Dafem-1</span>B; and (<b>F</b>) adult <span class="html-italic">Dafem-1</span>C. Formulas A, B, C, and D were designated as FA, FB, FC, and FD, respectively. Formula A served as a control, with additional protein, cellulose, and reducing sugar added to the feed in addition to the common components (see <a href="#sec4-ijms-25-10349" class="html-sec">Section 4</a> for details). The treatment groups were formulas B, C, and D, which were based on formula A with protein, cellulose, and reducing sugar removed, respectively. Relative expression levels were normalized to <span class="html-italic">β-actin</span>. All values are presented as mean ± SE (n = 3). Statistical analysis using one-way ANOVA and Tukey’s post hoc test revealed a significant difference at the <span class="html-italic">p</span> &lt; 0.05 level. Capital letters indicate inter-group differences. The asterisk indicates a significant difference between males and females (* <span class="html-italic">p</span> &lt; 0.05, ** <span class="html-italic">p</span> &lt; 0.01, and *** <span class="html-italic">p</span> &lt; 0.001, independent sample <span class="html-italic">t</span>-test).</p>
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<p>Relative expression of the <span class="html-italic">Dafem-1</span> genes in larvae and adults of <span class="html-italic">D. armandi</span> under feeding duration treatment. (<b>A</b>) Larvae <span class="html-italic">Dafem-1</span>A; (<b>B</b>) larvae <span class="html-italic">Dafem-1</span>B; (<b>C</b>) larvae <span class="html-italic">Dafem-1</span>C; (<b>D</b>) adult <span class="html-italic">Dafem-1</span>A; (<b>E</b>) adult <span class="html-italic">Dafem-1</span>B; and (<b>F</b>) adult <span class="html-italic">Dafem-1</span>C. Relative expression levels were normalized to <span class="html-italic">β-actin</span>. All values are presented as mean ± SE (n = 3). Statistical analysis using one-way ANOVA and Tukey’s post hoc test revealed a significant difference at the <span class="html-italic">p</span> &lt; 0.05 level. Capital letters indicate the degree of difference between groups. The asterisk indicates a significant difference between males and females (** <span class="html-italic">p</span> &lt; 0.01, and *** <span class="html-italic">p</span> &lt; 0.001, independent sample <span class="html-italic">t</span>-test).</p>
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<p>Relative expression of the <span class="html-italic">Dafem-1</span> genes in <span class="html-italic">D. armandi</span> under different terpenoid treatments. (<b>A</b>) Larvae <span class="html-italic">Dafem-1</span>A; (<b>B</b>) larvae <span class="html-italic">Dafem-1</span>B; (<b>C</b>) larvae <span class="html-italic">Dafem-1</span>C; (<b>D</b>) pupae <span class="html-italic">Dafem-1</span>A; (<b>E</b>) pupae <span class="html-italic">Dafem-1</span>B; (<b>F</b>) pupae <span class="html-italic">Dafem-1</span>C; (<b>G</b>) adult <span class="html-italic">Dafem-1</span>A; (<b>H</b>) adult <span class="html-italic">Dafem-1</span>B; and (<b>I</b>) adult <span class="html-italic">Dafem-1</span>C. CK for control check, +α for (+)-α-pinene, −α for (−)-α-pinene, +β for (+)-β-pinene, −β for (−)-β-pinene, Cam for (+)-camphene, Car for (+)-3-carene, and Lim for (±)-limone. The control check was not subjected to any terpenoid treatment. Relative expression levels were normalized to <span class="html-italic">β-actin</span>. All values are presented as mean ± SE (n = 3). Statistical analysis using one-way ANOVA and Tukey’s post hoc test revealed a significant difference at the <span class="html-italic">p</span> &lt; 0.05 level. Capital letters indicate the degree of between-group differences. The asterisk indicates a significant difference between males and females (** <span class="html-italic">p</span> &lt; 0.01, and *** <span class="html-italic">p</span> &lt; 0.001, independent sample <span class="html-italic">t</span>-test).</p>
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<p>RNA interference efficiency detection of the <span class="html-italic">Dafem-1</span> genes in <span class="html-italic">D. armandi</span>. (<b>A</b>) <span class="html-italic">Dafem-1</span>A; (<b>B</b>) <span class="html-italic">Dafem-1</span>B; and (<b>C</b>) <span class="html-italic">Dafem-1</span>C. The control group was administered an equal volume of diethyl pyrocarbonate (DEPC) in water. Relative expression levels were normalized to <span class="html-italic">β-actin</span>. All values are presented as mean ± SE (n = 3). Statistical analysis using one-way ANOVA and Tukey’s post hoc test revealed a significant difference at the <span class="html-italic">p</span> &lt; 0.05 level. Lowercase letters indicate the degree of within-group differences.</p>
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<p>Relative expression of the three <span class="html-italic">Dafem-1</span> genes after 72 h of RNA interference. (<b>A</b>) Gene expression of <span class="html-italic">Dafem-1</span>B after <span class="html-italic">Dafem-1</span>A interference; (<b>B</b>) gene expression of <span class="html-italic">Dafem-1</span>C after <span class="html-italic">Dafem-1</span>A interference; (<b>C</b>) gene expression of <span class="html-italic">Dafem-1</span>A after <span class="html-italic">Dafem-1</span>B interference; (<b>D</b>) gene expression of <span class="html-italic">Dafem-1</span>C after <span class="html-italic">Dafem-1</span>B interference; (<b>E</b>) gene expression of <span class="html-italic">Dafem-1</span>A after <span class="html-italic">Dafem-1</span>C interference; and (<b>F</b>) gene expression of <span class="html-italic">Dafem-1</span>B after <span class="html-italic">Dafem-1</span>C interference. Relative expressions were normalized to <span class="html-italic">β-actin</span>. All values are presented as mean ± SE (n = 3). The asterisk indicates the significant difference between the RNAi interference experimental group and the control group (* <span class="html-italic">p</span> &lt; 0.05, ** <span class="html-italic">p</span> &lt; 0.01, and *** <span class="html-italic">p</span> &lt; 0.001, independent sample <span class="html-italic">t</span>-test).</p>
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<p>Mortality and deformity rates of <span class="html-italic">D. armandi</span> at different developmental stages after RNA interference of the three <span class="html-italic">Dafem-1</span> genes. CK for control check. (<b>A</b>) Mortality rates of larvae, pupae, and adults after RNA interference with the <span class="html-italic">Dafem-1</span>A, <span class="html-italic">Dafem-1</span>B, and <span class="html-italic">Dafem-1</span>C genes; (<b>B</b>) abnormal morphology rate of larvae, pupae, and adults after RNA interference with the <span class="html-italic">Dafem-1</span>A, <span class="html-italic">Dafem-1</span>B, and <span class="html-italic">Dafem-1</span>C genes. The control check was not subjected to RNAi treatment. All values are presented as mean ± SE (n = 3). The asterisk indicates the significant difference between the RNAi interference experimental group and the control group (* <span class="html-italic">p</span> &lt; 0.05, ** <span class="html-italic">p</span> &lt; 0.01, and *** <span class="html-italic">p</span> &lt; 0.001, independent sample <span class="html-italic">t</span>-test).</p>
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21 pages, 11641 KiB  
Article
Study on Strain Field Reconstruction Method of Long-Span Hull Box Girder Based on iFEM
by Guocai Chen, Xueliang Wang, Nan Zhao, Zhentao Jiang, Fei Li, Haozheng Chen, Pengyu Wei and Tao Zhang
J. Mar. Sci. Eng. 2024, 12(9), 1482; https://doi.org/10.3390/jmse12091482 - 26 Aug 2024
Viewed by 851
Abstract
The box girder’s condition significantly impacts the safety and overall performance of the entire ship because it is the primary stress component of the hull construction. This work used experimental research on the long-span hull box girder based on IFEM (Inverse Finite Element [...] Read more.
The box girder’s condition significantly impacts the safety and overall performance of the entire ship because it is the primary stress component of the hull construction. This work used experimental research on the long-span hull box girder based on IFEM (Inverse Finite Element Method) technology to ensure the structural safety of the hull box girder. Due to the limitations of conventional experiments in this technical field, such as their reliance on finite element data and lack of input from physical tests, numerous research methods combining the strain sensing data from physical tests with the strain data from virtual sensors were conducted. The strain fields of the top plate, side plate, and bottom plate were each reconstructed in turn, and the verifier measuring points in the physical model test were used to assess the accuracy of the reconstruction results. The findings demonstrate that the top plate, side plate, and bottom plate reconstructions had relative errors of 0.24–7.86%, 0.75–8.13%, and 3.31–2.52%, respectively. This enables the reconstruction of the strain field of the long-span hull box girder using physical test data and promotes the use of iFEM technology in the field of structural health monitoring of large marine structures. Full article
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Figure 1

Figure 1
<p>Four-node inverse shell element iQS4; (<b>a</b>) iQS4 element showing global and local coordinate systems; (<b>b</b>) nodal DOF in the local coordinate system xyz.</p>
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<p>Discrete surface strain measured by strain flower in iQS4 element.</p>
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<p>Schematic diagram of four-point bending of hull box girder.</p>
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<p>Finite element model of long-span hull box girder.</p>
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<p>Nephogram of strain distribution in direction (longitudinal direction) of box girder.</p>
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<p>Test data acquisition–application process.</p>
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<p>Schematic diagram of compartments division.</p>
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<p>Schematic diagram of strain sensor arrangement.</p>
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<p>Layout of physical sensors on the test site; (<b>a</b>) layout of strain sensor on top surface; (<b>b</b>) layout diagram of shipboard physical sensor.</p>
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<p>Loading diagram of four-point bending test of long-span box girder.</p>
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<p>Test loading site of long-span box girder.</p>
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<p>Strain trend of measured values and reconstructed values of box girder top surface–side surface–bottom surface.</p>
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<p>Comparison and error between measured values and reconstructed values of measuring points on the top surface.</p>
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<p>Measured strain field on top plate surface.</p>
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<p>Reconstruction strain field of top plate surface.</p>
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<p>Comparison and error between measured values and reconstructed values of measuring points on the side.</p>
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<p>Measured strain field on the side.</p>
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<p>Strain field of the side reconstruction.</p>
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<p>Comparison and error between measured values and reconstructed values of measuring points on the bottom surface.</p>
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<p>Measured strain field on bottom plate surface.</p>
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<p>Reconstruction strain field of bottom plate surface.</p>
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18 pages, 8042 KiB  
Article
Discontinuous Deformation Monitoring of Smart Aerospace Structures Based on Hybrid Reconstruction Strategy and Fiber Bragg Grating
by Kangyu Chen, Hengzhen Fan and Hong Bao
Sensors 2024, 24(11), 3603; https://doi.org/10.3390/s24113603 - 3 Jun 2024
Cited by 1 | Viewed by 752
Abstract
A hybrid enhanced inverse finite element method (E-iFEM) is proposed for real-time intelligent sensing of discontinuous aerospace structures. The method can improve the flight performance of intelligent aircrafts by feeding back the structural shape information to the control system. Initially, the presented algorithm [...] Read more.
A hybrid enhanced inverse finite element method (E-iFEM) is proposed for real-time intelligent sensing of discontinuous aerospace structures. The method can improve the flight performance of intelligent aircrafts by feeding back the structural shape information to the control system. Initially, the presented algorithm combines rigid kinematics with the classical iFEM to discretize the aerospace structures into elastic parts and rigid parts, which will effectively overcome structural complexity due to fluctuating bending stiffness and a special aerodynamic section. Subsequently, the rigid parts provide geometric constraints for the iFEM in the shape reconstruction method. Meanwhile, utilizing the Fiber Bragg grating (FBG) strain sensor to obtain real-time strain information ensures lightweight and anti-interference of the monitoring system. Next, the strain data and the geometric constraints are processed by the iFEM for monitoring the full-field elastic deformation of the aerospace structures. The whole procedure can be interpreted as a piecewise sensing technology. Overall, the effectiveness and reliability of the proposed method are validated by employing a comprehensive numerical simulation and experiment. Full article
(This article belongs to the Section Intelligent Sensors)
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Figure 1
<p>Geometric and planar sketch of the deformed wing.</p>
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<p>A discrete surface strain sensor distribution scheme.</p>
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<p>Analysis of discontinuous structures.</p>
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<p>A specific description of the geometric relationship.</p>
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<p>Visual description of the reconstruction model.</p>
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<p>The flowchart of the E-iFEM.</p>
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<p>The geometric description of the airfoil frame.</p>
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<p>The visualizational stress cloud chart of the airfoil frame.</p>
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<p>Detailed strain distribution information of ribs and beams.</p>
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<p>Comparison of ABAQUS, T−iFEM, and E−iFEM calculations.</p>
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<p>Comparison of ABAQUS, T−iFEM, and E−iFEM calculations.</p>
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<p>Simulation model for hollow rectangular beam.</p>
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<p>Layout of the deformation verification points and loading points.</p>
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<p>Simulation strain curves for the hollow rectangular section.</p>
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<p>The inverse finite elements discretization of two methods.</p>
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<p>Comparison of displacement reconstruction errors between T-iFEM and E-iFEM.</p>
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<p>Experimental test platform for reconstruction algorithm E-iFEM.</p>
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<p>Experimental test platform for reconstruction algorithm E-iFEM.</p>
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<p>Experimental verification system.</p>
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<p>Comparison of reconstructed and measured displacements under different loads.</p>
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23 pages, 10457 KiB  
Article
A Critical Comparison of Shape Sensing Algorithms: The Calibration Matrix Method versus iFEM
by Cornelis de Mooij and Marcias Martinez
Sensors 2024, 24(11), 3562; https://doi.org/10.3390/s24113562 - 31 May 2024
Viewed by 752
Abstract
Two shape-sensing algorithms, the calibration matrix (CM) method and the inverse Finite Element Method (iFEM), were compared on their ability to accurately reconstruct displacements, strains, and loads and on their computational efficiency. CM reconstructs deformation through a linear combination of known load cases [...] Read more.
Two shape-sensing algorithms, the calibration matrix (CM) method and the inverse Finite Element Method (iFEM), were compared on their ability to accurately reconstruct displacements, strains, and loads and on their computational efficiency. CM reconstructs deformation through a linear combination of known load cases using the sensor data measured for each of these known load cases and the sensor data measured for the actual load case. iFEM reconstructs deformation by minimizing a least-squares error functional based on the difference between the measured and numerical values for displacement and/or strain. In this study, CM is covered in detail to determine the applicability and practicality of the method. The CM results for several benchmark problems from the literature were compared to the iFEM results. In addition, a representative aerospace structure consisting of a twisted and tapered blade with a NACA 6412 cross-sectional profile was evaluated using quadratic hexahedral solid elements with reduced integration. Both methods assumed linear elastic material conditions and used discrete displacement sensors, strain sensors, or a combination of both to reconstruct the full displacement and strain fields. In our study, surface-mounted and distributed sensors throughout the volume of the structure were considered. This comparative study was performed to support the growing demand for load monitoring, specifically for applications where the sensor data is obtained from discrete and irregularly distributed points on the structure. In this study, the CM method was shown to achieve greater accuracy than iFEM. Averaged over all the load cases examined, the CM algorithm achieved average displacement and strain errors of less than 0.01%, whereas the iFEM algorithm had an average displacement error of 21% and an average strain error of 99%. In addition, CM also achieved equal or better computational efficiency than iFEM after initial set-up, with similar first solution times and faster repeat solution times by a factor of approximately 100, for hundreds to thousands of sensors. Full article
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<p>Benchmarks problem from de Mooij et al. [<a href="#B37-sensors-24-03562" class="html-bibr">37</a>], based on a selection from MacNeal and Harder [<a href="#B16-sensors-24-03562" class="html-bibr">16</a>], augmented with sensors: displacement sensors are in red, strain sensors are in blue. Cantilever beams a-c are composed of regular, trapezoid and parallelogram elements, respectively.</p>
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<p>Isometric view of twisted and tapered blade with NACA 6412 cross-sectional profile, divided into 648 hexahedral elements. Fully constrained areas are highlighted in red and loaded areas are highlighted in blue.</p>
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<p>A 3D illustration of the mesh and the simulated strain sensors.</p>
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<p>CM algorithm flowchart for numerical approach.</p>
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<p>Example of a basic load case: a unit force along the z-axis, applied to a region on the top surface of the twisted beam, marked in blue. The boundary condition, the fully constrained root of the beam, is marked in red.</p>
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<p>iFEM algorithm flowchart for numerical approach.</p>
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<p>Comparison of analytical and FEM results for the MacNeal and Harder [<a href="#B55-sensors-24-03562" class="html-bibr">55</a>] benchmark problems.</p>
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<p>Example FEM result for the basic load case shown in <a href="#sensors-24-03562-f005" class="html-fig">Figure 5</a>. The value of the E11 strain component is plotted on the surface of the twisted beam.</p>
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<p>Example FEM result for basic load case 60 a unit force in the z-direction, applied with surface tractions. Shown are the loaded and constrained regions, the E<sub>11</sub> strains on the mesh, and the E<sub>11</sub> strains of the simulated sensors.</p>
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<p>Expected load case 1z, uniform load: expected tractions, reconstructed tractions, and relative traction errors.</p>
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<p>Load Cases vs. Richardson Error Bands.</p>
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<p>Force errors vs. number of sensors for the six reconstructed load cases in the X direction.</p>
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<p>Sensor strain errors vs. number of strain sensors for 6 reconstructed load cases in the X direction.</p>
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<p>Calculated errors between reconstructed and actual results for forces, coefficients, displacements, strains, and sensor strains per actual benchmark problem load case for the Calibration Matrix method.</p>
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<p>Comparison of surface loads (N) for a curved beam loaded in out-of-plane shear.</p>
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<p>Reconstructed surface loads (N) in x and y-directions for a curved beam loaded in out-of-plane shear.</p>
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<p>Comparison of relative errors of the displacements and strains for the CM and iFEM reconstructions of each benchmark problem.</p>
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<p>Computational efficiency: solution time vs. number of strain sensors (which was kept equal to the number of basic load cases here) for the first solution of CM and iFEM and the repeat solution of CM.</p>
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18 pages, 9116 KiB  
Article
Real-Time Digital Twin of Ship Structure Deformation Field Based on the Inverse Finite Element Method
by Pengyu Wei, Chuntong Li, Ze Jiang and Deyu Wang
J. Mar. Sci. Eng. 2024, 12(2), 257; https://doi.org/10.3390/jmse12020257 - 31 Jan 2024
Cited by 2 | Viewed by 1873
Abstract
Digital twins, an innovative technology propelled by data and models, play a seminal role in the digital transformation and intelligent upgrade of ships. This study introduces a digital twin methodology for the real-time monitoring of ship structure deformation fields, based on finite discrete [...] Read more.
Digital twins, an innovative technology propelled by data and models, play a seminal role in the digital transformation and intelligent upgrade of ships. This study introduces a digital twin methodology for the real-time monitoring of ship structure deformation fields, based on finite discrete strain data, and a visualization tool framework is developed using virtual reality technology. First, the inverse Finite Element Method (iFEM) is employed to derive the deformation field of the ship structure in real time using sensor strain data. Secondly, the deformation field data obtained based on the iFEM algorithm is converted into general visualization data conducive to interpretation within virtual reality (VR) applications. Lastly, a digital twin software tool is built to enable synchronous responses and interactions between the virtual scene and the physical scene, directly superposing particular virtual objects (data acquired by sensors, computer-aided design (CAD) virtual models, and deformation field cloud images) onto the physical scene in real time. The digital twin tool embodies a virtual reality visualization system framework integrating the physical data measurement, reconstruction, analysis, expression, storage, rendering, and interaction of deformation field data. Through practical application, the flexibility, effectiveness, and compatibility of the developed prototype tool are verified. According to the results, the system can enhance the efficiency of scientific communication, model validation, and interdisciplinary sharing during the analysis and evaluation of the mechanical properties of ship structures. Full article
(This article belongs to the Special Issue Advanced Analysis of Marine Structures—Edition II)
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<p>Overall process of real-time digital twin.</p>
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<p>Characteristics of the digital twin approach.</p>
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<p>Real-time digital twin technology flow chart for hull structure safety assessment.</p>
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<p>Four-node inverse-shell element. (<b>a</b>) iQS4 element showing global and local coordinate systems; (<b>b</b>) Nodal DOFs in the local coordinate system xyz.</p>
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<p>Strain rosettes instrumented on the top and bottom of iQS4 elements.</p>
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<p>Deformation field virtual visualization implementation process.</p>
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<p>Schematic of 3D model of stiffened plate.</p>
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<p>Stiffened plate model strain gauge and displacement gauge layout diagram.</p>
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<p>Loading scheme of the test model.</p>
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<p>The construction process of digital twin scenarios.</p>
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<p>The construction process of digital twin scenarios.</p>
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<p>Data transfer of digital twin models.</p>
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<p>Model reconstruction and rendering based on point arrays.</p>
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<p>Deformation field cloud diagram.</p>
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<p>Real-time digital twin platform for ship structure mechanic testing.</p>
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20 pages, 6457 KiB  
Article
Shape Sensing in Plate Structures through Inverse Finite Element Method Enhanced by Multi-Objective Genetic Optimization of Sensor Placement and Strain Pre-Extrapolation
by Emiliano Del Priore and Luca Lampani
Sensors 2024, 24(2), 608; https://doi.org/10.3390/s24020608 - 18 Jan 2024
Cited by 3 | Viewed by 1352
Abstract
The real-time reconstruction of the displacement field of a structure from a network of in situ strain sensors is commonly referred to as “shape sensing”. The inverse finite element method (iFEM) stands out as a highly effective and promising approach to perform this [...] Read more.
The real-time reconstruction of the displacement field of a structure from a network of in situ strain sensors is commonly referred to as “shape sensing”. The inverse finite element method (iFEM) stands out as a highly effective and promising approach to perform this task. In the current investigation, this technique is employed to monitor different plate structures experiencing flexural and torsional deformation fields. In order to reduce the number of installed sensors and obtain more accurate results, the iFEM is applied in synergy with smoothing element analysis (SEA), which allows the pre-extrapolation of the strain field over the entire structure from a limited number of measurement points. For the SEA extrapolation to be effective for a multitude of load cases, it is necessary to position the strain sensors appropriately. In this study, an innovative sensor placement strategy that relies on a multi-objective genetic algorithm (NSGA-II) is proposed. This approach aims to minimize the root mean square error of the pre-extrapolated strain field across a set of mode shapes for the examined plate structures. The optimized strain reconstruction is subsequently utilized as input for the iFEM technique. Comparisons are drawn between the displacement field reconstructions obtained using the proposed methodology and the conventional iFEM. In order to validate such methodology, two different numerical case studies, one involving a rectangular cantilevered plate and the other encompassing a square plate clamped at the edges, are investigated. For the considered case studies, the results obtained by the proposed approach reveal a significant improvement in the monitoring capabilities over the basic iFEM algorithm with the same number of sensors. Full article
(This article belongs to the Special Issue Sensors for Vibration Control and Structural Health Monitoring)
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<p>(<b>a</b>) iQS4 element; (<b>b</b>) discrete surface strains at i-th measurement point.</p>
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<p>Triangular smoothing element with nodes 1, 2, 3 and their corresponding DOF.</p>
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<p>(<b>a</b>) Example of FE mesh with element centroids; (<b>b</b>) example of evaluation points.</p>
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<p>(<b>a</b>) Uniform crossover: blue dots represent sensor positions coming from Parent 1 meanwhile red dots are those coming from Parent 2; (<b>b</b>) mutation process: yellow dots represent sensor positions of the original offspring meanwhile purple dots are the mutated ones.</p>
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<p>Flowchart of the optimization process.</p>
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<p>Cantilevered rectangular plate. FE mesh (<b>left</b>); SEA mesh (<b>center</b>); standard iFEM mesh (<b>right</b>).</p>
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<p>(<b>a</b>) Selected optimal sensor arrangement for the cantilevered rectangular plate; (<b>b</b>) uniform sensor placement used for validation.</p>
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<p>Pre-extrapolated <math display="inline"> <semantics> <mrow> <msub> <mi>k</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo> </mo> <mfenced> <mrow> <mfrac> <mn>1</mn> <mi mathvariant="normal">m</mi> </mfrac> </mrow> </mfenced> <mo> </mo> </mrow> </semantics> </math>on the left vs. reference on the right for the first 3 modes of the cantilevered plate.</p>
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<p>Cantilevered plate’s first three modes deflection <math display="inline"> <semantics> <mrow> <mi>w</mi> <mo> </mo> <mfenced> <mi mathvariant="normal">m</mi> </mfenced> </mrow> </semantics> </math> for iFEM aided by NSGA-II and SEA (<b>left</b>) and reference solution (<b>right</b>).</p>
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<p>Cantilevered plate’s reconstruction percentage error of the first three modes for standard iFEM (<b>left</b>), smoothed iFEM with uniformly distributed sensors (<b>center</b>), and smoothed iFEM with optimal layout (<b>right</b>).</p>
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<p>Clamped square plate. FE mesh (<b>left</b>); SEA mesh (<b>center</b>); standard iFEM mesh (<b>right</b>).</p>
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<p>(<b>a</b>) Selected optimal sensor arrangement for the clamped square plate; (<b>b</b>) uniform sensor placement used for validation.</p>
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<p>Pre-extrapolated <math display="inline"> <semantics> <mrow> <msub> <mi>k</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo> </mo> <mfenced> <mrow> <mfrac> <mn>1</mn> <mi mathvariant="normal">m</mi> </mfrac> </mrow> </mfenced> <mo> </mo> </mrow> </semantics> </math> on the left vs. reference on the right for the first four modes of the clamped plate.</p>
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<p>Clamped plate’s first four modes deflection <math display="inline"> <semantics> <mrow> <mi>w</mi> <mo> </mo> <mfenced> <mi mathvariant="normal">m</mi> </mfenced> </mrow> </semantics> </math> for iFEM aided by NSGA-II and SEA (<b>left</b>) and reference solution (<b>right</b>).</p>
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<p>Clamped plate’s reconstruction percentage error of the first four modes for standard iFEM (<b>left</b>), smoothed iFEM with uniformly distributed sensors (<b>center</b>), and smoothed iFEM with optimal layout (<b>right</b>).</p>
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18 pages, 13551 KiB  
Article
Enhanced Strain Field Reconstruction in Ship Stiffened Panels Using Optical Fiber Sensors and the Strain Function-Inverse Finite Element Method
by Qingfeng Zhu, Guoqing Wu, Jie Zeng, Zhentao Jiang, Yingping Yue, Chao Xiang, Jun Zhan and Bohan Zhao
Appl. Sci. 2024, 14(1), 370; https://doi.org/10.3390/app14010370 - 30 Dec 2023
Cited by 2 | Viewed by 1265
Abstract
Accurately reconstructing the strain field within stiffened ship panels is crucial for effective structural health monitoring. This study presents a groundbreaking approach to strain field reconstruction in such panels, utilizing optical fiber sensors in conjunction with the strain function-inverse finite element method (SF-iFEM). [...] Read more.
Accurately reconstructing the strain field within stiffened ship panels is crucial for effective structural health monitoring. This study presents a groundbreaking approach to strain field reconstruction in such panels, utilizing optical fiber sensors in conjunction with the strain function-inverse finite element method (SF-iFEM). A novel technique for solving nodal strain vectors, based on the element strain function, has been devised to improve the accuracy of strain reconstruction using the inverse finite element method (iFEM), addressing the limitations associated with traditional nodal displacement vector solutions. Moreover, the proposed method for determining the equivalent neutral layer of stiffened ship panels not only reduces the number of elements effectively but also establishes a strain function between the inner and outer surfaces of the structure. Using this function, a layout scheme for optical fiber sensors on the inner side of ship stiffened panels is provided, overcoming the symmetrical arrangement constraints of iFEM for sensor placement on both the inner and outer sides of the structure. The results demonstrate a significant improvement in strain reconstruction accuracy under bending and bending–torsion deformations compared to conventional iFEM. Consequently, the findings of this research will contribute to enhancing the engineering applicability of iFEM in ship structure health monitoring. Full article
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<p>Four-node shell element for ship stiffened panel division.</p>
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<p>Sensor layout based on equivalent neutral layer calculation for stiffened ship panel.</p>
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<p>Strain field reconstruction process for stiffened ship panels based on SF-iFEM.</p>
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<p>(<b>a</b>) Ship stiffened panel; (<b>b</b>) reconstructed strain verification point and loading point.</p>
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<p>(<b>a</b>) Four paths in the transverse T−section; (<b>b</b>) X−direction simulated strain curves for the four paths.</p>
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<p>Various mesh partitioning schemes for stiffened ship panels: (<b>a</b>) mesh partitioning scheme 1; (<b>b</b>) mesh partitioning scheme 2; (<b>c</b>) mesh partitioning scheme 3.</p>
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<p>Strain fields reconstruction accuracy for three different mesh partitioning schemes.</p>
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<p>Strain field reconstruction accuracy for three different load magnitudes.</p>
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<p>Strain cloud diagram of 1 # and 2 # transverse T−section in Case 1: (<b>a</b>) FEM results; (<b>b</b>) SF−iFEM results.</p>
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<p>Strain cloud diagram of 1 # transverse T−section in Case 2: (<b>a</b>) FEM results; (<b>b</b>) SF−iFEM results.</p>
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<p>Strain cloud diagram of 2 # transverse T−section in Case 2: (<b>a</b>) FEM results; (<b>b</b>) SF−iFEM results.</p>
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<p>Strain cloud diagram of plate for Case 1: (<b>a</b>) FEM results; (<b>b</b>) SF-iFEM results.</p>
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<p>Strain cloud diagram of the plate for Case 2: (<b>a</b>) FEM results; (<b>b</b>) SF-iFEM results.</p>
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<p>Comparison of strain reconstruction errors between iFEM and SF-iFEM in Case 1: (<b>a</b>) absolute error; (<b>b</b>) relative error.</p>
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<p>Comparison of strain reconstruction errors between iFEM and SF-iFEM in Case 2: (<b>a</b>) absolute error; (<b>b</b>) relative error.</p>
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<p>(<b>a</b>) Strain monitoring system for the ship stiffened panel; (<b>b</b>) loading point and strain reconstruction verification path position on the outer side of the ship stiffened panel.</p>
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<p>(<b>a</b>) OFDR fiber optic sensor layout for transverse T−section; (<b>b</b>) the X−direction measured strain curves of the six paths.</p>
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<p>(<b>a</b>) The layout of fiber optic sensors located inside the stiffened ship panel; (<b>b</b>) the integration of optical fiber sensors.</p>
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<p>The strain reconstruction cloud diagram of the monitoring area: (<b>a</b>) in Case 3; (<b>b</b>) in Case 4.</p>
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<p>Reconstruction of the strain on three paths in Case 3: (<b>a</b>) comparison; (<b>b</b>) relative error.</p>
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<p>Reconstruction of the strain on three paths in Case 4: (<b>a</b>) comparison; (<b>b</b>) relative error.</p>
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17 pages, 9082 KiB  
Article
A Two-Dimensional Eight-Node Quadrilateral Inverse Element for Shape Sensing and Structural Health Monitoring
by Mingyang Li, Erkan Oterkus and Selda Oterkus
Sensors 2023, 23(24), 9809; https://doi.org/10.3390/s23249809 - 14 Dec 2023
Cited by 2 | Viewed by 1124
Abstract
The inverse finite element method (iFEM) is a powerful tool for shape sensing and structural health monitoring and has several advantages with respect to some other existing approaches. In this study, a two-dimensional eight-node quadrilateral inverse finite element formulation is presented. The element [...] Read more.
The inverse finite element method (iFEM) is a powerful tool for shape sensing and structural health monitoring and has several advantages with respect to some other existing approaches. In this study, a two-dimensional eight-node quadrilateral inverse finite element formulation is presented. The element is suitable for thin structures under in-plane loading conditions. To validate the accuracy and demonstrate the capability of the inverse element, four different numerical cases are considered for different loading and boundary conditions. iFEM analysis results are compared with regular finite element analysis results as the reference solution and very good agreement is observed between the two solutions, demonstrating the capability of the iFEM approach. Full article
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<p>(<b>a</b>) Two-dimensional eight-node quadrilateral inverse element, (<b>b</b>) the master element in <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>,</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> space.</p>
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<p>Flowchart describing the iFEM analysis process.</p>
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<p>The loading of Case 1.</p>
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<p>Three different meshes of Case 1, (<b>a</b>) 16 elements, (<b>b</b>) 100 elements, and (<b>c</b>) 1600 elements.</p>
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<p>The reduced sensor locations of Case 1 with 1600 elements (iFEM-r).</p>
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<p>The plots of displacements of Case 1 with 16 elements: (<b>a</b>) x displacements of FEM, (<b>b</b>) x displacements of iFEM, (<b>c</b>) y displacements of FEM, (<b>d</b>) y displacements of iFEM.</p>
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<p>The plots of displacements of Case 1 with 100 elements: (<b>a</b>) x displacements of FEM, (<b>b</b>) x displacements of iFEM, (<b>c</b>) y displacements of FEM, (<b>d</b>) y displacements of iFEM.</p>
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<p>The plots of displacements of Case 1 with 1600 elements: (<b>a</b>) x displacements of FEM, (<b>b</b>) x displacements of iFEM, (<b>c</b>) x displacements of iFEM-r, (<b>d</b>) y displacements of FEM, (<b>e</b>) y displacements of iFEM, (<b>f</b>) y displacements of iFEM-r.</p>
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<p>The loading and displacement boundary conditions of Case 2.</p>
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<p>Two different meshes of Case 2: (<b>a</b>) 125 elements, (<b>b</b>) 2000 elements.</p>
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<p>The plots of displacements of Case 2 with 125 elements: (<b>a</b>) x displacements of FEM, (<b>b</b>) x displacements of iFEM, (<b>c</b>) y displacements of FEM, (<b>d</b>) y displacements of iFEM.</p>
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<p>The plots of displacements of Case 2 with 2000 elements: (<b>a</b>) x displacements of FEM, (<b>b</b>) x displacements of iFEM, (<b>c</b>) x displacements of iFEM-r, (<b>d</b>) y displacements of FEM, (<b>e</b>) y displacements of iFEM, (<b>f</b>) y displacements of iFEM-r.</p>
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<p>The sensor locations of Case 2 with 2000 elements.</p>
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<p>The loading and displacement boundary conditions of Case 3.</p>
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<p>The plots of displacements of Case 3 with 2000 elements: (<b>a</b>) x displacements of FEM, (<b>b</b>) x displacements of iFEM, (<b>c</b>) x displacements of iFEM-r, (<b>d</b>) y displacements of FEM, (<b>e</b>) y displacements of iFEM, (<b>f</b>) y displacements of iFEM-r.</p>
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<p>The loading and displacement boundary conditions of Case 4.</p>
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<p>The mesh for Case 4 (1293 elements).</p>
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<p>The reduced sensor locations for Case 4 with 304 elements (iFEM-r).</p>
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<p>The plots of displacements of Case 4: (<b>a</b>) x displacements of FEM, (<b>b</b>) x displacements of iFEM, (<b>c</b>) x displacements of iFEM-r, (<b>d</b>) y displacements of FEM, (<b>e</b>) y displacements of iFEM, (<b>f</b>) y displacements of iFEM-r.</p>
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<p>The plots of von Mises stress of Case 4: (<b>a</b>) FEM, (<b>b</b>) iFEM, (<b>c</b>) iFEM-r.</p>
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19 pages, 5688 KiB  
Article
Parametric Design and Shape Sensing of Geared Back Frame Shell Structure for Floating Cylindrical Reflector Antenna off the Coast
by Mengmei Mei, He Huang, Yugang Li and Zhe Zheng
Appl. Sci. 2023, 13(20), 11602; https://doi.org/10.3390/app132011602 - 23 Oct 2023
Cited by 1 | Viewed by 1094
Abstract
At present, numerous reflector antennas have been constructed worldwide on land. However, there are few applications of reflector antennas directly set off the coast. To expand the application region of reflector antennas, a floating cylindrical reflector antenna (FCRA) driven by the moving mass [...] Read more.
At present, numerous reflector antennas have been constructed worldwide on land. However, there are few applications of reflector antennas directly set off the coast. To expand the application region of reflector antennas, a floating cylindrical reflector antenna (FCRA) driven by the moving mass was developed to implement the elevation angle adjustment. Firstly, the structure design is introduced in detail. The design parameters are stated and analyzed to obtain the kinematic relationship while considering the water surface constraint. Then, the effects of each variable on the rotation capacity and structural stability are discussed. Further, the feasibility of the elevation angle adjustment process is demonstrated by using a prototype model test and software simulation. Finally, the deformation analyses and shape sensing of the back frame are carried out on the basis of the inverse finite element method (iFEM). We concluded that this new structure is feasible and expected to sit off the coast. In addition, the iFEM algorithm with sub-region reconstruction was proved to be suitable for the shape sensing of the over-constrained FCRA during the angle adjustment process via several quasi-static sampling moments. Full article
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<p>Structure design of FCRA. (<b>a</b>) Vertical view, (<b>b</b>) Axonometric view, (<b>c</b>) Axonometric view without semi-cylindrical shell.</p>
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<p>Parameter definitions (upright state of FCRA).</p>
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<p>Moving mass motion and structural rotation. (<b>a</b>) Stage 1, mass moving relative to the track, (<b>b</b>) Stage 2, structure rotation.</p>
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<p>Left view before and after heeling.</p>
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<p>Parameter analyses. (<b>a</b>) Curves of <span class="html-italic">a</span> − <span class="html-italic">β</span><sub>min</sub> and <span class="html-italic">a</span> − <span class="html-italic">S</span>, (<b>b</b>) Curves of <span class="html-italic">b</span> − <span class="html-italic">β</span><sub>min</sub> and <span class="html-italic">b</span> − <span class="html-italic">S</span>, (<b>c</b>) Curves of <span class="html-italic">c</span> − <span class="html-italic">β</span><sub>min</sub> and <span class="html-italic">c</span> − <span class="html-italic">S</span>, (<b>d</b>) Curves of <span class="html-italic">n</span> − <span class="html-italic">β</span><sub>min</sub> and <span class="html-italic">n</span> − <span class="html-italic">S</span>, (<b>e</b>) Curves of <span class="html-italic">a</span><sub>1</sub> − <span class="html-italic">β</span><sub>min</sub> and <span class="html-italic">a</span><sub>1</sub> − <span class="html-italic">S</span>, (<b>f</b>) Curves of <span class="html-italic">b</span><sub>1</sub> − <span class="html-italic">β</span><sub>min</sub> and <span class="html-italic">b</span><sub>1</sub> − <span class="html-italic">S</span>.</p>
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<p>Prototype model.</p>
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<p>Analysis model. (<b>a</b>) Model in axonometric view, (<b>b</b>) Model without shell.</p>
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<p>Rotation process of test. (<b>a</b>) <span class="html-italic">γ</span> = 0°, <span class="html-italic">β</span> = 90°, (<b>b</b>) <span class="html-italic">γ</span> = 12.5°, <span class="html-italic">β</span> = 80°, (<b>c</b>) <span class="html-italic">γ</span> = 20°, <span class="html-italic">β</span> = 75°, (<b>d</b>) <span class="html-italic">γ</span> = 32.5°, <span class="html-italic">β</span> = 65°.</p>
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<p>Rotation process of simulation. (<b>a</b>) <span class="html-italic">γ</span> = 0°, <span class="html-italic">β</span> = 90°, (<b>b</b>) γ = 12.5°, β = 80.13°, (<b>c</b>) <span class="html-italic">γ</span> = 20°, <span class="html-italic">β</span> = 74.21°, (<b>d</b>) <span class="html-italic">γ</span> = 32.5°, <span class="html-italic">β</span> = 64.18°.</p>
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<p>The four-node quadrilateral shell element (iQS4). (<b>a</b>) Element local coordinates, (<b>b</b>) Isoparametric coordinates.</p>
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<p>Discrete surface strain measurement.</p>
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<p>Finite element model of back frame.</p>
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<p>The FEM contours of total displacement in four rotation states. (<b>a</b>) <span class="html-italic">β</span> = 90°, <span class="html-italic">φ</span> = 0°, (<b>b</b>) <span class="html-italic">β</span> = 80°, <span class="html-italic">φ</span>= 10°, (<b>c</b>) <span class="html-italic">β</span> = 75°, <span class="html-italic">φ</span> = 15°, (<b>d</b>) <span class="html-italic">β</span> = 65°, <span class="html-italic">φ</span> = 25°.</p>
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<p>The iFEM contours of total displacement for fine mesh in four rotation states. (<b>a</b>) <span class="html-italic">β</span> = 90°, <span class="html-italic">φ</span> = 0°, (<b>b</b>) <span class="html-italic">β</span> = 80°, <span class="html-italic">φ</span> = 10°, (<b>c</b>) <span class="html-italic">β</span> = 75°, <span class="html-italic">φ</span> = 15°, (<b>d</b>) <span class="html-italic">β</span> = 65°, <span class="html-italic">φ</span> = 25°.</p>
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<p>The iFEM contour of total displacement for coarse mesh at <span class="html-italic">β</span> = 90°, <span class="html-italic">φ</span> = 0°.</p>
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19 pages, 2845 KiB  
Article
A Large-Scale Sensor Layout Optimization Algorithm for Improving the Accuracy of Inverse Finite Element Method
by Zhenyi Zhao, Kangyu Chen, Yimin Liu and Hong Bao
Sensors 2023, 23(19), 8176; https://doi.org/10.3390/s23198176 - 29 Sep 2023
Cited by 2 | Viewed by 1372
Abstract
The inverse finite element method (iFEM) based on fiber grating sensors has been demonstrated as a shape sensing method for health monitoring of large and complex engineering structures. However, the existing optimization algorithms cause the local optima and low computational efficiency for high-dimensional [...] Read more.
The inverse finite element method (iFEM) based on fiber grating sensors has been demonstrated as a shape sensing method for health monitoring of large and complex engineering structures. However, the existing optimization algorithms cause the local optima and low computational efficiency for high-dimensional strain sensor layout optimization problems of complex antenna truss models. This paper proposes the improved adaptive large-scale cooperative coevolution (IALSCC) algorithm to obtain the strain sensors deployment on iFEM, and the method includes the initialization strategy, adaptive region partitioning strategy, and gbest selection and particle updating strategies, enhancing the reconstruction accuracy of iFEM for antenna truss structure and algorithm efficiency. The strain sensors optimization deployment on the antenna truss model for different postures is achieved, and the numerical results show that the optimization algorithm IALSCC proposed in this paper can well handle the high-dimensional sensor layout optimization problem. Full article
(This article belongs to the Special Issue Feature Papers in Fault Diagnosis & Sensors 2023)
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<p>Schematic diagram of Timoshenko beam.</p>
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<p>Normalized coordinate establishment.</p>
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<p><math display="inline"><semantics> <mrow> <mi>G</mi> <mi>b</mi> <mi>e</mi> <mi>s</mi> <mi>t</mi> </mrow> </semantics></math> selection and particle update strategies (<math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>).</p>
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<p>Schematic diagram of the parameters of the CRD calculation method (<math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>).</p>
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<p>Algorithm Flowchart.</p>
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<p>Antenna Truss Model.</p>
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<p>Truss Meshing and Coordinate System.</p>
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<p>Forces and constraints of truss structures.</p>
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<p>Check point distribution.</p>
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<p>(<b>a</b>) PF of 10° working condition; (<b>b</b>) PF of 45° working condition; (<b>c</b>) PF of 80° working condition.</p>
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21 pages, 18309 KiB  
Article
Hybrid Shell-Beam Inverse Finite Element Method for the Shape Sensing of Stiffened Thin-Walled Structures: Formulation and Experimental Validation on a Composite Wing-Shaped Panel
by Marco Esposito, Rinto Roy, Cecilia Surace and Marco Gherlone
Sensors 2023, 23(13), 5962; https://doi.org/10.3390/s23135962 - 27 Jun 2023
Cited by 6 | Viewed by 2109
Abstract
This work presents a novel methodology for the accurate and efficient elastic deformation reconstruction of thin-walled and stiffened structures from discrete strains. It builds on the inverse finite element method (iFEM), a variationally-based shape-sensing approach that reconstructs structural displacements by matching a set [...] Read more.
This work presents a novel methodology for the accurate and efficient elastic deformation reconstruction of thin-walled and stiffened structures from discrete strains. It builds on the inverse finite element method (iFEM), a variationally-based shape-sensing approach that reconstructs structural displacements by matching a set of analytical and experimental strains in a least-squares sense. As iFEM employs the finite element framework to discretize the structural domain and as the displacements and strains are approximated using element shape functions, the kind of element used influences the accuracy and efficiency of the iFEM analysis. This problem is addressed in the present work through a novel discretization scheme that combines beam and shell inverse elements to develop an iFEM model of the structure. Such a hybrid discretization paradigm paves the way for more accurate shape-sensing of geometrically complex structures using fewer sensor measurements and lower computational effort than traditional approaches. The hybrid iFEM is experimentally demonstrated in this work for the shape sensing of bending and torsional deformations of a composite stiffened wing panel instrumented with strain rosettes and fiber-optic sensors. The experimental results are accurate, robust, and computationally efficient, demonstrating the potential of this hybrid scheme for developing an efficient digital twin for online structural monitoring and control. Full article
(This article belongs to the Special Issue Shape Sensing 2021-2024)
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<p>Definition of the variables used to describe Timoshenko beam kinematics; the location and component of strains measured by a sensor placed on the beam surface are also shown.</p>
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<p>Illustration of the plate structure: (<b>a</b>) kinematic variables used to describe the plate deformations, and (<b>b</b>) sensors mounted on the top and bottom plate surfaces.</p>
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<p>Geometry of the stiffened wing panel. All dimensions are expressed in mm.</p>
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<p>Reference directions for the panel’s lay-up orientations.</p>
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<p>Transverse section of the stringers. The figure shows the stacking sequence of the stringer’s web and how it is folded to obtain the two caps. Moreover, the strain sensor configuration for the web is illustrated. All dimensions are expressed in mm.</p>
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<p>Hybrid inverse model of the panel.</p>
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<p>Shell-only inverse model of the panel.</p>
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<p>Testing configuration of the panel with the boundary conditions indicated. Moreover, the transverse loads <math display="inline"><semantics><msub><mi>F</mi><mn>1</mn></msub></semantics></math> and <math display="inline"><semantics><msub><mi>F</mi><mn>2</mn></msub></semantics></math>, relative to the first and second loading conditions, are presented. The locations of the six transverse displacement transducers (<math display="inline"><semantics><msub><mi>w</mi><mrow><mn>1</mn><mo>−</mo><mn>6</mn></mrow></msub></semantics></math>) are also shown. All dimensions are expressed in mm.</p>
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<p>Torsional load case: contour plot of the transverse deformation (along <span class="html-italic">z</span>) for the first load case. All dimensions and displacements are expressed in mm.</p>
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<p>Primarily bending load case: contour plot of the transverse deformation (along <span class="html-italic">z</span>) for the second load case. All dimensions and displacements are expressed in mm.</p>
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<p>The optimal sensor configuration.</p>
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<p><b>Top</b> and <b>bottom</b> surfaces of the composite wing panel.</p>
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<p>Experimental test setup showing the location of the LVDTs, loads, and supports.</p>
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<p>The load application system for the first load case. The sphere that transmits the load can be moved along the bar to obtain the second load case.</p>
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25 pages, 6614 KiB  
Article
Coarse and Fine Two-Stage Calibration Method for Enhancing the Accuracy of Inverse Finite Element Method
by Jiewei Lu, Dahang He, Zhenyi Zhao and Hong Bao
Sensors 2023, 23(13), 5793; https://doi.org/10.3390/s23135793 - 21 Jun 2023
Viewed by 1426
Abstract
The inverse finite element method (iFEM) is a novel method for reconstructing the full-field displacement of structures by discrete measurement strain. In practical engineering applications, the accuracy of iFEM is reduced due to the positional offset of strain sensors during installation and errors [...] Read more.
The inverse finite element method (iFEM) is a novel method for reconstructing the full-field displacement of structures by discrete measurement strain. In practical engineering applications, the accuracy of iFEM is reduced due to the positional offset of strain sensors during installation and errors in structural installation. Therefore, a coarse and fine two-stage calibration (CFTSC) method is proposed to enhance the accuracy of the reconstruction of structures. Firstly, the coarse calibration is based on a single-objective particle swarm optimization algorithm (SOPSO) to optimize the displacement–strain transformation matrix related to the sensor position. Secondly, as selecting different training data can affect the training effect of self-constructed fuzzy networks (SCFN), this paper proposes to screen the appropriate training data based on residual analysis. Finally, the experiments of the wing-integrated antenna structure verify the efficiency of the method on the reconstruction accuracy of the structural body displacement field. Full article
(This article belongs to the Section Fault Diagnosis & Sensors)
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<p>Installation position of the strain sensor on the outer surface of the beam.</p>
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<p>Geometric and kinematic variables of the structure.</p>
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<p>The transfer process of particle swarm optimization.</p>
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<p>NURBS approximation diagram.</p>
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<p>Flow framework of CFTSC method.</p>
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<p>The whole experimental framework: (<b>a</b>) Experimental subject; (<b>b</b>) Position sensors and FBG; (<b>c</b>) Displacement-measuring instrument NDI; (<b>d</b>) Fiber Bragg grating demodulation system; (<b>e</b>) FBG sensor.</p>
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<p>Functional diagram of each component of the experiment.</p>
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<p>Model loading.</p>
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<p>Example of strain sensor optimization range.</p>
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<p>Comparison of displacement accuracy between coarse calibration and iFEM.</p>
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<p>Residual plot for each working condition.</p>
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<p>Comparison of CFTSC, Coarse calibration and iFEM displacements under test loads: (<b>a</b>) 80 N; (<b>b</b>) 120 N; (<b>c</b>) 200 N; (<b>d</b>) 240 N; (<b>e</b>)280 N; (<b>f</b>) 320 N; (<b>g</b>) 360 N.</p>
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<p>Comparison of CFTSC, Coarse calibration and iFEM displacements under test loads: (<b>a</b>) 80 N; (<b>b</b>) 120 N; (<b>c</b>) 200 N; (<b>d</b>) 240 N; (<b>e</b>)280 N; (<b>f</b>) 320 N; (<b>g</b>) 360 N.</p>
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<p>Comparison of RMSE under test data for CFTSC, Coarse calibration and iFEM.</p>
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<p>Comparison of MER under test data for CFTSC, Coarse calibration and iFEM.</p>
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17 pages, 3417 KiB  
Article
An Online Measurement and Calibration Method for a Radio Telescope Sub-Reflector Support Structure Using Fiber Bragg Grating
by Qian Xu and Hong Bao
Micromachines 2023, 14(5), 1093; https://doi.org/10.3390/mi14051093 - 22 May 2023
Cited by 1 | Viewed by 1525
Abstract
The position and altitude of a sub-reflector have an important influence on the pointing accuracy of a radio telescope. With the increase of the antenna aperture, the stiffness of the support structure for the sub-reflector decreases. This causes deformation of the support structure [...] Read more.
The position and altitude of a sub-reflector have an important influence on the pointing accuracy of a radio telescope. With the increase of the antenna aperture, the stiffness of the support structure for the sub-reflector decreases. This causes deformation of the support structure when environmental loads, such as gravity, temperature, and wind load, are applied to the sub-reflector, which will seriously influence antenna pointing accuracy. This paper proposes an online measurement and calibration method for assessing the deformation of the sub-reflector support structure based on the Fiber Bragg Grating (FBG) sensors. Firstly, a reconstruction model between the strain measurements and the deformation displacements of a sub-reflector support structure is established based on the inverse finite element method (iFEM). In addition, a temperature-compensating device with an FBG sensor is designed to eliminate the effects of temperature variations on strain measurements. Considering the lack of the trained original correction, a non-uniform rational B spline (NURBS) curve is built to extend the sample data set. Next, a self-structuring fuzzy network (SSFN) is designed for calibrating the reconstruction model, which can further improve the displacement reconstruction accuracy of the support structure. Finally, a full-day experiment was carried out using a sub-reflector support model to verify the effectiveness of the proposed method. Full article
(This article belongs to the Special Issue Advanced Antenna System: Structural Analysis, Design and Application)
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<p>Radio telescope and the sub-reflector.</p>
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<p>Deformation field of a Timoshenko Round Beam.</p>
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<p>Position coordinates of the beam surface strain sensor.</p>
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<p>Measurement devices: (<b>a</b>) Strain sensor and temperature compensation device; (<b>b</b>) Demodulation instrument.</p>
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<p>Schematic diagram of cubic B-spline curve.</p>
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<p>Flowchart of the self-structuring fuzzy network algorithm.</p>
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<p>Scaled-down model for the radio telescope test antenna.</p>
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<p>Deformation measurement system.</p>
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<p>Global and local coordinate systems.</p>
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<p>Test antenna model at two different elevation angles.</p>
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29 pages, 7953 KiB  
Article
Towards Automatic Crack Size Estimation with iFEM for Structural Health Monitoring
by Daniele Oboe, Dario Poloni, Claudio Sbarufatti and Marco Giglio
Sensors 2023, 23(7), 3406; https://doi.org/10.3390/s23073406 - 23 Mar 2023
Cited by 5 | Viewed by 2424
Abstract
The inverse finite element method (iFEM) is a model-based technique to compute the displacement (and then the strain) field of a structure from strain measurements and a geometrical discretization of the same. Different literature works exploit the error between the numerically reconstructed strains [...] Read more.
The inverse finite element method (iFEM) is a model-based technique to compute the displacement (and then the strain) field of a structure from strain measurements and a geometrical discretization of the same. Different literature works exploit the error between the numerically reconstructed strains and the experimental measurements to perform damage identification in a structural health monitoring framework. However, only damage detection and localization are performed, without attempting a proper damage size estimation. The latter could be based on machine learning techniques; however, an a priori definition of the damage conditions would be required. To overcome these limitations, the present work proposes a new approach in which the damage is systematically introduced in the iFEM model to minimize its discrepancy with respect to the physical structure. This is performed with a maximum likelihood estimation framework, where the most accurate damage scenario is selected among a series of different models. The proposed approach was experimentally verified on an aluminum plate subjected to fatigue crack propagation, which enables the creation of a digital twin of the structure itself. The strain field fed to the iFEM routine was experimentally measured with an optical backscatter reflectometry fiber and the methodology was validated with independent observations of lasers and the digital image correlation. Full article
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Graphical abstract
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<p>Discrete sensor location both on the top and on the bottom surface of the shell structure.</p>
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<p>iQS4 element with global (X,Y,Z) and local (x,y,z) reference systems and the related degrees of freedoms (dof). Numbers 1 to 4 refer to the element nodes.</p>
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<p>Crack propagation routine: (<b>a</b>) identification of the crack center (red node); (<b>b</b>) crack introduction; (<b>c</b>) crack propagation.</p>
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<p>Portion of the shell structure with input and test sensor networks definition.</p>
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<p>Crack size estimation framework.</p>
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<p>Strain pre-extrapolation framework.</p>
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<p>Stress field near a crack in an infinite plate subjected to a general biaxial loading condition.</p>
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<p>Specimen with the sensor network.</p>
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<p>Test rig with the main acquisition systems.</p>
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<p>iFEM model with boundary conditions and the crack center positions.</p>
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<p>Mesh discretization (3 mm and 1 mm meshes) with input and test sensor networks.</p>
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<p>Likelihood trend for experimental strain acquired at a <math display="inline"><semantics> <mrow> <mn>24.3</mn> <mo> </mo> <mi>mm</mi> </mrow> </semantics></math> crack length. Strain field computed by the iFEM for four different models; deformed shape with scale factor <math display="inline"><semantics> <mrow> <mn>100</mn> </mrow> </semantics></math>.</p>
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<p>Likelihood trend for experimental strain acquired at 32.5 mm crack length.</p>
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<p>Comparison between iFEM strain reconstruction and the DIC strain field with the same color scale. Acquisition performed at 20,000 cycles: real crack length (DIC) of <math display="inline"><semantics> <mrow> <mn>24.30</mn> <mo> </mo> <mi>mm</mi> </mrow> </semantics></math> and iFEM crack length of <math display="inline"><semantics> <mrow> <mn>24</mn> <mo> </mo> <mi>mm</mi> </mrow> </semantics></math>. Inverse FEM deformed shape with a scale factor 100.</p>
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<p>Crack size estimation with the 3 mm iFEM model for the whole acquisition history.</p>
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<p>Total number of iFEM iterations and matrix inversions for the whole fatigue propagation.</p>
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<p>Likelihood trend with the 1 mm model: (<b>a</b>) experimental strain acquired at <math display="inline"><semantics> <mrow> <mn>24.3</mn> <mo> </mo> <mi>mm</mi> </mrow> </semantics></math> (20,000 cycles); (<b>b</b>) experimental strain acquired at <math display="inline"><semantics> <mrow> <mn>32.5</mn> <mo> </mo> <mi>mm</mi> </mrow> </semantics></math> crack length (32,000 cycles).</p>
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<p>Crack size estimation routine results: (<b>a</b>) crack size estimated with respect to the target value of each acquisition; (<b>b</b>) the number of iFEM iterations and matrix inversions performed by the real-time algorithm.</p>
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