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Global sensitivity analysis for fiber reinforced composite fiber path based on D-MORPH-HDMR algorithm

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Abstract

This study presents a quantitative sensitivity analysis for the assessment of fiber reinforced composites (FRCs). Global sensitivity analysis (GSA) approach is based on the variance based method incorporating Random Sampling-High Dimensional Model Representation (RS-HDMR) expansion in which component functions are determined by diffeomorphic modulation under observable response preserving homotopy (D-MORPH) regression. The advantage of the D-MORPH regression lies in its capability to solve linear algebraic equations with a limited number of sample points. The main purpose is to investigate the influence of fiber path, regarded as the design variable, on the formability and structural performance of FRCs. Wherein, spring-back and load-carrying capacity are two meaningful problems to be addressed. Two typical FRCs are included that an L-shaped part with straight fiber path using autoclave manufacturing process and a variable stiffness composite cylindrical shell under pure bending. The work not only focuses on the ranking of design variables but also hopes to find out their interactions represented by the second order global sensitivity indexes. After being tested by three typical numerical functions, the GSA algorithm highlights that spring-back of FRC using autoclave manufacturing process is most sensitive to fiber orientation angles on plies close to the tool. And buckling performance of the VS cylinder is dominated by fiber orientation angles at compression/tension regions.

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Abbreviations

GSA:

Global sensitivity analysis

GSI:

Global sensitivity index

D-MORPH:

Diffeomorphic Modulation Under Observable Response Preserving Homotopy

RS-HDMR:

Random Sampling High Dimensional Model Representation

χ i :

ith input variable

χ :

Input variable vector,χ = (χ 1, χ 2,  ⋯ , χ d )

ρ(χ):

Output response

K d :

K d = {(χ 1, χ 2,  ⋯ , χ d )|0 ≤ χ i  ≤ 1, i = 1, 2,  ⋯ , d}

k , l , m , t :

Integers

N :

Total number of sample points

\( {S}_i/{S}_{i j},{S}_i^T \) :

First/second order GSI and total effect index

φ(χ):

Orthonormal polynomial basis

α , β :

Undetermined coefficients in HDMR expansion’s component function

Q :

Cost function

c :

Vector containing all undetermined coefficients

B :

Weight matrix of the cost function

γ :

Fiber orientation angle for VS composite cylinder

κ :

Angle

T :

Design variable of fiber orientation angle

ρ spring − back :

Spring-back for the L-shaped composite part

ρ cr :

Critical buckling load for the VS composite cylinder

References

  • Abdelal GF, Robotham A, Cantwell W (2013) Autoclave cure simulation of composite structures applying implicit and explicit FE techniques. Int J Mech Mater Des 9(1):55–63

    Article  Google Scholar 

  • Banichuk NV, Saurin VV, Barsuk AA (1995) Optimal orientation of orthotropic materials for plates designed against buckling. Structural optimization 10(3–4):191–196

    Article  Google Scholar 

  • Blom AW, Stickler PB, Gürdal Z (2010) Optimization of a composite cylinder under bending by tailoring stiffness properties in circumferential direction. Compos Part B 41(2):157–165

    Article  Google Scholar 

  • Chakraborty S, Chowdhury R (2014) Polynomial correlated function expansion for nonlinear stochastic dynamic analysis. J Eng Mech 141(3):04014132

    Article  Google Scholar 

  • Chakraborty S, Chowdhury R (2015) Multivariate function approximations using the D-MORPH algorithm. Appl Math Model 39(23):7155–7180

    Article  MathSciNet  Google Scholar 

  • Chen S, Brune WH, Oluwole OO et al (2012) Global sensitivity analysis of the regional atmospheric chemical mechanism: an application of random sampling-high dimensional model representation to urban oxidation chemistry. Environmental science & technology 46(20):11162–11170

    Article  Google Scholar 

  • Dey S, Mukhopadhyay T, Sahu SK et al (2015) Thermal uncertainty quantification in frequency responses of laminated composite plates. Compos Part B 80:186–197

    Article  Google Scholar 

  • Galanti S, Jung AR (1997) Low-discrepancy sequences: Monte Carlo simulation of option prices. J Deriv 5(1):63–83

    Article  Google Scholar 

  • Gürdal Z, Tatting BF, Wu CK (2008) Variable stiffness composite panels: effects of stiffness variation on the in-plane and buckling response. Compos A: Appl Sci Manuf 39(5):911–922

    Article  Google Scholar 

  • Haftka RT, Mroz Z (1986) First-and second-order sensitivity analysis of linear and nonlinear structures. AIAA J 24(7):1187–1192

    Article  MathSciNet  MATH  Google Scholar 

  • Homma T, Saltelli A (1996) Importance measures in global sensitivity analysis of nonlinear models. Reliability Engineering & System Safety 52(1):1–17

    Article  Google Scholar 

  • Hu Z, Mahadevan S (2016) Global sensitivity analysis-enhanced surrogate (GSAS) modeling for reliability analysis. Struct Multidiscip Optim 53(3):501–521

    Article  MathSciNet  Google Scholar 

  • Hyer MW, Lee HH (1991) The use of curvilinear fiber format to improve buckling resistance of composite plates with central circular holes. Compos Struct 18(3):239–261

    Article  Google Scholar 

  • Jin R, Chen W, Sudjianto A et al (2005) An efficient algorithm for constructing optimal design of computer experiments. Journal of Statistical Planning and Inference 134(1):268–287

    Article  MathSciNet  MATH  Google Scholar 

  • Jung WK, Chu WS, Ahn SH et al (2007) Measurement and compensation of spring-back of a hybrid composite beam. J Compos Mater 41(7):851–864

    Article  Google Scholar 

  • Khani A, Abdalla MM, Gürdal Z (2012) Circumferential stiffness tailoring of general cross section cylinders for maximum buckling load with strength constraints. Compos Struct 94(9):2851–2860

    Article  Google Scholar 

  • Li G, Rabitz H (2010) D-MORPH regression: application to modeling with unknown parameters more than observation data. J Math Chem 48(4):1010–1035

    Article  MathSciNet  MATH  Google Scholar 

  • Li G, Wang SW, Rabitz H (2002a) Practical approaches to construct RS-HDMR component functions. J Phys Chem A 106(37):8721–8733

    Article  Google Scholar 

  • Li G, Wang S, Rabitz H et al (2002b) Global uncertainty assessments by high dimensional model representations (HDMR). Chem Eng Sci 57(21):4445–4460

    Article  Google Scholar 

  • Li J, Yao XF, Liu YH et al (2008) Curing deformation analysis for the composite T-shaped integrated structures. Appl Compos Mater 15(4–6):207–225

    Article  Google Scholar 

  • Li G, Rabitz H, Yelvington PE et al (2010) Global sensitivity analysis for systems with independent and/or correlated inputs. J Phys Chem A 114(19):6022–6032

    Article  Google Scholar 

  • Li E, Wang H, Li G (2012a) High dimensional model representation (HDMR) coupled intelligent sampling strategy for nonlinear problems. Comput Phys Commun 183(9):1947–1955

    Article  MathSciNet  Google Scholar 

  • Li G, Rey-de-Castro R, Rabitz H (2012b) D-MORPH regression for modeling with fewer unknown parameters than observation data. J Math Chem 50(7):1747–1764

    Article  MathSciNet  MATH  Google Scholar 

  • Li G, Bastian C, Welsh W et al (2015a) Experimental Design of Formulations Utilizing High Dimensional Model Representation. J Phys Chem A 119(29):8237–8249

    Article  Google Scholar 

  • Li G, Rey-de-Castro R, Xing X et al (2015b) Sparse and nonnegative sparse D-MORPH regression. J Math Chem 53(8):1885–1914

    Article  MathSciNet  MATH  Google Scholar 

  • Nasir MNM, Aminanda Y, Mezeix L et al (2016) Spring-back simulation of flat symmetrical laminates with angled plies manufactured through autoclave processing//IOP conference series: materials science and engineering. IOP Publishing 152(1):012046

    Google Scholar 

  • Oakeshott JL (2003) Warpage of carbon–epoxy composite channels. Plastics, rubber and composites 32(3):104–113

    Article  Google Scholar 

  • Parlevliet PP, Bersee HEN, Beukers A (2006) Residual stresses in thermoplastic composites—a study of the literature—part I: formation of residual stresses. Compos A: Appl Sci Manuf 37(11):1847–1857

    Article  Google Scholar 

  • Rabitz H, Aliş ÖF (1999) General foundations of high-dimensional model representations. J Math Chem 25(2–3):197–233

    Article  MathSciNet  MATH  Google Scholar 

  • Rouhi M, Ghayoor H, Hoa SV et al (2014) Effect of structural parameters on design of variable-stiffness composite cylinders made by fiber steering. Compos Struct 118:472–481

    Article  Google Scholar 

  • Rouhi M, Ghayoor H, Hoa SV et al (2015) The effect of the percentage of steered plies on the bending-induced buckling performance of a variable stiffness composite cylinder. Sci Eng Compos Mater 22(2):149–156

    Article  Google Scholar 

  • Setoodeh S, Abdalla M, Gürdal Z (2006a) Design of variable–stiffness laminates using lamination parameters. Composites Part B-engineering 301-309

  • Setoodeh S, Abdalla M, Ijsselmuiden ST et al (2006b) Design of Variable-Stiffness Composite Panels for maximum buckling load. Compos Struct 87(1):109–117

    Article  Google Scholar 

  • Shan S, Wang GG (2011) Turning black-box functions into white functions. J Mech Des 133(3):031003

    Article  Google Scholar 

  • Sobol IM (2001) Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math Comput Simul 55(1):271–280

    Article  MathSciNet  MATH  Google Scholar 

  • Sudret B (2008) Global sensitivity analysis using polynomial chaos expansions. Reliability Engineering & System Safety 93(7):964–979

    Article  Google Scholar 

  • Sudret B, Mai CV (2015) Computing derivative-based global sensitivity measures using polynomial chaos expansions. Reliability Engineering & System Safety 241-250

  • Tang L, Wang H, Li G (2013) Advanced high strength steel springback optimization by projection-based heuristic global search algorithm. Mater Des 43:426–437

    Article  Google Scholar 

  • Tatting BF. (1998) Analysis and design of variable stiffness composite cylinders. Ph.D. Thesis. Virginia Polytechnic Institute and State University.

  • Twigg G, Poursartip A, Fernlund G (2003) An experimental method for quantifying tool–part shear interaction during composites processing. Compos Sci Technol 63(13):1985–2002

    Article  Google Scholar 

  • Vescovini R, Dozio L (2016) A variable-kinematic model for variable stiffness plates: vibration and buckling analysis. Compos Struct 15-26

  • Wang H, Tang L, Li GY (2011) Adaptive MLS-HDMR metamodeling techniques for high dimensional problems. Expert Syst Appl 38(11):14117–14126

    Google Scholar 

  • Wang P, Lu Z, Tang Z (2013) An application of the kriging method in global sensitivity analysis with parameter uncertainty. Appl Math Model 37(9):6543–6555

    Article  MathSciNet  Google Scholar 

  • Wu Z, Wang D (2016) Okolo P, et al. Global sensitivity analysis using a Gaussian radial basis function metamodel, Reliability Engineering & System Safety

    Google Scholar 

  • Wu Z, Weaver PM, Raju G et al (2012) Buckling analysis and optimization of variable angle tow composite plates. Thin-Walled Struct 60:163–172

    Article  Google Scholar 

  • Ye F, Wang H, Li G (2017) Variable stiffness composite material design by using support vector regression assisted efficient global optimization method. Struct Multidiscip Optim 1-17

  • Zuo W, Bai J, Yu J (2016) Sensitivity reanalysis of static displacement using Taylor series expansion and combined approximate method. Struct Multidiscip Optim 53(5):953–959

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work has been supported by Project of the Key Program of National Natural Science Foundation of China under the Grant Numbers 11572120, 11302266.

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Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Hunan, China

    Hu Wang, Liming Chen, Fan Ye & Lei Chen

  2. Joint Center for Intelligent New Energy Vehicle, Shanghai, China

    Hu Wang, Liming Chen, Fan Ye & Lei Chen

Authors
  1. Hu Wang
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  2. Liming Chen
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  3. Fan Ye
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  4. Lei Chen
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Corresponding author

Correspondence to Hu Wang.

Appendix

Appendix

Penrose conditions

The Penrose conditions for the matrix Hwith the generalized inverse H +:

$$ \begin{array}{l}(1){\mathbf{H}\mathbf{H}}^{+}\mathbf{H}=\mathbf{H}\kern4.599998em (2){\mathbf{H}}^{+}{\mathbf{H}\mathbf{H}}^{+}={\mathbf{H}}^{+}\\ {}(3)\kern0.2em {\left({\mathbf{H}\mathbf{H}}^{+}\right)}^T={\mathbf{H}\mathbf{H}}^{+}\kern2.3em (4)\kern0.1em {\left({\mathbf{H}}^{+}\mathbf{H}\right)}^T={\mathbf{H}}^{+}\mathbf{H}\end{array} $$
(A.1)

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Wang, H., Chen, L., Ye, F. et al. Global sensitivity analysis for fiber reinforced composite fiber path based on D-MORPH-HDMR algorithm. Struct Multidisc Optim 56, 697–712 (2017). https://doi.org/10.1007/s00158-017-1681-9

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  • DOI: https://doi.org/10.1007/s00158-017-1681-9

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