# Constrained-manufacturable stacking sequence design optimization using an improved global shared-layer blending method and its 98-line Matlab code

- 308 Downloads

## Abstract

An improved global shared-layer blending method (GSLB) is suggested to address the constrained-manufacturable stacking sequence design optimization problem of tapered composite structures. First, the mathematical model for tapered composite structures design problem is constructed and the effect of blending constraint on the design space is analyzed. By introducing the set theory, the original GSLB method is improved by aggregating a shape prediction algorithm and a thickness evaluation procedure. The shape prediction algorithm takes advantage of the set computation procedure, which simplifies the process for detecting the shared layers’ boundaries. The maximum blending shared layers are evaluated by the improved GSLB in terms of the thickness distribution of multiple ply orientations. Subsequently, the obtained shared-layers are served as integrated variables for stacking sequence design, in which complex manufacturing constraints are involved. Three multi-panel structures and a wing box structure are adopted to verify the improved GSLB method and stacking sequence design strategy, and perfectly blended solutions are found without violation of manufacturing constraints and mechanical requirements. Finally, the 98 line Matlab code of the improved GSLB method is provided for the convenience of engineering application. This research has two purposes: providing a technique for tailoring design of tapered composite structures and giving reference solutions for constrained-manufacturable stacking sequence design optimization problem.

## Keywords

Global shared-layer blending method Tapered composite structures Stacking sequence Shared layers Manufacturing constraints## Notation

*A*A set to denote the appeared panels

*b*number of subregions corresponding to a specific ply drop matrix

**G***C*A set to denote the checked panels

- CTi
The

*i*th manufacturing constraint*d*Number of contiguity ply drops

*D*A region set with panels whose ply number is bigger than 1

*DS*_{T}Total design space

*DS*_{G}Design space within the domain of all panels

*DS*_{TB}Design space with maximum blending constraint

*D*_{ij}Bending stiffness coefficients of the optimal stacking sequence (

*i*,*j*= 1, 2, 6)- \( {D}_{ij}^{\ast } \)
Bending stiffness coefficients of the initial superply bundles (

*i*,*j*= 1, 2, 6)**f**_{j}Vector to denote a shared layer dropped at panel

*j***F**_{P × P}Matrix to denote a shared layer dropped at some panels

**g**_{j}Vector to denote a shared layer drops or covers panel

*j***G**_{P × P}Ply drop matrix

**G**^{k}Ply drop matrix of the

*k*th shared layer*h*Ply thickness

*k*Stacking position

*k =*1,2,...,*Q**m*Total weight of the structure

*M*Total number of candidate ply orientations

*n*_{i}Total ply number of panel

*i*- \( {n}_j^{\theta_r} \)
Ply number of ply orientation

*θ*_{r}in panel*j*(*r =*1,2,...,*M*and*j =*1,2,...,*P*)*P*Total number of panels

*Q*Total number of shared layers

*Ratio*Ratio of perfectly blending design space to total design space

*R*_{j}*j*th subregion**R**Region matrix

*S*_{j}The set to record the thickness distribution of

*j*th subregion*t*Number of subregions with corresponding ply drop matrix

**G**_{t}**T**_{r}Thickness distribution vector of ply orientation

*θ*_{r}**T**_{M × P}Thickness distribution matrix

*U*number of stacking positions in a panel

*v*_{ij}Element 0 or 1 to denote the adjacent relationship of panel

*i*and*j***V**^{l}Adjacent matrix of a shared layer

**V**^{s}Structural adjacent matrix

*W*A set to denote unchecked panels

*Y*A set to denote the connect panels

**z**_{j}Vector to denote panel

*j*is covered by a shared layer**Z**_{P × P}Matrix to denote all panels are covered by a shared layer

*θ*A ply orientation

*θ*_{r}The

*r*th candidate ply orientation- \( {\theta}_{r_k} \)
The ply orientation of the

*k*th shared layer (*k =*1,2,...,*Q*)**θ**_{M × P}Ply orientation matrix

**ζ**_{i}Stacking sequence of panel

*i**ω*_{rj}The elements of same shape shared-layer-matrix \( \overline{\omega} \)

**ω**_{r}Vector to denote the a same shape shared-layer with ply orientation

*θ*_{r}- \( {\omega}_{r_k}^k \)
The

*k*th (*k =*1,2,...,*Q*) shared layer with ply orientation \( {\theta}_{r_k} \)- \( \overline{\omega} \)
_{M × P} Same shape shared-layer-matrix

**Ω**_{, j}Column vector to denote stacking sequence of panel

*j*with ply drops**Ω**_{i,}Row vector to the

*i*th shared layer**Ω**Stacking sequence matrix

*λ*Critical buckling load factor

*i*,*j*,*r*,*t*Subscripts

## Notes

### Acknowledgments

This work is supported by the National Natural Science Foundation of China (Nos. 11572134, 51375386) and the Project funded by China Postdoctoral Science Foundation (No. 2017M612443). Thanks to the anonymous reviewers for their efforts and constructive advice to improve the study.

## References

- Bruyneel M (2006) A general and effective approach for the optimal design of fiber reinforced composite structures. Compos Sci Technol 66(10):1303–1314CrossRefGoogle Scholar
- Bóna M (2012) Combinatorics of permutations.Chapman & Hall, CRC press, Boca Raton London New York Washington, D.C.Google Scholar
- Lund E (2009) Buckling topology optimization of laminated multi-material composite shell structures. Compos Struct 91(2):158–167CrossRefGoogle Scholar
- Lindgaard E, Lund E (2010) Nonlinear buckling optimization of composite structures. Comput Methods Appl Mech Eng 199(37–40):2319–2330MathSciNetCrossRefzbMATHGoogle Scholar
- Gan KW, Allegri G, Hallett SR (2016) A simplified layered beam approach for predicting ply drop delamination in thick composite laminates. Mater Des 108:570–580CrossRefGoogle Scholar
- Graeme JK, Martins JRRA (2013) A laminate parametrization technique for discrete ply-angle problems with manufacturing constraints. Struct Multidiscip Optim 48(2):379–393MathSciNetCrossRefGoogle Scholar
- Haftka RT, Waish JL (1992) Stacking-sequence optimization for buckling of laminated plates by integer programming. AIAA J 30(3):814–818CrossRefGoogle Scholar
- Irisarri F-X, Lasseigne A, Leroy FH, Riche RL (2014) Optimal design of laminated composite structures with ply drops using stacking sequence tables. Compos Struct 107(1):559–569CrossRefGoogle Scholar
- Meddaikar YM, Irisarri F-X, Abdalla MM (2016) Laminate optimization of blended composite structures using a modified Shepard’s method and stacking sequence tables. Struct Multidiscip Optim 55(2):535–546CrossRefGoogle Scholar
- Ijsselmuiden ST, Abdalla MM, Seresta O, Gürdal Z (2009) Multi-step blended stacking sequence design of panel assemblies with buckling constraints. Compos B: Eng 40:329–336CrossRefGoogle Scholar
- Jech T (1978) Set theory. Academic, New YorkzbMATHGoogle Scholar
- Jing Z, Fan XL, Sun Q (2015a) Stacking sequence optimization of composite laminates for maximum buckling load using permutation search algorithm. Compos Struct 121(121):225–236CrossRefGoogle Scholar
- Jing Z, Sun Q, Silberschmidt VV (2016a) Sequential permutation table method for optimization of stacking sequence in composite laminates. Compos Struct 2016(141):240–252CrossRefGoogle Scholar
- Jing Z, Fan XL, Sun Q (2015b) Global shared layer blending method for stacking sequence optimization design and blending of composite structures. Compos B: Eng 69:181–190CrossRefGoogle Scholar
- Jing Z, Sun Q, Silberschmidt VV (2016b) A framework for design and optimization of tapered composite structures part I: from individual panel to global blending structure. Compos Struct 154:106–128.26CrossRefGoogle Scholar
- Jing Z, Sun Q, Chen JQ, Silberschmidt VV (2018) A framework for design and optimization of tapered composite structures part II: enhanced design framework with a global blending model. Compos Struct 188:531–552CrossRefGoogle Scholar
- Le Riche R, Haftka RT (1993) Optimization of laminate stacking sequence for buckling load maximization by genetic algorithm. AIAA J 31(5):951–956CrossRefzbMATHGoogle Scholar
- Liu BY, Haftka RT (2001) Composite wing structural design optimization with continuity constraints. 42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA Paper 2001–1205Google Scholar
- Liu BY, Haftka RT, Akgun MA, Todoroki A (2000a) Permutation genetic algorithm for stacking sequence design of composite laminates. Comput Methods Appl Mech Eng 186:357–372MathSciNetCrossRefzbMATHGoogle Scholar
- Liu BY, Haftka RT, Akgün MA (2000b) Two-level composite wing structural optimization using response surfaces. Struct Multidiscip Optim 20(2):87–96CrossRefGoogle Scholar
- Liu BY (2001) Two-level optimization of composite wing structures based on panel genetic optimization. Phd Thesis. University of FloridaGoogle Scholar
- Liu DZ, Toropov VV, Querin OM, Barton DC (2009) Bi-level optimization of blended composite panels. 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA Paper, 2009–2182Google Scholar
- Liu DZ, Toropov VV, Querin OM, Barton DC (2011) Bi-level optimization of blended composite wing panels. J Aircr 48(1):107–118CrossRefGoogle Scholar
- Liu DZ, Toropov VV (2013) A lamination parameter-based strategy for solving an integer-continuous problem arising in composite optimization. Comput Struct 128(8):170–174CrossRefGoogle Scholar
- Liu WL, Richard B (2007) Optimum buckling design of composite wing cover panels with manufacturing constraints. 48th AIAA SDM Conference, AIAA Paper 2007–2215Google Scholar
- Liu WL, Richard B, Kim HA (2008) Optimization of composite stiffened panels subject to compression and lateral pressure using a bi-level approach. Struct Multidiscip Optim 36(3):235–245CrossRefGoogle Scholar
- Merris R (2003) Combinatorics.2nd ed. John Wiley & Sons, Inc., Hoboken, New Jersey.Google Scholar
- Nagendra S, Jestin D, Gurdal Z, Haftka RT, Watson LT (1996) Improved genetic algorithm for the design of stiffened composite panels. Comput Struct 58(3):543–555CrossRefzbMATHGoogle Scholar
- Park CH, Lee WI, Han WS, Vautrin A (2008) Improved genetic algorithm for multidisciplinary optimization of composite laminates. Comput Struct 86(19):1894–1903CrossRefGoogle Scholar
- Seresta O, Gürdal Z, Adams DB, Watson LT (2007) Optimal design of composite wing structures with blended laminates. Compos B: Eng 38:469–480CrossRefGoogle Scholar
- Shen S, Vereshchagin NK, Shen A (2002) Basic set theory. American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA.Google Scholar
- Vergara JPC (1997) Sorting by bounded permutations. Ph.D. Dissertation, Computer Science and Applications Dept, Virginia Tech Univ, Blacksburg VirginiaGoogle Scholar
- Watkins RI, Morris AJ (1987) A multicriteria objective function optimization scheme for laminated composites for use in multilevel structural optimization schemes. Comput Methods Appl Mech Eng 60(2):233–251CrossRefzbMATHGoogle Scholar
- Xia L, Breitkopf P (2015) Design of materials using topology optimization and energy-based homogenization approach in Matlab. Struct Multidiscip Optim 52(6):1229–1241MathSciNetCrossRefGoogle Scholar
- Xia L, Fritzen F, Breitkopf P (2016) Evolutionary topology optimization of elastoplastic structures. Struct Multidiscip Optim 55(2):1–13MathSciNetGoogle Scholar
- Zein S, Basso P, Grihon S (2014) A constraint satisfaction programming approach for computing manufacturable stacking sequences. Comput Struct 136(7):56–63CrossRefGoogle Scholar
- Zein S, Colson B, Grihon S (2012) A primal-dual backtracking optimization method for blended composite structures. Struct Multidiscip Optim 45(5):669–680MathSciNetCrossRefzbMATHGoogle Scholar
- Zein S, Bruyneel M (2015) A bilevel integer programming method for blended composite structures. Adv Eng Softw 79(C):1–12CrossRefGoogle Scholar
- Zein S, Madhavan V, Dumas D, Ravier L, Yague I (2016) From stacking sequences to ply layouts: an algorithm to design manufacturable composite structures. Compos Struct 141:32–38CrossRefGoogle Scholar
- Zhou M, Fleury R, Kemp M (2010) Optimization of composite: recent advances and application. 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, AIAA Paper, 2010–9212Google Scholar