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

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## 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.

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