Optimization of variable stiffness laminates with gap-overlap and curvature constraints

doi:10.1016/j.compstruct.2019.111494

Composite Structures

Volume 230, 15 December 2019, 111494

https://doi.org/10.1016/j.compstruct.2019.111494 Get rights and content

Abstract

In the present study, the gap-overlap and curvature constraints on fiber tows are considered in the design optimization of variable stiffness laminates. The optimization problem is formulated in a framework proposed in our previous studies in which the fiber angle arrangement of a laminate is described by a continuous function constructed through the Shepard interpolation. In order to deal with the gap-overlap constraint, a gap-overlap-free rectangle is defined for each finite element. The fiber angles of the elements within this rectangle are constrained to be equal to each other, thus ensuring the fiber tows that pass through this rectangle are parallel. In order to control the curvature, a curvature-constrained rectangle is defined for each finite element. Within this rectangle the differences between fiber angles of the elements are constrained to be smaller than a user-specified upper bound. The compliance minimization with manufacturability constraints is considered, and it is solved with the MMA optimization algorithm. The results of numerical examples prove that the proposed method is effective.

Introduction

With the advent of advanced manufacturing technologies, for instance the automated tape laying (ATL) and the automated fiber placement (AFP), composite laminates can be manufactured with curvilinear fiber, i.e., the fiber angle is allowed to change continuously [1], [2]. Such curvilinear fiber leads to variable stiffness at different locations of a laminate, yielding the so-called variable stiffness laminate (VSL). As compared to conventional constant stiffness laminate, the VSL offers larger design freedom, hence having more potential for improving the performance [3], [4]. Therefore, it has been recognized as an important direction for future development.

In the manufacturing of VSL with curvilinear fiber, there are several constraints that the design of a VSL should satisfy [2], [5]. First, it is required that neighboring tows should be parallel to each other [2], [6], [7], [8], [9]. In the design of a laminate, if the placement paths for neighbouring tows are not parallel, it will be difficult to avoid gaps and overlaps between neighboring tows after they are laid to the laminate, since the ATL and AFP have very limited capability to tailor the fiber tows. Gap or overlap will generate defects in the laminate, thus adversely affecting the performance of VSL. Therefore, there should be gap-overlap constraints in the design optimization of VSL. The second constraint is about the curvature of tows [7], [8]. When the curvature is larger than the maximum allowable curvature permitted by the ATL or AFP [2], [13], the tensile stress at the outer edge of tow will be high, thus delamination may happen as the tow is laid. At the same time, large curvature also results in high compressive stress at the inner edge of tow, thus wrinkling may happen. Therefore, the curvature of tows should be constrained. In a word, with the consideration of these constraints, the optimal design of a VSL will have better manufacturability. Furthermore, it can ensure that the manufactured laminate has the expected mechanical performance.

In recent years, the integration of manufacturability constraints into the design optimization of VSL has caught much attention. Nik et al. [10] investigated the effects of design variables and manufacturing parameters on the optimized laminates. Peeters et al. [11] proposed a steering constraint to control the spatial variation of fiber angles. Montemurro et al. [12] proposed to constrain the minimum curvature radius of the tow by the B-spline surfaces that represent the variation of polar parameters. Lozano et al. [5] developed a method to control the gap-overlap, maximum curvature, minimum cut length and tow width. Brook et al. [13] proposed to present the tow-steered pattern through a vector field, and by constraining the curl and divergence of the vector field the gap-overlap and curvature constraints can be satisfied.

In the present study, the constraints of gap-overlap and curvature are also considered in the design optimization of VSL. The optimization problem is formulated in a framework proposed in our previous studies [14], [15], [16], in which the fiber angle arrangement of a laminate is described by a continuous function constructed through the Shepard interpolation. The fiber angles at scattered interpolation points are taken as the design variables. The Shepard interpolation inherently guarantees spatial continuity of fiber angle. In order to deal with the gap-overlap constraint, a gap-overlap-free rectangle whose long side is perpendicular to the fiber orientation is defined for each finite element. The fiber angles of the elements within this rectangle are constrained to be equal to each other, thus ensuring the fiber tows passing through this rectangle are parallel. In order to control the curvature, another rectangle whose long side is along the fiber orientation is defined for each finite element. Within this rectangle the differences between fiber angles of the elements are constrained to be smaller than a user-specified upper bound.

The paper is organized as follows. In Section 2 the representation of fiber angle arrangement through Shepard interpolation is introduced. In Section 3 the manufacturability constraints are described. In Section 4 the design optimization problem and the derivative analysis are described. Section 5 gives numerical examples and discussions. Section 6 concludes this paper.

Section snippets

Representation of fiber angle through Shepard interpolation

Kang and Wang [17], [18] made a remarkable progress by introducing the Shepard interpolation into the structural topology optimization. Inspired by this research work, several refinements were developed [19], [20], [21]. The Shepard interpolation enables topology optimization to be performed on density points that are independent of finite element mesh. Therefore, such a method is called iPDI (independent Point-wise Density Interpolation) method [20]. An additional important benefit of such

Gap-overlap and curvature constraints

Before we define the gap-overlap and curvature constraints, let us first take a look at the fiber arrangement that satisfies these constraints. As shown in Fig. 3, there are three fiber paths in the $x - y$ plane. First, these fiber paths are parallel to each other, thus there will be no gap or overlap between neighboring tows. Second, the curvatures of these fiber paths are not too large, thus the tows will not wrinkle and delaminate.

As shown in Fig. 3, suppose that on the middle path there is a

Problem formulation

In the present study, the minimum compliance problem is considered. The optimization problem with manufacturability constraints is defined as $\begin{matrix} find & θ_{i} (i = 1, 2, \dots, n) \\ \min & c = f^{T} u \\ s . t . & Ku = f \\ \sum_{a \in R_{e}} {(θ_{a} - θ_{e})}^{2} ⩽ ε, e = 1, 2, \dots, N_{e} \\ {(θ_{a} - θ_{e})}^{2} ⩽ ζ^{2}, a \in T_{e}, e = 1, 2, \dots, N_{e} \\ θ_{\min} ⩽ θ_{i} ⩽ θ_{\max} \end{matrix}$ where design variables $θ_{i}$ are the angles at design points; n is the number of design points; c is the compliance; $f$ is the force vector; $u$ is the displacement vector; $K$ is the global stiffness matrix. The gap-overlap constraint and the curvature constraint are

Numerical examples

This section gives several 2D examples with in-plane load to verify the effectiveness of the proposed method. For all the numerical examples, the mechanical parameters of fiber reinforced material are assumed that $E_{x} = 1, E_{y} = 0.05, G_{xy} = 0.03, ν_{xy} = 0.3$ , and plane-stress state is assumed. Self-weight of structure is not considered. When the structure does not include interior holes or cracks, 4-node square elements are used. When the structure has non-convex shape, the structure is meshed by 4-node

Conclusions

In the manufacturing of VSL with curvilinear fiber, there are several constraints that the design of a VSL should satisfy. In the present study, the constraints of gap-overlap and curvature of fiber tows are considered in the design optimization of VSL. The optimization problem is formulated in a framework in which the arrangement of fiber angle of a laminate is described by a continuous function constructed through the Shepard interpolation. The gap-overlap and curvature constraints are

Declaration of Competing Interest

None.

Acknowledgements

This research work is supported by the National Natural Science Foundation of China (Grant No. 51975227, 51575203) and the Natural Science Foundation for Distinguished Young Scholars of Hubei Province (Grant No. 2017CFA044). Also, the authors gratefully acknowledge the support from Professor Krister Svanberg who provided the MMA code and allowed us to use it in the present study. The insightful comments of the reviewers are cordially appreciated.

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Optimization of variable stiffness laminates with gap-overlap and curvature constraints

Abstract

Introduction

Section snippets

Representation of fiber angle through Shepard interpolation

Gap-overlap and curvature constraints

Problem formulation

Numerical examples

Conclusions

Declaration of Competing Interest

Acknowledgements

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Comput Methods Appl Mech Eng

The engineering aspects of automated prepreg layup: history, present and future

Compos Partt B

A review on design for manufacture of variable stiffness composite laminates

Proc Inst Mech Eng Part B