February 19, 2018

# Optimization of Laminated Composites with Adaptive Hierarchical Zoning

*This Tech Tip is based on "Towards an Automated Optimization of Laminated Composite Structures: Hierarchical Zoning Approach with Exact Blending Rules" publication.*

Laminated composites are widely used in the industry, including aerospace, automotive, shipbuilding and many others, due to their extraordinary stiffness properties. Among different types of composite materials, there is a distinguished family of laminated composites which resemble well-known ply-woods, but nowadays are composed of a large number of modern ply materials (for example, carbon fibers) and glued together with innovative and high-quality resins (matrix).

Generically, the stiffness of fibers in longitudinal and transversal directions, as well as the stiffness of matrix, is quite different so that the structural properties of laminated composite packet strongly depend upon used specific fiber orientation sequence. Therefore, the actual problem in laminated composites design is the optimal selection of fiber orientations which results in the minimal mass of the structure provided that given load constraints remain feasible. Note that to achieve an optimal solution, different compositions in distinct parts (zones) of the structure are to be admitted.

*Fig. 1 Different zones of composite structural elements*

The mentioned problem is in fact not fairly solvable: corresponding mathematical formulation belongs to the class of non-linear integer-valued optimization problem, which is known to be NP-hard. Therefore, some simplifications are to be accepted.

First, physical considerations allow concluding that only symmetric (with respect to mid-plane) laminates are worth considering. Moreover, it turns out that the design space is to be restricted to a particular class of balanced compositions, for which no significant unwanted coupling between in- and out- of plane stresses arise. Second, extensive investigations of microscopic fiber mechanics allowed formulating several composition rules, which ensure the stability of fibers at small scales. It is crucial that these rules do not refer to microscopic properties of each ply material. Instead, they are formulated entirely with respect to relative neighboring fibers orientations and therefore admit macroscopic treatment.

Generically, these crucial blending rules might be divided into two different groups: the ones, referring to particular patch (zone) of laminate with constant composition, and the rules which explicitly constraint allowed fiber orientations at the boundaries of neighboring zones. Therefore, the problem of optimal laminated composites design might be formulated entirely in terms of macroscopic degrees of freedom (fiber orientations), subject to a large variety of blending rules.

*Fig. 2 Fiber orientations in a laminated composite*

However, the optimization task yet remains hard to solve even approximately: design variables are still integer-valued, and their large number prohibits enumerative approaches, while structural properties are given in terms of non-linear response functions.

In this Tech Tip, we discuss a novel approach to optimize multi-zoned laminated composite structures of variable stiffness. It follows from above that the problem is notoriously difficult due to its large-scale non-linear combinatorial nature. However, we argue that there is a small natural parameter in the problem, which might be exploited to get reasonable approximation scheme. Indeed, in practically relevant applications the number of plies is sufficiently large so that the thickness of one ply might be considered small compared to the total thickness of the laminate.

Then the particular sequences of fiber orientations (stacking sequences) constitute the small-scale (microscopic) variables essentially, while mechanical properties of the laminate structure, which are related to total fiber stack, characterize large-scale (macroscopic) degrees of freedom. Exploitation of this idea leads to well-known bi-level optimization strategy, which attempts to separate these macro- and microscopic variables via the introduction of explicit integral characteristics of the laminate (lamination parameters). The problem is solved in terms of macro-parameters only, and representative stacking sequence is reconstructed once at the very end. Splitting of scales is known to be of invaluable importance in numerous fields, and bi-level approach was proven to be superior when applicable. Unfortunately, explicit separation of scales is rarely possible in industrial applications, where the majority of in-house developed modeling tools operate in terms of microscopic variables (stacking sequences) only. The bi-level strategy is not applicable in this case and requires particular generalization.

We solve the issue via the development of “fast and greedy” algorithms to reconstruct specific stacking sequence which best fits the given values of macroscopic parameters. Moreover, reconstructed fiber orientations identically satisfy a conventional set of blending and composition rules. Therefore, our algorithms essentially solve the problem posed at the beginning. Moreover, it is computationally inexpensive and thus could be embedded into the majority of existing applications. Although our approach ceases to be explicitly bi-level, it yet retains prime advantages of bi-level approach: optimization is performed in terms of macroscopic parameters only, representative stacking sequences are reconstructed “on the fly” essentially utilizing the fact that there is an exponentially large number of appropriate solutions.

*Fig. 3 Generic scheme of proposed approach to laminated composites optimization*

An important ingredient of our approach is the automated hierarchical zoning. Indeed, best performance could ever be achieved only if macroscopic parameters (lamina stiffnesses) are allowed to be spatially inhomogeneous. In many cases, the design of large composite structures is based on its subdivision into local panel (zone) design problems in which the material composition of each panel is determined under assumed fixed loads. Since different zones are structurally interrelated, the process is to be iterated: refined material composition in particular zone affects loads of other panels and hence requires to solve the panel-based problem again.

Even within such iterative settings, it remains unclear how to select the number and locations of different zones properly. The inadequate choice might result in a vast number of design variables, for which the problem might become intractable. We propose an adaptive multi-scale approach that has proven to be extremely useful in various contexts. It allows not only selecting the appropriate number of zones automatically but also significantly facilitates the solution process.

Namely, one starts with only a few zones (or, perhaps, a single one), for which a solution might be found easily although it is expected to be far from being optimal. Then the obtained zones are subdivided further, and the problem is solved again with smaller spatial resolution using the previous result as the initial guess. The process stops when either smallest allowed resolution is reached or sequentially achieved, a gain in the mass of the structure becomes negligible.

Despite its ideological simplicity, the hierarchical zoning outlined above is technically involved. For instance, in real-life applications, the number of zones to be considered simultaneously might be very large. The corresponding total number of macroscopic parameters might reach hundreds, making the optimization problem difficult to solve.

To circumvent this, we use locality of zones interaction in physically motivated problems. Indeed, the prime reason of rapidly rising dimensionality is a structural interconnection of different zones. If there would be no loads redistribution, the problem factorized into the set of independent optimization tasks for each zone.

On the other hand, a change in the properties of a particular zone mainly affects only its neighbors, for sufficiently distant panels’ loads remain almost the same. This suggests the possibility to develop a specific optimization approach which exploits the locality of zone interactions and remains efficient even when the number of zones is large.

*Fig. 4 Stacking sequence optimization of laminated composites in pSeven*

The above ideas and algorithms were implemented within the pSeven integration platform aimed to automate the solution of various engineering problems. In more details, pSeven provides an easy way to integrate various commercial and in-house developed tools into a united execution workflow, quantitatively describing multi-disciplinary properties of the studied design. A resulting model of the designed product could then be investigated with various mathematical methods ranging from Design of Experiments (DoE) studies to modern design optimization and data analysis algorithms.

It seems natural to implement the above laminated composites optimization approach as a separate module of pSeven. Indeed, pSeven allows avoiding the notorious task related to the proper interaction of different modeling tools and concentrating instead on essential algorithmic issues of our approach. As a consequence, the developed approach becomes universally applicable in virtually any context because it is not tied anymore with particular modeling software (vendor-neutral).

As far as the capabilities of pSeven are concerned, they are significantly boosted with an implemented module for laminated composites optimization. pSeven then becomes the tool of choice in mission-critical applications, where the quality and reliability of a design are of crucial importance.

Many engineering projects within aerospace, automotive and shipbuilding industries, where manufacturing requirements are critical and development time constraints are particularly acute, may benefit from this approach. It demonstrated its’ efficiency in improving the computational performance of the stacking sequence optimization problem while achieving the required mechanical characteristics and the best manufacturability of the composite element. It also guarantees the minimal mass of the structure, while satisfying structural integrity and manufacturing constraints.

Contact us, if you are interested in implementing our laminated composites optimization approach and pSeven design optimization capabilities in your company:

*By Dinara Shvarts, Application Engineer, DATADVANCE*