Buckling topology optimization of laminated multi-material composite shell structures

Abstract

The design problem of maximizing the buckling load factor of laminated multi-material composite shell structures is investigated using the so-called Discrete Material Optimization (DMO) approach. The design optimization method is based on ideas from multi-phase topology optimization where the material stiffness is computed as a weighted sum of candidate materials, thus making it possible to solve discrete optimization problems using gradient based techniques and mathematical programming. The potential of the DMO method to solve the combinatorial problem of proper choice of material and fiber orientation simultaneously is illustrated for multilayered plate examples and a simplified shell model of a spar cap of a wind turbine blade.

Introduction

The use of laminated composite structures with unidirectional Glass or Carbon Fiber Reinforced Polymers (GFRP/CFRP) is popular for lightweight constructions due to their superior strength and stiffness characteristics. In order to fully exploit the weight saving potential of these multilayered structures, it is necessary to tailor the laminate layup and behavior to the given structural needs which calls for the use of advanced structural optimization tools.

An example of a challenging lightweight structure made of multi-material laminates is a wind turbine blade. The size of wind turbines has increased dramatically in the last decade, and today a standard wind turbine blade has a length between 40 and 60 m. These high performance, multi-material shell structures may exhibit maximum tip displacements of up to about 25% of the length before they fail due to local buckling on the compressive side of the blade. Thus, in order to improve the structural performance of a wind turbine blade the design objective is to increase the buckling load factor, taking weight considerations into account.

The outer shape of a wind turbine blade is determined by aerodynamic considerations and therefore in general not subject to change. These structures consist of stiff fiber reinforced polymers such as GFRP and CFRP together with foam and different types of wood stacked in a number of layers and bonded together by a resin. The design problem is then to determine the best stacking sequence by proper choice of material and fiber orientation of each FRP layer in order to obtain the desired structural performance, taking weight constraints into account.

This paper deals with this buckling design problem for wind turbine blades specifically, but the methodology can be applied to any laminated multi-material composite shell structure. The parametrization used in this work is the so-called Discrete Material Optimization (DMO) approach [1], [2], [3], that was originally developed for minimum compliance problems.

Laminate design is a specific area of major importance, as shown in the book [4] that contains extensive references. A survey of optimal design of laminated plates and shells can be found in Abrate [5].

One of the major challenges in design optimization of laminated composite structures is the non-convexity of the design space, i.e. the risk of ending up with a local optimum solution is high. Several different approaches to design optimization of composite structures have been proposed, and they typically fall within the categories of analytical methods (optimality criterion methods), gradient based optimization, parametrization methods and evolutionary algorithms (and combinations of these).

The analytical methods rely on the closed form formulation of an optimality criterion, see Prager [6] and Masur [7], and based on the analytical optimality criterion heuristic optimization schemes can be developed. In minimum compliance problems with only a volume constraint, the necessary optimality criterion is uniform energy density, see Wasiutynski [8]. Pedersen [9], [10], [11] has derived optimality criteria in order to minimize the elastic strain energy of laminates w.r.t. material orientation and thickness subject to a volume constraint, see also earlier work by Banichuk [12] and the work by Sacchi-Landriani and Rovati [13]. However, for general shell problems the optimality criteria approaches have not yet been successfully applied.

Gradient based optimization using mathematical programming techniques has been investigated by numerous researchers, see for example [14], [15]. Some approaches like [15] are concerned with tuning the optimizer itself rather than reformulating the problem or changing the parametrization of the laminate design problem. These methods do not ensure convergence to the global optimum solution.

Another approach is parametrization methods where the parametrization is changed to obtain a convex design space. Such methods include the lamination parameter method by Tsai and Pagano [16] used by e.g., Miki and Sugiyama [17] and Hammer et al. [18] for orientational stiffness optimization of plates. Fukunaga and Vanderplaats [19] studied buckling optimization of orthotropic laminated cylindrical shells under combined loadings using a mathematical programming method. Foldager et al. [20] studied buckling optimization of plates and cylindrical shells taking thermal effects into account. The lamination parameter method requires closed-form analytical formulation of appropriate lamination parameters which has so far only been achieved for relatively simple geometries, i.e., not in case of general shell structures. In Foldager et al. [21] a general approach of forcing convexity of ply angle optimization of single material composite laminates based on lamination parameters is presented. However, lamination parameters do not allow for design with multiple materials and material properties that vary spatially over the structure and convexity is lost for problems with multiple design criteria.

Evolutionary algorithms such as genetic algorithms that rely on the application of Darwin’s principle of survival of the fittest, such that they mimic the natural selection process and evolution, see [4] and references therein, have been used by many researchers. In such evolutionary algorithms the performance of each design has to be evaluated, for example using finite element methods like in this work, and in case of large real life structures with many design variables such approaches thus become very computationally expensive.

The problem of combined design for orientation and topology has been studied using several different approaches. Varying the fiber volume fraction and changing the fibers orientation at each point of the structure result in a variable stiffness composite. Leissa and Martin [22] have solved the vibration and buckling problem of a rectangular composite ply composed of variably spaced straight and parallel fibers. In the work by Thomsen and Olhoff [23], [24] a two-step method is used to optimize the compliance of plates with in-plane loading (2D). Their modeling includes design parameters in each element; a fiber orientation, two volumetric fiber concentrations and finally a ratio between thicknesses. In each step of the optimization the fiber orientation is first determined using the optimality criteria derived in [9], [10] and then the values of the three other design variables are determined using mathematical programming. In [25] a heuristic optimization algorithm for minimum weight design of laminated composites is proposed based on layerwise removal of elements with low stress measure. The compliance minimization problem was solved by Duvaut et al. [26] for optimal fiber orientations and fiber volume fraction subjected to a fiber cost function constraint. Combined topology and fiber path design of composite laminae has been studied by Setoodeh et al. in [27] using the solid isotropic material penalization (SIMP) approach of topology design, see [28], in a cellular automata framework.

The idea of the DMO parametrization applied in this work for multi-material laminate design is to define a number of candidate materials, each with a given orientation in case of anisotropic materials, such that the problem initially is formulated as a discrete material selection problem. In case of laminates with FRP, it is convenient to limit the choice of fiber material orientation to a few discrete angles, e.g., 0°, $\pm 45°$ and 90°, and the layer thicknesses are unchanged. This also ensures low-cost manufacturing of the laminate. Interpolation functions are introduced by extending the SIMP approach known from topology optimization, such that the discrete material selection problem is converted into a continuous problem that can be solved using traditional gradient based optimization techniques, see [1], [2], [3] for details about the DMO approach for compliance problems. The result is a strong preprocessing tool that yields the topology of the laminated structure, and in this paper the objective is to maximize the buckling load factor.

The paper first presents the finite element based analysis tools used for the linearized buckling problem considered together with derivations of the design sensitivity analysis expressions and the mathematical programming formulation used for the minimax problem of maximizing the buckling load factor. The DMO parametrization method is described, and finally examples of the DMO method to solve the combinatorial problem of proper choice of material and fiber orientation simultaneously is illustrated for multilayered plate examples and a simplified shell model of a spar cap of a wind turbine blade.