Der Artikel vertieft die kritischen Aspekte moderner Offshore-Windenergieanlagen und konzentriert sich dabei auf das Gurit98m-Modell, ein Open-Source-Strukturmodell, das für Optimierungsanwendungen entwickelt wurde. Er beginnt mit der Diskussion über die Bedeutung von Windkraftanlagen auf horizontaler Achse für die erneuerbaren Energien und beleuchtet ihre Stärken und Herausforderungen, vor denen sie stehen, insbesondere im Bereich der materiellen Nachhaltigkeit. Das Gurit98m-Modell wird als repräsentatives und zeitgemäßes Werkzeug für Forscher eingeführt, das einen Ausgangspunkt für Optimierungsstudien bietet, die auf den Übergang zu nachhaltigeren Materialien abzielen. Der Artikel untersucht verschiedene Optimierungsmethoden - von frühen Mixed-Integer-Formulierungen bis hin zu neueren gradientenbasierten Ansätzen - und betont die Recheneffizienz, die für eine großflächige Blade-Optimierung erforderlich ist. Außerdem werden die Grenzen von Zero-Order-Methoden und Ansätzen der künstlichen Intelligenz bei der Strukturoptimierung diskutiert, was die Voraussetzungen für die detaillierte Strukturoptimierung der Gurit98m-Klinge schafft. Die Entwicklung des Modells, einschließlich Materiallayout, Lasthülle und Finite-Elemente-Netz, ist ausführlich beschrieben und bildet eine solide Grundlage für den Optimierungsprozess. Die Optimierung selbst ist eine gradientenbasierte Dickenoptimierung, die darauf abzielt, die Materialkosten zu minimieren und gleichzeitig mehrere Lastfälle und Beschränkungen einzuhalten. Die Ergebnisse zeigen eine signifikante Kosten- und Massensenkung und zeigen das Potenzial des Gurit98-Modells zur Verbesserung der Leistung und Nachhaltigkeit von Windkraftanlagen. Der Artikel schließt mit einer kritischen Diskussion der Optimierungsstrategie, die Wege für zukünftige Forschung und Entwicklung aufzeigt. Insgesamt bietet der Artikel einen tiefen Einblick in die Komplexität der Rotorblattoptimierung von Windkraftanlagen und liefert wertvolle Erkenntnisse und Werkzeuge, um das Feld voranzubringen.
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Abstract
One of the primary challenges in wind turbine blade optimization research is creating a model that is representative of current state-of-the-art blade structures. This task is complex and time-consuming, given the multiple disciplines involved in blade design and the large-scale of such blades, which now exceed 100 meters in length. Moreover, the procedures used to establish the models are typically not scientific, implying a significant risk associated with blade research, as substantial resources spent on developing models are wasted if the actual research ideas are ineffective in practice. To reduce the risk and accelerate research efforts in the scientific community, this work introduces an open-source large offshore wind turbine blade model and demonstrates application in structural optimization research. A detailed thickness optimization of the blade’s constituent material layers is performed, with the objective of minimizing cost while accounting for buckling, tip displacement, and static failure constraints, which are many of the key design criteria according to design certification guidelines. A semi-analytical adjoint design sensitivity analysis approach is used to efficiently compute problem sensitivities, allowing inclusion of the constraints for up to twelve extreme load cases each. Application of the presented optimization strategy reduces cost by 17% and mass by 25%, while maintaining all constraints within allowable limits. The change from initial to optimized laminate thickness distribution is shown, and the optimized function response is demonstrated on the blade, showing the material is at its load-carrying limit throughout the entire blade, highlighting the efficiency of the achieved design.
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1 Introduction
The horizontal-axis wind turbine is one of the foremost renewable energy technologies (GWEC 2023). It performs well on the levelized cost of electricity metric (IRENA 2022; Lazard 2023) and increases energy independence for countries with limited fossil fuel resources. However, the technology has several weaknesses, and significant research and development efforts have to be undertaken to reduce their impact or even resolve them entirely. One key challenge is the use of materials that are not sustainable, and a promising tool to accelerate the transition to sustainable materials is structural optimization. This paper supports this objective by introducing the Gurit98m blade model to provide researchers with a representative and contemporary wind turbine blade model that can be used as a starting point for optimization studies.
Work on optimization of wind turbine blades has accumulated over the past two decades, matching the increase in demand for wind energy. The pioneering study on structural optimization of full blade models was conducted in Jureczko et al. (2005), which used a mixed-integer formulation and a genetic algorithm to solve the problems. More recent publications have also applied zero-order methods to solve different blade optimization problems, see Chen et al. (2013); Monte et al. (2013); Nicholas et al. (2015); Barnes and Morozov (2016); Albanesi et al. (2018a, b); Zhu et al. (2022); Jureczko and Mrówka (2022). However, the usefulness of zero-order methods in structural optimization remains limited due to their computational inefficiency, which severely restricts the number of design variables that can be treated in a problem (Sigmund 2011). Similar limitations are present with artificial intelligence approaches for structural optimization (Woldseth et al. 2022), which has also recently been demonstrated for wind turbine blade design (Jordi et al. 2024). For state-of-the-art wind turbine blades, exceeding 100 m in length, computational efficiency is crucial, as the combination of size and complex double-curved shape introduces many local effects that can lead to failure (Broberg 2023), and these effects have to be addressed in the optimization to some extent to provide useful results.
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Gradient-based methods offer the efficiency needed for large-scale wind turbine blade optimization, but are not straightforward to apply. The problem has to be reformulated such that all design variables and functions are continuous for sensitivities to be computed. In addition, the efficiency depends on how sensitivities are computed, which must be carefully considered for each particular problem (Tortorelli and Michaleris 1994; Christensen and Klarbring 2009). Nevertheless, several proposals for gradient-based methods and strategies have been suggested for computationally efficient wind turbine blade optimization.
Buckney et al. (2012) and Forcier and Joncas (2012) take a multi-step approach, first using topology optimization to design the internal layout of the blade, followed by thickness optimization of the composite laminates in the outer shell and shear webs. In these studies, the material is assumed isotropic during topology optimization and criteria only relevant for such materials are included, e.g., von Mises stress, but these do not capture the anisotropic behavior of the composite laminates. Moreover, the usefulness of topology optimization is limited in wind turbine blade optimization, as the outer shell is a closed-cell structure with shear webs being straight and thin-walled, allowing no holes due to manufacturing limitations. Instead, a thickness parametrization of the composite laminates is typically adopted, and several studies have demonstrated its potential for wind turbine blade optimization.
In the series of works by Bottasso et al. (2012, 2014); Bortolotti et al. (2016); Bottasso et al. (2016) design frameworks are proposed to include the different aspects and disciplines involved in wind turbine blade design, and to perform optimization at different levels of model detail and accuracy. Aerodynamic, aero-elastic, structural, and control considerations are all addressed for the separate phases in typical blade design development. However, despite using a gradient-based solver, the considered structural optimization problems remain relatively small because of the many disciplines simultaneously considered, and the lack of efficient optimization methods and solvers in the frameworks.
The work by Hu et al. (2016) demonstrates wind turbine blade design in a reliability framework, first using a deterministic optimization to provide an initial design for the subsequent reliability optimization. This framework is useful to address the inherently uncertain nature of the wind loads imposed on a particular blade. However, only small-scale problems are demonstrated because of the computational cost of carrying out Monte-Carlo-based simulations being too extensive.
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In Sjølund and Lund (2018), thickness optimization of a 73.5 m blade is performed under twelve load cases, with buckling, tip displacement, and failure constraints. The model utilizes solid-shell elements, which capture out-of-plane stress that is critical for addressing inter-laminar failure. As a consequence of using solid-shell elements, thickness optimization involves a shape parametrization, where the nodes have to be moved when modifying element thickness during optimization. Node relocation is carried out using a surface normal update scheme that ensures elements do not collapse onto each other.
Simultaneous structural and aerodynamic optimizations have been addressed in Scott et al. (2020), starting with an aerodynamic optimization to maximize blade energy production, followed by a structural optimization to minimize mass. The two optimizations are not coupled, but a loop is included around structural optimization where the load envelope is updated after achieving convergence. The structural optimization is then repeated until convergence of the loads is achieved. Fully coupled aero-structural approaches have also been demonstrated in Mangano et al. (2022); Scott et al. (2022), with both studies highlighting a potential for achieving improved designs over non-coupled approaches. The key challenge, however, is determining an ideal trade-off between two competing objectives, which necessitates heuristic weighting when formulating an objective function. Furthermore, managing computational expense is challenging due to the additional simulation required for each iteration.
The work by Hermansen and Lund (2024) has shown multi-material and thickness optimization of a wind turbine blade root section, simultaneously optimizing material choice, fiber orientation, stack sequence, and layer thickness. The root section is defined as the first 35 m of the blade, and is subject to buckling, static, and fatigue constraints for four load cases. Although relying on materials commonly used in existing blade designs, the presented approach facilitates challenging these typical material choices. However, such changes could significantly alter the elastic behavior of the blade, which can influence the aerodynamic performance, and this factor is not considered in the presented optimization.
Recently, Hermansen et al. (2024b) demonstrated a detailed thickness optimization of a wind turbine blade root section. Here, the root section is subject to twelve load cases, with buckling, static, as well as fatigue failure constraints. A significant mass reduction was achieved, with several active constraints of each type throughout the root section. Notably, several buckling constraints are active at convergence, where in full-blade optimizations the critical buckling modes tend to occur at further distance from the root connection. This observation suggests potential benefits through isolated root design optimization. Specifically, in the context of gradient-based methods, a smaller model with fewer variables can increase the probability of locating a good optimum in the complex, non-convex design space.
The presented literature survey documents the majority of works published in the field of structural optimization of wind turbine blades, and evidently, it is still quite limited. One of the significant barriers to entry to research on wind turbine blades is the substantial resource investment to set up an adequate blade model that can be used for, e.g., optimization. Designing a model that is representative of state-of-the-art blades is challenging, as it involves aerodynamic, aero-elastic, structural, control, and manufacturing considerations which are competing criteria implying non-trivial trade-offs have to be made. Moreover, if the intended research idea turns out ineffective, the modeling work may be wasted, as it often lacks scientific novelty.
Fortunately, blade manufacturers have been actively engaged in academic research to facilitate collaborative Research and Development (R&D) efforts for new innovative design methods and tools, and have therefore provided models for use in scientific publications. However, with the increased pressure to publish code, data, and models, it becomes undesirable for manufacturers to share confidential designs, emphasizing the need for high-quality open-source models that can fill this gap.
This paper aims to encourage increased academic R&D on wind turbine blade structural design, with the goal of enhancing wind turbine performance. It is pursued through two main objectives; the first is to introduce the Gurit98m blade model, a detailed open-source finite element model of a 98 m wind turbine blade with multiple use cases, which has been made publicly available on Hermansen et al. (2024a) in both Abaqus and ANSYS input file formats. The second objective of the paper is to demonstrate applicability in structural optimization research by performing a structural sizing optimization of the blade model. The purpose is to investigate optimal distribution of material under buckling, tip displacement, and static failure constraints for twelve load cases. Additionally, the optimization will treat a transition of core material from balsa wood to recyclable PET to enhance the sustainability profile of the blade. Thus, the main contribution of the paper is twofold; first, providing a reference to aid future use of the Gurit98m model in research; and second, a demonstration of detailed structural optimization of large-scale wind turbine blade structures under multiple load cases and many constraints to accelerate transition to more sustainable materials.
The remaining paper is structured as follows: Sect. 2 will describe how the model has been generated, the material layout of the blade, the extreme load envelope applied, and the resulting structural finite element model. In Sect. 3, the optimization use case is presented, containing a description of the parametrization, finite element analyses, optimization approach, results, and a discussion. Concluding remarks finalize the paper in Sect. 4.
2 Description of the Gurit98m model
The Gurit98m blade is an open-source model of a 98 m offshore horizontal-axis wind turbine blade jointly developed by Gurit Wind Systems A/S and blade3 ApS. The model is intended for use in the detailed structural design phase, where the outer aerodynamic shape has been fixed. Gurit is committed to maintaining and updating this model, and different versions have been generated, which are available on an accompanying GitHub repository (Hermansen et al. 2024a). For the purpose of this work, a version containing a single shear web and a simplified layup is used to ensure efficient structural optimization. Note that the approach in this paper is equally applicable to onshore wind turbine blades.
The input parameters required to create a blade model are an outer shape, material layup, and loads, which are all provided with the Gurit98m model. The outer shape of the blade is created by interpolating between aerodynamically designed airfoils derived in Bortolotti et al. (2019b), using the b3p software (Ruijter 2024). The designer, blade3 ApS, has provided ultimate flap- and edgewise load cases, which are applied with equal magnitudes in both directions. Typically, the magnitude of the load and its distribution will vary between the different sides, hence this is a conservative assumption. In operation, a blade is subject to more than 100 load cases; however, in the detailed structural design phase, a smaller envelope of design-driving critical extreme load cases is identified and used for the remaining design modifications. Although four extreme load cases in flap- and edgewise directions are sufficient for this design stage, using a load envelope with twelve load cases, containing mixed flap- and edgewise load cases, is more accurate and therefore allows a reduction in the design safety factors prescribed by certification guidelines (DNV-GL 2015). The twelve-load-case envelope presented in Hermansen et al. (2024b) is used to create eight additional load cases, which are scaled from the existing flap- and edgewise loads. The resulting load envelope applied to the model is shown in Fig. 1. The loads are applied as nodal forces onto the spar caps of the blade.
Fig. 1
The load envelope applied to the blade model, illustrated at the blade root. The loads taper off towards the tip of the blade
The structural design uses industry-standard materials, which are high-performance laminated composites with fiber-reinforced polymers for stiffness and strength, and low-density core materials to improve blade bending stiffness at minimal weight and cost. While the initial material layup was developed for the single shear web model, additional versions of the model with two and three shear webs have been made available on Hermansen et al. (2024a). Nevertheless, the overall material distribution can be described by dividing the blade airfoil cross section into a number of regions, each region with distinct layup characteristics. Crudely, these different regions are categorized as fiber- or core-dominated, with further division based on the airfoil characteristics. This regional division, shown in Fig. 2, forms the basis for the parametrization discussed in Subsect. 3.1.
Fig. 2
Sketch of a cross section of a single-shear-web blade at around the maximum chord width, showing how the different materials are used in the blade
The fiber-dominated regions are the edges, Leading Edge (LE), Trailing Edge (TE), and the Spar Caps (SC), while the core-dominated regions are the Trailing Edge Core (TEC) and Leading Edge Core (LEC). The cross section is further divided into DownWind (DW) and UpWind (UW), allowing for varying laminates on both sides of the blade. Moreover, the blade is designed with a pre-bend (Olesen et al. 2012), which is standard in current state-of-the-art offshore wind turbine blades and adds geometrical tolerance to prevent blade-tower collision, reducing the amount of material needed to provide the necessary amount of stiffness.
Table 1
Elastic and strength properties used in the structural optimization
Property
GFRP
CFRP
GFRP Biax
GFRP Triax
PET Kerdyn 150
PET Kerdyn 250
\(E_1\) [GPa]
38.000
125.000
24.000
26.472
0.112
0.244
\(E_2\) [GPa]
9.000
8.000
24.000
10.073
0.112
0.244
\(\nu _{12}\) [-]
0.280
0.300
0.110
0.430
0.400
0.400
\(G_{12}\) [GPa]
3.600
5.000
3.600
5.873
0.036
0.072
\(G_{13}\) [GPa]
3.500
3.080
3.500
3.500
0.039
0.079
\(G_{23}\) [GPa]
3.600
5.000
3.600
4.340
0.036
0.072
\(e_{1t}\) [mm/mm]
0.0245
0.0143
0.0108
0.0204
–
–
\(e_{1c}\) [mm/mm]
0.0150
0.0097
0.0108
0.0137
–
–
\(e_{2t}\) [mm/mm]
0.0037
0.0050
0.0108
0.0058
–
–
\(e_{2c}\) [mm/mm]
0.0122
0.0146
0.0108
0.0118
–
–
\(\gamma _{12}\) [mm/mm]
0.0194
0.0160
0.0166
0.0186
–
–
\(\gamma _{13}\) [mm/mm]
0.0194
0.0160
0.0100
0.0166
–
–
\(\gamma _{23}\) [mm/mm]
0.0120
0.0120
0.0100
0.0114
–
–
\(X_t\) [MPa]
–
–
–
–
1.63
2.61
\(X_c\) [MPa]
–
–
–
–
2.12
4.50
\(Y_t\) [MPa]
–
–
–
–
1.63
2.61
\(Y_c\) [MPa]
–
–
–
–
2.12
4.50
\(S_{12}\) [MPa]
–
–
–
–
1.26
1.76
\(S_{13}\) [MPa]
–
–
–
–
1.19
1.76
\(S_{23}\) [MPa]
–
–
–
–
1.19
1.76
Properties are from ESAComp (2016); ANSYS (2024); Gurit (2024)
The materials used are Carbon Fiber Reinforced Polymer (CFRP), E-Glass Fiber Reinforced Polymer (GFRP), recyclable PET foam, with densities of 150 kg/\(\hbox {m}^3\) and 250 kg/\(\hbox {m}^3\), and an epoxy adhesive. The higher density foam is used solely in the root, hence the majority of core material is PET150. Both CFRP and GFRP are used in Uni-Directional (UD) non-crimp fabrics, where fibers are oriented majorly uni-directionally with a small amount of perpendicular backing fiber to stabilize the fabric during handling (typically less than 5% of total fiber content). GFRP is also applied in bi-axial (biax) and tri-axial (triax) configurations, which are two and three UD plies, respectively, arranged as a -\(45^{\circ }\)/\(45^{\circ }\)/\(0^{\circ }\)-oriented laminate. UD layers are used primarily in the spar caps to provide the necessary stiffness against critical flapwise load cases i.e., LC4 and LC10 in Fig. 1, while biax and triax are particularly important for resisting the multi-directional load cases. Triax plies are additionally used in the root to provide necessary multi-directional strength and are tapered off from the root to the position of maximum chord width. Actual root sections also contain metal inserts for the bolted connection to the turbine hub, which adds additional thickness to resist the imposed large bending moments. In the model, the presence of these inserts is accounted for by increasing the thickness of the triax laminate and is not directly modeled.
The published model contains all elastic properties, which are also provided in Table 1 along with the utilized strength properties. The elastic and strength properties of the of the foam is based on Gurit Kerdyn foams, which are certified and are found at Gurit (2024). For the remaining materials, the material properties are compiled from the libraries provided by ESAComp (ESAComp 2016) and ANSYS Workbench (ANSYS 2024). The blade has initially been designed with simplified ultimate strain criteria in the axial direction for the outer shell, ultimate shear strain in the shear webs, and linear buckling, which has formed the basis for the initial layup. As this work will consider failure in multiple directions, the layup has been adjusted to provide a better starting point for the subsequent optimization. Hence, the initial layup aims to be feasible and representative of existing blades, but not optimized.
Note that fatigue properties are not provided, as fatigue will not be considered in the optimization. Although fatigue is a key failure criterion, it is intended to keep the study as simple as possible such that reproduction of results is kept as straightforward as possible. Fatigue is challenging to handle in optimization, material fatigue properties are difficult to acquire, and work on gradient-based fatigue optimization of laminated composites remains limited, see Hermansen (2023) for a review. Hence, fatigue optimization of Gurit98m is left for future work.
The objective function in the optimization is based on material cost. Although mass is commonly used as the objective function due to its correlation with cost, it is insufficient for multi-material structures that use both CFRP and GFRP. While CFRP is superior to GFRP on most structural metrics, it is also more expensive. A mass-optimized design may hence prove more costly, as the optimizer will prefer CFRP over GFRP to lower laminate thicknesses. However, cost is challenging to accurately estimate due to fluctuating market prices. For this study, material costs have been estimated in-house, and the result can be found in Table 2. More detailed cost models are available in literature (Bortolotti et al. 2019a), if inclusion of other factors such as manufacturing costs, salaries, and other operational costs, is desired.
Table 2
Cost per kg and density of the materials used in the structural optimization. Cost values are based on Bortolotti et al. (2019a) and are adjusted to be representative of current market. However, the values are generic and do not reflect competitive prices. The same cost is applied to all foams in the model, as it is most important to differentiate between GFRP and CFRP in the optimization
Material
Cost/mass [EUR/kg]
Density [kg/\(\hbox {m}^3\)]
GFRP
3
1870
CFRP
15
1500
PET foam core
5
150 and 250
Fig. 3
The finite element mesh used in the blade, illustrated from three different perspectives. The average aspect ratio of the mesh is 4.3. The top view of the blade shown at the bottom of the figure illustrates the slenderness of the blade
The industry-stable balsa wood core material has been replaced by structural PET foam in this study, to explore the transition to recyclable alternatives through structural optimization. Core material increases the thickness of the laminates in targeted regions, enhancing the bending stiffness geometrically. Used in a sandwich format with fiber-reinforced composites as face sheets to stabilize the core, such configuration lowers the cost compared to using purely fiber composite laminate. However, sandwich laminates have exclusive failure mechanisms (Zenkert 1997), which can be assessed using specialized failure criteria. Applying such criteria is not mandatory in blade design, but it can be requested if deemed relevant for certifying a particular design (DNV-GL 2015). In this work, sandwich criteria are not included to keep the optimization simple; however, the interested reader is referred to Löffelmann (2021) for inclusion of sandwich criteria in a multi-material and thickness optimization context.
The trailing edge is a notoriously difficult region of the blade to model and design. Typically, the upwind and downwind sides of the blade are manufactured in separate molds and are afterwards joined using a structural adhesive. To model the glue section, solid elements are included in the trailing edge of the Gurit98m model, adding additional stiffness to the region. For further details and discussion on accurately modeling the trailing edge see Haselbach (2017). Glue joints may also be present in the leading edge and to adhere shear webs to the outer shell of the blade. Simulations of these particular areas require detailed models to assess failure, which is out of scope in this global model. Hence, glue is not considered elsewhere than in the trailing edge, where it provides significant stiffness to the cross section.
The baseline models are meshed using linear four-noded shell elements, which are favored over quadratic elements to lower the number of degrees of freedom, hence maximizing efficiency in optimization. However, linear shell elements require stabilization techniques to remove locking issues and ensure correct deformation behavior. In the present implementation, Mixed Interpolation of Tensorial Components (MITC) (Dvorkin and Bathe 1984) is used to prevent in-plane shear locking, while Enhanced Assumed Strain (EAS) (Simo and Rifai 1990) prevents out-of-plane shear locking. These techniques are also utilized in commercial software such as ANSYS and Abaqus (ANSYS 2024; Smith 2020). The glue section is meshed using eight-noded solid-shell elements using similar stabilization techniques (Johansen and Lund 2009). An inwards-offset strategy is used in the shell element formulation to correctly represent the location of outer shell and layup in relation to each other. The mesh is fully structured, making mesh refinement straightforward in length- and width-wise directions if desirable. In total, the finite element mesh consists of 20,208 shell elements, 769 solid-shell elements, and 20,250 nodes. The mesh is illustrated in Fig. 3.
Note, the model has two main limitations. First is its fixed outer shape and size, which is challenging to modify on a finite element basis. The authors are actively working on creating blade models that represent the future offshore wind turbine scale, however, this is left for future work. Second is that the blade is created without an associated turbine, which limits the number of criteria that can be taken into account. For instance, a turbine is necessary in order to address key criteria such as blade-to-tower clearance and eigenfrequency failure (DNV-GL 2015), see e.g., Scott et al. (2020) for their inclusion. Tower clearence is considered to some extent by comparing the initial tip deflection to blades of similar scale and ensuring it is of comparable magnitude, see Brøndsted and Nijssen (2013); Mikkelsen (2016).
3 Structural optimization
To demonstrate how the model can be utilized for structural optimization studies, a thickness optimization is carried out to lower cost by finding an optimized material layout in the blade. This section details how the optimization model is established, the results attained by applying optimization, and a discussion of the results and overall approach.
3.1 Parametrization
Fig. 4
The distribution of laminate patches in the blade. Each patch corresponds to a region with an independent laminate design, and the patch distribution is selected to account for the thickness variation in complex regions. However, more patches will increase the number of thickness design variables, hence a trade-off is made to manage computational cost
Wind turbine blade design involves an initial design phase, where an ideal balance between aerodynamic performance and structural integrity is established. The initial design phase yields a crude layup and serves as the starting point for a subsequent optimization of the thickness distribution to minimize cost. However, the optimization of thickness has to account for the blade manufacturing process to be useful. The fiber-reinforced polymers are manufactured in rolls with constant thickness and are placed in a continuous, layer-by-layer process into a mold to create the laminates (Krogh et al. 2023). Core materials, likewise of constant thickness, are scored into gridded panels to conform to the complex doubly-curved shape of the blade, and are placed onto the fiber material in the mold. Thus, the achieved optimized laminate thickness must round to a discrete number of constant thickness layers to be suitable for manufacturing.
Given the constant thickness format of the materials, a discrete formulation of an optimization problem is straightforward. Unfortunately, solving discrete problems effectively is challenging, particularly given the size of the problem considered. The optimization problem is therefore relaxed and formulated as continuous instead, i.e., where ply thickness can vary continuously between an upper and lower bound. The drawback of such reformulation is that the resulting layup has to be post-processed to comply with actual material ply thickness. However, it is important to note current optimization tools are not yet able to include all relevant details and criteria in the wind turbine blade optimization. Hence, post-optimization modifications of the design are necessary nonetheless to account for all failure modes and mechanisms, particularly those associated to high complexity and computational cost, e.g., delamination-driven fatigue (Bak et al. 2014). The goal of the optimization is therefore to provide a better starting point for achieving an optimal complex variable-thickness distribution in the blade. By utilizing continuous variables, gradient-based optimizers can be employed, which are capable of solving problems of the treated size and beyond, as demonstrated in several studies, e.g., Aage et al. (2017); Baandrup et al. (2020); Träff et al. (2021).
To align with manufacturing and manage computational cost, it is prudent to divide the structure into a number of patches. In previous works (Sjølund and Lund 2018; Mangano et al. 2022; Hermansen et al. 2024b), patches are distributed into a regular, grid-like pattern. Such a distribution may not accurately capture the complex variation in thickness between laminates, particularly near the trailing edge, where tapering is required to fit the laminates between the narrowing top and bottom sides of the cross section. The Gurit98m model uses a similar division based on the characteristic regions in Fig. 2, but includes more detail to account for key local thickness variations, particularly in the trailing edge region, avoiding laminates collapsing onto each other. An overview of the patch distribution applied to Gurit98m is given in Fig. 4.
Fig. 5
An illustration of condensing the number of layers in the model. The laminate in (a) shows the original detailed layup, with up to 35 layers in the spar cap, which is reduced to a total of three layers of biax GFRP and UD CFRP in (b). (c) shows the equivalent simplified laminate for a core region with a PET foam core and biax GFRP face sheets
In total, the model contains 563 distinct patches, each with an independent laminate and associated design variables. Further reduction in computational cost can be achieved by condensing the number of individual layers in the model. This is carried out by combining multiples of the same material, and same orientation for fiber materials, into fewer layers with increased thickness. For state-of-the-art blades, the number of layers can exceed 150, with stress and strain being evaluated at the top and bottom of each ply at minimum. Hence, condensation can result in a drastic reduction in both the number of design variables and the computations in the analyses. Completing this process for Gurit98m reduces the number of design variables from 22,060 to 1,585, when plies are divided into nominal thickness. In the model file, a design variable is assigned each layer in every shell section occurring, excluding the solid glue section placed at the bottom of the composite section definition. The condensation process is exemplified in Fig. 5.
3.2 Finite element analysis
All applied structural criteria are evaluated based on linear elastic Finite Element Analysis (FEA). The point of departure is solving the equilibrium equations given in Eq. (1).
Here, \(\varvec{K}\) is the global stiffness matrix, \(\varvec{U}\) is a vector of admissible nodal displacements, and \(\varvec{F}\) is the consistent nodal load vector. To assess the buckling and static failure, strains and stresses are necessary, and these can be estimated using the calculated structural displacements. The strain vector in structural coordinates (\(\varepsilon _{xyz}\)) is determined from the element nodal displacements \(\varvec{u}^{(e)}\), see Eq. (2).
Here, \(\varvec{B}^{(e,l,m)}\) is the strain–displacement matrix for the particular element formulation, e indicates the element number, l is the layer number, and m is the thickness-wise position of strain evaluation in the element. Strains (and stresses) are evaluated at the top and bottom of each layer, hence \(m=1,2\). The stress vector in structural coordinates \(\varvec{\sigma }_{xyz}\) is then determined using the constitutive law as shown in Eq. (3).
Here, \(\varvec{C}\) is the material constitutive matrix.
3.3 Buckling analysis
The linear buckling problem is an eigenvalue problem, where the buckling load factors \(\lambda _{BLF}\) are the eigenvalues and buckling modes \(\varvec{\Phi }\) are the eigenvectors. Hence, buckling is evaluated from the governing equation shown in Eq. (4).
Here, \(\varvec{K}_\sigma\) is the global stress-stiffness matrix, j is a particular buckling load factor and corresponding mode, and \(N_j\) is the number of extracted eigenvalues and -modes. In the context of optimization, constraining several of the buckling load factors for each load case mitigates the risk of mode switching of the critical mode. Yet, mode switching can still occur with excessive design changes. However, if including a sufficient number of modes and using reasonable move limits, this strategy has been demonstrated to work well in practice (Sørensen et al. 2014; Sjølund and Lund 2018; Hermansen and Lund 2024). In addition, it is assumed that multiple eigenvalues do not occur during the optimization, which would warrant using directional derivatives to calculate buckling sensitivities (Seyranian et al. 1994).
The element stress-stiffness matrix is calculated according to Eq. (5).
Here, \(\varvec{S}_{Mat}\) is a matrix containing element stresses, \(\varvec{G}\) is a matrix of element shape functions, appropriately paired with \(\varvec{S}_{Mat}\) (Zienkiewicz and Taylor 2005; Bathe 2006; De Borst et al. 2012), and \(N_l^{(e)}\) is the number of layers in element e. The integral is evaluated numerically with Gauss quadrature.
3.4 Failure analysis
Failure analysis is carried out using the well-known max stress/strain criteria, where max stress is applied to the PET foam and max strain to all fiber materials. To apply these criteria, the strains and stresses have to be comparable to material strain and strength allowables, as provided in Table 1, which implies rotation from structural coordinates (x, y, z) to the material coordinate system (1, 2, 3). The stress vector is rotated according to Eq. (6).
Here, \(\varvec{\sigma }_{123}\) is the stress vector in material coordinates, and \(\varvec{T}\) is the tensorial stress transformation matrix. Transformation of the strain vector to material coordinates requires additional steps because of the relation between tensorial and engineering strain (Jones 1999), and the resulting expression is given in Eq. (7).
Here, \(\varvec{\varepsilon }_{123}\) is the strain vector in material coordinates. Finally, the failure index \((\varvec{\text {FI}})\) is computed as the maximum fraction of stress/strain component to its corresponding strength as shown in Eq. (8).
Here, \(\varvec{S}_{123}\) represents either the strain or stress vector in material coordinates, \(\oslash\) indicates element-wise division, and \({{\textbf {US}}}\) is the ultimate strain or strength vector.
Although being simple, max stress and strain are encouraged by design guidelines (DNV-GL 2015) for assessing intra-fiber failure, which is the main motivation for adopting them in this work. Inter-fiber failure is typically assessed by Puck’s criterion (Puck and Schürmann 2002; Deuschle and Puck 2013); however, the computational expense involved in its evaluation makes inclusion in optimization challenging. Furthermore, the Puck criterion involves calibration of inclination parameters, which is partially done based on experience. Hence, inter-fiber failure is also assessed using max strain and stress.
The initial and optimized designs are additionally verified using the sandwich-specific criteria to assess shear crimping failure and face wrinkling failure (Zenkert 1997), and these criteria are not violated in the initial design. Fatigue is another key failure type, which is not included in the present study; however, reference is made to Hermansen et al. (2024b), concerning inclusion of fatigue constraints in wind turbine blade optimization, for readers interested in incorporating these.
3.5 Safety factors
The choice of criteria is motivated by design guidelines for blade certification from DNV-GL (2015). The guidelines specify the required analyses and provide a set of partial scaling factors given the accuracy of the adopted methods. The total safety factor is calculated as the product of all partial factors, see Eq. (9).
Here, \(\gamma _{m0}\) is a constant base factor, \(\gamma _{mc}\) accounts for the criticality of the type of failure, \(\gamma _{m1}\) is long-term degradation of materials in the blade, \(\gamma _{m2}\) adjusts for temperature effects, \(\gamma _{m3}\) accounts for material and manufacturing effects/defects, \(\gamma _{m4}\) corrects for the accuracy of the approach in each analysis, and \(\gamma _{m5}\) is adjusted for the number of load cases considered in the load envelope. All partial and total safety factors are provided in Table 3.
Table 3
The safety factors resulting from adopting the presented analysis approach (DNV-GL 2015)
Factor
Buckling
Fiber failure
Inter-fiber failure
Description
\(\gamma _{m0}\)
1.20
1.20
1.20
Base factor
\(\gamma _{mc}\)
1.08
1.08
1.00
Failure criticality
\(\gamma _{m1}\)
1.05
1.20
1.10
Long-term degradation
\(\gamma _{m2}\)
1.05
1.10
1.00
Temperature
\(\gamma _{m3}\)
1.10
1.30
1.00
Material and manufacturing
\(\gamma _{m4}\)
1.25
1.00
1.15
Analysis accuracy
\(\gamma _{m5}\)
1.00
1.00
1.00
Load cases
\(\gamma _m\)
1.96
2.22
1.52
Total safety factor
There are also partial safety factors pertaining to the loads, estimated according to DNV-GL (2016), which are used earlier in the design process. The extreme load envelope should be understood as the twelve most critical loads cases of those simulated in earlier design phases including load safety factors.
3.6 Optimization problem, techniques, and settings
Prior to setting up the optimization problem, additional techniques are necessary to address computational challenge posed by a large number of constraints. As failure indices are determined from strain- and stress-based criteria, which are local values defined twice for each layer in every element, an intractable amount of constraints are imposed on the problem. To reduce the number of constraints, aggregation techniques (Duysinx and Sigmund 1998; Kennedy and Hicken 2015; Norato et al. 2022) are commonly applied, which formulate an approximated maximum constraint value that is differentiable. This work uses the well-known P-norm method due to its simplicity and proven effectiveness in previous studies, e.g., Duysinx and Sigmund (1998); Le et al. (2010); Zhang et al. (2019). The P-norm approximation of failure indices is calculated according to Eq. (10).
Here, \(N_e\) is the number of elements and P is an exponent that determines the accuracy of the P-norm approximation. As a consequence of using the exponent P, the expression becomes highly non-linear when requiring an accurate approximation of the maximum value. To mitigate, the implementation makes use of adaptive constraint scaling (Le et al. 2010; Oest and Lund 2017), which normalizes the constraint with values of previous iterations, achieving more accurate constraint values with negligible increase in computational cost and non-linearity. The approach is chosen due to its simplicity, and because it has been documented to work well in literature, despite the non-differentiability it introduces in the optimization. However, applying adaptive constraint scaling does not make the problem independent of P, so \(P = 12\) is chosen as a trade-off between approximation accuracy and a manageable degree of non-linearity. More sophisticated methods have been proposed more recently, see Kennedy and Hicken (2015); Norato et al. (2022).
The optimization problem can now be stated, see Eq. (11).
Here, c is the cost of materials, \(u_{y,Tip}\) is the tip displacement in the y-direction, \(u_{y,Tip,initial}\) is the initial tip displacement before optimization, \(t_{min}^{(l)}\) is the lower design variable limit, and \(t_{max}^{(l)}\) is the upper limit. The design variable limits are set as \(10^{-3}\) and twice the original layer thickness for lower and upper bounds, respectively. In total, the problem is subject to 25 structural constraints based on linear buckling, tip displacement, and static failure criteria.
The computational cost of solving the stated optimization problem largely depends on the efficiency of sensitivity computation. To minimize computational cost of the considered problem, sensitivities are computed using adjoint Design Sensitivity Analysis (DSA) (Tortorelli and Michaleris 1994; Christensen and Klarbring 2009), which is particularly effective for problems with many design variables and few constraints. For a general structural function f differentiated with respect to a design variable \(x^{(j)}\), the resulting adjoint sensitivity is as given in Eq. (12).
Here, \(\varvec{\lambda }\) is the adjoint vector. Derivation of the adjoint sensitivity equations, including partial derivative terms, for all functions used in this work can be found in Sørensen et al. (2014); Sjølund and Lund (2018); Hermansen et al. (2024b). The sensitivities are computed using a semi-analytical approach. Specifically, the partial derivatives \(\frac{{\partial }f}{{\partial }x}\) are forward finite difference approximated, \(\frac{\textrm{d}\varvec{K}}{\textrm{d}x}\) are central difference approximated, and the adjoint vector \(\lambda\) is computed analytically. A perturbation step of \(\Delta x = 10^{-5}\) is used for both finite difference approximations and has been observed to work well for the treated problem.
The optimization problem is solved using a Sequential Linear Programming (SLP) approach, incorporating adaptive move limits, a merit function reformulation, and an SLP filter (Sørensen et al. 2014; Sjølund and Lund 2018; Hermansen and Lund 2023; Hermansen et al. 2024b). Adaptive move limits are important when using a linear approach, as they prevent the optimizer from oscillating around an optimum and never converging. The move limits are selected based on numerical experimentation and are calculated as 10% of the difference in upper and lower bound of each design variable. The merit function allows constraints to enter infeasibility during the optimization process, relaxing the design space and increasing the probability of finding a strong local optimum. To ensure the converged solution is feasible, the merit function \(\tilde{f}\) adds penalization to the objective function f by introducing artificial design variables \(y^{(k)}\) as shown in Eq. (13).
Here, k indicates a constraint number, \(N_k\) is the total number of constraints, a and c are penalization factors. The artificial design variables are further used to achieve unconditional feasibility in constraints provided to the optimizer by reformulating the constraints as shown in Eq. (14).
Here, \(\tilde{g}^{(k)}\) is the \(k^{th}\) merit constraint, \(g^{(k)}\) is the original constraints stated in the problem of Eq. (11), and \(g_{lim}^{(k)}\) is the constraint limit.
The purpose of the SLP filter is to accelerate convergence by ensuring a newly-found solution is superior discarding poor ones (Chin and Fletcher 2003; Sørensen and Lund 2015). The main criterion is the objective function value, i.e., if the achieved objective function is an improvement, the solution is accepted and added to the filter. A limit \(u_{lim}\) is imposed on allowed infeasibility, which is defined as maximum absolute infeasibility of any right-hand-side-normalized constraint. If the limit is violated, the solution is rejected, and the problem is solved again with reduced move limits until an improved solution is found. The limit imposed in this work is \(u_{lim} = 0.01\). If a solution is better in both objective and constraint metrics, the poor solutions are discarded from the filter.
Two convergence criteria are used in the optimization; the first is minimum allowable change in the objective function of 0.1%, and the second is a 0.01% minimum allowable change in the norm of design variables. Although these move limits seem tight, small changes in the objective function can yield significant improvements in key performance indices. For instance, a 0.1% change in initial objective function value would result in a cost reduction of approximately €250 per blade.
The optimization problem is solved using an in-house Fortran program MUST (the MUltidisciplinary Synthesis Tool). The implementation is parallelized using OpenMP and MPI for all analyses and corresponding sensitivity calculations, substantially accelerating the solution process. The SLP-subproblem is solved using the IBM ILOG CPLEX V12.10.0 optimizer (IBM 2019), which is interfaced with MUST. Finally, a summary of the various settings and parameters used in the optimization is provided in Table 4.
Table 4
An overview of the various parameters involved in solving the optimization problem
The optimization problem converges after 135 iterations, with the convergence behavior illustrated in Fig. 6. The majority of improvements are attained in the first 20 iterations. After, the current move limits lead to constraint violation in excess of the SLP filter limit \(u_{lim}\), requiring the optimizer to adjust accordingly. Instead of terminating the optimization, the SLP filter identifies a feasible set of move limits to continue optimization. The most substantial improvement in objective is achieved in the first 70 iterations, with the remaining iterations reducing the objective function only by approximately 1% of its initial value. It is noteworthy that the substantial reduction is achieved despite the tip displacement constraint remaining stable at its initial value throughout the optimization. The potential for drastic improvements in key performance indices, such as cost and mass, despite a criterion appearing at its limit, underscores the importance of applying structural optimization in design.
Fig. 7
The initial laminate thickness distribution (top) and optimized thickness distribution (bottom) shown from the downwind side of the blade
A 17% reduction in cost is achieved, corresponding to a mass reduction of approximately 25%, with a final mass of 42.6 tons. The substantial reduction is possible due to the initial layup having been adjusted, allowing for a good starting point for the optimization, given the increased detail in failure consideration from how the original blade was designed. Moreover, manufacturing constraints such as ply-drops and ply overlaps have not been included in the optimization, which is expected to further restrict the design space, leading to higher-cost for a manufacturing-ready design. Ideally, the blade laminates should increase in thickness from the root to a point of maximum thickness, and then taper down towards the tip, preventing any local thickness valleys or peaks. Including manufacturing constraints in optimization can reduce the tendency of sudden thickness change, but no constraints have been proposed to ensure actual manufacturing-appropriate thickness distribution for the layer thickness parametrization used. Thus, a post-processing will be necessary to comply entirely with manufacturing rules, nonetheless.
At convergence, the buckling constraints for eight different load cases are either active or very close to, see Fig. 6b. Overall, the buckling constraints converge smoothly with the objective function with limited oscillation, indicating that constraining multiple modes per load case has worked as intended. In contrast, the failure constraints exhibit more oscillation during the optimization, which is attributed to their local definition, the higher degree of non-linearity involved, and how these factors are handled by the P-norm approximation, see Fig. 6c. Four failure constraints are active at convergence, including the failure constraint for load case 6, which is slightly infeasible, albeit within the allowable limit \(u_{lim}\) used with the SLP filter. Moreover, the failure constraint of load case 6 is initially slightly infeasible due to using simplified strength criteria in development of the model. Despite this, the optimizer manages to quickly achieve total feasibility, while simultaneously improving the objective function significantly. Evidently, these constraints are the most challenging to handle in the considered optimization problem, consistent with findings in Hermansen et al. (2024b); Hermansen and Lund (2024). Nevertheless, the smallest failure constraint is 0.7, indicating a good distribution of failure across the blade for all load cases.
The initial and optimized thickness distribution on the downwind side of the blade is shown in Fig. 7. Overall, the thickness is reduced in all regions, particularly in the spar cap that utilizes the higher-cost CFRP material. In the leading and trailing edge regions, as well as near the tip, the laminate thickness has also been notably reduced relative to initial thickness, with the lower bound on layer thickness of \(10^{-3}\) mm being achieved in some areas. In general, the optimizer prioritizes removal of the expensive fiber material over core material, and the distribution of core material therefore remains largely unchanged from initial to optimized design in the outer shell. An exception is observed in the shear web (not shown on figure), which is parametrized with five taperings. The shear web shows reductions from root-to-tip of approximately 5 mm, 10 mm, 10 mm, 7 mm, and 2 mm in core material, along with around 50% reductions in all biax layer thicknesses.
Buckling modes for load cases 3 and 5 are shown in Fig. 8. These modes have corresponding buckling load factors at the limit and span a significant portion of the blade. The location of the mode shapes varies between the two load cases to match the direction of load, see Fig. 1 for reference. For instance, in load case 3 the loads act as pressure on the downwind side and leading edge, resulting in a mode shape placed in the leading edge core region. Furthermore, several buckling modes are active at different lengthwise positions in the same load case.
An envelope plot of the blade failure index distribution, showing the maximum failure index of all computed positions combined for all load cases, is shown in Fig. 9. Overall, a good distribution is observed, with the majority of the downwind side showing element failure indices in excess of 0.65. The largest failure index is concentrated around transition regions between patches of differing laminate thickness. Some regions with low failure indices are instead buckling-driven, such as near blade tip, where an active buckling mode occurs in load case 4.
Fig. 9
Envelope plot of the failure criteria showing the maximum values for all stress evaluation points in all load cases. A good overall distribution across the blade is observed, with particular concentrations in transitions between patches, i.e., where thickness varies. Note the solid-shell elements representing glue have been removed in this visualization, as they have not been included in the failure assessment
A significant mass reduction of 25% is achieved through optimization and such large reduction is neither expected nor desirable at the detailed design stage. Such drastic change in blade mass, mass moment of inertia, and center of mass will affect the acting aerodynamic performance and loads, necessitating further back-tracking iterations to ensure turbine performance criteria are met. Future work could explore incorporating an update scheme in the optimization, e.g., by updating the loads and repeating the optimization as demonstrated in Scott et al. (2020), or coupling aero-structural optimization as shown in Scott et al. (2022); Mangano et al. (2022). Nevertheless, the obtained blade mass of 42.6 ton at 22 m tip deflection seems plausible for an optimized design at the scale of Gurit98m (Brøndsted and Nijssen 2013; Mikkelsen 2016), despite incorporation of recyclable PET core in the design.
When observing the behavior of the response functions, the failure constraints exhibit the most oscillations. The oscillatory behavior can be attributed to the many local constraints, the degree of non-linearity involved, and the use of the non-differentiable adaptive constraint scaling factor. It is therefore of interest to study alternatives to the P-norm-based approach for local optimization of composite structures, such as the improved methods already mentioned (Kennedy and Hicken 2015; Norato et al. 2022). Alternatively, augmented Lagrangian methods have been shown capable of solving problems involving millions of local stress constraints (da Silva et al. 2021), but have not been tested with composite structures in literature.
A significant aspect not accounted for in the optimization problem is manufacturing. Factors such as fiber misalignment, fiber wrinkling, allowable ply-drops, etc. are crucial to the structural integrity of the manufactured blade, and it is therefore desirable to include them in optimization. Fiber misalignment (Krogh et al. 2021) and fiber wrinkling (Broberg 2023) should be addressed by extending the analyses used in optimization, incorporating e.g., reliability-based approaches, while ply-drops can be explicitly included in the optimization problem. Simple ply-drop constraints have been proposed for layer thickness optimization in Sjølund and Lund (2018) and help limit thickness variations between adjacent laminates, but they do not eliminate the local thickness peaks and valleys that are undesirable to manufacture. In the alternative two-step optimization approach (Albazzan et al. 2019), ply-drop (or blending) constraints are treated in a dedicated manufacturing step. The process involves solving an initial gradient-based problem using a lamination or polar parametrization, providing an initial design for the following manufacturing step. In this step, a discrete programming problem is solved, where detailed ply-drop constraints can be included Macquart et al. (2016); Panettieri et al. (2019). However, there is no coupling with the initial optimization problem that aims to ensure structural integrity, and the final design may consequently end up infeasible (Zein and Bruyneel 2015), necessitating post-processing.
With the availability of the model to the research community, the authors hope to encourage innovations in blade parametrization, optimization, and design. One of the major challenges in blade design is that failure occurs locally, and often in vicinity to complex structural details. Such details include the glue connections between shear webs and outer shell, as well as connections joining the upwind and downwind sides in the leading and trailing edges, depending on the manufacturing. Validation and verification of glue connections are prescribed by design verification guidelines (DNV-GL 2015), but direct inclusion of criteria for assessment in optimization has not yet been demonstrated in literature. Formulating and including adhesive failure constraints could lead to improved results, potentially reducing the extent of post-processing after optimization.
Another important factor is inter-fiber failure, which is failure in the matrix surrounding the fibers, and it is mandated by DNV-GL (2015) that in-plane transverse inter-fiber failure is verified using appropriate analyses. In this work, failure analysis is carried out using the max strain and stress criteria to address both intra- and inter-fiber failure. However, DNV-GL (2015) recommends using the Puck criterion (Puck and Schürmann 2002; Deuschle and Puck 2013) for inter-fiber failure, which better predicts this failure mode (Hinton 2004). Integrating Puck’s criterion in optimization is challenging, as it has two main disadvantages. First, the criterion involves searching all possible orientations to identify a critical plane of damage, which is a computationally expensive procedure that must be repeated at every stress evaluation position, for every iteration in the optimization. Moreover, this process makes the criterion non-differentiable since the failure mode can change, creating a non-smooth failure envelope. However, Puck may be handled similar to the max strain and stress criteria used in this work, assuming constant failure mode in DSA, which works well in practice. Second, the criterion relies on inclination parameters, which are calibrated based on experience. Nevertheless, inter-fiber failure is driving failure in several areas of the blade and addressing it could potentially lead to a more reliable design, reducing necessary post-processing after optimization. Other criteria such as Tsai-Wu and Hashin are also available to enhance failure prediction accuracy, however, these are rarely applied in the wind industry, as it is not possible to lower safety factors according to DNV-GL (2015).
A reduction of 0.2 in the buckling safety factor, as shown in Table 3, is attainable if the blade is designed using non-linear buckling analysis. The reason for allowing a reduction in safety factor is the significant influence of geometric non-linearities during the extreme load cases, leading to a pronounced difference between linear and non-linear buckling load factors, being taken into account. The work by Lindgaard and Lund (2010, 2011) lays out general strategies for non-linear buckling optimization of laminated composite structures. Extending these strategies to large wind turbine blades should be a priority, as streamlining the analysis and optimization procedures is expected to significantly decrease lead time required for developing high quality, desirable large blade designs.
4 Conclusion
This paper introduces the Gurit98m model, an open-source wind turbine blade, which can support structural optimization research. The applicability of the model is demonstrated by defining and solving a gradient-based layer thickness optimization of its constituent laminates. Key model details are discussed including used materials, layup, loads, and finite element mesh. The process of defining the optimization problem is then explained, beginning with parametrization of the model using layer thickness design variables. A presentation of the finite element model is then given, followed by a description of the linear buckling and static strength analyses used to assess structural integrity along with determination of the criterion-specific safety factors from design guidelines. The optimization problem is then established, and auxiliary techniques are described, including P-norm aggregation for the failure constraints, and the merit function reformulation of the problem functions to improve convergence. The semi-analytical adjoint design sensitivity analysis strategy for efficient computation of sensitivities is also briefly outlined. Solving the optimization problem yielded a significant 17% reduction in cost and a 25% reduction in mass. The resulting laminate thickness distribution and structural response is shown, and the overall results as well as strategy are critically discussed, leading to suggestions for future work. It is our hope that the Gurit98m model will serve as a valuable resource for the scientific community to address both existing and emerging challenges for optimizing the wind turbine technology.
Acknowledgements
This work has been financially supported by the Innovation Fund Denmark under the “Industrial Researcher”-Programme, Grant “Future Core Materials for Wind Turbine Blades” no. 3195-00001B. This support is gratefully acknowledged.
Declarations
Conflict of interest
The authors declare that they have no Conflict of interest.
Replication of results
To replicate the results, go to Hermansen et al. (2024a) and download the model used in this study. The repeated layers are combined to a single representative, which layer thickness is used as design variable. The properties provided in Sect. 2 are assigned to the used materials. The analyses described in Sect. 3 should then be implemented and used to set up the optimization problem in Eq. (11). Finally, the optimization techniques from Subsect. 3.6 should be implemented to complement the optimizer.
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