The need for cost-effective support structure designs for offshore wind turbines has led to continued interest in the development of design optimization methods. So far, almost no studies have considered the effect of uncertainty, and hence probabilistic constraints, on the support structure design optimization problem. In this work, we present a general methodology that implements recent developments in gradient-based design optimization, in particular the use of analytical gradients, within the context of reliability-based design optimization methods. Gradient-based optimization is typically more efficient and has more well-defined convergence properties than gradient-free methods, making this the preferred paradigm for reliability-based optimization where possible. By an assumed factorization of the uncertain response into a design-independent, probabilistic part and a design-dependent but completely deterministic part, it is possible to computationally decouple the reliability analysis from the design optimization. Furthermore, this decoupling makes no further assumption about the functional nature of the stochastic response, meaning that high-fidelity surrogate modeling through Gaussian process regression of the probabilistic part can be performed while using analytical gradient-based methods for the design optimization. We apply this methodology to several different cases based around a uniform cantilever beam and the OC3 Monopile and different loading and constraint scenarios. The results demonstrate the viability of the approach in terms of obtaining reliable, optimal support structure designs and furthermore show that in practice only a limited amount of additional computational effort is required compared to deterministic design optimization. While there are some limitations in the applied cases, and some further refinement might be necessary for applications to high-fidelity design scenarios, the demonstrated capabilities of the proposed methodology show that efficient reliability-based optimization for offshore wind turbine support structures is feasible.

Offshore wind energy is becoming an increasingly competitive alternative to the traditional land-based wind farms. However, there remains a level of additional cost which, together with some practical challenges, ensures that offshore wind is still a secondary consideration in many markets. Hence, the reduction of this cost is a primary objective in current research and development. Cost reduction is generally a multidisciplinary issue, including turbine components like rotor blades and the drivetrain, wind farm layout, the electrical grid and the design of the support structure (including the tower). Methods to derive cost-effective, optimal support structure designs – balancing minimal use of materials (and potentially other cost-driving design aspects) with the ability to safely withstand the loads required by design standards – have been an active area of research for many years. However, very few studies have taken into account the probabilistic, fundamentally uncertain aspects of the design process. This includes, for example, uncertainties in the environment and the modeling of the environment, affecting the loads experienced by the structure, as well as uncertainties about the details of the design itself, affecting the response to the applied loads. Taking such uncertainties into account generally requires the use of probabilistic mathematical methods that severely complicate the design optimization problem that needs to be solved, both formally and numerically. Hence, deterministic safety factors tend to be used. This is also true even for single design assessments. The present study aims to address these issues by proposing a methodology that allows the use of both state-of-the-art optimization methods recently developed for support structure design and probabilistic assessments of the structural response to both fatigue and extreme loads.

Design optimization of structures subject to probabilistic problem variables and parameters, sometimes called
optimization under uncertainty, is in general a large field of research at the intersection of two larger fields,
optimization and probabilistic design. One main distinction is between robust design optimization (RBO)

A substantial amount of the literature for both structural reliability analyses and RBDO of offshore wind turbines
(OWTs) has focused on aspects other than support structure design. Areas such as blade design

Only a limited number of studies applying RBDO to OWT support structure design have been made. In a series of studies,
Yang and collaborators investigated optimization of a tripod support structure with probabilistic constraints. In

The previous work on RBDO for OWTs (for both support structures and otherwise) demonstrates that these methods can obtain
optimal designs that are both more robust/safe with respect to uncertainties than designs optimized under deterministic
criteria and more tailored to specific design conditions than deterministic designs using safety factors. However, so
far (and this is particularly true for support structure designs), no studies have taken advantage of recent advances in
deterministic structural optimization methodology. Optimization methods in general can be divided into gradient-based
and gradient-free (often heuristic) methods. Both approaches have been applied to support structure design and other
wind turbine components (both on- and offshore). With some exceptions (e.g.,

As seen in several of the cited studies above, the use of surrogate modeling to simplify the response analysis has
become more common recently. For optimization and reliability analysis, and all the more so for RBDO, this is a natural
way to make the problems more computationally tractable when faced with having to perform a large number of
time-consuming simulations. However, surrogate modeling is increasingly also proposed for basic structural analysis due
to the large number of environmental states that need to be checked for certification according to design standards
(e.g.,

In summary, while considerable work has gone into improving the various analyses and methods involved in RBDO for support structures, there is a very limited amount of studies that connect these pieces together. In particular, the work on analytical design sensitivities has not been implemented into RBDO, nor has it been combined with surrogate modeling approaches that make more comprehensive structural analysis and/or reliability analysis computationally feasible. These gaps are what we intend to explore in the present study. By a very particular formulation of the probabilistic constraints (limit state functions) used for the support structure design optimization, we demonstrate how these constraints can remain analytically differentiable with respect to the design variables while at the same time using a surrogate model for the stochastic variation of the response. By doing so, we retain the advantages of the state-of-the-art deterministic optimization formulations while ensuring that the uncertainties are propagated through the system in a way that makes less simplifications than the commonly used factorization approaches and without incurring substantial additional computational effort. By assuming that some kind of factorization of the response is valid locally in design space, a standard double-loop RBDO formulation can be applied together with a design-independent Gaussian process surrogate model that makes the inner loop used to solve the reliability problem computationally insignificant. Retraining the surrogate model and repeating the optimization a few additional times then leads to convergence and an optimal design that is feasible with respect to uncertainties in both the loads and the structural modeling. In addition to incorporating more advanced optimization methods to the RBDO problem than has been done previously for OWT support structure design, the current approach can also be seen as a natural middle ground between, on the one hand, the simplified analytical limit state formulations and, on the other hand, the completely surrogate-model-based limit state formulations, the two most commonly used approaches in reliability analysis and RBDO for OWTs previously.

The structure of the paper from this point on is as follows. In the first section (methodology), the general theoretical background is presented first, with focus on optimization, reliability analysis and RBDO, but some details about surrogate modeling are also included. The section is concluded with a motivation and presentation of the proposed method from a general point of view. The next section (testing and implementation details) describes the setup for how we have chosen to test the method in practice. This includes specific models and what kinds of loads are included, the type of constraints included in the optimization, sensitivity analysis and uncertainty modeling. Additionally, some particular practical details of how the method has been implemented are discussed. The remaining sections of the paper include a presentation and discussion of the results, in the results section; more detailed treatment of a few points of interest, in the further discussion section; and a summary and final thoughts, in the conclusions.

In the following, we present the basic framework of (deterministic) design optimization for OWT support structures. Then, some aspects of RBDO and surrogate modeling are explained. Finally, the synthesis of these aspects resulting in the proposed RBDO methodology is motivated and presented.

For the task of finding the minimum structural mass

The optimization problem in Eq. (

It is a well-known result (see, e.g.,

The main distinguishing feature, with respect to the problem structure defined in Eq. (

However, for any but the most trivial limit state functions, the determination of the values of

The objective of FORM is to approximate the non-linear failure surface, the set of points such that

However, one problem with the definition of FORM given in Eq. (

The main idea of PMA is to reverse the role of objective and constraint in Eq. (

The difference between the solutions provided by RIA

The RBDO problem using PMA can be stated as

As is evident from Eq. (

Surrogate modeling is generally a vast topic and the interested reader is referred to

Focusing our attention to wind turbine applications, it has been common for quite some time to use surrogate modeling
due to the computationally demanding simulations required for time-domain analysis. This is especially true for reliability analysis, optimization and RBDO, due to the drastically increased computational effort involved. The most commonly applied types of surrogate
models in wind energy have been response surface models (typically second-order polynomials), Taylor expansions and
(especially more recently) GPR. GPR has many advantages, including the ability to capture non-linearities with higher
fidelity and providing an estimate of its own uncertainty by default but generally requires a larger number of samples
to gain a significant advantage over response surface methods

The essentials of GPR are quite similar to conventional regression methods. We wish to construct a model

GPR demonstrated on two related test functions: one with no noise

As noted previously, GPR can require a large number of samples to attain its desired fidelity. For this reason, it is
common to apply specialized sampling techniques, together usually referred to as the design of experiment (DOE), that sample
the input space more efficiently and thereby require less samples than, e.g., uniform random sampling. Depending on the
desired outcome, one could, for instance, opt for importance sampling (most useful in this case if it is known that only a
certain region of the input space is of interest, e.g., for a reliability analysis where one mainly wishes to use the
surrogate model around the failure surface) or a space-filling approach like Latin hypercube sampling or quasi-Monte
Carlo sampling (these are most useful when as wide coverage of the input space as possible is needed, e.g., for
optimization where the region of interest is likely to shift dynamically). A comparison between Latin hypercube sampling
and quasi-Monte Carlo sampling was performed in

In the following, we will explain the details of our proposed framework for RBDO of OWT support structures. However, we begin with a few remarks that serve to motivate this approach.

Considering the state of the art for reliability analysis and RBDO for OWTs more generally, we can make a few summary observations based on the previous discussion. Firstly, the vast majority of studies make use of either simplified analytical limit state functions (allowing more easily the use of FORM and making the probabilistic constraints easier to combine with design optimization) or surrogate models that completely replace simulation output (usually combined with sampling-based reliability analysis). Secondly, when not based on heuristic optimization methods (as has been the case for all RBDO studies concerning the design of support structures specifically), gradient-based design optimization as part of RBDO has not utilized analytical sensitivities. Thirdly, little to no use of PMA for reliability analysis or more advanced RBDO methods like SORA or SLA has been made, despite their notable advantages.

What can be concluded from this? Simply put, considerable progress could be made by making the state of the art for OWT RBDO, and for support structure design in particular, more in line with the general state of the art. However, this should be done in a way that maintains some of the OWT-specific developments made in previous optimization studies. Furthermore, by combining elements from all these sources, it could be possible to obtain a synthesized methodology that retains many of the individual advantages. However, this requires a new approach because of the ways in which the previous methods seem incompatible. It is, e.g., seemingly not possible to use analytical sensitivities if the simulation output is replaced by surrogate models.

Suppose that all relevant limit state functions

An example of factoring out the dependence of one variable from a non-separable expression by using GPR to
fit this unknown factor. The figure shows the relative error when approximating

Our overall proposed framework is based on the previously stated PMA-based RBDO problem in
Eq. (

Flowchart representation of Algorithm 1.

The design optimization performed in this study will in all cases be based on output from time-domain simulations of
finite element models. These have been implemented in an in-house, MATLAB finite element code as assembled Timoschenko
beam elements with 6 degrees of freedom at each end of each element. The analysis is based on Newmark integration and
uses a consistent mass matrix and a Rayleigh damping matrix with mass and stiffness proportionality scaled according to
the first two eigenmodes. A typical finite element, including variables, is shown in Fig.

Beam element with design variables

To test the proposed methodology, two main cases will be used. The first of these cases is a simplified model based on a
uniform section of a monopile support structure, initially uniform in its cross-sectional dimensions and with uniform
lengths for each element. This is meant to demonstrate the basic idea of the method without having to consider realistic
designs. This model will be referred to below as the “Simple Beam”. The second case is a simplified but
reasonably realistic representation of the OC3 Monopile

Properties of the two models used in the study.

n/a – not applicable.

Properties of the loading scenarios. International Electrotechnical Commission (IEC) design load cases (DLCs) refer to

As indicated in Eq. (

The nominal expression for the Kreisselmeier–Steinhauser function does not actually include the global
maximum as it does here, but for improved numerical performance it has been added (first term) and subtracted
(exponential term) as suggested in the original study

In principle, many other types of constraints should be included in order to ensure that the structure adheres to safety standards (e.g., buckling) and having constraints on eigenfrequency (to avoid dynamic amplification) is also common. However, in order to focus our attention on how the basic support structure optimization problem is affected by the presence of probabilistic constraints, and see the effect of each probabilistic constraint more clearly, these other safety checks have been left out.

The estimation of gradients for the objective function as given in Eq. (

For the RBDO formulation based on PMA, the limit state functions represented by Eqs. (

The uncertainty in stiffness is assumed to come from uncertainty in soil stiffness. Though no soil modeling is included
in the present analysis, this mainly introduces a shift of the mean stiffness, and hence one may still consider the
impact of a stochastic uncertainty in the soil stiffness. The effect of soil pile stiffness on the fundamental
eigenfrequency of a monopile was discussed in

For the uncertainty in global damping, the two main contributions are assumed to come from aerodynamic damping and soil
damping. Expected ranges of the damping coefficients corresponding to these two sources can be obtained from

As a small comment, we have so far assumed both stiffness and damping to follow normal distributions. It has been common
practice in previous studies to model uncertainties related to soil and aerodynamic damping as log-normally distributed
(sometimes other skewed distributions). However, at a CoV of 0.05, there is almost no difference between the
corresponding normal and log-normal distributions. Even with a CoV of 0.12, the differences are fairly small. See
Fig.

Normal distribution vs. log-normal distribution sharing mean and standard deviation: mean 1.0 and standard
deviation 0.05

The turbulence intensity is modeled as log-normally distributed with a wind speed-dependent mean and a CoV of 0.05,
i.e.,

The fatigue resistance is modeled as log-normally distributed with a mean of 1.0 and a CoV of 0.3, consistent with,
e.g.,

The yield strength is modeled as log-normally distributed with a mean of 288 MPa and a CoV of 0.1, consistent with,
e.g.,

The uncertainty modeling is summarized in Table

Uncertainty modeling details. Quantities marked with

So far, some of the details of the proposed methodology have been left unspecified in order to suggest an overall framework for RBDO rather than a very specific set of methods. The literature contains a large amount of choice in regards to optimization algorithms (both at the design optimization level and in the inner optimization loop solving the reliability problem), surrogate modeling and DOE. While these details have to be fixed in order to demonstrate the method in practice, the optimal selection of algorithms is not considered within the scope of the present study. Such optimality will in any case be both application specific and depend on the personal preferences of the designer.

With regards to both levels of the optimization, we use a combination of SQP and interior-point methods

For the surrogate modeling, we have chosen GPR due to the benefits stated previously. After some initial trial and
error, the Matérn class kernel with

The DOE was done using Sobol sequences, a quasi-Monte Carlo method. This has the advantage of being more space-filling, covering a larger range of the space while still having some clustering to account for local variations, compared to many ordinary Monte Carlo methods. While not made use of here, Sobol sequences also have the advantage compared to the commonly used Latin hypercube sampling method that it is much easier to interactively add new samples to the old set. To this last point, the use of an adaptive DOE was not used here, despite this increasingly becoming the common approach for GPR. The main reason why was a practical one, having to do with the way the loading input was sampled, which made interactively adding samples during a fitting procedure difficult for our implementation. A total number of 500 samples were pre-generated, with the number of samples actually used increasing for each iteration of the outer loop in the following way. For the initial surrogate model, 50 samples were used. The new model at the solution of the first RBDO procedure was then trained with 100 samples and compared with the old model using 25 additional samples, for a total of 125 samples used. All subsequent iterations use the full set of 500 samples, with 400 used for training and 100 for comparing the current model with the previous one. In a sense, the DOE is thus somewhat dynamic, even if it is not adaptive.

Finally, the outer loop needs termination criteria, as indicated in Algorithm 1. One such criterion was
chosen to be simply the convergence of the objective function value. Once this value changes less than a certain small
tolerance, the outer loop was halted. However, it is possible to terminate slightly earlier if the surrogate model is
seen to converge, since in that case the objective function will not change significantly or at all during the next
iteration. As implied above, the new surrogate models trained at the solution of the current RBDO loop were thus
compared with the models used during that loop. Due to the use of noisy regression models, the surrogate models will in
practice never converge entirely (or will at least do so very slowly) as long as there are small changes in the design
(and small changes in the surrogate model give further small changes in the design, etc.). Hence, a more relaxed
convergence criterion was developed for the surrogate models. Specifically, if we denote by

To illustrate both the basic workings of the RBDO method and the effect of certain modeling choices and constraints, a
number of different cases are studied. For easy reference, these have been given names and will be referred to as such
from now on. The names and properties of each of these cases are listed in Table

Testing cases for RBDO. Loading scenario numbers refer to the values in Table

The objective function for case BEAM-PA-CON is shown in Fig.

The optimization process for case BEAM-PA-CON: the objective function

Summary results of cases BEAM-PA-CON and BEAM-DA-CON.

Selected summary results of cases BEAM-PA and BEAM-DA.

The results for BEAM-PA and BEAM-DA are shown in Figs.

All in all, the results so far show that the method works well for these simple systems. The convergence behavior is more or less as for the deterministic case, with the addition of a few short extra loops to achieve overall convergence with respect to the updated GPR-based surrogate model. However, the results obtained from the first outer loop are likely good enough for practical purposes. The fatigue constraints dominate over the extreme load constraints, which is not unexpected. Furthermore, the system seems driven by the thickness(es) both with respect to the objective (structure mass) and the (fatigue) constraint, and the solutions reflect this (with minimal thicknesses and increased diameters where necessary to compensate). This can mostly be understood as a result of the fact that the contribution of the thickness to the cross-sectional areas and second moments of area is of higher order than that of the diameter.

The optimization process for case BEAM-PA: the objective function

The optimization process for case BEAM-DA: the objective function

Beginning with the two basic cases for the OC3 Monopile, OC3-PA and OC3-DA, displayed in Figs.

Selected summary results of cases OC3-PA and OC3-DA. Design variable numbers run from 1 (bottom element) to 14 (top element).

The optimization process for case OC3-PA: the objective function

The optimization process for case OC3-DA: the objective function

Next, the effect of including

The optimization process for case OC3-PA-DT: the objective function

Randomizing the initial OC3 design gives the results displayed for OC3-PA-RND in Fig.

The optimization process for case OC3-PA-RND: the objective function

Finally, the effects of no wave loads (OC3-PA-NW), only fatigue constraints (OC3-PF) and only extreme load constraints
(OC3-PU) are shown in the results in Figs.

More detailed results for OC3-PA-DT, OC3-PA-RND, OC3-PA-NW, OC3-PF and OC3-PU can be found in Table

The optimization process for case OC3-PA-NW: the objective function

The optimization process for case OC3-PF: the objective function

The optimization process for case OC3-PU: the objective function

Selected summary results of cases OC3-PA-DT, OC3-PA-RND, OC3-PA-NW, OC3-PF and OC3-PU. Design variable numbers run from 1 (bottom element) to 14 (top element).

The results demonstrate quite clearly the capability of the proposed methodology to obtain reliable optimal support structure designs without making the optimization process itself much more computationally complex than in the deterministic case. In fact, the initial outer iteration of the RBDO approach requires about the same number of iterations as the corresponding deterministic optimization cases. The small amount of changes to the design that occur in the additional outer iterations indicate that, even with the simplifications involved in the response factorization, the surrogate model is a fairly accurate global approximation. Final convergence of the outer loop is then mostly necessary for convergence in a mathematical sense, and the added computational effort required is of lesser practical importance. Tightening the non-linear constraints slightly would ensure that feasible solutions were obtained after only one round of RBDO. The 50 samples used to train the surrogate model for the initial RBDO loop represent a very small additional computational effort compared to what is required for the optimization in general. Since the minimum number of function evaluations (and thus simulations) required for a single iteration is one base evaluation plus one additional evaluation for every design variable, 50 simulations becomes rather negligible (for the OC3 Monopile models, this number is surpassed after only two iterations). Even when using all 500 samples, the added computational effort is not particularly large when compared with a full optimization procedure (it is equivalent to at most 18 iterations of OC3 Monopile models or at most 46 iterations of unconnected Simple Beam models). All in all, this makes the proposed RBDO framework a realistic option if gradient-based deterministic optimization is computationally feasible for the desired application. We note that the computation time on a single workstation (16 cores at 2.7 GHz; 128 GB RAM) for one full evaluation of the constraints (including all simulations required for four loading scenarios and the computation of all design sensitivities) was about 40 s for the non-connected Simple Beam designs and about 100 s for the OC3 designs. The total solution time was consequently on the order of hours: at most 10–12 h for all outer iterations to complete but generally only a few hours for the initial outer iteration.

The results obtained from RBDO do not appear functionally or systematically different than those obtained with
deterministic optimization, producing designs that are similar and only slightly heavier. Note, for example, the large
differences in maximum probability of failure compared to the small differences in total mass. The designs are driven by
fatigue on the load side and the element thicknesses on the structural side, leading in general to designs with small
thicknesses and large diameters. The OC3 Monopile designs tend to be quite a bit stiffer than the initial design, except
when the loads or constraints are relaxed enough to allow for very light designs (as in the case of OC3-PA-NW and
OC3-PU). The overall exception to these trends is the case with a

Some further simplifications have been made in the present analysis compared with more realistic applications. The main
examples are the system model (with no soil model or detailed hydrodynamic modeling), the load analysis (simplified wave
modeling, small number of environmental states considered) and the uncertainty modeling (potentially a much larger set
of uncertainties might have been considered and a more detailed approach could have been used to obtain the specific
uncertainty models). None of these simplifications are negligible but are not expected to affect the viability of the
results dramatically either. The system and load modeling are not necessarily so far away from approaches commonly used
for industrial applications, nor do they affect the system response in a way that would cause large deviations from the
behavior seen in this study. The simplified (or lack of) soil structure interaction and hydrodynamic properties mostly
serve to increase the global stiffness, reduce global damping and change the self weight of the system. These are
systematic effects that may change the amplitudes of the response but are not expected to change the relative response
to specific scenarios and so change, e.g., the complexity required to fit the surrogate model with respect to design
changes. Similarly, the simplified load analysis is also not expected to affect the relative responses very much,
especially for the fatigue analysis, where recent studies have shown that the distribution of fatigue damage over a
comprehensive set of environmental states does not change drastically when the design changes, particularly as long as
the eigenfrequency does not change too much (

In this work, we have presented a general methodology for performing RBDO of OWT support structures. The fundamental idea is that if the stochastic system response can be factorized into a design-dependent, deterministic (mean) response and a design-independent, probabilistic response, then it becomes possible to implement state-of-the-art RBDO, including state-of-the-art support structure design optimization methods, without adding much computational effort compared to deterministic optimization. The further advantages of the approach are that no assumptions about the functional representation of the probabilistic response are necessary, and since all design dependence is found in the deterministic part of the response, high-fidelity surrogate models can be fit for the probabilistic response while simultaneously making use of analytical methods for the estimation of design sensitivities. Together, this makes it possible to utilize recently developed gradient-based methods without having to make further adaptations of more general RBDO methods.

For the range of considered cases, the results show the feasibility of the proposed methodology. Although the overall approach includes an additional outer loop to ensure local fidelity of the surrogate model at the solution, these additional iterations are only necessary to ensure convergence in a stricter sense. For practical purposes, a single surrogate model fit and a single RBDO procedure suffices. Furthermore, the number of iterations of the RBDO procedure (not counting the solution of each reliability subproblem, which is computationally negligible when using a surrogate model), and hence the number of simulations required during optimization, is very close to that of the equivalent deterministic cases. The only additional computational effort is then found in the training of the surrogate model. However, this effort is comparable to that of a small number of additional iterations of the design optimization, especially for a larger number of design variables. Hence, the overall added computational complexity is small and makes the RBDO problem comparable to the equivalent deterministic optimization problem. The results also indicate that the RBDO framework does not change anything significantly about the kind of optimal designs that are obtained, as compared with deterministic design optimization. The same properties (fatigue and element thickness) seem to drive the designs, and the main differences are that probabilistically constrained designs are more conservative than their deterministic counterparts, as one would expect.

The current study is somewhat preliminary, in the sense that only a limited number of loading scenarios and constraints are considered, as well as the fact that the structural and environmental models are simplified and that limited effort has been put into refining, or otherwise optimizing, the methods used in the implementation of the overall framework. With regards to the simplifications, this is not expected to be a very limiting factor, though future work with higher fidelity is needed to ensure the practical viability of the proposed approach. As for the lack of refinement, this would indicate at least some potential for improving the methodology presented herein, which already works fairly well. It is likely that at the very least a more efficient design of experiment will be crucial if a larger amount of loading scenarios and higher-fidelity system modeling is to be made practical. Considering that many of the underlying optimization procedures used were originally developed for jacket support structures, it is expected that the current results, derived for monopiles, should be applicable with only minor modifications. Since very few studies of RBDO for OWTs have been done so far, in particular for support structure design, the current developments will hopefully open up new avenues for further research.

The data used for creating the figures and tables displaying the results are available in the Supplement. The code used to generate the results is very comprehensive and is, in its current form, not suitable for publication.

The supplement related to this article is available online at:

LESS formulated the main idea and implemented the method, conducted the analysis, created the figures and wrote the manuscript. MM provided essential input and suggestions throughout the process, aided in the formulation of the scope of the work and helped with the composition of the manuscript.

The authors declare that they have no conflict of interest.

This work has been partly supported by NOWITECH FME (Research Council of Norway, contract no. 193823) and by the Danish Council for Strategic Research through the project “Advancing BeYond Shallow waterS (ABYSS) – Optimal design of offshore wind turbine support structures”.

This research has been supported by the Norges Forskningsråd (grant no. 193823) and the Strategiske Forskningsråd (grant no. 1305-00020B).

This paper was edited by Athanasios Kolios and reviewed by two anonymous referees.