For the task of finding the minimum structural mass fmass of a topologically fixed design consisting of N circular cross sections, the following optimization problem can be formulated: 1min⁡xfmass(x)such thatAlinx≤bx≤xux≥xlcj(x)≤0∀j∈J. Here, x are the design variables, Alin and b give rise to a system of linear inequality constraints, xu and xl are upper and lower bounds, respectively, and cj represents a non-linear constraint function indexed according to some set J. The design variables for this problem will be the diameters Di and thicknesses ti of each cross section i∈{1,…,N}. The total mass of all N cross sections is calculated as 2fmass(x=(D;t))=πρ∑i=1NLi(Diti-ti2), where Li represents the (constant) length of each structural element with cross sections given by Di and ti, and ρ is the material density (assuming a uniform density throughout the structure). Examples of the type of linear constraints that can be represented by Alinx≤b are limits on the ratio of each Di to each ti (the D-t ratio). The non-linear constraint cj typically corresponds to safety criteria for ULS and the fatigue limit state (FLS) but often also includes constraints on the first eigenfrequency of the structure.

The optimization problem in Eq. () can be solved either by gradient-based or gradient-free (heuristic) methods. All gradient-based methods require, as the name suggests, the calculation of the gradients of the problem. In a constrained problem like Eq. (), that means estimating the gradients of both the objective function fmass and all the constraints. For an objective function like the one stated in Eq. () and for any linear constraints, this is a trivial problem. For non-linear constraints, the calculation of gradients (often called sensitivities in the optimization field) can be very difficult, especially when the value of these constraints depends on output from simulations, as is generally the case for support structure optimization. This difficulty can in principle be accommodated by the use of finite difference methods, where the function values around the current design point are used to get an estimate of the gradient. However, the use of finite difference methods can lead to inaccurate solutions or failure to converge, or will at least often require a larger number of function evaluations (computationally costly when simulations are needed for each such evaluation) to obtain the same solution as one would using the exact gradients (see, e.g., ). Additionally, the accuracy of finite difference estimates depends strongly on the chosen step size, the optimal value of which again depends strongly on the (possibly local) properties of the function in question (see, e.g., for a general discussion and for a demonstration of this effect for support structure design). Hence, it is desirable to use analytical sensitivities whenever possible. Examples of common heuristic methods are genetic algorithms, particle swarm algorithms and random search. The reason that these methods might be used over gradient-based methods is that no estimation of sensitivities is necessary in that case, seemingly avoiding the problem described above related to gradient estimation. However, as a trade-off, these gradient-free methods generally require a much larger number of iterations to convergence to the solution, since the methodology is typically founded on some kind of loosely guided (possibly random) search of the parameter space. While being able to overcome some of the weaknesses of gradient-based optimization, for simulation-based problems, the resulting added computational expense of heuristic methods might not be acceptable in practice. Since the gradient-based methods using analytical sensitivities are able to avoid the numerical issues associated with finite differences and obtain accurate gradient information, these methods are consequently preferable. Hence, we shall focus our attention on gradient-based methods from this point onwards.

It is a well-known result (see, e.g., ) that when the displacements u(t) of the structural system under dynamic loading S(t) are found by time integration of the equation of motion, given as 3M(x)u¨(t)+C(x)u˙(t)+K(x)u(t)=S(t), for mass matrix M, damping matrix C and stiffness matrix K, then the sensitivities of the displacements can be found by time integration of the following equation: 4M(x)du¨(t)dx+C(x)du˙(t)dx+K(x)du(t)dx=dS(t)dx-dM(x)dxu¨(t)+dC(x)dxu˙(t)+dK(x)dxu(t). Hence, if the non-linear constraints can be expressed as analytical functions of the displacements, the sensitivities are obtainable via (possibly repeated) application of the chain rule and finally the solution of Eq. (). It is presently assumed that the system matrices (M, C and K) are known analytical functions of the design variables, as is the case when the structural analysis is based on finite element modeling with beam elements defined according to Euler or Timoschenko beam theory. If this is not the case, the use of semi-analytical methods (where the gradients of the system matrices are estimated with finite differences) must be used. For OWT support structures, it was shown in how the sensitivities of both ULS and FLS constraints could be obtained using the analytical approach described above.

The main distinguishing feature, with respect to the problem structure defined in Eq. (), of optimization under uncertainty, is the addition of a new set of stochastic variables θ that in general can enter both the objective function and the constraints. In fact, some or all of the design variables in x could be replaced (or depend on) variables in θ. However, in our case, we shall restrict the discussion to cases where all the design variables are deterministic. It follows that the only θ dependence must then be in the so far to be determined non-linear constraints cj. In RBDO, the main idea is that we seek to constrain (and/or, in some formulations, optimize) the reliability of the system. The reliability of a structural system is a probabilistic measure of its ability to resist loads. In the most straightforward mathematical representation, this is expressed as the extent to which the load effect Q (usually depending on the response) does not exceed the resistance R (usually depending on the capacity or structural strength). In a probabilistic setting, this is quantified by the probability of failure Pf, defined as 5Pf=Prob(Q-R>0). Formally, the reliability is the probability of non-failure, 1-Pf, though commonly one tends to use Pf rather than the actual reliability in analysis and calculations. Furthermore, since the analogy of load effect and resistance is not always applicable, the notion of failure is usually represented by a limit state function g, encoding failure as positive function values, with the probability of failure as 6Pf=Prob(g>0). In general, g is a function of both x and θ, and to calculate Pf requires knowledge of the joint probability distribution hθ of all the stochastic variables in θ. An exact estimate of Pf is then given by the integral of hθ over the part of its domain where g>0, i.e., 7Pf=∫g(x,θ)>0hθ(θ′)dθ′. The general RBDO problem may then be formalized as 8min⁡xfmass(x)such thatAlinx≤bx≤xux≥xlcj(x)≤0∀j∈JdetPf,j(x)≤Pf,jmax∀j∈Jprob, where Jdet and Jprob represent the indices of deterministic and probabilistic constraints, respectively, and Pf,jmax are the desired upper bounds on the probabilities of failure Pf,j.

However, for any but the most trivial limit state functions, the determination of the values of x and θ giving g>0, and hence the determination of the integral in Eq. (), cannot be done exactly. The most straightforward and robust way to accommodate this is through the use of sampling methods. In particular, the family of Monte Carlo and quasi-Monte Carlo methods is typically used. These methods generally have the property that for a large enough sample size, the resulting estimate Pf^ tends towards the exact value of Pf. Unfortunately, large enough can be an intractable requirement. While the use of variance reduction techniques can speed up the convergence, as the dimensionality and complexity of the problem grows, the number of samples does too. This can be particularly problematic when one or more simulations are required for each sample. Furthermore, sampling methods do not naturally lend themselves well to gradient-based optimization due to the additional effort involved in the calculation of the gradients of a quantity estimated by sampling. In some cases, when the design variables are stochastic, the use of what is called score functions for the estimation of design sensitivity is possible, in which case no additional samples are needed (see, e.g., ). At the very least, no analytical gradients can be obtained. Hence, it is common to make use of first- and second-order approximations of the limit state function, making integration over the g>0 region feasible.

Surrogate modeling is generally a vast topic and the interested reader is referred to and for more general overviews, as well as to for the high-dimensional model representation approach and and for more detailed looks at Gaussian process regression (GPR). For applications to RBDO in general, and are instructive.

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 . We note that GPR is often referred to as kriging in the engineering literature. Although, for most practical purposes, the two terms can be used interchangeably, GPR is more general. Hence, to avoid specificity where it is not needed, we will use the term GPR.

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.

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 with the cross-sectional dimensions of each segment initially corresponding to the OC3 design, i.e., with a uniform monopile segment and a linearly tapered tower segment. The element lengths are consistent within each major segment but differ between the tower and monopile. Furthermore, this model also includes a point mass on the top of the tower, with mass and inertia properties meant to represent the National Renewable Energy Laboratory (NREL) 5 MW turbine . This model will be referred to as the “OC3 Monopile”. Some of the basic properties of these two models are listed in Table , and the material properties are consistent with the ones in . The models are fixed (clamped) at one end (at a location that corresponds to the mudline for the OC3 Monopile); i.e., there is no modeling of soil included. This will affect the global stiffness and change the dynamics of the structural models somewhat but is not expected to have a large effect on how these models function in terms of testing the RBDO method. Both models are loaded at the top with force and moment time series extracted from fixed rotor simulations of the NREL 5 MW turbine subject to turbulent wind fields within the aeroelastic Fedem Windpower software . Note that the externally input rotor loads are only calculated once and are taken as design independent, though the response to these loads is calculated for every design. These forces and moments are input into the dynamic simulation as loads on each of the 6 degrees of freedom on the top node of the tower. Wave loads are represented by a horizontal force time series, applied at a location corresponding to the bottom of the tower in the OC3 Monopile and at an analog location for the Simple Beam. In the dynamic simulation, this is input as a load on the degree of freedom corresponding to displacement in the mean wind direction, but no other degrees of freedom are loaded by this force. This force has been tuned to give an equivalent moment at the lower end of the structures as the integrated contribution of all horizontal wave forces along the height of the water column at each instant in time. These forces are calculated from the Morison equation, based on wave kinematics sampled from the Joint North Sea Wave Project (JONSWAP) spectrum and including Wheeler stretching. The inertia and drag coefficients are 2.0 and 0.8, respectively. Since the wave loads depend on the diameter of the relevant members, these loads are recalculated for every design before being input to the dynamic simulation. The duration of the applied loads is 600 s after the removal of initial transients. Up to four different loading scenarios are included in the analysis, with environmental data based on the Ijmuiden Shallow Water Site and using different random seeds for each realization of wind and wave conditions. The probabilities of occurrence for fatigue estimation have been renormalized so that they sum to 1 for the reduced set of cases. These loading scenarios are summarized in Table .

As indicated in Eq. (), the optimization is constrained with upper and lower bounds on the design variables. This is done from a theoretical point of view in order for the final designs to be somewhat realistic with respect to practical constraints related to manufacturing, transportation and installation that are not specifically accounted for in the modeling. From a more practical point of view, some decreased numerical performance was observed, during initial testing, once the design variables went outside a certain range, especially for certain combinations of values for some variables. For this reason, the upper and lower bounds were set stricter than normal practice would dictate. In particular, the lower bounds are for both models set at 70 % of the smallest initial diameter and thickness, respectively; the upper bounds are set at 150 % of the largest diameter and thickness, respectively. We refer to Table  for the smallest and largest values for each model. While these bounds do not directly correspond to any specific real limits, the restriction is not expected to rule out any design of practical interest. As for linear constraints, we will in some cases include an upper limit on the D-t ratio of 120, consistent with the NORSOK standard . In terms of non-linear constraints, we consider limits on the accumulated 20-year fatigue damage and on the maximum bending moment, to be detailed below. Fatigue calculations are done as follows. From the displacements obtained as the solutions to Eq. (), the internal forces and moments Sin are obtained by multiplication with the element stiffness matrices Ke as 22Sin=Keue. From the components of Sin corresponding to the axial force Sax, in-plane and out-of-plane moments Sip and Sop, the normal stress σn is then identified as 23σn=SaxAc+D2Ic(Sip-Sop), where Ac=π((D2)2-(D2-t)2) is the cross-sectional area and Ic=π4((D2)4-(D2-t)4) is the second moment of area. Rainflow counting is then applied to the normal stress time series, resulting in a set of amplitudes Δσi that correspond to the differences between stresses at particular times (a stress cycle), encoded in the vectors tpeak and tvalley, i.e., 24Δσi=σn(tpeak(i))-σn(tvalley(i)). Unlike what is otherwise common practice, these amplitudes are not binned. This is in order to facilitate the sensitivity analysis. The incurred fatigue damage F is then estimated by use of the Palmgren–Miner linear summation rule, where the contribution from each stress cycle is given by application of the appropriate SN curves and thickness correction : 25F=∑iniaiΔσi-wittref-kwi, where ni is either 0.5 or 1.0 depending on whether the given cycle is a half or full cycle, ai is a constant, wi is the Wöhler exponent, tref is the reference thickness below which no correction is necessary, and k is the thickness correction exponent. The constants ai, wi, tref and k are set using the SN curves for welds in tubular joints (in air and in water with corrosion protection, respectively, depending on the element) found in . The total lifetime fatigue damage, Ftot, from all considered environmental states, E, is then 26Ftot=Tlf∑EF(E)Pocc(E), where Tlf is a factor scaling up from simulation time to a 20-year lifetime, F(E) evaluates Eq. () for each state, E and Pocc are the corresponding probabilities of occurrence for these states. The limit state function for fatigue, measuring the extent to which the lifetime fatigue damage exceeds the fatigue resistance ΔF (usually set to 1), used as a constraint in the deterministic optimization problem, is hence 27Ftot-ΔF≤0. For the maximum bending moment, the calculation is based on Sip as obtained from Eq. (). However, since the use of global maxima can cause problems with smoothness (the global maximum may not always change smoothly) and since checking the bending moment at every time step would be very time consuming, a compromise is made. In particular, the Kreisselmeier–Steinhauser function is used to smoothly aggregate the bending moment time series into an upper envelope of the maximum: 28MKS=Sip,max+1rlog⁡∑iexp⁡(Sip(ti)-Sip,max)r, where the subscript max denotes the global maximum, and r is a constant controlling the accuracy of the approximation. The above expression approaches Sip,max from above as r approaches infinity. For computations, r is typically taken to be 50–200 (set to 200 here) depending on desired accuracy (see for a discussion of this for OWT applications). Note that, algebraically speaking, Sip,max cancels out

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 .

The estimation of gradients for the objective function as given in Eq. () and the linear constraints is trivial. For the non-linear constraints, it is mainly a question of repeated application of the chain rule as well as the rule of total derivatives for multivariate functions. See, e.g., for details of how this can be done (the only modification in our case being the additional level added by Eq. , which is easily differentiated). However, note that, due to how the displacement sensitivity is calculated in Eq. (), regardless of location in the structure, there is always a dependence on each design variable for every non-linear constraint. Hence, none of these derivatives are zero in general.

For the RBDO formulation based on PMA, the limit state functions represented by Eqs. () and () can be directly used, with the understanding that Ftot, Δf, MKS and fy become probabilistic quantities. Using the notation in Eq. (), we have 30gfls=yfls(θq)F̃tot(x)-ΔF31guls=yuls(θq)M̃KS(x)-Z(x)γM0.94fy-0.76120Efy2, where, for simplicity, we define ΔF:=θΔF and fy:=θfy. The sensitivities then follow from Eqs. (), () and the above discussion. In this study, a target reliability index of βmax=3.3, corresponding to Pf=4.8×10-4(≈5×10-4), will be used for the solution of the PMA subproblem in Eq. () as part of Algorithm 1. The uncertainties that are included in the response are global stiffness, global damping and turbulence intensity. The uncertainty modeling will be detailed below.

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 , both of which are common examples of gradient-based non-linear constrained optimization algorithms. In principle, the PMA-based reliability problem can be solved more efficiently by use of the hybrid mean-value method or related approaches, but due to the use of the surrogate model, this is not deemed necessary for the current application. Convergence of the optimization is based on fairly standard criteria, with termination of the algorithms when either relative first-order optimality (see, e.g., ) is achieved with a tolerance of 10-6 or the relative changes in the design variables are less than 10-6. Solutions are required to be feasible with a tolerance of 10-6.

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 ν=3/2 was chosen, including the use of individual length scale hyperparameters for each input variable (this implements what is known as automatic relevance determination, in principle de-emphasizing less relevant input variables in the regression problem; see, e.g., ). Overall, this was found to be the most robust for the regression problem in this study, especially when considering repeated regression for additional iterations of the outer loop in Algorithm 1. The Matérn class of kernels was also used for OWT support structures in . We also note here that in order to simplify the simultaneous regression with respect to all of the three parameters in θq, these parameters were input to the fitting problem in such a way that the surrogate model became co-monotonic in every variable (an increase in one or more variables giving always an increase in the output). In this case, that meant inputting the inverse (1/θi) of the parameters controlling damping and stiffness. Furthermore, these parameters were implemented as scaling variables with means of 1.0, such that the actual variables as input to the simulations were a product of the deterministic values and the respective stochastic scaling parameters in θq. The hyperparameters of the Gaussian process model were fit using Bayesian global optimization methods (expected improvement) . The noise standard deviation was taken to be non-zero and also fit during this procedure, even though the simulation outputs used in the fitting are in a certain sense noise-free. This was done because it was seen to give more robust surrogates with respect to changes in the design.

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 y‾new and y‾old, the mean prediction of the new and old surrogate models, respectively, and by σy,old, the predicted standard deviation of the old model, then if 32|y‾new-y‾old|≤σy,old, for every test sample point, the surrogate model is not updated. If no surrogate models are updated after an outer iteration, the procedure terminates. In order for this relaxed tolerance not to give infeasible results with respect to the new surrogate model (which is not used when it is within the above tolerance), the surrogate predictions for y(θq), used to compute the numerical values for the limit state functions in Eqs. () and (), are based on the mean plus the standard deviation, y‾+σy, rather than just the mean. This guarantees that when y‾old is used instead of the updated model, the derived results remain strictly feasible with respect to mean of the more accurate prediction (which would otherwise have been used). Since the standard deviations tend to be ∼[10-3,10-2], this does not have a large impact on the results.

The objective function for case BEAM-PA-CON is shown in Fig. a. Note how there are only very minor changes after the first loop. The small modifications to the design variables in the second and third loops are caused by the updates in the probabilistic constraints, seen in Fig. b. The final design is characterized by an overall minimization of thickness while the diameter is increased, as seen in Fig. c. Comparing with the corresponding deterministic case BEAM-DA-CON, the main difference is a slightly more conservative design, as would be expected. The corresponding plots are not shown, as they are almost identical, but results for both cases are summarized in Table . Note that the amount of outer iterations for loop 1 of BEAM-PA-CON is about the same as the total number of iterations for BEAM-DA-CON.

The results for BEAM-PA and BEAM-DA are shown in Figs.  and , respectively. Compared to the cases with connected design variables, there is an (expected) increase in the number of iterations required to solve the problem, and the resulting designs are different in the way that the dimensions are reduced for elements higher in the structure. This is a natural consequence of the fact that the loads are higher towards the bottom and the constraints there will be stricter in terms of allowable cross-sectional dimensions. Otherwise, the results are similar. The non-linear constraints are somewhat closer to being active at the solution of the RBDO problem compared to the deterministic case, which was true previously but is more apparent for these cases. Detailed summary results are for these cases displayed in Table .

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.

Beginning with the two basic cases for the OC3 Monopile, OC3-PA and OC3-DA, displayed in Figs.  and , respectively, we see that the behavior is fairly similar to the Simple Beam cases without connected design variables. Despite having more than twice the number of design variables, convergence is achieved in about the same number of iterations. The main new detail in the solution is that the second element from the bottom does not follow the otherwise apparent pattern of monotonically increasing diameters from top to bottom. In fact, this element has a smaller diameter than the element above, with a comparatively increased thickness to compensate. This is expected to be due to the wave loads, which are driven more by the diameter. The reason this does not happen for the Simple Beam cases is most likely because the smaller number of elements cannot resolve this effect. Otherwise, there is in the probabilistic case a much larger constraint violation at some intermediate points and the objective function initially increases above its starting value, but this does not seem to have much of an effect on the overall solution. More detailed results are shown in Table .

Next, the effect of including D-t ratio constraints for all elements is shown in the results from case OC3-PA-DT in Fig. . With this constraint in place, the low-thickness, high-diameter solution obtained previously is no longer feasible and the result is a solution which balances the reduction more evenly among the thicknesses and diameters. The result is in a sense more pleasing from a practical point of view, since it is more in line with a design that would actually be manufactured; both due to the lack of very large diameters and because one avoids the wave-load-induced “hourglass shape” seen in the previous two cases. The convergence is also faster (73 vs. 140 iterations), though there is one additional (very short) outer iteration required. On the other hand, the D-t ratio constraint is a lot more strict overall and less than 10 % reduction in mass is possible. In fact, the initial OC3 design is not feasible with respect to this constraint, which is why the objective function is increased by quite some amount at the beginning of the first loop. Note also that, as opposed to other comparable cases, the final design is softer (at 0.24 Hz) than the initial design (at 0.28 Hz).

Randomizing the initial OC3 design gives the results displayed for OC3-PA-RND in Fig. . This initial design is both heavier and much less feasible (a probability of failure of 1 essentially in several locations) than the initial OC3 design, which seems to make the convergence a bit slower in this case but not by too much. The solution is not exactly the same as for OC3-PA, but the difference is negligible (1 % or less in the design variables and less than 0.005 % in the objective). This is within the expected variation caused by the small inherent randomness in the surrogate modeling.

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. , and , respectively. As would be expected, the removal of the wave loads leads to a slightly lighter design and one where the element diameters consistently decrease from bottom to top. The difference in convergence behavior is likely negligible and mostly due to the randomness in the surrogate modeling. The resulting design has a slightly higher utilization of extreme loads. Since the fatigue constraints generally dominate over the extreme load constraints, the results for OC3-PF are as expected, with negligible differences in the final solution compared with OC3-PA and OC3-PA-RND. Conversely, using only extreme load constraints as in OC3-PU results in a final design that has about 16 % less mass than OC3-PA. This case also removes the visible effect of the wave loads on the solution, most likely because (at least for this loading scenario) the extreme loads caused by the waves are much less significant than the corresponding fatigue loads. Otherwise, the behavior is more or less as in the other cases.

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

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 D-t ratio constraint, though the thickness is also driving in this case. However, since the thickness cannot be arbitrarily smaller than the diameters in this case, the result is an effective upper bound on the thickness corresponding to the values where the overall design (with much smaller diameters than the other cases) is at the boundary imposed by the non-linear constraints. All in all, this is beneficial for the design process, since reliability-based constraints do not seem to change anything fundamental about the problem or introduce anything phenomenologically new from the design point of view. By and large, this indicates that as long as the structural and load models can be successfully adapted to the probabilistic setting, e.g., in the manner done in this study, then most if not all of previous knowledge and experience from deterministic design optimization is still valid and useful. On the other hand, this should not be taken to mean that probabilistic constraints can be easily replaced by more conservative deterministic ones. There is no way to determine sensible limits for such constraints – sensible here in the sense of being sufficiently safe while not being overly conservative – without performing some kind of non-deterministic analysis. Such analyses, for example, probabilistically tuned partial safety factors as basis for deterministically constrained optimization, are not necessarily more efficient than the present RBDO framework and cannot account for any potentially design-dependent changes that would be naturally accounted for with our methodology.

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 (). Finally, the uncertainty analysis is mostly consistent with previous work in terms of the specific modeling but uses a smaller number of uncertainties than has typically been part of reliability studies. This can be seen as somewhat limiting in regards to the above points about the lack of added computational complexity, but it should be noted that it is usually possible to reduce the number of uncertainties down to a level closer to that of the present study by careful preliminary studies of the sensitivity to each uncertain parameter and subsequent elimination of all but the most important parameters. The automatic relevance determination of the GPR approach used presently is also advantageous for such a purpose.