Reliability Updating of Offshore Structures Subjected to Marine Growth

: Marine growth is a known problem for oceanic infrastructure and has been shown to negatively impact the reliability of bottom-ﬁxed or ﬂoating offshore structures submitted to fatigue or extreme loading. Among other effects, it has been shown to change drag forces by increasing member diameters and modifying the roughness. Bio-colonization being highly random, the objective of this paper is to show how one-site inspection data increases reliability by decreasing uncertainties. This can be introduced in a reliability-based inspection framework for optimizing inspection and maintenance (here, cleaning). The modeling and computation are illustrated through the reliability analysis of a monopile in the European Atlantic area subjected to marine growth and according to the plastic collapse limit state. Based on surveys of structures in the North Sea, long-term stochastic modeling (space and time) of the marine growth thickness is ﬁrst suggested. A Dynamic Bayesian Network is then developed for reliability updating from the inspection data. Finally, several realistic (10–20 measurements) inspection strategies are compared in terms of reliability improvement and the accuracy of reliability assessment.


Introduction
Several weeks after immersion, a non-protected offshore structure is covered first by a bio-film and second by macro fouling [1].Macro fouling acts on the dynamic loading by increasing the mass of the structure, on the hydrodynamics regime by altering the relative roughness [2], on the vibratory response by changing the geometry and mass [3], on drag forces by increasing the diameter of structural members [4][5][6], and on drag coefficients due to changes in roughness [5,7].In the Gulf of Guinea, for instance, the thickness of corals up to 25 cm have been reported [8].This biological phenomenon is affected by the nature (material) and shape of the colonized surface, the age of the structure and the season of installation, the hydrodynamic conditions that affect the fixing and arrival of larvae, the local bio-diversity and the extent of opportunities for colonization and competition [4], and long-term or short-term physio-chemical changes (death or development of the fouling) [8,9].Understanding this complexity remains challenging because it is site-specific, and it requires on-site measurements of the marine growth with time and the monitoring of environmental parameters.Ongoing projects such as SURFFEOL [10], ABIOP [11], LEHERO-MG [2,[12][13][14][15], and ABIOP+ [16] attempt to achieve this objective by installing specific buoys [10,17].Moreover, assessing the evolution of marine growth with time requires accurate non-destructive testing tools in underwater conditions.Recently, the development of specific image processing algorithms is promising [18][19][20], especially if coupled with underwater drones to increase the safety of divers and reduce costs.Based on these studies and expanding industry knowledge, the probabilistic modeling (stochastic processes in space and time) of macro fouling appears reachable.Updating models with inspection data for uncertainty reduction is required.Risk-based inspection [21,22] offers a theoretical framework that links reliability, inspection, and risk.This paper focuses on the random effect of marine growth on drag forces and coefficients.It addresses potential significant increases in the screen effect and modifications to the flow regime around oceanic structures.
First, the paper reviews the major effects of marine growth on offshore structures and the key parameters that should be modeled for a structural reliability assessment.Then, a long-term a priori model of growth with time is provided based on observations of an existing database.A Dynamic Bayesian Network is then introduced for modeling the structural reliability and updating marine growth parameters from inspection data.It is built and calibrated from the reliability assessment of a monopile subjected to marine growth.Finally, reliability updating is performed and several inspection strategies are compared on a study case in the North Sea where measurements of marine growth come from real field data.

Modeling of Marine Growth Thickness Stochastic Process
Most studies analyzing the hydrodynamic effects of biofouling deal with the behavior of small bodies, usually cylinder components, compared with wavelength; that is the case of offshore monopiles and the components (beams) of offshore jacket structures.Usually, papers aim at modeling the changes in the Morison equation [23].The present paper is well within this field of modeling.Generally, biofouling has several effects on loading to cylindrical structures and the flow regime.Increasing the diameter of the cylinder is the first direct effect of biofouling on the structural characteristics.The drag and inertia loading on the body are thus modified due to the increase in the projected area, displaced volume, and changes to the hydrodynamic coefficients [4,24].Furthermore, flow instability will increase along with increases in effective diameter [25].The second effect reduces the shedding frequency, which directly affects the hydrodynamic loads [4].Moreover, the growth of biofouling on the structure will increase its structural weight [26].From a maintenance point of view, biofouling has some negative structural consequences.It obscures the structural surface and needs to be removed prior to any visual inspection or non-destructive testing and maintenance (welding) of the structure by divers or remotely operated vehicles [27].As another impact, biofouling increases surface roughness.The relative roughness ratio (e = k/D, where k is the average peak-to-valley and D is the structure equivalent diameter) from hard biofouling on offshore tubular structures is greater than 10 −3 and may exceed 10 −2 [28].These are significantly higher than those for normal cylinders, which changes the Strouhal number (St) [29,30].It is worth mentioning that many of the aforementioned studies used surface roughness as an instrument to promote the early transition of the boundary layer in order to simulate high-Reynolds flow phenomena at physically low Reynolds numbers [31][32][33][34][35].A smaller number of these studies were, in fact, addressing the marine fouling effect.
Bio-colonization is a complex process depending on several environmental variables and biological processes with many interactions between them and with the structural surface [27,37].The process is so complex it would be unrealistic to envisage a complete model involving a multilayer of various marine organisms that have complex interactions for survival, growth, and reproduction.Therefore, this paper focuses on the growth of a single dominant species on Atlantic coasts [38,39], the blue mussel Mytilus edulis, which was observed to be in the top three most frequent species.So far, there is neither a longterm model nor a database for the growth of mussels on offshore structures.So, this paper proposes prior stochastic modeling of a mussel's growth with available average and variance values from a sparse, available database of the North Sea and Atlantic coasts.For the modeling of the average thickness trend, a simplified growth model inspired by the model of Bayne [40] and Bayne and Worrall [41] is considered (1).It is derived from more general growth models [42,43] whose application fields have been shown to be various (see the review of Vincenzi et al. [44]).It was shown to be consistent with the average value of a mussel's length S (cm) for the French Atlantic coast [45].
where t is the time (in years).The aforementioned equation is used for the modeling of the trend of the average mussel's thickness value.In fact, it is shown [7,8] that the thickness reaches an asymptotic value after about a decade due to several factors: competition, diseases, chocks in environmental conditions, detachment due to storms, or self-weight.
A report of Veritec in the North Sea [46] is used in the present paper in view of obtaining a set of experimental data.The measurements of marine growth were reported from the inspection of 18 structures installed between 1970 and 1985.Each structure was inspected 2 or 3 times during this period.The earliest inspection was after 1 year and the latest after 15 years.In absence of a protocol, we know that there is heterogeneity in data collection: the depth of the inspection is reported but not the component or even its orientation (vertical, horizontal, . . .).So, for characterizing the prior thickness, the nominal value is defined as the maximum value obtained at a given age between 0 and −10 m depth among the set of structures inspected at this time.Note that this does not mean that it is the maximum value of the process at a given depth, only the maximum value measured in the area of the structure investigated by the diver.In the following, for conservative considerations, this nominal value is considered as the mean value for the model.Table 1 presents the thickness measurements collected after 15 years of inspection.These values are in line with the measurements published for the southern North Sea [47].Only the maximum measurements at each inspection time are retained for further analysis and it is observed that the marine growth thickness stabilizes at around 40 mm after 10 years.Due to the lack of prior information on a given site, some assumptions about the prior knowledge regarding parameters in (2) (α and β) must be based on available data.In this study, the initial growth velocity is assumed to be equal to 27.5 (mm/year) when the time t reaches 0, according to data obtained from [9].Hence, prior boundary conditions are established on (2) as follows: By solving the boundary conditions, we obtain α = 40 and β = 0.6875.The prior evolution of marine growth thickness is thus given in (3) and plotted in Figure 1.Parameter β in (1) characterizes the evolution trend of a specific mussel and environment.Therefore, in probabilistic modeling, β is considered a constant parameter.Parameter α in (1) represents the magnitude of the thickness of the marine growth.This should be adapted to each site.In this study, α is assumed to be normally distributed with a Coefficient of Variation (COV) equal to 0.3 [4].Based on these assumptions, the prior mussel's thickness curve Th prior is given in (4) and the evolution of statistics is plotted in Figure 1.
Field Measurement (mm)Measurement (mm) By solving the boundary conditions, we obtain α = 40 and β = 0.6875.The prior evolution of marine growth thickness is thus given in (3) and plotted in Figure 1.Parameter β in (1) characterizes the evolution trend of a specific mussel and environment.Therefore, in probabilistic modeling, β is considered a constant parameter.Parameter α in (1) represents the magnitude of the thickness of the marine growth.This should be adapted to each site.In this study, α is assumed to be normally distributed with a Coefficient of Variation (COV) equal to 0.3 [4].Based on these assumptions, the prior mussel's thickness curve Thprior is given in (4) and the evolution of statistics is plotted in Figure 1.For structures in other areas, a similar a priori distribution can be exploited from prior thickness on fixed structures or metocean buoys such as Biocolmar ® with the same material [48].

Dynamic Bayesian Networks for Probability Updating
Stochastic loading computation is required for the time-dependent reliability assessment of offshore structures.However, this is time-consuming, especially if changes in the For structures in other areas, a similar a priori distribution can be exploited from prior thickness on fixed structures or metocean buoys such as Biocolmar ® with the same material [48].

Dynamic Bayesian Networks for Probability Updating
Stochastic loading computation is required for the time-dependent reliability assessment of offshore structures.However, this is time-consuming, especially if changes in the process are observed (inspection cleaning) and because wave loading is a nonlinear process of the random input variables.In this paper, a metamodel is developed for uncertainty propagation in view of reducing the computational costs when implementing several strategies for maintenance.A Bayesian Network (BN) is an efficient approach to graphically modeling the probabilistic dependency between random variables using a directed acyclic graph (DAG).In the DAG, random variables are symbolized by nodes and are connected by edges to illustrate their dependencies.A conditional Probability Density Function (PDF), f(X|pa(X)) or Probability Mass Function (PMF), p(X|pa(X)) is assigned to each child node, where pa(X) is the parent of X in the DAG.An edge may represent causal relationships between the variables (nodes), but this is not a requirement.The graphical structure of a BN encodes conditional independence assumptions among the random variables.Hence, a BN is a compact model representing the joint PDF or PMF among random variables.In this Energies 2022, 15, 414 5 of 15 study, only BNs with discrete random variables are considered.The joint PMF of the BN with X 1 , X 2 , . . .X N is formed using the chain rule as: where P(X i |pa(X i )) is the conditional PMF of variable X 1 given its parent variables.Figure 2 illustrates a simple BN that consists of three nodes representing three discrete random variables, X 1 , X 2 , and X 3 , in which X 2 and X 3 are children of the parent node X 1 .The joint PMF of the BN present in Figure 2 is expressed as a product of the PMFs of all nodes in the BN: where P(X i |X j ) denotes the conditional probability of X i given X j .
graphically modeling the probabilistic dependency between random variables using a directed acyclic graph (DAG).In the DAG, random variables are symbolized by nodes and are connected by edges to illustrate their dependencies.A conditional Probability Density Function (PDF), f(X|pa(X)) or Probability Mass Function (PMF), p(X|pa(X)) is assigned to each child node, where pa(X) is the parent of X in the DAG.An edge may represent causal relationships between the variables (nodes), but this is not a requirement.The graphical structure of a BN encodes conditional independence assumptions among the random variables.Hence, a BN is a compact model representing the joint PDF or PMF among random variables.In this study, only BNs with discrete random variables are considered.The joint PMF of the BN with X1, X2,… XN is formed using the chain rule as: where P(Xi│pa(Xi)) is the conditional PMF of variable X1 given its parent variables.Figure 2 illustrates a simple BN that consists of three nodes representing three discrete random variables, X1, X2, and X3, in which X2 and X3 are children of the parent node X1.The joint PMF of the BN present in Figure 2 is expressed as a product of the PMFs of all nodes in the BN: where P(Xi |Xj) denotes the conditional probability of Xi given Xj.BNs allow for the introduction of new information (evidence) from the observed nodes (here, inspection data) to update the probabilities in the network.For example, if we have some evidence ο to introduce to node X2 (X2 = o), this information propagates through the network, and the joint PMF of the two other nodes can be recalculated as: A DBN is an extended form of a BN, which is often used to present random processes and can be used for modeling a system's performance over its lifetime.Figure 3 presents a simple form of a DBN consisting of T slices.Slices in the DBN are connected by links from nodes in slice (i − 1) to nodes in slice i.At each slice is a BN that represents the system performance at that time.The state of the system at slice i depends only on its state in slice (i − 1); hence, this DBN could be seen as a Markov Chain.Each slice i of the DBN presented in Figure 3 consists of two nodes, Xi and Yi, in which their joint probability could be computed as: BNs allow for the introduction of new information (evidence) from the observed nodes (here, inspection data) to update the probabilities in the network.For example, if we have some evidence o to introduce to node X 2 (X 2 = o), this information propagates through the network, and the joint PMF of the two other nodes can be recalculated as: A DBN is an extended form of a BN, which is often used to present random processes and can be used for modeling a system's performance over its lifetime.Figure 3 presents a simple form of a DBN consisting of T slices.Slices in the DBN are connected by links from nodes in slice (i − 1) to nodes in slice i.At each slice is a BN that represents the system performance at that time.The state of the system at slice i depends only on its state in slice (i − 1); hence, this DBN could be seen as a Markov Chain.Each slice i of the DBN presented in Figure 3 consists of two nodes, X i and Y i , in which their joint probability could be computed as: where P(X i−1 ) is the marginal probability distribution of node X i−1 in slice (i − 1), P(X i |X i−1 ) is a transition matrix presenting the conditional probability between the two adjacent slices of node X, and P(Y i |X i ) defines the conditional probability of node Y i given X i in slice i.
( ,  ) = ( )( | ) ( | ) where P(Xi−1) is the marginal probability distribution of node Xi−1 in slice (i − 1), P(Xi│Xi−1) is a transition matrix presenting the conditional probability between the two adjacent slices of node X, and P(Yi│Xi) defines the conditional probability of node Yi given Xi in slice i.

DBN for Structural Reliability in the Presence of Marine Growth
This section presents how to construct the DBN for reliability modeling and updating the marine growth process.The analysis is performed on a simple structure that is common (81% of grid-connected foundations in Europe in 2019 [49]) in the offshore wind energy industry: the monopile in steel.It is perfectly embedded in the soil and the water depth is 30 m, which is a typical figure for offshore wind turbines (average water depth of online offshore wind farms in 2018 [49]).
Only the wave loading is considered in this paper, as the wind loading is not affected by marine growth; the upper part of the turbine (mast, turbine) is thus not modeled.In view of accounting for the loading of all extreme waves, the total height of the monopile above the soil level is 40 m (Figure 4, left).Moreover, this foundation is shown to be affected by mussels [50,51]; the model presented in Section 2.1 will thus be used.Only one loading case and limit state are considered for this study: the ultimate limit state in the presence of a storm.First, a beam finite element model (BFEM) of a monopile foundation subjected to marine growth colonization was developed (Figure 4, left).Five beams were selected because this was shown to be accurate for representing the loading [52].A flowchart describing the algorithms for load computation according to Morison equations on the BFEM is presented in Figure 5.The wave loading on the monopile is computed from the well-known Morison equations because the diameter of the monopile is small compared to the wave length [23].Because the maximum wave load is obtained when the crest of the wave reaches the monopile, only the drag force is considered.The marine growth data is characterized by the growing thickness (Th), generated by employing Latin Hypercube simulations in (1).According to the actual knowledge [8,53], a decreasing profile is modeled by using a piecewise-constant model: from 0 to −5 m with the full thickness (4), from 5 to −10 m with half this thickness, and from −10 to bottom without marine growth.Another key parameter [2,7,15] is the roughness of the colonization surface (k): this is simulated by using the yearly growth using the gamma process according to [9,54].The marine growth is assumed to be homogeneous around the pile.As a consequence, the wave is unidirectional (Figure 4 left).The metocean data, represented by a couple of values of significant wave height (H) and extreme wave period (T), are introduced to calculate the instantaneous flow velocity according to the Airy theory at each node.These values come from the paper by Ameryoun [9] and are derived from the study in [55].Their joint PDF is plotted in Figure 4 on the right.

DBN for Structural Reliability in the Presence of Marine Growth
This section presents how to construct the DBN for reliability modeling and updating the marine growth process.The analysis is performed on a simple structure that is common (81% of grid-connected foundations in Europe in 2019 [49]) in the offshore wind energy industry: the monopile in steel.It is perfectly embedded in the soil and the water depth is 30 m, which is a typical figure for offshore wind turbines (average water depth of online offshore wind farms in 2018 [49]).
Only the wave loading is considered in this paper, as the wind loading is not affected by marine growth; the upper part of the turbine (mast, turbine) is thus not modeled.In view of accounting for the loading of all extreme waves, the total height of the monopile above the soil level is 40 m (Figure 4, left).Moreover, this foundation is shown to be affected by mussels [50,51]; the model presented in Section 2.1 will thus be used.Only one loading case and limit state are considered for this study: the ultimate limit state in the presence of a storm.First, a beam finite element model (BFEM) of a monopile foundation subjected to marine growth colonization was developed (Figure 4, left).Five beams were selected because this was shown to be accurate for representing the loading [52].A flowchart describing the algorithms for load computation according to Morison equations on the BFEM is presented in Figure 5.The wave loading on the monopile is computed from the well-known Morison equations because the diameter of the monopile is small compared to the wave length [23].Because the maximum wave load is obtained when the crest of the wave reaches the monopile, only the drag force is considered.The marine growth data is characterized by the growing thickness (Th), generated by employing Latin Hypercube simulations in (1).According to the actual knowledge [8,53], a decreasing profile is modeled by using a piecewise-constant model: from 0 to −5 m with the full thickness (4), from 5 to −10 m with half this thickness, and from −10 to bottom without marine growth.Another key parameter [2,7,15] is the roughness of the colonization surface (k): this is simulated by using the yearly growth using the gamma process according to [9,54].The marine growth is assumed to be homogeneous around the pile.As a consequence, the wave is unidirectional (Figure 4 left).The metocean data, represented by a couple of values of significant wave height (H) and extreme wave period (T), are introduced to calculate the instantaneous flow velocity according to the Airy theory at each node.These values come from the paper by Ameryoun [9] and are derived from the study in [55].Their joint PDF is plotted in Figure 4 on the right.
Knowing this kinematic field and marine growth, hydraulic parameters (Reynolds Re and Keulegan-Carpenter KC numbers) are estimated following the procedure already published in [5,52].The hydrodynamic coefficients (drag coefficient C D and inertia coefficient C m ), which are useful for modeling the fluid-structure interaction, can then be calculated from these hydraulic parameters.
The output from BFEM is the maximum (at the sea bed) elasticity stress σ, which can be used to define the limit state g(X) of the structure, as follows: where σ e is the yield stress of the steel and X is the vector of random variables.Knowing this kinematic field and marine growth, hydraulic parameters (Reynolds Re and Keulegan-Carpenter KC numbers) are estimated following the procedure already published in [5,52].The hydrodynamic coefficients (drag coefficient CD and inertia coefficient Cm), which are useful for modeling the fluid-structure interaction, can then be calculated from these hydraulic parameters.
The output from BFEM is the maximum (at the sea bed) elasticity stress σ, which can be used to define the limit state g(X) of the structure, as follows: where σe is the yield stress of the steel and X is the vector of random variables.Table 2 gathers the input parameters and random variables used for numerical simulations.Uncertainties are propagated throughout the BFEM by generating 30,000 samples based on Latin Hypercube Simulations for each random variable.Data obtained from the BFEM analysis are used to construct the DBN (Figure 6) by computing the Conditional Probability Tables (CPTs) for all nodes in the network.The DBN configuration consists of T slices in which a BN at each slice represents the structural performance at that time.There are 4 nodes in each slice, representing 4 basic random variables: coefficient α, thickness (th), maximum stress (σ), and the limit state (g).The CPTs P(thi│αi), P(σi│thi), and P(gi│σi) define the relationships between parent and child nodes within a slice, and the transition matrices P(αi│αi−1) and P(thi│αi, thi−1) describing the dependencies between two neighbor slices are determined from the FEA data.Note that P(αi│αi−1) is a unit matrix since α is a time-invariant parameter.For this limit state function, the probability of failure −P f is determined by integrating the joint density function f(X) of the random vector over the failure domain: Table 2 gathers the input parameters and random variables used for numerical simulations.Uncertainties are propagated throughout the BFEM by generating 30,000 samples based on Latin Hypercube Simulations for each random variable.Data obtained from the BFEM analysis are used to construct the DBN (Figure 6) by computing the Conditional Probability Tables (CPTs) for all nodes in the network.The DBN configuration consists of T slices in which a BN at each slice represents the structural performance at that time.There are 4 nodes in each slice, representing 4 basic random variables: coefficient α, thickness (th), maximum stress (σ), and the limit state (g).The CPTs P(th i |α i ), P(σ i |th i ), and P(g i |σ i ) define the relationships between parent and child nodes within a slice, and the transition matrices P(α i |α i−1 ) and P(th i |α i , th i−1 ) describing the dependencies between two neighbor slices are determined from the FEA data.Note that P(α i |α i−1 ) is a unit matrix since α is a time-invariant parameter.Table 2 gathers the input parameters and random variables used for numerical simulations.Uncertainties are propagated throughout the BFEM by generating 30,000 samples based on Latin Hypercube Simulations for each random variable.Data obtained from the BFEM analysis are used to construct the DBN (Figure 6) by computing the Conditional Probability Tables (CPTs) for all nodes in the network.The DBN configuration consists of T slices in which a BN at each slice represents the structural performance at that time.There are 4 nodes in each slice, representing 4 basic random variables: coefficient α, thickness (th), maximum stress (σ), and the limit state (g).The CPTs P(thi│αi), P(σi│thi), and P(gi│σi) define the relationships between parent and child nodes within a slice, and the transition matrices P(αi│αi−1) and P(thi│αi, thi−1) describing the dependencies between two neighbor slices are determined from the FEA data.Note that P(αi│αi−1) is a unit matrix since α is a time-invariant parameter.In this study, we only consider the DBN with discrete nodes; therefore, all nodes are discretized into a number of finite states (Table 3).Each node is divided into a number of states in pre-defined boundaries, which could cover possible values.The discretization of Water Particle Velocity (u((x, z),H,T)  In this study, we only consider the DBN with discrete nodes; therefore, all nodes are discretized into a number of finite states (Table 3).Each node is divided into a number of states in pre-defined boundaries, which could cover possible values.The discretization of nodes in the DBN could introduce approximation errors [56]; however, this issue is beyond the scope of this study.

Results
This section is devoted to performing a reliability assessment of a monopile foundation subjected to marine growth from inspection data, i.e., the measurement of marine growth thickness (th).Numerical measurements are generated from Monte Carlo simulations by assuming that observations are simulated from Equation ( 4), where α obs follows a normal PDF with mean µ obs α = 0.027 and a Coefficient of Variation COV obs α = 0.2.Note that the mean value and CoV of the observations are selected to be lower than the a priori conservative one.Section 3.1 analyses the effect of the formulation of a limit state, condition-based or performance-based.The Section 3.2 investigates the importance of the number of measurements during an inspection for reliability updating.Then, the following section analyses the effects of combining measurement data from two inspection dates for updating.

Comparison between Two Limit States
We now assess the reliability by considering two formulations of the ultimate limit state.Considering the limit state based on structural performance, i.e., defined from elasticity stress ( 9), the prior probability of failure after 25 years, estimated from the DBN, is 1 × 10 −3 (Figure 7).Note that reassessment from a structural reliability assessment could be time-consuming and that the stakeholder could be interested in basing their decision-making on a second formulation of the limit state g' relying on the condition state (marine growth thickness).This condition-based limit state writes (11):

Results
This section is devoted to performing a reliability assessment of a monopile foundation subjected to marine growth from inspection data, i.e., the measurement of marine growth thickness (th).Numerical measurements are generated from Monte Carlo simulations by assuming that observations are simulated from Equation ( 4), where αobs follows a normal PDF with mean μobs α = 0.027 and a Coefficient of Variation COVobs α = 0.2.Note that the mean value and CoV of the observations are selected to be lower than the a priori conservative one.Section 3.1 analyses the effect of the formulation of a limit state, condition-based or performance-based.The Section 3.2 investigates the importance of the number of measurements during an inspection for reliability updating.Then, the following section analyses the effects of combining measurement data from two inspection dates for updating.

Comparison between Two Limit States
We now assess the reliability by considering two formulations of the ultimate limit state.Considering the limit state based on structural performance, i.e., defined from elasticity stress (9), the prior probability of failure after 25 years, estimated from the DBN, is 1 × 10 −3 (Figure 7).Note that reassessment from a structural reliability assessment could be time-consuming and that the stakeholder could be interested in basing their decisionmaking on a second formulation of the limit state g' relying on the condition state (marine growth thickness).This condition-based limit state writes (11):  The thickness threshold th th should be defined assuming that an acceptable probability of failure P fa after 25 years is the probability reached by the previous formulation of the limit state: P fa = 1 × 10 −3 , so the threshold is then th th = 0.078 (m).Based on this value, the prior probability of failure defined from this second limit state is computed and reported in Figure 7.
The results in Figure 7 show that, given a similar probability of failure at 25 years, the evolution of P f computed from the two limit states presents significant discrepancies.At a given time, the prediction of P f from elasticity stress is higher than the one computed from the thickness measurement.The reliability assessment of structures subjected to marine growth from only thickness measurements is, thus, not sufficiently conservative and a full structural reliability assessment is mandatory.In the following, only limit state ( 9) is used.

Reliability Updating from One Inspection Time
Let us focus now on the effect of the number of measurements (evidence) on reliability updating.It is assumed that the first inspection time is performed after the bio-colonization is assumed to be mature, here, five years.This strategy is justified by the fact that an inspection should be carried out neither too early (in the first two years) nor too late in view of capturing the key information about trends and planning preventive maintenance.The assessment considers a typical maintenance ship (mean velocity of 10 knots, https: //atlantique-scaphandre.fr/pdf/miniplon.pdf,accessed on 29 December 2021) and a field 30 km from the coast (mean value of bottom-fixed offshore wind-turbines [49]), and 10 h of work, including 3 h of traveling, 2 h of deployment and retrieval, and a 5 h dive.Due to the diving time restrictions, it is assumed that a single diver is carrying out the measurements at a given time.That means that about 20 measurements can be performed (four components per hour).This figure corresponds to the actual practice of the stakeholders [8,46].First, the reliability updating from 20 measurements is analyzed; these are used as evidence for the DBN update.The updating being affected by the statistical distribution of these 20 measurements, 1000 inspection data are simulated.Figure 8 (left) presents the prior probability of failure and the mean and 95% and 5% quantiles for the updated probability of failure P f with limit state g (9).
the evolution of Pf computed from the two limit states presents significant discrepancies.At a given time, the prediction of Pf from elasticity stress is higher than the one computed from the thickness measurement.The reliability assessment of structures subjected to marine growth from only thickness measurements is, thus, not sufficiently conservative and a full structural reliability assessment is mandatory.In the following, only limit state ( 9) is used.

Reliability Updating from One Inspection Time
Let us focus now on the effect of the number of measurements (evidence) on reliability updating.It is assumed that the first inspection time is performed after the bio-colonization is assumed to be mature, here, five years.This strategy is justified by the fact that an inspection should be carried out neither too early (in the first two years) nor too late in view of capturing the key information about trends and planning preventive maintenance.The assessment considers a typical maintenance ship (mean velocity of 10 knots, https://atlantique-scaphandre.fr/pdf/miniplon.pdf,accessed on 1 January 2022) and a field 30 km from the coast (mean value of bottom-fixed offshore wind-turbines [49]), and 10 h of work, including 3 h of traveling, 2 h of deployment and retrieval, and a 5 h dive.Due to the diving time restrictions, it is assumed that a single diver is carrying out the measurements at a given time.That means that about 20 measurements can be performed (four components per hour).This figure corresponds to the actual practice of the stakeholders [8,46].First, the reliability updating from 20 measurements is analyzed; these are used as evidence for the DBN update.The updating being affected by the statistical distribution of these 20 measurements, 1000 inspection data are simulated.Figure 8 (left) presents the prior probability of failure and the mean and 95% and 5% quantiles for the updated probability of failure Pf with limit state g (9).The extent of observed marine growth is smaller on average than that which was a priori in (4).Thus, the mean of Pf is smaller with a difference of 1 × 10 −4 , i.e., 10% of Pfa.It is interesting to underline that the difference between 5% and 95% quantiles and the mean The extent of observed marine growth is smaller on average than that which was a priori in (4).Thus, the mean of P f is smaller with a difference of 1 × 10 −4 , i.e., 10% of P fa .It is interesting to underline that the difference between 5% and 95% quantiles and the mean trend P f (Pf-mean in Figure 8) is not significant and reaches 1 × 10 −4 , i.e., 12% of P f .Twenty measurements appear to give a good update with good accuracy on P f .Bayesian updating is influenced by the amount of available evidence.Let us now analyze the effect of the number of measurements.In Figure 8, right, P f is updated when only five measurements are taken at 5 years.It is clear that when fewer measurements are available, the gap between 5% and 95% of the posterior of P f increases, reaching 17 × 10 −5 , i.e., 20% compared to P f , and even the interest of the updating is questionable: the 5% fractile is 7.6 × 10 −4 .By varying the number of measurements for updating between 3 and 100, Figure 9 presents different quantiles of the posterior of P f at 5 years with a convergence trend when the number of measurements increases.Table 4 presents the relative errors computed with 5% and 95% with respect to the mean value of P f .In real practice, the structure owners require the minimization of the number of measurements in order to reduce inspection costs.For offshore structures, due to the high cost of transportation, labor, and hired vessels, 20 measurements appear to be a good compromise with an error in the assessment of P f of less than 6%, which could be an acceptable error level (Table 4).100, Figure 9 presents different quantiles of the posterior of Pf at 5 years with a convergence trend when the number of measurements increases.Table 4 presents the relative errors computed with 5% and 95% with respect to the mean value of Pf.In real practice, the structure owners require the minimization of the number of measurements in order to reduce inspection costs.For offshore structures, due to the high cost of transportation, labor, and hired vessels, 20 measurements appear to be a good compromise with an error in the assessment of Pf of less than 6%, which could be an acceptable error level (Table 4).

Reliability Updating from Two Inspection Dates
This section studies the effect of combining measurements from two inspection dates for reliability assessment.The objective of such an analysis is to study the efficiency of different inspection campaigns.
The same budget is defined for each strategy in terms of the number of measurements: 20 total measurements, which could be conducted either 5 or 10 years after the structure's installation (for example, 10 at 5 years and 10 at 10 years equals 20 total inspection data).Three inspection schedules are considered, as shown in Table 5. Figure 10 presents the updating of Pf for each of these schedules.Note that the hidden value of Pf, which is calculated from the assumed value of α (μobs α = 0.027 and COVobs α = 0.2), is also plotted in Figure 10.The Pf updated in three cases is lower than its prior one because of the lower

Reliability Updating from Two Inspection Dates
This section studies the effect of combining measurements from two inspection dates for reliability assessment.The objective of such an analysis is to study the efficiency of different inspection campaigns.
The same budget is defined for each strategy in terms of the number of measurements: 20 total measurements, which could be conducted either 5 or 10 years after the structure's installation (for example, 10 at 5 years and 10 at 10 years equals 20 total inspection data).Three inspection schedules are considered, as shown in Table 5. Figure 10 presents the updating of P f for each of these schedules.Note that the hidden value of P f , which is calculated from the assumed value of α (µ obs α = 0.027 and COV obs α = 0.2), is also plotted in Figure 10.The P f updated in three cases is lower than its prior one because of the lower mean assumption of α_obs.However, there are significant differences between the updated values of P f and their hidden values.This trend could be explained by the fact that the thickness measurements for the prior model were not categorized in horizontal and vertical components.This limitation leads to important differences between the updated and hidden values of P f .By using evidence from the measurements from two inspection dates (case 2), the mean of the updated probability of failure is lower than those in the case of using measurements from only one inspection time (case 1 and case 3) (Figure 10).This behavior highlights the importance of having more inspection dates during the service life of structures to obtain better predictions.

Discussion
Marine growth is experiencing a renewed interest in the reliability assessment of existing offshore wind turbines or the optimization of the maintenance of these turbines.This phenomenon being stochastic, this paper has proposed a long-term evolution of thicknesses based on the literature and some available data for the bio-colonization of mussels which is one of the dominant species in the North Sea and on the European Atlantic Coasts.Until now, the link between the long-term modeling of the marine growth, its effect on the structural reliability, and the updating from inspections have not been investigated.This paper offers a complete framework and illustrates its interest with three studies: the effects of the limit state formulation, updating with one inspection, and updating with two inspections with a limited budget along the service lifetime.
If the conclusions are site-specific, the paper provides an efficient framework for updating marine growth evolution with time that can be used at other sites.This approach can thus be transferred to other sites once the extreme values of marine growth are known.This highlights the need for on-site observatories and the share of knowledge through open science and open databases.

Discussion
Marine growth is experiencing a renewed interest in the reliability assessment of existing offshore wind turbines or the optimization of the maintenance of these turbines.This phenomenon being stochastic, this paper has proposed a long-term evolution of thicknesses based on the literature and some available data for the bio-colonization of mussels which is one of the dominant species in the North Sea and on the European Atlantic Coasts.Until now, the link between the long-term modeling of the marine growth, its effect on the structural reliability, and the updating from inspections have not been investigated.This paper offers a complete framework and illustrates its interest with three studies: the effects of the limit state formulation, updating with one inspection, and updating with two inspections with a limited budget along the service lifetime.
If the conclusions are site-specific, the paper provides an efficient framework for updating marine growth evolution with time that can be used at other sites.This approach can thus be transferred to other sites once the extreme values of marine growth are known.This highlights the need for on-site observatories and the share of knowledge through open science and open databases.

Conclusions
The paper focuses on an efficient structural reliability updating based on a Dynamic Bayesian Network to update the probability of failure with evidence (measurements) of marine growth.This network is built and calibrated on the basis of a structural finite element model of a monopile where the loading integrates, through the Morison equation, the marine growth effect on both the diameter and the change in hydrodynamical coefficients.
A real study case is then analyzed, using real data from the North Sea.Two limit states, one based on a performance criterion and assessed through full structural reliability and the second based on a condition assessment, are first compared.It was shown that the condition assessment is not sufficiently conservative and cannot be considered as a criterion for decision making (cleaning).
Then, several strategies for the inspection of marine growth thicknesses are compared with single or double inspection times.It was shown that, given a fixed budget for inspection, two inspection dates at 5 and 10 years can sufficiently improve reliability updating.
The key input required for using this framework is a long-term model of marine growth thickness.This paper provides, on the basis of some preliminary data and biological knowledge, a long-term stochastic model of thickness increase with time.
Future studies should integrate a cleaning strategy with more or less efficient cleaning methods.

Figure 1 .
Figure 1.Mussel's thickness evolution based on a database of the North Sea; the blue line represents the mean value trend and the red dashed line represents the mean ± standard deviation.

Figure 1 .
Figure 1.Mussel's thickness evolution based on a database of the North Sea; the blue line represents the mean value trend and the red dashed line represents the mean ± standard deviation.

Figure 4 .
Figure 4. General geometry and finite element model (left) and joint PDF of H and T (right).

From − 5
(m) to −19 (m) From −10 (m) to bottom (m) Th2 = 0.5 × Th Th3 = 0 × Th For this limit state function, the probability of failure −Pf is determined by integrating the joint density function f(X) of the random vector over the failure domain:  =  () 0 = () ()

Figure 5 .
Figure 5.Proposed flowchart for loading computation in the presence of marine growth (adapted from[9]).

Figure 5 .
Figure5.Proposed flowchart for loading computation in the presence of marine growth (adapted from[9]).

Figure 6 .
Figure 6.DBN modeling of the marine growth process and limit state assessment.

Figure 6 .
Figure 6.DBN modeling of the marine growth process and limit state assessment.

Figure 7 .
Figure 7. Probability of failure with the two limit states (blue: g; dashed green: g′).Figure 7. Probability of failure with the two limit states (blue: g; dashed green: g ).

Figure 7 .
Figure 7. Probability of failure with the two limit states (blue: g; dashed green: g′).Figure 7. Probability of failure with the two limit states (blue: g; dashed green: g ).

Figure 8 .
Figure 8. Updated probability of failure with: (left) 20 inspection data at 5 years and (right) 5 inspection data after 5 years.

Figure 8 .
Figure 8. Updated probability of failure with: (left) 20 inspection data at 5 years and (right) 5 inspection data after 5 years.

Figure 9 .
Figure 9. Mean, 5% and 95% confidence interval of the updated probability of failure at 5 years.

Figure 9 .
Figure 9. Mean, 5% and 95% confidence interval of the updated probability of failure at 5 years.

Table 5 .Figure 10 .
Figure 10.Updated probability of failure with 20 inspections and different inspection dates.

Figure 10 .
Figure 10.Updated probability of failure with 20 inspections and different inspection dates.

Table 2
presents the random variables considered in this study: X = [H; T; α; β]

Table 2 .
Input parameters and random variables.

Table 2 .
Input parameters and random variables.

Table 3 .
Discretization of nodes in the DBN.

Table 4 .
Errors of 5% and 95% fractiles of Pf with different numbers of simulated inspection data.

Table 4 .
Errors of 5% and 95% fractiles of P f with different numbers of simulated inspection data.