3. Example of a more complex structural analysis by Monte Carlo simulation

In the following, an application of a physics-based digital twin model is illustrated for the analysis and the structural integrity management optimization of a specific jacket structure, also in combination with Monte Carlo simulation techniques. The loading is represented by a response surface with the basic environmental parameters as input. The control points are based on physics-based load models. The structural response is obtained by means of a numerical model, which is able to account for large deformations and plastic behavior. This implies that the load-displacement curve is characterized by a maximum value, which is followed by a rapid decline of load-carrying ability similar to the previous simplified example.

### 3.1 System modeling and reliability formulation

The failure of a structural system, e.g., offshore jacket platform is often defined as the total collapse of the structure. The collapse event can be modeled as a series system of several parallel subsystems as follows [2]:

$$\mathbf{g}\_{F\_S}(\ldots) = \bigcup\_{j=1}^{N} \bigcap\_{i=1}^{n} \left( \mathbf{g}\_{F\_{\vec{y}}}(\ldots) \le \mathbf{0} \right) \tag{6}$$

where n is the number of components in the system, N is the number of failure modes, gFij ð Þ … is the limit state function of component i for failure mode j. The system failure probability for systems like offshore jacket platforms can be accurately estimated by considering a single failure mode and expressing the system resistance R and the system load S in terms of base shear [3, 4]. The system resistance R is the ultimate capacity base shear, which is a function of system damage state's matrix D. The system load S is the base shear load for a given environmental variable E. The probability of system overload failure for a given system damage state D is calculated shown in Eq. (7).

$$P(F\_{\mathbb{S}\_2O}|\mathbf{D}) = P[R(\mathbf{D}) - L(\mathbf{E}) \le \mathbf{0}] \tag{7}$$

The performance of structural components in the system deteriorates over time due to, e.g., fatigue damage or corrosion. The system damage state's matrix D contains the (fatigue) damage state of each component at time t, i.e.,

Utilization of Digital Twins and Other Numerical Relatives for Efficient Monte Carlo… DOI: http://dx.doi.org/10.5772/intechopen.89144

D ¼ ½ � HF<sup>1</sup> ð Þ Y, t , ::HFi ð Þ Y, t , ::HFn ð Þ Y, t , where HFi is an indicator function that equals one if the component i fails (i.e., gFij ð Þ Y, t ≤0) and zero if otherwise. Y is a vector of random variables that influences the fatigue damage (see Chapter 3.2). The total probability theorem is then utilized to calculate the probability of system failure due to both overload and fatigue failures as follows:

$$P(F\_{\mathcal{S}}) = \int\_{\mathbf{D}} P(F\_{\mathcal{S}}|\mathbf{D} = \mathbf{g}\_{\mathbf{F}}(\mathbf{Y}, t)) f\_{\mathbf{Y}}(\mathbf{y}) d\mathbf{y} \tag{8}$$

Following [3], Eq. (8) can be approximated as follows:

$$P(F\_S) \approx P\left(F\_{S\_\bullet O}^{intant}\right) + \sum\_{i=1}^n P(F\_i)P\left(F\_{S\_\bullet O}|F\_i\right) + \sum\_{i=1}^n \sum\_{j=1}^{n-1} P\left(F\_i \cap F\_j\right) P\left(F\_{S\_\bullet O}|F\_i \cap F\_j\right) + \dots \tag{9}$$

where P Fintact S, <sup>o</sup> � � is the system failure probability due to overload in the intact condition, P Fð Þ<sup>i</sup> is the fatigue failure probability for component i, P FS, <sup>o</sup>jFi � � is the conditional system failure probability due to overload after fatigue failure occurs at component i, and P Fi ∩ Fj � � is the probability that fatigue failures occurs at components i and j before the overload failure. Eq. (9) is often referred as annual probability of system failure in the context of structural integrity management, where P Fð Þ<sup>i</sup> is defined as the probability of failure at component i given survival up until year t [5]. As a first approximation, the annual probability of system failure can be calculated by keeping only the first two terms [5]:

$$P(F\_S) \approx P\left(F\_{\mathbb{S},O}^{intact}\right) + \sum\_{i=1}^{n} P(F\_i) P\left(F\_{\mathbb{S},O} | F\_i\right) \tag{10}$$

#### 3.2 Response surface

model for the low loading regime could at best be referred to as a more distant

Numerical representation Probability of failure Exact (digital twin) 0.0719 Response surface (physics-based), quadratic 0.0719 Response surface (physics-based), MSE quadratic 0.0671 Data-driven, cubic regression, low loading 1.0000 Data-driven, cubic regression, intermediate loading 0.9102

Theory, Application, and Implementation of Monte Carlo Method in Science and Technology

3. Example of a more complex structural analysis by Monte Carlo

In the following, an application of a physics-based digital twin model is illustrated for the analysis and the structural integrity management optimization of a specific jacket structure, also in combination with Monte Carlo simulation techniques. The loading is represented by a response surface with the basic environmental parameters as input. The control points are based on physics-based load models. The structural response is obtained by means of a numerical model, which is able to account for large deformations and plastic behavior. This implies that the load-displacement curve is characterized by a maximum value, which is followed by a rapid decline of load-carrying ability similar to the previous simplified example.

The failure of a structural system, e.g., offshore jacket platform is often defined as the total collapse of the structure. The collapse event can be modeled as a series

where n is the number of components in the system, N is the number of failure

The performance of structural components in the system deteriorates over time

due to, e.g., fatigue damage or corrosion. The system damage state's matrix D

contains the (fatigue) damage state of each component at time t, i.e.,

failure probability for systems like offshore jacket platforms can be accurately estimated by considering a single failure mode and expressing the system resistance R and the system load S in terms of base shear [3, 4]. The system resistance R is the ultimate capacity base shear, which is a function of system damage state's matrix D. The system load S is the base shear load for a given environmental variable E. The probability of system overload failure for a given system damage state D is calcu-

gFij

ð Þ … is the limit state function of component i for failure mode j. The system

ð Þ … ≤0 

P FS, <sup>O</sup>j<sup>D</sup> <sup>¼</sup> P R½ � ð Þ� <sup>D</sup> <sup>L</sup>ð Þ <sup>E</sup> <sup>≤</sup><sup>0</sup> (7)

(6)

relative (e.g., a half-brother or a cousin).

Failure probabilities corresponding to different numerical representations.

3.1 System modeling and reliability formulation

system of several parallel subsystems as follows [2]:

gFS

ð Þ¼ … ⋃ N j¼1 ⋂ n i¼1

simulation

Table 1.

modes, gFij

162

lated shown in Eq. (7).

The system load L is a function of environmental variable vector E. In this work, the wave height H and wave period T are considered as the environmental random variables, i.e., E = [H,T]. The system load L is expressed as the base shear for a given combination of wave height and wave period. The response surface method with quadratic polynomial function is utilized to estimate the system load as follows [6]:

$$L(H,T) = \mathbf{a}\_0 + \mathbf{a}\_1 H + \mathbf{a}\_2 T + \mathbf{a}\_3 H\_2 + \mathbf{a}\_4 T\_2 + \mathbf{a}\_5 HT \tag{11}$$

where a0….a5 are the coefficients to be determined. Probabilistic linear regression analysis is employed to obtain the coefficients and the predictive distribution of the system load L. The linear model is written as follows:

$$L = X\beta + \varepsilon \tag{12}$$

where L is a (1 � m) vector of "responses" (i.e., which here is the load), β is a (1 � m) vector of the regression coefficients (see Eq. (11)), and ε is the 1 � m vector containing the error terms. The error is assumed Normal-distributed with zero expected value and variance σ<sup>2</sup> <sup>e</sup>. X is a m � p design matrix which consists of p combinations of individual terms (see Eq. (11)) and m number of samples as follows:

$$X = \begin{bmatrix} \mathbf{1} & H\_1 & T\_1 & H\_1^2 & T\_1^2 & H\_1 T\_1 \\\\ \mathbf{1} & H\_2 & T\_2 & H\_2^2 & T\_2^2 & H\_2 T\_2 \\\\ \vdots & & & & \begin{array}{c} \\\\ \mathbf{1} & H\_m & T\_m & H\_m^2 & T\_m^2 & H\_m T\_m \end{array} \end{bmatrix} \tag{13}$$

The unconditional predictive distribution is given by the multivariate noncentral Students t-distribution i.e., <sup>L</sup>~∣L^ � tm�<sup>p</sup> <sup>E</sup> <sup>L</sup>~jL^ � �,COV <sup>L</sup>~jL^ � � � � with parameters as given as follows [7]:

$$\mathbf{E}\left[\tilde{\mathbf{L}}|\hat{\mathbf{L}}\right] = \tilde{\mathbf{X}}\mathbf{E}[\beta] \tag{14}$$

gFi

crack growth is modeled using Paris' Law as follows:

DOI: http://dx.doi.org/10.5772/intechopen.89144

δð Þ¼ Y, t δ

ability of fatigue failure is calculated as follows:

calculated as follows:

updated as given in [14].

165

where δ<sup>c</sup> is the critical crack size and δ(Y,t) is the crack size at time t. Here, failure of the structure will occur if this failure function becomes negative. The

Utilization of Digital Twins and Other Numerical Relatives for Efficient Monte Carlo…

where m and C are the empirical model parameters, Ns is the number of stress cycles, and ΔK is the stress range intensity factor. For through-thickness cracks on

� �C BSIFBΔSΔSe

where δ<sup>0</sup> is the initial crack size, and ν is the annual cycle rate. BSIF and BΔ<sup>S</sup> are the model uncertainties of the stress intensity factor and for the stress range calculation, respectively (see e.g., [11]). ΔSe is the so-called equivalent stress range and

Y is a vector of random variables i.e., Y ¼ ½ � δ0;C; BSIF; BΔS; ln γ; λ , where γ and λ is the scale and shape parameter of the Weibull distributed stress range. The prob-

gFið Þ <sup>Y</sup>, <sup>t</sup> <sup>≤</sup><sup>0</sup>

P Fð Þ <sup>i</sup>ð Þt is defined as the probability of annual fatigue failure given survival up until year t. Statistical dependencies between fatigue hotspots are modeled using correlation coefficients of the random variables in the Y vector. There are 6 correlation coefficients: ρδ<sup>0</sup> ;ρC;ρBSIF ;ρ<sup>B</sup>Δ<sup>S</sup> ;ργ ;ρλ. The coefficient ρδ<sup>0</sup> represents the statistical dependencies due to the same fabrication process. ρ<sup>C</sup> indicates the dependencies due to common material characteristics. ργ and ρλ describe the statistical dependencies due to the similar loading patterns. ρBSIF and ρ<sup>B</sup>Δ<sup>S</sup> depict the dependencies due to common stress intensity factor and stress range calculation. Probabilistic models that are able to represent inspection activities in a proper

� � <sup>1</sup>

m λ h i � � <sup>1</sup>

m

f <sup>Y</sup> y

2

ΔSe ¼ γ Γ 1 þ

ð

way are also required. Information regarding structural performance can be obtained by carrying out inspection or structural monitoring. There are two outcomes of an inspection: no damage indication (I1) or damage indication (I2). The objective of inspection modeling is to obtain the marginal probability of indication (and no indication) followed by an update of the probability of system failure. By utilizing detection theory, the probability of an indication can be derived from the noise and signal distributions (see e.g., [9, 12]). Signal and noise characteristics are typically modeled by means of a Normal distribution (see e.g., [9, 13]). The updating of component fatigue failure probability is performed by utilizing Bayes' law. Given no indication after an inspection, the probability of fatigue failure is

Furthermore, models for statistical representation of the structural monitoring methods are required. Structural health monitoring (SHM) systems can be installed to monitor specific structural properties such as, e.g., vibration or strain in the

P Fð Þ¼ <sup>i</sup>ð Þt

an infinite panel, the solution to Eq. (22) can be written as follows [10]:

<sup>1</sup>�<sup>m</sup> 2 <sup>0</sup> <sup>þ</sup> <sup>1</sup> � <sup>m</sup>

ð Þ¼ Y, t δ<sup>c</sup> � δð Þ Y, t (21)

<sup>d</sup>δð Þ Ns <sup>=</sup>dNs <sup>¼</sup> <sup>C</sup>ð Þ <sup>Δ</sup><sup>K</sup> <sup>m</sup> (22)

ffiffiffi

<sup>1</sup>�<sup>m</sup>

� �dy (25)

<sup>2</sup> (23)

(24)

<sup>π</sup> � � <sup>p</sup> <sup>m</sup>ν<sup>t</sup>

$$\text{COV}[\tilde{\mathbf{L}}|\hat{\mathbf{L}}] = \hat{\sigma}\_{\text{e}}^{2} \left( \tilde{\mathbf{X}} \left( \hat{\mathbf{X}}^{\text{T}} \hat{\mathbf{X}} \right)^{-1} \tilde{\mathbf{X}}^{\text{T}} + \mathbf{I} \right) \tag{15}$$

$$\mathbf{E}[\mathfrak{B}] = \left(\hat{\mathbf{X}}^{\mathrm{T}}\hat{\mathbf{X}}\right)^{-1}\hat{\mathbf{X}}^{\mathrm{T}}\hat{\mathbf{L}}\tag{16}$$

$$
\hat{\sigma}\_{\mathbf{e}} = \frac{1}{\mathbf{m} - \mathbf{r}} \left(\hat{\mathbf{L}} - \hat{\mathbf{X}} \mathbf{E}[\mathfrak{H}]\right)^{T} \left(\hat{\mathbf{L}} - \hat{\mathbf{X}} \mathbf{E}[\mathfrak{H}]\right) \quad \text{and} \quad \mathbf{r} = \text{rank}\left(\hat{\mathbf{X}}\right) \tag{17}
$$

X^ and L^ are matrices that contain the pre-computed load points from, e.g., finite element analysis. L~ is a vector of load predictions from regression analysis for given X~ , which is calculated, e.g., from the samples of wave height and wave period. The predictive distribution of the load, p <sup>L</sup>~jL^ � �, can be seen as a measure of the model uncertainty associated with the response surface.

The wave height is assumed to be Weibull distribution. A special type of a conditional Weibull distribution proposed by Forristal is utilized and written as follows [8, 9]:

$$F\_{H|H\_s}(h|h\_s) = 1 - \exp\left(-2.263\left(\frac{h}{h\_s}\right)^{2.126}\right) \tag{18}$$

where hs is the significant wave height. Probabilistic models for the wave period are less studied compared to wave height. In the present work, the wave period is assumed to follow a Lognormal distribution, which is conditional on wave height, and the parameters are defined as follows:

$$
\mu\_T(H) = E[\ln T] = b\_1 + b\_2(\mathbf{0}.\mathbf{5}H)^{b\_3} \tag{19}
$$

$$\sigma\_T(H) = \text{Std}[\ln T] = c\_1 + c\_2 \exp\left(\mathbf{0}.\mathbf{5}H \cdot \mathbf{c}\_3\right) \tag{20}$$

where b1; b2; b<sup>3</sup> and c1; c2; c<sup>3</sup> are the coefficients to be determined. Eqs. (19) and (20) ensure that the wave period is dependent on the wave height in order to avoid drawing unrealistic samples of wave height and period (e.g., very large wave heights with very small wave periods).

## 3.3 Structural fatigue and reliability updating with monitoring and inspection information

Fatigue failure occurs if the crack size exceeds a critical crack size, and this can be modeled by means of a limit state function as follows:

Utilization of Digital Twins and Other Numerical Relatives for Efficient Monte Carlo… DOI: http://dx.doi.org/10.5772/intechopen.89144

$$\mathbf{g}\_{F\_i}(\mathbf{Y}, t) = \delta\_c - \delta(\mathbf{Y}, t) \tag{21}$$

where δ<sup>c</sup> is the critical crack size and δ(Y,t) is the crack size at time t. Here, failure of the structure will occur if this failure function becomes negative. The crack growth is modeled using Paris' Law as follows:

$$d\delta(\mathbf{N}\_{\epsilon})/d\mathbf{N}\_{\epsilon} = \mathbf{C}(\Delta \mathbf{K})^{\mathsf{m}} \tag{22}$$

where m and C are the empirical model parameters, Ns is the number of stress cycles, and ΔK is the stress range intensity factor. For through-thickness cracks on an infinite panel, the solution to Eq. (22) can be written as follows [10]:

$$\delta(\mathbf{Y},t) = \left(\delta\_0^{1-\frac{m}{2}} + \left(1 - \frac{m}{2}\right) \mathbf{C} \left(\mathbf{B}\_{\text{SIF}} \mathbf{B}\_{\Delta S} \Delta \mathbf{S}\_{\epsilon} \sqrt{\pi}\right)^m \nu t\right)^{\frac{1}{1-\frac{m}{2}}} \tag{23}$$

where δ<sup>0</sup> is the initial crack size, and ν is the annual cycle rate. BSIF and BΔ<sup>S</sup> are the model uncertainties of the stress intensity factor and for the stress range calculation, respectively (see e.g., [11]). ΔSe is the so-called equivalent stress range and calculated as follows:

$$
\Delta \mathbf{S}\_{\mathbf{e}} = \chi \left[ \Gamma \left( \mathbf{1} + \frac{m}{\lambda} \right) \right]^{\frac{1}{m}} \tag{24}
$$

Y is a vector of random variables i.e., Y ¼ ½ � δ0;C; BSIF; BΔS; ln γ; λ , where γ and λ is the scale and shape parameter of the Weibull distributed stress range. The probability of fatigue failure is calculated as follows:

$$P(F\_i(t)) = \int\_{\mathcal{g}\_{\text{fi}}(\mathbf{Y}, t) \le 0} f\_{\mathbf{Y}}(\mathbf{y}) d\mathbf{y} \tag{25}$$

P Fð Þ <sup>i</sup>ð Þt is defined as the probability of annual fatigue failure given survival up until year t. Statistical dependencies between fatigue hotspots are modeled using correlation coefficients of the random variables in the Y vector. There are 6 correlation coefficients: ρδ<sup>0</sup> ;ρC;ρBSIF ;ρ<sup>B</sup>Δ<sup>S</sup> ;ργ ;ρλ. The coefficient ρδ<sup>0</sup> represents the statistical dependencies due to the same fabrication process. ρ<sup>C</sup> indicates the dependencies due to common material characteristics. ργ and ρλ describe the statistical dependencies due to the similar loading patterns. ρBSIF and ρ<sup>B</sup>Δ<sup>S</sup> depict the dependencies due to common stress intensity factor and stress range calculation.

Probabilistic models that are able to represent inspection activities in a proper way are also required. Information regarding structural performance can be obtained by carrying out inspection or structural monitoring. There are two outcomes of an inspection: no damage indication (I1) or damage indication (I2). The objective of inspection modeling is to obtain the marginal probability of indication (and no indication) followed by an update of the probability of system failure. By utilizing detection theory, the probability of an indication can be derived from the noise and signal distributions (see e.g., [9, 12]). Signal and noise characteristics are typically modeled by means of a Normal distribution (see e.g., [9, 13]). The updating of component fatigue failure probability is performed by utilizing Bayes' law. Given no indication after an inspection, the probability of fatigue failure is updated as given in [14].

Furthermore, models for statistical representation of the structural monitoring methods are required. Structural health monitoring (SHM) systems can be installed to monitor specific structural properties such as, e.g., vibration or strain in the

X ¼

as given as follows [7]:

<sup>σ</sup>^<sup>e</sup> <sup>¼</sup> <sup>1</sup> m � r

follows [8, 9]:

: :

COV <sup>L</sup>~jL^ � � <sup>¼</sup> <sup>σ</sup>^<sup>2</sup>

uncertainty associated with the response surface.

FH∣Hs

and the parameters are defined as follows:

heights with very small wave periods).

be modeled by means of a limit state function as follows:

information

164

1 H<sup>1</sup> T<sup>1</sup> H<sup>2</sup>

Theory, Application, and Implementation of Monte Carlo Method in Science and Technology

1 H<sup>2</sup> T<sup>2</sup> H<sup>2</sup>

1 Hm Tm H<sup>2</sup>

<sup>E</sup>½ �¼ <sup>β</sup> <sup>X</sup>^ <sup>T</sup>

The unconditional predictive distribution is given by the multivariate noncentral Students t-distribution i.e., <sup>L</sup>~∣L^ � tm�<sup>p</sup> <sup>E</sup> <sup>L</sup>~jL^ � �,COV <sup>L</sup>~jL^ � � � � with parameters

<sup>e</sup> <sup>X</sup><sup>~</sup> <sup>X</sup>^ <sup>T</sup>

<sup>X</sup>^ � ��<sup>1</sup>

X^ and L^ are matrices that contain the pre-computed load points from, e.g., finite element analysis. L~ is a vector of load predictions from regression analysis for given X~ , which is calculated, e.g., from the samples of wave height and wave period. The predictive distribution of the load, p <sup>L</sup>~jL^ � �, can be seen as a measure of the model

The wave height is assumed to be Weibull distribution. A special type of a conditional Weibull distribution proposed by Forristal is utilized and written as

ð Þ¼ <sup>h</sup>jhs <sup>1</sup> � exp �2:<sup>263</sup> <sup>h</sup>

where hs is the significant wave height. Probabilistic models for the wave period are less studied compared to wave height. In the present work, the wave period is assumed to follow a Lognormal distribution, which is conditional on wave height,

where b1; b2; b<sup>3</sup> and c1; c2; c<sup>3</sup> are the coefficients to be determined. Eqs. (19) and (20) ensure that the wave period is dependent on the wave height in order to avoid drawing unrealistic samples of wave height and period (e.g., very large wave

3.3 Structural fatigue and reliability updating with monitoring and inspection

Fatigue failure occurs if the crack size exceeds a critical crack size, and this can

<sup>1</sup> T<sup>2</sup>

<sup>2</sup> T<sup>2</sup>

<sup>m</sup> T<sup>2</sup>

<sup>X</sup>^ � ��<sup>1</sup>

� �

X^ T

<sup>L</sup>^ � <sup>X</sup>^E½ � <sup>β</sup> � �<sup>T</sup> <sup>L</sup>^ � <sup>X</sup>^E½ � <sup>β</sup> � � and r <sup>¼</sup> rank <sup>X</sup>^� � (17)

hs � �2:126 !

<sup>μ</sup>Tð Þ¼ <sup>H</sup> <sup>E</sup>½ �¼ ln <sup>T</sup> <sup>b</sup><sup>1</sup> <sup>þ</sup> <sup>b</sup>2ð Þ <sup>0</sup>:5<sup>H</sup> <sup>b</sup><sup>3</sup> (19) σTð Þ¼ H Std½ �¼ ln T c<sup>1</sup> þ c<sup>2</sup> exp 0ð Þ :5H � c<sup>3</sup> (20)

<sup>1</sup> H1T<sup>1</sup>

(13)

(15)

(18)

<sup>2</sup> H2T<sup>2</sup>

<sup>m</sup> HmTm:

<sup>E</sup> <sup>L</sup>~jL^ � � <sup>¼</sup> <sup>X</sup>~E½ � <sup>β</sup> (14)

L^ (16)

X~ T þ I structural system. Information from a SHM can be viewed as one of the possible realizations of the model uncertainty (see e.g., [15, 16]), which is associated with the measured property such as, e.g., stress ranges. In the present work, the SHM modeling proposed by [17] is employed, i.e., three different possible SHM outcomes are considered: The outcome Z1 corresponds to the case where monitoring indicates lower stress ranges than expected and indicates that the monitored component has a high performance. Outcome Z3 indicates that the monitoring component has a low performance due to higher than expected stress ranges. Outcome Z2 indicates that the monitored component performs as expected. Calculation of the updated probability of system failure is carried out as described in [14].

where NIC is the number of inspected components. E[CFS] and E[CF,i] in Eq. (27) are calculated by considering the updated probability of system and fatigue failure, respectively. E[CR,i] is the expected repair costs over the service with NIC

Utilization of Digital Twins and Other Numerical Relatives for Efficient Monte Carlo…

The expected total costs for the second strategy is calculated as follows:

� � ! <sup>þ</sup> E C½ �þ SHM <sup>X</sup><sup>n</sup>

S, <sup>A</sup>

A typical deepwater offshore jacket platform with 25 years of service life and located at 190 m waterdepth is utilized in the present work (see Figure 6a). The jacket platform has 200 components and each component is subjected to fatigue deterioration. In this study, each component is assumed to have exactly one hotspot for which a trough-thickness crack will result in fatigue failure. The incoming wave direction is taken as 135° and the 100-year significant wave height HS;100y equals to

(a) Deepwater finite element jacket model used in the case study. (b) The response surface of L with the black crosses signify the training data set. (c) The expected value of the load prediction (blue line) with

where E C½ � SHM is the expected SHM costs. Further details are given in [14]. The

� � ! <sup>þ</sup> E CFS ½ � " #

i�1

Cð Þ� S, A C<sup>0</sup> (29)

E CF,i

(28)

repaired components.

C Sð Þ¼ <sup>2</sup>, A EZ

3.5 Case study

24.3 m.

Figure 6.

167

95%-confidence interval (black lines) for T = 20.8 s.

X NIC

DOI: http://dx.doi.org/10.5772/intechopen.89144

E CI,i

� � <sup>þ</sup> E CR,i

VOIA ¼ max

i�1

VOIA is then calculated as follows:

### 3.4 Quantification of the value of SIM strategies

The quantification of the value of SIM strategies builds upon the Bayesian preposterior decision analysis framework as formulated by Benjamin and Cornell, [18]. A SIM strategy decision problem can be modeled by a decision tree in pre-posterior form as shown in Figure 5. The information space S consists of available information acquirement strategies i (e.g., inspection and monitoring). The outcome space O comprises the possible outcomes of a given information acquirement strategy i. The action space A consists of the possible actions that can be taken such as e.g. repair. The state space θ contains possible states such as, e.g., failure or survival.

The value of SIM strategies is quantified by utilizing the value of information and action (VOIA) analysis (see [19]). A VOIA analysis consists of a base and an enhancement scenario. The base scenario is defined as the scenario without any SHM/inspection and risk-mitigating action such as e.g., repair. There are two states considered in this system state analysis: the (collapse/no collapse) and the component state (failure/no failure). Therefore, the expected cost C0 in the base scenario is the sum of the expected system E[CFS] and component E[CF,i] failure costs over the service life TSL:

$$\mathbf{C}\_{0} = \left(\sum\_{i=1}^{n} E\left[\mathbf{C}\_{F\_{\mathbf{s}}} i\right]\right) + E[\mathbf{C}\_{F\_{\mathbf{S}}}] \tag{26}$$

where n is the number of structural components. Procedures for calculation of E[CFS] and E[CF,i] are described in [14]. The failure probabilities, which are required in order to calculate these costs, are computed by means of Monte Carlo simulation.

Two different SIM strategies are analyzed as enhancement scenarios. The first strategy is to perform inspections and repair if required during the service life. The second strategy is to install the SHM system for 1 year at one component and to perform inspections and repair if required. For both strategies, a repair action is performed if the inspection indicates a damage. The expected total cost of the first strategy is calculated as the sum of expected costs of inspection E[Ci], repair E[CR,i], system failure E[CFS], and fatigue failure E[CF,i] costs over the service life as follows:

Utilization of Digital Twins and Other Numerical Relatives for Efficient Monte Carlo… DOI: http://dx.doi.org/10.5772/intechopen.89144

where NIC is the number of inspected components. E[CFS] and E[CF,i] in Eq. (27) are calculated by considering the updated probability of system and fatigue failure, respectively. E[CR,i] is the expected repair costs over the service with NIC repaired components.

The expected total costs for the second strategy is calculated as follows:

$$\mathbf{C}(\mathbf{S}\_2, \mathbf{A}) = \mathbf{E}\_Z \left[ \left( \sum\_{i=1}^{N\_{\text{IC}}} E[\mathbf{C}\_{I\_i}] + E[\mathbf{C}\_{R\_i}] \right) + E[\mathbf{C}\_{\text{SHM}}] + \left( \sum\_{i=1}^{n} E[\mathbf{C}\_{F\_i}] \right) + E[\mathbf{C}\_{F\_\mathcal{S}}] \right] \tag{28}$$

where E C½ � SHM is the expected SHM costs. Further details are given in [14]. The VOIA is then calculated as follows:

$$\text{VOIA} = \max\_{\mathbf{S}, \mathbf{A}} \mathbf{C}(\mathbf{S}, \mathbf{A}) - \mathbf{C}\_0 \tag{29}$$

#### 3.5 Case study

structural system. Information from a SHM can be viewed as one of the possible realizations of the model uncertainty (see e.g., [15, 16]), which is associated with the measured property such as, e.g., stress ranges. In the present work, the SHM modeling proposed by [17] is employed, i.e., three different possible SHM outcomes are considered: The outcome Z1 corresponds to the case where monitoring indicates lower stress ranges than expected and indicates that the monitored component has a high performance. Outcome Z3 indicates that the monitoring component has a low performance due to higher than expected stress ranges. Outcome Z2 indicates that the monitored component performs as expected. Calculation of the updated proba-

Theory, Application, and Implementation of Monte Carlo Method in Science and Technology

The quantification of the value of SIM strategies builds upon the Bayesian preposterior decision analysis framework as formulated by Benjamin and Cornell, [18]. A SIM strategy decision problem can be modeled by a decision tree in pre-posterior form as shown in Figure 5. The information space S consists of available information acquirement strategies i (e.g., inspection and monitoring). The outcome space O comprises the possible outcomes of a given information acquirement strategy i. The action space A consists of the possible actions that can be taken such as e.g. repair. The state space θ contains possible states such as, e.g., failure or survival. The value of SIM strategies is quantified by utilizing the value of information and action (VOIA) analysis (see [19]). A VOIA analysis consists of a base and an enhancement scenario. The base scenario is defined as the scenario without any SHM/inspection and risk-mitigating action such as e.g., repair. There are two states considered in this system state analysis: the (collapse/no collapse) and the component state (failure/no failure). Therefore, the expected cost C0 in the base scenario is the sum of the expected system E[CFS] and component E[CF,i] failure costs over

bility of system failure is carried out as described in [14].

<sup>C</sup><sup>0</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup>

i�1

E CF,i � � !

where n is the number of structural components. Procedures for calculation of E[CFS] and E[CF,i] are described in [14]. The failure probabilities, which are required in order to calculate these costs, are computed by means of Monte Carlo simulation. Two different SIM strategies are analyzed as enhancement scenarios. The first strategy is to perform inspections and repair if required during the service life. The second strategy is to install the SHM system for 1 year at one component and to perform inspections and repair if required. For both strategies, a repair action is performed if the inspection indicates a damage. The expected total cost of the first strategy is calculated as the sum of expected costs of inspection E[Ci], repair E[CR,i], system failure E[CFS], and fatigue failure E[CF,i] costs over the service life as follows:

> <sup>þ</sup> <sup>X</sup><sup>n</sup> i�1

E CF,i � � !

þ E CFS ½ � (26)

þ E CFS ½ � (27)

3.4 Quantification of the value of SIM strategies

the service life TSL:

C Sð Þ¼ <sup>1</sup>, A <sup>X</sup>

Figure 5.

166

NIC

i�1

E CI,i

� � <sup>þ</sup> E CR,i � � !

Illustration of a decision tree with rectangular modeling decision and ellipses for chance nodes.

A typical deepwater offshore jacket platform with 25 years of service life and located at 190 m waterdepth is utilized in the present work (see Figure 6a). The jacket platform has 200 components and each component is subjected to fatigue deterioration. In this study, each component is assumed to have exactly one hotspot for which a trough-thickness crack will result in fatigue failure. The incoming wave direction is taken as 135° and the 100-year significant wave height HS;100y equals to 24.3 m.

#### Figure 6.

(a) Deepwater finite element jacket model used in the case study. (b) The response surface of L with the black crosses signify the training data set. (c) The expected value of the load prediction (blue line) with 95%-confidence interval (black lines) for T = 20.8 s.

System failure is defined as the collapse of the jacket platform due to overload and fatigue deterioration. In this example, it is assumed that only single component failure is possible before the overload failure and the probability of system failure is calculated by Eq. (10). The resistance R is the ultimate base shear for given damage state matrix D and is calculated by performing pushover analysis with software USFOS [20]. The system load L is approximated by utilizing response surface analysis outlined in Section 3.2 with 248 pre-computed load points. The samples of L are drawn from the predictive distribution for a given H and T. The coefficients in Eq. (19) and (20) are assumed to have the following values (based on a typical North Sea environment): b = [1.322; 0.8; 0.242] and c = [0.005; 0.09374; 0.32]. The conditional probability of system collapse (see Eq. (7)) is calculated by utilizing Monte Carlo simulation with 10<sup>6</sup> samples. The response surface of the system load L is shown in Figure 6b with the coefficient of determination R<sup>2</sup> = 0.9905. Figure 6c shows the predictive distribution of L for T = 20.8 s.

The system failure probability in intact condition is 5E6. The conditional system failure probabilities for different damaged components and its associated ultimate base shears are given in Table 2. The components in Table 2 are located on the jacket's legs.

3.5.2 Structural integrity management (SIM)

Summary of the random variables for fatigue modeling.

DOI: http://dx.doi.org/10.5772/intechopen.89144

uted with the following parameters:

(Fi)=1�10�<sup>3</sup>

listic models is shown in Table 4.

where δ(t) is the crack size at year t.

measure.

Table 3.

and P2 T

169

Two SIM strategies are considered: inspection and repair and inspection with SHM and repair. For the first strategy, inspections are performed at 3 critical components 1 year before the annual system failure probability P(FS) is estimated to exceed the threshold Pth(FS) (i.e., constant threshold approach). The minimum system failure probability threshold during operation is set equal to 10�<sup>4</sup> which corresponds to target system failure probability recommended by JCSS [15] for structures with large consequences of failure and large relative cost of safety

Variable Dimension Distribution Expected value St. deviation TSL year — 25 δ<sup>0</sup> mm Exponential 0.11 δ<sup>C</sup> mm — 8 lnC N and mm Normal �29.97 0.5095 m — — 3.0 — BSIF — Lognormal 1.0 0.1 BΔ<sup>S</sup> — Lognormal 1.0 0.2 lnγ N and mm Normal 2.1 0.22 λ — Normal 0.8 0.08 ν 1/year — 10<sup>7</sup> —

Utilization of Digital Twins and Other Numerical Relatives for Efficient Monte Carlo…

In order to simplify the decision analysis, a repair action is performed only if any damage is detected by inspections. This decision rule is practical and VoI-optimal, see [22]. Repaired components are assumed to behave as components with no damage indication. The probability of indication is derived from the noise and signal distributions. The noise SR is assumed to follow a Normal distribution with zero mean and a standard deviation of 0.5. The signal threshold ths is calibrated to the probability of false indication (PFI) of 0.01. The signal S is also Normal distrib-

In the second SIM strategy, a SHM system for stress range monitoring is installed 1 year before the first inspection is performed, with a monitoring duration of 1 year (i.e., up to the time of the inspection itself). SHM performance is modeled as proposed in [17] by utilizing the stress range model uncertainty BΔ<sup>S</sup>. Two thresholds distinguishing the outcomes Z1 (low stress ranges), Z2 (stress ranges as designed)

and Z3 (high stress ranges) are calibrated to target probabilities of P1

for structures with minor consequences of failure with normal and large relative cost of safety measure. The time dependent threshold's calibration is illustrated by Figure 7. The measurement uncertainty U is assumed Normal distributed with the expected value of 1.0 and a standard deviation of 0.05. A summary of the probabi-

μSðÞ¼ t 0:8 þ 0:1 � δð Þt , σSðÞ¼ t 0:3 � 0:01 � δð Þt (30)

, respectively. The target reliabilities are selected following [13]

T

(Fi)=1�10�<sup>4</sup>
