3.5.1 Fatigue model

All components are subjected to fatigue deterioration over the service life with one critical hotspot for each component. The contributions from the fatigue failure probabilities to the system failure are weighted w.r.t. the conditional system failure probabilities. Due to the high number of structural components, only the 10 most critical components with the highest conditional system failure probabilities (see Table 2) are presently considered. Fatigue failure contributions from other components are considerably smaller and can hence be neglected. Fatigue failure probability at time t is calculated by application of Monte Carlo simulation. The probability is calculated for a period of 1 year, where survival until year t is given. All fatigue hotspots are modeled with the same probabilistic models, which are shown in Table 3. BΔ<sup>S</sup> and BSIF are assumed fully correlated between components following [21]. The other random variables are assumed to have a correlation coefficient of 0.8 [21].


#### Table 2.

Ten highest conditional probabilities of system failure.

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


#### Table 3.

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

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

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

All components are subjected to fatigue deterioration over the service life with one critical hotspot for each component. The contributions from the fatigue failure probabilities to the system failure are weighted w.r.t. the conditional system failure probabilities. Due to the high number of structural components, only the 10 most critical components with the highest conditional system failure probabilities (see Table 2) are presently considered. Fatigue failure contributions from other components are considerably smaller and can hence be neglected. Fatigue failure probability at time t is calculated by application of Monte Carlo simulation. The probability is calculated for a period of 1 year, where survival until year t is given. All fatigue hotspots are modeled with the same probabilistic models, which are shown in Table 3. BΔ<sup>S</sup> and BSIF are assumed fully correlated between components following [21]. The other random variables are assumed to have a correlation

Damaged component Ultimate base shear (N) P(FS,O|D) All intact 1.51E+08 5.00E06 7.05E+07 3.54E03 7.19E+07 3.16E03 7.42E+07 2.63E03 7.98E+07 1.67E03 8.61E+07 1.01E03 8.68E+07 9.60E04 8.71E+07 9.37E04 8.99E+07 7.55E04 1.03E+08 2.93E04 1.06E+08 1.90E04

shows the predictive distribution of L for T = 20.8 s.

the jacket's legs.

3.5.1 Fatigue model

coefficient of 0.8 [21].

Table 2.

168

Ten highest conditional probabilities of system failure.

Summary of the random variables for fatigue modeling.

#### 3.5.2 Structural integrity management (SIM)

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 measure.

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 distributed with the following parameters:

$$
\mu\_{\rm S}(t) = \mathbf{0.8} + \mathbf{0.1} \cdot \delta(t), \quad \sigma\_{\rm S}(t) = \mathbf{0.3} - \mathbf{0.01} \cdot \delta(t) \tag{30}
$$

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

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 T (Fi)=1�10�<sup>4</sup> and P2 T (Fi)=1�10�<sup>3</sup> , respectively. The target reliabilities are selected following [13] 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 probabilistic models is shown in Table 4.

#### Figure 7.

Illustration of thresholds calibration for two different SHM installation times for SHM modeling. η<sup>1</sup> and η<sup>2</sup> are the thresholds of stress range model uncertainty BΔ<sup>S</sup> associated with P<sup>T</sup> <sup>1</sup> ð Þ Fi and <sup>P</sup><sup>T</sup> <sup>2</sup> ð Þ Fi , respectively. The integration of the colored regions results in the probabilities of (Z1, Z2, and Z3), which are modeled as three indication events.


#### Table 4.

Summary of the random variables for inspection and SHM modeling.

The costs considered in this case study are consisting of inspection costs CI, SHM costs CSHM, repair costs CR, system failure costs CFS, and component failure costs CFi. SHM costs are further divided into investment costs CInv SHM, installation costs CInst SHM, and operational costs Cop SHM. The cost model used in this example is shown in Table 5 based on [23, 24].

#### 3.5.3 Results

The annual component and system failure for t = 1....25 years are shown in Figure 8. The system failure probability for the intact condition is P(FS,O| D = 0) = 5E�6. The annual system failure probability at the end of service life is 9.5E�5, which is less than the minimum operational threshold, i.e., no SIM implementations are required to achieve the minimum operational requirement. However, the decision-maker may wish to increase the structural reliability above the minimum requirement and presumably enhance the value of SIM. In this work, three different annual system failure probability thresholds are studied: 6E�5, 7E�5, and 8E�5.

The first inspection time is at t = 8 years. Increasing the threshold means that the inspections are performed later during the service life. Because of this, the inspection frequency during the service life is decreasing with a higher threshold in

(a) Annual system failure probability with inspections and repairs for one specific threshold, Pth(FS) = 6E5. (b) Annual system failure probability with SHM, inspections and repairs for one specific threshold,

Pth(FS) = 6E5. The inspections and repairs are performed at three components while monitoring is performed

Type Cost CI 0.001 CInvSHM 1.33<sup>10</sup><sup>4</sup>

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

CInstSHM 1.33<sup>10</sup><sup>4</sup>

CopSHM <sup>2</sup><sup>10</sup><sup>4</sup>

CR 0.01 CF,i 1 CFS 100

Table 5.

Figure 8.

Figure 9.

171

at one component.

Cost models used in the case study.

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

/channel

/channel

/year

The second SIM strategy (S = S2) is based on installation of a SHM system at one component to monitor stress ranges for 1 year before the first inspection. There are three possible outcomes for monitoring based on the component performance. Figure 9b shows the annual system failure probability with monitoring and inspections for a system failure probability threshold of 6E-5. Compared to Figure 9a, it is observed that the outcome of monitoring can influence the future inspection

exchange of a higher annual system failure probability.

Annual component and system failure probability over the service life.

For the inspection-only strategy (S = S1), inspections and repairs are performed at three components 1 year before the system failure probability threshold Pth(FS) is predicted to be reached. After each inspection, probabilities of fatigue and system failure are updated. Figure 9a shows the annual system failure probability for the inspection-only strategy as a function of time for a specific threshold value.

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


#### Table 5.

Cost models used in the case study.

Figure 8. Annual component and system failure probability over the service life.

#### Figure 9.

The costs considered in this case study are consisting of inspection costs CI, SHM costs CSHM, repair costs CR, system failure costs CFS, and component failure costs

Illustration of thresholds calibration for two different SHM installation times for SHM modeling. η<sup>1</sup> and η<sup>2</sup> are

Variable Dimension Distribution Expected value Std. deviation SR — Normal 0 0.5 S — Normal μS(t) σS(t) PFI — — 0.01 —

(Fi) — — <sup>5</sup>�10�<sup>4</sup> —

(Fi) — — <sup>5</sup>�10�<sup>3</sup> — U — Normal 1.0 0.05

integration of the colored regions results in the probabilities of (Z1, Z2, and Z3), which are modeled as three

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

The annual component and system failure for t = 1....25 years are shown in

For the inspection-only strategy (S = S1), inspections and repairs are performed at three components 1 year before the system failure probability threshold Pth(FS) is predicted to be reached. After each inspection, probabilities of fatigue and system failure are updated. Figure 9a shows the annual system failure probability for the inspection-only strategy as a function of time for a specific threshold value.

Figure 8. The system failure probability for the intact condition is P(FS,O| D = 0) = 5E�6. The annual system failure probability at the end of service life is 9.5E�5, which is less than the minimum operational threshold, i.e., no SIM implementations are required to achieve the minimum operational requirement. However, the decision-maker may wish to increase the structural reliability above the minimum requirement and presumably enhance the value of SIM. In this work, three different annual system failure probability thresholds are studied: 6E�5,

SHM. The cost model used in this example is

<sup>1</sup> ð Þ Fi and <sup>P</sup><sup>T</sup>

SHM, installation costs

<sup>2</sup> ð Þ Fi , respectively. The

CFi. SHM costs are further divided into investment costs CInv

Summary of the random variables for inspection and SHM modeling.

the thresholds of stress range model uncertainty BΔ<sup>S</sup> associated with P<sup>T</sup>

SHM, and operational costs Cop

shown in Table 5 based on [23, 24].

CInst

Figure 7.

P1 T

P2 T

Table 4.

indication events.

3.5.3 Results

7E�5, and 8E�5.

170

(a) Annual system failure probability with inspections and repairs for one specific threshold, Pth(FS) = 6E5. (b) Annual system failure probability with SHM, inspections and repairs for one specific threshold, Pth(FS) = 6E5. The inspections and repairs are performed at three components while monitoring is performed at one component.

The first inspection time is at t = 8 years. Increasing the threshold means that the inspections are performed later during the service life. Because of this, the inspection frequency during the service life is decreasing with a higher threshold in exchange of a higher annual system failure probability.

The second SIM strategy (S = S2) is based on installation of a SHM system at one component to monitor stress ranges for 1 year before the first inspection. There are three possible outcomes for monitoring based on the component performance. Figure 9b shows the annual system failure probability with monitoring and inspections for a system failure probability threshold of 6E-5. Compared to Figure 9a, it is observed that the outcome of monitoring can influence the future inspection


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

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

expected costs and structural risks is associated to the lowest annual system failure probability threshold. It is further demonstrated that structural systems with a high reliability requirement will benefit more from a SHM system implementation.

The authors acknowledge the funding received from the Center for Oil and Gas-DTU/Danish Hydrocarbon Research and Technology Center (DHRTC). The authors are also grateful to Professor Jørgen Amdahl for his support regarding

corresponding to normal operation of the structures.

the Finite element analysis with USFOS.

There is no conflict of interest.

It is believed that for the present type of analysis, which involves large structural deformations and structural failure behavior, data-driven models will not be adequate due to an insufficient amount of relevant data. Clearly, this belief is also based on the assumption that the model uncertainties associated with the physics-based numerical models can be adequately controlled. This can be achieved by collecting data from laboratory (destructive) testing and full-scale measurements including failure records. In this way, data-calibrated and physics-based numerical models can be developed, rather than relying on data-driven models based on conditions

A framework has been developed to plan and to optimize the structural integrity management (SIM) by utilizing the physics-based digital twin model. By extending the concept of the value of information to a value of information and action analysis, the value of inspection and monitoring information and repair actions is quantified. A novel approach of SHM modeling introduced by Agusta and Thӧns [17] has been employed in conjunction with inspection modeling based on a probabilistic representation of inspections. The optimal SIM strategy leading to the least

plastic behavior.

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

Acknowledgements

Conflict of interest

173

#### Table 6.

The probability of SHM outcomes for different annual system probability thresholds.

#### Figure 10.

(a) Expected total costs for base scenario, inspection and repairs strategy (S1), and SHM, inspections, and repairs (S3). (b) Value of information and action based on SIM strategies for different annual system failure probability thresholds.

schedule. The SHM installation time differs depending on the thresholds (see Table 6). A higher threshold means that the SHM system is installed closer to the end of service life. With increasing annual system failure probability threshold, the probability of obtaining low performance outcome (Z3) becomes higher.

The values of information and action (VOIA) of the two SIM strategies are shown in Figure 10. It is observed that increasing the system failure probability threshold will reduce the value of information and action. With a higher threshold, inspection and monitoring are performed later in the service life, which reduces the benefits due to a higher annual system failure probability during the remaining service life compared to a lower system failure probability threshold. It is also observed that the VOIA of the SIM strategy SHM, inspection and repair (S2) is higher than the inspection-only strategy (S1) for all investigated system failure probability thresholds. This shows that information from SHM system can enhance the value of the SIM, i.e., reduce the expected total cost. In this example, the cost of system failure is dominating the expected total cost over the service life.

### 4. Summary and conclusions

In the present analysis, physics-based numerical models of the load, structural behavior and for the integrity management have been utilized in combination with response surface techniques and Monte Carlo simulation. An application of a physics-based digital twin model is illustrated for the structural and integrity management analysis of a specific jacket structure. The loading is represented by a response surface with the basic environmental parameters as input. The control

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

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.

A framework has been developed to plan and to optimize the structural integrity management (SIM) by utilizing the physics-based digital twin model. By extending the concept of the value of information to a value of information and action analysis, the value of inspection and monitoring information and repair actions is quantified. A novel approach of SHM modeling introduced by Agusta and Thӧns [17] has been employed in conjunction with inspection modeling based on a probabilistic representation of inspections. The optimal SIM strategy leading to the least expected costs and structural risks is associated to the lowest annual system failure probability threshold. It is further demonstrated that structural systems with a high reliability requirement will benefit more from a SHM system implementation.

It is believed that for the present type of analysis, which involves large structural deformations and structural failure behavior, data-driven models will not be adequate due to an insufficient amount of relevant data. Clearly, this belief is also based on the assumption that the model uncertainties associated with the physics-based numerical models can be adequately controlled. This can be achieved by collecting data from laboratory (destructive) testing and full-scale measurements including failure records. In this way, data-calibrated and physics-based numerical models can be developed, rather than relying on data-driven models based on conditions corresponding to normal operation of the structures.
