2.2 More distant numerical relatives based on different kinds of simplified physics-based models

One way of being able to reduce the computation time for even larger and more complex numerical models, is to introduce a simplified representation which is no longer a twin but some more distant numerical relative. As an example, a so-called response surface model can be applied in order to overcome excessive computational efforts. Such models are frequently also referred to as "meta-models" or "cyber-physical" models. One possible approach is to use a response surface representation based, e.g., on first- or second-order polynomials as approximating functions. The parameters of these functions and their weighting coefficients are then determined, e.g., based on minimization of the mean square error. The "control points" for the approximate model are then established based on application of the physics-based model at just these points (i.e., for given input parameter values). By a proper selection of control points, the prediction error associated with the entire range of structural displacement levels can be limited in magnitude.

As examples, we consider approximation of the exact load-displacement relationship with a quadratic and also an alternative quadratic response surface model. For the former, the control points are selected as (0, 0); (0.5, 3/16) and (1.0, 0.0), where 3/16 represents the exact maximum value of the cubic function (but with the location of the maximum point shifted to an abscissa value of 0.5). For the latter, a minimum mean square error approximation within the interval 0.0–1.0. The first of these approximations is compared to the "exact physics-based model" in Figure 3.

The error associated with the second order approximations over the range from r/h = 0 to 1 is seen to be acceptable, while for the less interesting range (within the present context) from 1 to 2 it is highly inaccurate and of little use.

> curve. As a second approximation, the 10 data points (including noise) are next taken to lie in the range from r/h = 0 to 0.2 (i.e., into the slightly more nonlinear regime). The corresponding second order approximation is shown in Figure 4b. The maximum load is somewhat closer to the true value, but still a significant

Measured data points and second order approximations: (a) data range is in the interval from 0 to 0.15 r/h

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

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

2.4 Comparison of failure probabilities calculated by application of the

By introducing a structural failure criterion for the truss in addition to joint statistical models for the inherent random variables, the failure probability corresponding to a given reference period can be computed. Presently, the failure function is expressed in terms of the maximum allowable load (i.e., Rmax), and the only random variable is the external extreme environmental load (i.e., Rex) which follows a Gumbel distribution with a mean value of 0.9 Rmax and a coefficient of variation of 10% (i.e., a standard deviation of 0.09 Rmax). In the present section, a comparison is made between structural failure probabilities, which are obtained by application of the different structural representations that were considered above

Not unexpectedly, the accuracy of the physics-based representations is significantly higher than the data-driven models for the present example. While the cubic response surface almost corresponds to the twin representation, the data driven

These results are intended to illustrate the limitations of extrapolations based on data driven models unless the measurement points are available in the region with "high nonlinearity." For structural systems, such data points are generally scarce as they represent rare events that may even correspond to failure of the structure.

underprediction is observed.

and (b) data range is from 0 to 0.15 r/h.

Figure 4.

different numerical models

(Table 1).

161

### 2.3 Data-driven simplified models

A numerical representation of the load-displacement relationship based on a data-driven simplified model is next considered. First, it is assumed that 10 data points in the range from r/h = 0 to 0.15 are available, which is mainly in the weakly non-linear regime. These are, e.g., obtained during normal operation of the structure. A measurement noise with a standard deviation of 10% of the measured signal is also introduced. The extrapolated second order approximation (based on regression analysis) is shown in Figure 4a together with the data points themselves. It is seen that the maximum value of the load R is significantly underpredicted by this

Figure 3. Comparison of second order response surface with exact relationship.

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

#### Figure 4.

2.2 More distant numerical relatives based on different kinds of simplified

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

range of structural displacement levels can be limited in magnitude.

present context) from 1 to 2 it is highly inaccurate and of little use.

2.3 Data-driven simplified models

Comparison of second order response surface with exact relationship.

Figure 3.

160

One way of being able to reduce the computation time for even larger and more complex numerical models, is to introduce a simplified representation which is no longer a twin but some more distant numerical relative. As an example, a so-called response surface model can be applied in order to overcome excessive computational efforts. Such models are frequently also referred to as "meta-models" or "cyber-physical" models. One possible approach is to use a response surface representation based, e.g., on first- or second-order polynomials as approximating functions. The parameters of these functions and their weighting coefficients are then determined, e.g., based on minimization of the mean square error. The "control points" for the approximate model are then established based on application of the physics-based model at just these points (i.e., for given input parameter values). By a proper selection of control points, the prediction error associated with the entire

As examples, we consider approximation of the exact load-displacement relationship with a quadratic and also an alternative quadratic response surface model. For the former, the control points are selected as (0, 0); (0.5, 3/16) and (1.0, 0.0), where 3/16 represents the exact maximum value of the cubic function (but with the location of the maximum point shifted to an abscissa value of 0.5). For the latter, a minimum mean square error approximation within the interval 0.0–1.0. The first of these approximations is compared to the "exact physics-based model" in Figure 3. The error associated with the second order approximations over the range from r/h = 0 to 1 is seen to be acceptable, while for the less interesting range (within the

A numerical representation of the load-displacement relationship based on a data-driven simplified model is next considered. First, it is assumed that 10 data points in the range from r/h = 0 to 0.15 are available, which is mainly in the weakly non-linear regime. These are, e.g., obtained during normal operation of the structure. A measurement noise with a standard deviation of 10% of the measured signal is also introduced. The extrapolated second order approximation (based on regression analysis) is shown in Figure 4a together with the data points themselves. It is seen that the maximum value of the load R is significantly underpredicted by this

physics-based models

Measured data points and second order approximations: (a) data range is in the interval from 0 to 0.15 r/h and (b) data range is from 0 to 0.15 r/h.

curve. As a second approximation, the 10 data points (including noise) are next taken to lie in the range from r/h = 0 to 0.2 (i.e., into the slightly more nonlinear regime). The corresponding second order approximation is shown in Figure 4b. The maximum load is somewhat closer to the true value, but still a significant underprediction is observed.

These results are intended to illustrate the limitations of extrapolations based on data driven models unless the measurement points are available in the region with "high nonlinearity." For structural systems, such data points are generally scarce as they represent rare events that may even correspond to failure of the structure.

### 2.4 Comparison of failure probabilities calculated by application of the different numerical models

By introducing a structural failure criterion for the truss in addition to joint statistical models for the inherent random variables, the failure probability corresponding to a given reference period can be computed. Presently, the failure function is expressed in terms of the maximum allowable load (i.e., Rmax), and the only random variable is the external extreme environmental load (i.e., Rex) which follows a Gumbel distribution with a mean value of 0.9 Rmax and a coefficient of variation of 10% (i.e., a standard deviation of 0.09 Rmax). In the present section, a comparison is made between structural failure probabilities, which are obtained by application of the different structural representations that were considered above (Table 1).

Not unexpectedly, the accuracy of the physics-based representations is significantly higher than the data-driven models for the present example. While the cubic response surface almost corresponds to the twin representation, the data driven


D ¼ ½ � HF<sup>1</sup> ð Þ Y, t , ::HFi

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

to both overload and fatigue failures as follows:

i¼1

P Fð Þ¼ <sup>S</sup>

ð D

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

can be calculated by keeping only the first two terms [5]:

P Fð Þ<sup>S</sup> <sup>≈</sup> P Fintact

of the system load L. The linear model is written as follows:

S, <sup>O</sup> � � þX<sup>n</sup>

P Fð Þ<sup>i</sup> P FS, <sup>O</sup>jFi

one if the component i fails (i.e., gFij

P Fð Þ<sup>S</sup> <sup>≈</sup> P Fintact

where P Fintact

3.2 Response surface

expected value and variance σ<sup>2</sup>

as follows [6]:

follows:

163

S, O � � þX<sup>n</sup>

S, <sup>o</sup>

at component i, and P Fi ∩ Fj

ð Þ Y, t , ::HFn ð Þ Y, t , where HFi is an indicator function that equals

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

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

� � þX<sup>n</sup>

condition, P Fð Þ<sup>i</sup> is the fatigue failure probability for component i, P FS, <sup>o</sup>jFi

conditional system failure probability due to overload after fatigue failure occurs

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

i¼1

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

where a0….a5 are the coefficients to be determined. Probabilistic linear regression analysis is employed to obtain the coefficients and the predictive distribution

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

combinations of individual terms (see Eq. (11)) and m number of samples as

P FSj<sup>D</sup> <sup>¼</sup> <sup>g</sup>Fð Þ <sup>Y</sup>, t � �<sup>f</sup> <sup>Y</sup> <sup>y</sup>

i¼1

� � is the system failure probability due to overload in the intact

Xn�1 j¼1

� � is the probability that fatigue failures occurs at

P Fð Þ<sup>i</sup> P FS, <sup>O</sup>jFi

L H, T ð Þ¼ a0 þ a1H þ a2T þ a3H<sup>2</sup> þ a4T<sup>2</sup> þ a5HT (11)

L ¼ Xβ þ ε (12)

<sup>e</sup>. X is a m � p design matrix which consists of p

P Fi ⋂ Fj

ð Þ Y, t ≤0) and zero if otherwise. Y is a vector of

� �dy (8)

� � <sup>þ</sup> …

� � is the

(9)

� �P FS, <sup>O</sup>jFi <sup>⋂</sup> Fj

� � (10)

Table 1.

Failure probabilities corresponding to different numerical representations.

model for the low loading regime could at best be referred to as a more distant relative (e.g., a half-brother or a cousin).
