2. Numerical representations of physical structures

#### 2.1 The (near to) perfect twin based on multi-physics models

In the present text, the concept of a digital twin is understood in the following sense: A digital twin is a numerical model capable of reproducing the state and behavior of a unique real asset in real time (or faster), with this model also being able to represent the performance of the asset for new and artificially generated conditions (i.e., in connection with extrapolated predictions). As a primary candidate for a digital twin, a complete numerical model based on first principles in terms of multi-physics modeling seems to be most relevant. Such a model will also be able to represent non-linear features of the structural behavior of the asset.

As an example, a relatively simple structure with pronounced non-linear behavior is considered: Figure 1 shows a structure composed of two truss members. The structure is subjected to a vertical load R.

If the geometry is assumed to be non-deformed, the relationship between the vertical load R and the vertical displacement is obtained as:

$$R = \frac{2EA}{l} (\sin a\_0)^2 (\cos a\_0) r \tag{1}$$

which for small angles can be approximated by

$$R = \frac{2EA}{l}a\_0^2r\tag{2}$$

"digital twin" of the structure (implying that, e.g., buckling of the members

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

This highly nonlinear expression in Eq. (3) can be closely approximated by a

<sup>0</sup> <sup>1</sup> � <sup>r</sup> h <sup>1</sup> � <sup>r</sup>

<sup>0</sup> <sup>1</sup> � <sup>r</sup> h <sup>1</sup> � <sup>r</sup>

Both of the R-r (i.e., load and displacement) relationships according to Eqs. (3) and (5) are shown in Figure 2 for a slope angle of α<sup>0</sup> = π/15. It is seen that they can barely be distinguished from another. (This implies that the third order representation can also be regarded as a digital twin, although not of the one-egg kind).

Both of the curves are characterized by a very non-linear behavior, where a so-

completely horizontal position. After snap-through has occurred, a second equilibrium configuration is obtained for which a further increase of the vertical load can take place. However, this second equilibrium configuration will in most cases represent a "failed condition" in the sense that the structure will survive but such that an unwanted large displacement has taken place (which would, e.g., be the case if

Up to around one quarter of the maximum load point, the load-displacement curve is quite close to being linear. Accordingly, if only empirical load-displacement data points for this interval are available, this would typically lead to the assumption that the structural behavior is linear for any load level (unless the physical behavior of the system is taken into consideration). Having available data sets for many different structures of the same type, it is very unlikely that any of the sets contain information about the post-snap interval if all the structures are still in operation. For structures of the present type in cases where also the stress-strain behavior of the material is nonlinear, numerical solution methods will generally be required in order to compute the load-displacement curve. This will increase the computation time significantly, which will be particularly cumbersome in connection with Monte Carlo simulation procedures where a large number of repeated calculations is typically required (e.g., of the order of millions and upwards). In any case, simpli-

called snap-through occurs when the two truss members are displaced to a

the structure represents a load-carrying roof structure or an arch system).

fied but "adequate" models need to be introduced.

Load-displacement curves according to Eqs. (3) and (5) α<sup>0</sup> = π/15.

Figure 2.

159

By inserting α<sup>0</sup> = h/l (also assuming small angles), this can be written as

2h

2h r

<sup>r</sup> (4)

<sup>h</sup> (5)

themselves is not relevant due to their non-slender characteristics).

<sup>R</sup> <sup>¼</sup> <sup>2</sup>EA <sup>l</sup> <sup>α</sup><sup>2</sup>

<sup>R</sup> <sup>¼</sup> <sup>2</sup>EA � <sup>α</sup><sup>3</sup>

third order polynomial as follows, [1]:

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

where α<sup>0</sup> is the slope angle of both truss members.

However, by accounting for changing geometry due to the vertical load, a different relationship between the vertical load, R, and displacement, r, is obtained. By consideration of geometric compatibility, equilibrium conditions and a linear stress-strain relationship, the expression for the load-displacement curve can then be derived as:

$$R = \frac{2EA}{l} \left(\frac{h}{r} - 1\right) \left(\frac{1}{\sqrt{l^2 + \left(h - r\right)^2}} - \frac{1}{\sqrt{l^2 + h^2}}\right) r \tag{3}$$

where h is the height of the truss and l is half the horizontal span length. E is the modulus of elasticity for the relevant material and A is the cross-section area of both truss members. The model uncertainty associated with this relationship is presently considered to be negligible, such that it can be assumed to represent a

Figure 1. Truss structure subjected to vertical load R.

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

"digital twin" of the structure (implying that, e.g., buckling of the members themselves is not relevant due to their non-slender characteristics).

This highly nonlinear expression in Eq. (3) can be closely approximated by a third order polynomial as follows, [1]:

$$R = \frac{2EA}{l} a\_0^2 \left(1 - \frac{r}{h}\right) \left(1 - \frac{r}{2h}\right) r \tag{4}$$

By inserting α<sup>0</sup> = h/l (also assuming small angles), this can be written as

$$R = 2EA \cdot a\_0^3 \left(1 - \frac{r}{h}\right) \left(1 - \frac{r}{2h}\right) \frac{r}{h} \tag{5}$$

Both of the R-r (i.e., load and displacement) relationships according to Eqs. (3) and (5) are shown in Figure 2 for a slope angle of α<sup>0</sup> = π/15. It is seen that they can barely be distinguished from another. (This implies that the third order representation can also be regarded as a digital twin, although not of the one-egg kind).

Both of the curves are characterized by a very non-linear behavior, where a socalled snap-through occurs when the two truss members are displaced to a completely horizontal position. After snap-through has occurred, a second equilibrium configuration is obtained for which a further increase of the vertical load can take place. However, this second equilibrium configuration will in most cases represent a "failed condition" in the sense that the structure will survive but such that an unwanted large displacement has taken place (which would, e.g., be the case if the structure represents a load-carrying roof structure or an arch system).

Up to around one quarter of the maximum load point, the load-displacement curve is quite close to being linear. Accordingly, if only empirical load-displacement data points for this interval are available, this would typically lead to the assumption that the structural behavior is linear for any load level (unless the physical behavior of the system is taken into consideration). Having available data sets for many different structures of the same type, it is very unlikely that any of the sets contain information about the post-snap interval if all the structures are still in operation.

For structures of the present type in cases where also the stress-strain behavior of the material is nonlinear, numerical solution methods will generally be required in order to compute the load-displacement curve. This will increase the computation time significantly, which will be particularly cumbersome in connection with Monte Carlo simulation procedures where a large number of repeated calculations is typically required (e.g., of the order of millions and upwards). In any case, simplified but "adequate" models need to be introduced.

Figure 2. Load-displacement curves according to Eqs. (3) and (5) α<sup>0</sup> = π/15.

2. Numerical representations of physical structures

vertical load R and the vertical displacement is obtained as:

which for small angles can be approximated by

where α<sup>0</sup> is the slope angle of both truss members.

h r � 1 � � 1

<sup>R</sup> <sup>¼</sup> <sup>2</sup>EA l

<sup>R</sup> <sup>¼</sup> <sup>2</sup>EA

structure is subjected to a vertical load R.

be derived as:

Figure 1.

158

Truss structure subjected to vertical load R.

2.1 The (near to) perfect twin based on multi-physics models

In the present text, the concept of a digital twin is understood in the following sense: A digital twin is a numerical model capable of reproducing the state and behavior of a unique real asset in real time (or faster), with this model also being able to represent the performance of the asset for new and artificially generated conditions (i.e., in connection with extrapolated predictions). As a primary candidate for a digital twin, a complete numerical model based on first principles in terms of multi-physics modeling seems to be most relevant. Such a model will also be able to represent non-linear features of the structural behavior of the asset.

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

As an example, a relatively simple structure with pronounced non-linear behavior is considered: Figure 1 shows a structure composed of two truss members. The

If the geometry is assumed to be non-deformed, the relationship between the

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 <sup>q</sup> � <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

l <sup>2</sup> <sup>þ</sup> <sup>h</sup><sup>2</sup> <sup>p</sup>

<sup>2</sup> <sup>þ</sup> ð Þ <sup>h</sup> � <sup>r</sup>

where h is the height of the truss and l is half the horizontal span length. E is the modulus of elasticity for the relevant material and A is the cross-section area of both truss members. The model uncertainty associated with this relationship is presently considered to be negligible, such that it can be assumed to represent a

ð Þ cos α<sup>0</sup> r (1)

<sup>0</sup>r (2)

1

CA

r (3)

<sup>l</sup> ð Þ sin <sup>α</sup><sup>0</sup>

<sup>R</sup> <sup>¼</sup> <sup>2</sup>EA <sup>l</sup> <sup>α</sup><sup>2</sup>

However, by accounting for changing geometry due to the vertical load, a different relationship between the vertical load, R, and displacement, r, is obtained. By consideration of geometric compatibility, equilibrium conditions and a linear stress-strain relationship, the expression for the load-displacement curve can then

l

0

B@
