# Engineering, Emulators, Digital Twins, and Performance Engineering

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Methodology: The Design and Analysis of Emulators

#### 2.1. Design of Emulators of Systems

_{1}, x

_{2}, …, x

_{p})

_{1}, x

_{2}, …, x

_{p}) ∈ $D$ is the input variable, y ∈ R is the output variable, and $D$ is the input variable space and a subset of R

^{p}. The deterministic function f does not need to have an analytic representation. Frequently, the solution of such systems of equations is an approximation of model (1), g, where:

_{1}, x

_{2}, …, x

_{p})

#### 2.2. Computer Experiments from Simulations

_{n}= {x

_{1}, x

_{2}, …, x

_{n}} that ensure the deviation between model (1) and metamodel (2):

^{p}(0 ≤ x

_{j}≤ 1, j = 1, 2, …, p), where p is the number of the variables. Emulators or surrogate models are used to minimise (3). An example is the Kriging model that assumes that the experimental responses are realisations of zero-mean Gaussian random field plus a regression term:

^{p}, $f\left(x\right)={\left({f}_{1}\left(x\right),{f}_{2}\left(x\right),\dots ,{f}_{m}\left(x\right)\right)}^{T}$ is a set of specified trend functions and $\beta ={\left({\beta}_{1},{\beta}_{2},\dots ,{\beta}_{m}\right)}^{T}$ is a set of coefficients. Z(x) is a Gaussian random process with zero mean and stationary covariance over D so that $E\left[Y\left(x\right)\right]={\beta}^{T}f\left(x\right)$ and $E\left[Y\left(x\right)\right]={\beta}^{T}f\left(x\right)$, where ${\sigma}_{Z}^{2}$ is the process variance, and R is a correlation function depending on the displacement vector h between any pair of points in D and on a parameter θ. A typical correlation function is:

_{1},θ

_{2},…,θ

_{p},s)

^{T}, θ

_{j}are positive scale parameters, and s is a common smoothing parameter. Parameter θ

_{j}describes how rapidly correlation decays in direction j with increasing distance h

_{j}. The positive correlation between outputs in (6) diminishes with increasing distance between input locations. If θ

_{j}= θ ∀j, the correlation depends only on the distance |h| between any pair of points (x and x + h). Parameter s describes the pattern of the correlation decay. When s = 2, the correlation function is Gaussian. The power exponential family: s = 2 gives infinitely differentiable trajectories for the Gaussian process realisations. Values s < 2 correspond to non-differentiable trajectories. The prediction of the response at a new point x

_{0}is based on prior information in a set of experimental points ${x}^{n}={\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)}^{T}$, with x

_{i}∈D for I = 1, 2, …, n. A common prediction criterion is the mean squared prediction error. Under the hypothesis that the joint random variable $\left(Y\left({x}_{0}\right),Y\left({x}_{1}\right),Y\left({x}_{2}\right),\dots ,Y\left({x}_{n}\right)\right)$ is a multivariate normal, denoted as $N\left({\left({f}_{0}^{T},\mathrm{F}\right)}^{T}\beta ,{\sigma}_{Z}^{2}\right)$, with $=\left(\begin{array}{cc}1& {\mathrm{r}}_{0}^{T}\\ {\mathrm{r}}_{0}& R\end{array}\right)$, the conditional mean of Y(x) at the new point x

_{0,}given the process data, ${\widehat{Y}}_{0}=E\left[Y\left({x}_{0}\right)|{Y}^{n}\right]$, is:

_{0}is the m × 1 vector of the trend functions in x

_{0}; F is the n × m matrix ${\left\{{f}_{j}\left({x}_{i}\right)\right\}}_{\begin{array}{c}i=1,\dots ,n\\ j=1,\dots ,m\end{array}}$ of the trend functions computed in $\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)$; r

_{0}is the correlation vector ${\left(R\left({x}_{0}-{x}_{1}\right),\dots ,R\left({x}_{0}-{x}_{n}\right)\right)}^{T}$; and R is the n × n correlation matrix whose (i,j) element is $R\left({{h}_{ij}=x}_{i}-{x}_{j}\right)$.

_{0}) because it minimizes the mean squared prediction error, $E\left[{\left({\widehat{Y}}_{0}-{Y}_{0}\right)}^{2}\right]$, and is also unique. For more information on computer experiments, see [8,13,14].

#### 2.3. Optimising Products and Processes

- Modelling: This can be derived from the results of an initial experiment, purely on theoretical grounds, or by a combination of the two.
- Uncertainty: Characterising uncertainty in the system is describing how the input factors vary.
- Computer experiment design: Plan a computer experimental design of the input factors.
- Generate simulated data: Apply the noise distributions in the computer experimental design.
- Stochastic emulator: Construct a model that relates response variables to the design factor settings.
- Optimisation: Determine a setup that ensures optimisation of both target performance and robustness.

_{F}corresponding to a degradation variable or faults. The stochastic emulator provides prescriptive and optimisation capabilities.

#### 2.4. Other Methods

## 3. Case Study 1: The Piston Simulator

## 4. Case Study 2: The PENSIM Simulator

## 5. Discussion and Conclusions

- A description of emulators that can be derived from computationally intensive models using Gaussian processes.
- Consideration of hybrid models combining physical and simulations-based data.
- Applications of emulators for enhanced monitoring and diagnostics.
- Incorporation of emulators in digital twin platforms.
- An introduction of stochastic emulators to optimise performance and robustness.
- Case studies demonstrating the above.

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Control charts of cycle time (

**left**) for 30 and another 30 (

**right**) consecutive observations (JMP 17.0).

**Figure 7.**P versus X scatterplots in various pH categories. The colour of points indicates the level of feed (JMP 17.0).

**Figure 8.**Contour plots of pH versus feed, for P (

**left**) and X (

**right**), with observations used to fit the Gaussian model marked as dots (JMP 17.0).

Factor | Lower Level | Higher Level |
---|---|---|

S0 | 5 | 15 |

X0 | 0.05 | 0.1 |

pH | 4 | 5 |

T | 293 | 298 |

air | 6 | 8.6 |

agitation | 15 | 29.9 |

time | 250 | 350 |

feed | 0.0226 | 0.0426 |

S0 | X0 | pH | T | Air | Agitation | Time | Feed |
---|---|---|---|---|---|---|---|

9.9994526 | 0.0778006 | 4.8308467 | 297.99973 | 7.2998112 | 15.000027 | 350 | 0.0417674 |

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Kenett, R.S.
Engineering, Emulators, Digital Twins, and Performance Engineering. *Electronics* **2024**, *13*, 1829.
https://doi.org/10.3390/electronics13101829

**AMA Style**

Kenett RS.
Engineering, Emulators, Digital Twins, and Performance Engineering. *Electronics*. 2024; 13(10):1829.
https://doi.org/10.3390/electronics13101829

**Chicago/Turabian Style**

Kenett, Ron S.
2024. "Engineering, Emulators, Digital Twins, and Performance Engineering" *Electronics* 13, no. 10: 1829.
https://doi.org/10.3390/electronics13101829