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A novel, finite-element-based framework for sparse data solution reconstruction and multiple choices

Wiera Bielajewa, Michelle Baxter, Perumal Nithiarasu

Abstract

Digital twinning offers a capability of effective real-time monitoring and control, which are vital for cost-intensive experimental facilities, particularly the ones where extreme conditions exist. Sparse experimental measurements collected by various diagnostic sensors are usually the only source of information available during the course of a physical experiment. Consequently, in order to enable monitoring and control of the experiment (digital twinning), the ability to perform inverse analysis, facilitating the full field solution reconstruction from the sparse experimental data in real time, is crucial. This paper shows for the first time that it is possible to directly solve inverse problems, such as solution reconstruction, where some or all boundary conditions (BCs) are unknown, by purely using a finite-element (FE) approach, without needing to employ any traditional inverse analysis techniques or any machine learning models, as is normally done in the field. This novel and efficient FE-based inverse analysis framework employs a conventional FE discretisation, splits the loading vector into two parts corresponding to the known and unknown BCs, and then defines a loss function based on that split. In spite of the loading vector split, the loss function preserves the element connectivity. This function is minimised using a gradient-based optimisation. Furthermore, this paper presents a novel modification of this approach, which allows it to generate a range of different solutions satisfying given requirements in a controlled manner. Controlled multiple solution generation in the context of inverse problems and their intrinsic ill-posedness is a novel notion, which has not been explored before. This is done in order to potentially introduce the capability of semi-autonomous system control with intermittent human intervention to the workflow.

A novel, finite-element-based framework for sparse data solution reconstruction and multiple choices

Abstract

Digital twinning offers a capability of effective real-time monitoring and control, which are vital for cost-intensive experimental facilities, particularly the ones where extreme conditions exist. Sparse experimental measurements collected by various diagnostic sensors are usually the only source of information available during the course of a physical experiment. Consequently, in order to enable monitoring and control of the experiment (digital twinning), the ability to perform inverse analysis, facilitating the full field solution reconstruction from the sparse experimental data in real time, is crucial. This paper shows for the first time that it is possible to directly solve inverse problems, such as solution reconstruction, where some or all boundary conditions (BCs) are unknown, by purely using a finite-element (FE) approach, without needing to employ any traditional inverse analysis techniques or any machine learning models, as is normally done in the field. This novel and efficient FE-based inverse analysis framework employs a conventional FE discretisation, splits the loading vector into two parts corresponding to the known and unknown BCs, and then defines a loss function based on that split. In spite of the loading vector split, the loss function preserves the element connectivity. This function is minimised using a gradient-based optimisation. Furthermore, this paper presents a novel modification of this approach, which allows it to generate a range of different solutions satisfying given requirements in a controlled manner. Controlled multiple solution generation in the context of inverse problems and their intrinsic ill-posedness is a novel notion, which has not been explored before. This is done in order to potentially introduce the capability of semi-autonomous system control with intermittent human intervention to the workflow.
Paper Structure (20 sections, 15 equations, 20 figures, 7 tables)

This paper contains 20 sections, 15 equations, 20 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Linear transient problem
  4. Inverse analysis
  5. Minimisation
  6. Multiple choice generation
  7. Results and discussion
  8. Thermal field reconstruction
  9. Number of measurements and errors
  10. $\Delta t_{\mathrm{ref}}$, $\Delta t_{\mathrm{rec}}$, and errors
  11. Runtimes
  12. Multiple choices
  13. Total heat
  14. Runtimes
  15. Conclusions
  16. ...and 5 more sections

Figures (20)

  • Figure 1: Summary of the workflow for the solution reconstruction and multiple solution generation. For the steady-state problem only $\left[ \bm{K} \right]$ global matrix and $\left\{ \bm{f} \right\}_{h}$ global vector are computed, and the whole procedure is repeated only once; whereas for the transient problem, it is repeated every time step and then $\left\{ \bm{T} \right\}_{\mathrm{final}}$ refers to the solution at one time step. For solution reconstruction the initial temperature distribution $\left\{ \bm{T} \right\}_{\mathrm{init}}$ is a uniform temperature equal to the average measurement value for steady state; $\left\{ \bm{T} \right\}_{\mathrm{init}}$ used for each time step is a uniform temperature equal to the average time-step measurement value.
  • Figure 2: Labels for the applied BCs (Table \ref{['table0']}).
  • Figure 3: Two options for the measurement placement considered: 15 (left) and 9 (right) measurements.
  • Figure 4: Average and maximum time step runtimes for all cases, compared with the time step size $\Delta t_{\mathrm{rec}}$ used; one GPU is used for the local and supercomputer calculations.
  • Figure 5: The dependence of relative and absolute errors on time for Case No. 1.
  • ...and 15 more figures