Periodic Reporting for period 3 - DIMENSION (Real-time Data-Informed Multi-scale Computational Methods for Material Design and Processing)
Période du rapport: 2023-01-01 au 2024-06-30
In multi-scale systems --- such as those occurring in the context of materials and production engineering --- inverse analyses (e.g. parameter estimation, inverse problems, optimization, etc.) is extremely challenging. In such systems, the high computational cost of a single multi-scale forward simulation renders inverse analyses prohibitive, and taking an inexpensive but inaccurate forward model is generally not an option. Inaccurate forward models will lead to sub-optimal designs, increased risk and uncertainty, or wrong predictions. There is thus a great need for computational methods that provide accurate and predictive yet (relatively) inexpensive approximations to complex problems in multi-scale systems. With multi-scale systems so ubiquitous in nature and in our lives --- not just in materials but also, for example, in climate systems --- such research developments have huge societal importance.
To address this important issue, this project has two main objectives geared towards developing predictive computational methods for inverse analysis of multi-scale systems. We aim to increase the quality of our predictions by addressing two main obstacles: the prohibitively high cost of forward multi-scale simulations, and the underlying error in the mathematical model. The project’s first objective is thus to develop predictive dimension reduction techniques to enable rapid and efficient multi-scale simulations. The developed techniques should deal with deterministic as well as stochastic system parameters. The second objective is to develop methods to efficiently assimilate measurement data into multi-scale models. Although the project proposal mainly focused on multi-scale systems in materials and process engineering, such methods are also relevant in other multi-scale applications such as climate systems.
To address this important issue, this project has two main objectives geared towards developing predictive computational methods for inverse analysis of multi-scale systems. We aim to increase the quality of our predictions by addressing two main obstacles: the prohibitively high cost of forward multi-scale simulations, and the underlying error in the mathematical model. The project’s first objective is thus to develop predictive dimension reduction techniques to enable rapid and efficient multi-scale simulations. The developed techniques should deal with deterministic as well as stochastic system parameters. The second objective is to develop methods to efficiently assimilate measurement data into multi-scale models. Although the project proposal mainly focused on multi-scale systems in materials and process engineering, such methods are also relevant in other multi-scale applications such as climate systems.
Part 1 - Dimension Reduction for Rapid Localization of Parametrized Microstructures
In the first period, we successfully developed new approaches for (non-intrusive) model order reduction (MOR) capable of providing inexpensive surrogates for parametrized microscale representative volume elements as required in multi-scale simulations. This is a significant step towards our goal of developing dimension reduction methods capable of flexibly and efficiently dealing with complex, parametrized microstructures exhibiting nonlinear behaviour. With such methods, we are able to drastically reduce the computational cost to simulate a multi-scale component’s response for any parameter value within a prescribed range.
In particular, we developed a non-intrusive reduced basis method based on proper orthogonal decomposition and gaussian process regression capable of constructing inexpensive surrogates for parametrized microscale problems. The proposed method has three key features. First, the microscopic stress field can be fully recovered. Second, the method can accurately predict the stress field for a wide range of material parameters; furthermore, the derivatives of the effective stress with respect to the material parameters are available and can be readily utilized in solving optimization problems. Finally, it is more data efficient, i.e. requiring less training data, as compared to directly performing a regression on the effective stress. The method has been successfully applied to material and loading parametrizations (see [Guo, Rokos & Veroy, 2021]); a further publication applying the method to the more difficult case of geometric parametrizations has also been submitted to a top journal and is now in the final stages of revision. These works have been presented and disseminated in several contributed and invited talks in conferences and workshops.
Part 2 - Data Assimilation and Optimal Experimental Design in the Multi-Scale Setting
In this part, we have three main achievements in the first period.
(1) We successfully developed a reduced order methodology for ensemble Kalman inversion that even makes corrections for the errors introduced by the reduced basis surrogate. The method is applicable and relevant not only in materials, but also in other fields such as geophysics. This work has also been presented in several workshops and conferences. A manuscript describing this method is in the final stages of preparation and will be submitted to a journal by the end of the calendar year.
(2) In this period, we built upon our previous work on reduced basis and optimal experimental design methods for 3D-VAR (for stationary problems) and extended the methods to 4D-VAR (for time-dependent problems). We were able to develop efficiently computable reduced order models for space-time discretizations, which then allowed extension of the the 3D-VAR methodology also for time-dependent problems. Two manuscripts on this work are in the final stages of preparation and will be submitted to journals by the end of the calendar year.
(3) Building upon our previous work on optimal experimental design for a simpler setting, we developed a sequential method for optimal experimental design in the Bayesian (probabilistic) setting. In this first work, we focus on a single scale problem. The developed algorithm allows for correlated noise models even for large sets of admissible sensors and remains computationally efficient for high-dimensional forward models through MOR. A manuscript describing this work is in the final stages of preparation and will be submitted by the end of the year. This work has been presented by in several invited talks.
In the first period, we successfully developed new approaches for (non-intrusive) model order reduction (MOR) capable of providing inexpensive surrogates for parametrized microscale representative volume elements as required in multi-scale simulations. This is a significant step towards our goal of developing dimension reduction methods capable of flexibly and efficiently dealing with complex, parametrized microstructures exhibiting nonlinear behaviour. With such methods, we are able to drastically reduce the computational cost to simulate a multi-scale component’s response for any parameter value within a prescribed range.
In particular, we developed a non-intrusive reduced basis method based on proper orthogonal decomposition and gaussian process regression capable of constructing inexpensive surrogates for parametrized microscale problems. The proposed method has three key features. First, the microscopic stress field can be fully recovered. Second, the method can accurately predict the stress field for a wide range of material parameters; furthermore, the derivatives of the effective stress with respect to the material parameters are available and can be readily utilized in solving optimization problems. Finally, it is more data efficient, i.e. requiring less training data, as compared to directly performing a regression on the effective stress. The method has been successfully applied to material and loading parametrizations (see [Guo, Rokos & Veroy, 2021]); a further publication applying the method to the more difficult case of geometric parametrizations has also been submitted to a top journal and is now in the final stages of revision. These works have been presented and disseminated in several contributed and invited talks in conferences and workshops.
Part 2 - Data Assimilation and Optimal Experimental Design in the Multi-Scale Setting
In this part, we have three main achievements in the first period.
(1) We successfully developed a reduced order methodology for ensemble Kalman inversion that even makes corrections for the errors introduced by the reduced basis surrogate. The method is applicable and relevant not only in materials, but also in other fields such as geophysics. This work has also been presented in several workshops and conferences. A manuscript describing this method is in the final stages of preparation and will be submitted to a journal by the end of the calendar year.
(2) In this period, we built upon our previous work on reduced basis and optimal experimental design methods for 3D-VAR (for stationary problems) and extended the methods to 4D-VAR (for time-dependent problems). We were able to develop efficiently computable reduced order models for space-time discretizations, which then allowed extension of the the 3D-VAR methodology also for time-dependent problems. Two manuscripts on this work are in the final stages of preparation and will be submitted to journals by the end of the calendar year.
(3) Building upon our previous work on optimal experimental design for a simpler setting, we developed a sequential method for optimal experimental design in the Bayesian (probabilistic) setting. In this first work, we focus on a single scale problem. The developed algorithm allows for correlated noise models even for large sets of admissible sensors and remains computationally efficient for high-dimensional forward models through MOR. A manuscript describing this work is in the final stages of preparation and will be submitted by the end of the year. This work has been presented by in several invited talks.
(1) New methods for model order reduction for nonlinear microstructure models were developed that are capable of dealing with (deterministic) material, loading, and geometric parameters. In the next phase, we will deal with the more difficult case of stochastic parameters.
(2) We successfully developed a reduced order methodology for ensemble Kalman inversion that corrects for the errors introduced by the RB approximation. In the next phase, multi-level and adaptive methods for the ensemble Kalman filter will be developed to further improve efficiency particularly in the multi-scale context.
(3) Computationally efficient RB methods for space-time discretisations of parabolic problems were developed and were subsequently exploited within a variational data assimilation framework.
(4) Through the use of model order reduction, a computationally efficient sequential method for optimal experimental design in a single-scale Bayesian setting was developed that allows for correlated noise models even for large sets of admissible sensors.
In the next phase, our research will address several key difficulties. First, the problem of stochastic parameters in Item (1) presents very fundamental difficulties especially in the high-dimensionality of the parameters. Model error --- often neglected in existing methods due to the difficulties they pose --- must also be addressed. Furthermore, the developed methods will be extended to the multi-scale setting, and will be tested particularly in realistic (single and multi-scale, possibly coupled) problems in mechanics. This will inevitably pose additional challenges to the methodology.
(2) We successfully developed a reduced order methodology for ensemble Kalman inversion that corrects for the errors introduced by the RB approximation. In the next phase, multi-level and adaptive methods for the ensemble Kalman filter will be developed to further improve efficiency particularly in the multi-scale context.
(3) Computationally efficient RB methods for space-time discretisations of parabolic problems were developed and were subsequently exploited within a variational data assimilation framework.
(4) Through the use of model order reduction, a computationally efficient sequential method for optimal experimental design in a single-scale Bayesian setting was developed that allows for correlated noise models even for large sets of admissible sensors.
In the next phase, our research will address several key difficulties. First, the problem of stochastic parameters in Item (1) presents very fundamental difficulties especially in the high-dimensionality of the parameters. Model error --- often neglected in existing methods due to the difficulties they pose --- must also be addressed. Furthermore, the developed methods will be extended to the multi-scale setting, and will be tested particularly in realistic (single and multi-scale, possibly coupled) problems in mechanics. This will inevitably pose additional challenges to the methodology.