Conference Agenda

We recently proposed an approach for Efficient Unsupervised Constitutive Law Identification and Discovery (EUCLID), which exploits machine learning tools such as sparse regression [1–3], Bayesian learning [4], or neural networks [5] to automatically discover material laws independent of stress data, but solely based on full-field displacement and global force data obtained from mechanical testing. The displacement field can be measured on the surface of a target specimen via digital image correlation (DIC).

An important feature of the approach is that, in principle, the discovery of the material law can be performed in a one-shot fashion, i.e., using only one experiment. However, this capability heavily relies upon the richness of the measured displacement data, i.e., their ability to probe the stress-strain space (where the stresses depend on the constitutive law being sought) to an extent sufficient for an accurate and robust discovery process. The richness of the displacement data is in turn directly governed by the specimen geometry.

In the present study, we aim to optimally design the geometry of the target specimen in order to maximise the richness of the deformation field obtained by the DIC method. We seek to excite various deformation modes (from tension/compression to shear) in a single optimised specimen, to maximise the performance of EUCLID. To this aim, we utilise density-based topology optimisation driven by an objective function deduced from EUCLID itself, which essentially aims at enhancing the identification robustness of material parameters (especially in noisy measurements). In this contribution, we shed light on the objective function, the topology optimisation framework, and the initial results.

The intricate and nonlinear nature of material behavior, characterized by a progressively changing stress-strain relationship, is a fundamental and indispensable property governing the behavior of numerous mechanical systems. Examples of these mechanical systems include rubber components in automobile suspensions and engine mounts, soft tissues and organs in bio-mechanics and medical engineering, as well as packaging materials such as foam, paper, and plastics. Components used for the construction of these mechanical systems often need to meet specific stiffness requirements which can be influenced by the composition of the employed material, see Steinbrecher et al. [Computational Mechanics, 69 (2022)].

On the macro or micro level, such materials can often be classified as fiber-reinforced materials, i.e., thin and long fibers embedded inside a matrix material. One way to control the stress-strain relationship of fiber-reinforced materials is to alter the geometry of the reinforcements, thus creating passive materials with a highly nonlinear stress-strain response. This can be a viable method for the development of optimized system components or meta-materials.

This method can be explored with a single beam embedded in a softer matrix. If the embedded beam is straight, the stress increase would be approximately linear with increasing strain. Bending the beam inside the matrix will lower the starting stress rate. The stress rate increases until the encased beam is straight, at which point the stress rate will not increase further. By manipulating the initial geometry of the beam, the evolution of the strain rate can be influenced.

Previously, Reinforcement Learning based shape optimization has been used to optimize structures in the context of fluid dynamics, see Fricke et al. [Advances in Computational Science and Engineering, 1 (2023)]. This approach is different from classical optimization methods, as it trains an agent to solve a specific task inside a defined problem set. While the training is computationally more expensive than a single optimization, the trained agent is able to optimize a problem inside the learned problem set with less effort.

Applying the RL-based shape optimization method to the beam geometry, an agent is trained to identify optimal beam geometries for a set of starting stress rates and ending stress rates.

In injection molding, shrinkage and warpage lead to shape deviations of the produced parts with respect to the cavity. Caused by shrinkage and warpage, these deviations occur due to uneven cooling and internal stresses inside the part. One method to mitigate this effect is to adapt the cavity shape to the expected deformation. This deformation can be determined using appropriate simulation models, which then also serve as a basis for determining the optimal cavity shape.

Shape optimization usually requires a sequence of forward simulations, which can be computationally expensive. To reduce this computational cost, we use Gaussian Process Regression (GPR) as a surrogate model. The GPR learns the objective function, which measures the shape difference. This difference is computed by taking the average Euclidean distance of sample points on the surface of the deformed product on the one hand and the desired shape on the other hand. Additionally, GPR has the benefit that it allows to account for uncertainty in the model parameters and thus provides a means to investigate their influence on the optimization result. We present a GPR trained with samples from a finite-element solid-body model. It predicts the deformation of the product after solidification and, together with GPR, allows for efficient cavity shape optimization. The optimization parameters are the position of spline points representing the geometry.

To further improve the learning efficiency, we use Bayesian optimization. This approach selects the next training points by utilizing an acquisition function that balances exploration and exploitation. Here, training points with low function values and high uncertainty are given priority. This achieves the goal of finding the optimal solution with the least number of training points. The Bayesian optimization framework was implemented in Python code.

The material parameters can underlie fluctuations because of different batches or when using recycled material. To account for these uncertainties in the material parameters, a new formulation of the objective function is proposed to find the optimal shape resulting in low shrinkage and warpage as well as low variance with respect to the input parameters.

The nasal cavity is one of the most important organs of the human body. Its various functionalities are essential for the well-being of the individual person. It is responsible for the sense of smell, supports degustation, and filters, tempers, and moistens the inhaled air to provide optimal conditions for the lung. Diseases of the nasal cavity like chronic rhinosinusitis, septal deviation, or nasal polyps may lead to restrictions or complete loss of these functionalities. A decreased respiratory capability, the development of irritations and inflammations, and lung diseases can be the consequences.

The shape of the nasal cavity varies from person to person with stronger changes being present in pathological cases. A decent analysis on a per-patient basis is hence crucial to plan for a surgery with a successful outcome. Nowadays diagnostic methods rely on morphological analyses of the shape of the nasal cavity. They employ methods of medical imaging such as computed tomography (CT) or magnetic resonance imaging (MRI), and nasal endoscopy. Such methods, however, do not cover the fluid mechanics of respiration, which are essential to understand the impact of a pathology on the quality of respiration, and to plan for a surgery. Only a meaningful and physics-based diagnosis can help to adequately understand the functional efficiency of the nasal cavity, to quantify the impact of different pathologies on respiration, and to support surgeons in decision making.

Physics-based methods to diagnose pathologies in the human respiratory system have recently evolved to include results of computational fluid dynamics (CFD) simulations. In the current study, a reinforcement learning (RL) algorithm that learns to optimize the nasal cavity shape based on feedback from CFD simulations is developed. The final structure of the airway then functions as the proposed surgery. It is investigated how the algorithm finds the optimal structural modification based on various optimization criteria:

(i) the capability of inhaling and supplying the lung with air, expressed by the pressure difference between the nostrils and the pharynx;

(ii) the functionality of the nose for heating up incoming air, represented by the temperature difference between the nostrils and the pharynx;

(iii) the possibility for a balanced air supply between the left and right nasal passages, realized by equal mass flow rates through both nasal passages;

Furthermore, different types of RL algorithms are employed and their computational efficiency is analyzed. It is the aim to further advance these techniques to include them into clinical pathways, thereby allowing personalized analyses on a per-patient basis.