Locally optimal unstructured finite element meshes in 3 dimensions

This paper investigates the adaptive finite element solution of a general class of variational problems in three dimensions using a combination of node movement, edge swapping, face swapping and node insertion. The adaptive strategy proposed is a generalization of previous work in two dimensions and is based upon the construction of a hierarchy of locally optimal meshes. Results presented, both for a single equation and a system of coupled equations, suggest that this approach is able to produce better meshes of tetrahedra than those obtained by more conventional adaptive strategies and in a relatively efficient manner

An optimal control method for time-dependent fluid-structure interaction problems

In this article, we derive an adjoint fluid-structure interaction (FSI) system in an arbitrary Lagrangian-Eulerian (ALE) framework, based upon a one-field finite element method. A key feature of this approach is that the interface condition is automatically satisfied and the problem size is reduced since we only solve for one velocity field for both the primary and adjoint system. A velocity (and/or displacement)-matching optimisation problem is considered by controlling a distributed force. The optimisation problem is solved using a gradient descent method, and a stabilised Barzilai-Borwein method is adopted to accelerate the convergence, which does not need additional evaluations of the objective functional. The proposed control method is validated and assessed against a series of static and dynamic benchmark FSI problems, before being applied successfully to solve a highly challenging FSI control problem

Towards the development and application of an optimal solver for continuum models of tumour growth

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On spatial adaptivity and interpolation when using the method of lines

The solution of time-dependent partial differential equations with discrete time static remeshing is considered within a method of lines framework. Numerical examples in one and two space dimensions are used to show that spatial interpolation error may have an important impact on the efficiency of integration. Analysis of a simple problem and of the time integration method is used to confirm the experimental results and a computational test for monitoring the impact of this error is derived and tested

Developing a cost-effective emulator for groundwater flow modeling using deep neural operators

Current groundwater models face a significant challenge in their implementation due to heavy computational burdens. To overcome this, our work proposes a cost-effective emulator that efficiently and accurately forecasts the impact of abstraction in an aquifer. Our approach uses a deep neural operator (DeepONet) to learn operators that map between infinite-dimensional function spaces via deep neural networks. The goal is to infer the distribution of hydraulic head in a confined aquifer in the presence of a pumping well. We successfully tested the DeepONet on four problems, including two forward problems, an inverse analysis, and a nonlinear system. Additionally, we propose a novel extension of the DeepONet-based architecture to generate accurate predictions for varied hydraulic conductivity fields and pumping well locations that are unseen during training. Our emulator's predictions match the target data with excellent performance, demonstrating that the proposed model can act as an efficient and fast tool to support a range of tasks that require repetitive forward numerical simulations or inverse simulations of groundwater flow problems. Overall, our work provides a promising avenue for developing cost-effective and accurate groundwater models

Understanding the Efficacy of U-Net & Vision Transformer for Groundwater Numerical Modelling

This paper presents a comprehensive comparison of various machine learning models, namely U-Net, U-Net integrated with Vision Transformers (ViT), and Fourier Neural Operator (FNO), for time-dependent forward modelling in groundwater systems. Through testing on synthetic datasets, it is demonstrated that U-Net and U-Net + ViT models outperform FNO in accuracy and efficiency, especially in sparse data scenarios. These findings underscore the potential of U-Net-based models for groundwater modelling in real-world applications where data scarcity is prevalent