Related papers: G-Adaptivity: optimised graph-based mesh relocatio…
This work proposes an $r$-adaptive finite element method (FEM) using neural networks (NNs). The method employs the Ritz energy functional as the loss function, currently limiting its applicability to symmetric and coercive problems, such as…
Adaptive mesh refinement is central to the efficient solution of partial differential equations (PDEs) via the finite element method (FEM). Classical $r$-adaptivity optimizes vertex positions but requires solving expensive auxiliary PDEs…
Traditional simulation of complex mechanical systems relies on numerical solvers of Partial Differential Equations (PDEs), e.g., using the Finite Element Method (FEM). The FEM solvers frequently suffer from intensive computation cost and…
Accurate prediction of structural displacements under external loading is fundamental to structural health monitoring and seismic safety assessment. Although the finite element method (FEM) remains the prevailing approach because of its…
Correctly capturing intraoperative brain shift in image-guided neurosurgical procedures is a critical task for aligning preoperative data with intraoperative geometry for ensuring accurate surgical navigation. While the finite element…
It has become a popular paradigm to transfer the knowledge of large-scale pre-trained models to various downstream tasks via fine-tuning the entire model parameters. However, with the growth of model scale and the rising number of…
Fracture is one of the main causes of failure in engineering structures. Phase field methods coupled with adaptive mesh refinement (AMR) techniques have been widely used to model crack propagation due to their ease of implementation and…
Complex mechanic systems simulation is important in many real-world applications. The de-facto numeric solver using Finite Element Method (FEM) suffers from computationally intensive overhead. Though with many progress on the reduction of…
Mesh-based Graph Neural Networks (GNNs) have recently shown capabilities to simulate complex multiphysics problems with accelerated performance times. However, mesh-based GNNs require a large number of message-passing (MP) steps and suffer…
Computer representations of three-dimensional (3D) geometries are crucial for simulating systems and processes in engineering and science. In medicine, and more specifically, biomechanics and orthopaedics, obtaining and using 3D geometries…
This work surveys an r-adaptive moving mesh finite element method for the numerical solution of premixed laminar flame problems. Since the model of chemically reacting flow involves many different modes with diverse length scales, the…
Given a partial differential equation (PDE), goal-oriented error estimation allows us to understand how errors in a diagnostic quantity of interest (QoI), or goal, occur and accumulate in a numerical approximation, for example using the…
We analyze optimal complexity of adaptive finite element methods (AFEMs) for general second-order linear elliptic partial differential equations (PDEs) in the Lax-Milgram setting. To this end, we formulate an adaptive algorithm which steers…
We present the first rigorous convergence analysis of the smoothed adaptive finite element method (S-AFEM) proposed in [Mulita, Giani, Heltai: SIAM J. Sci. Comput. 43, 2021]. S-AFEM modifies the classical adaptive finite element method…
Computational Fluid Dynamics (CFD) is widely used in different engineering fields, but accurate simulations are dependent upon proper meshing of the simulation domain. While highly refined meshes may ensure precision, they come with high…
The cost and accuracy of simulating complex physical systems using the Finite Element Method (FEM) scales with the resolution of the underlying mesh. Adaptive meshes improve computational efficiency by refining resolution in critical…
Mesh-based simulation using Graph Neural Networks (GNNs) has been recognized as a promising approach for modeling fluid dynamics. However, the mesh refinement techniques which allocate finer resolution to regions with steep gradients can…
Graph neural networks (GNNs) naturally align with sparse operators and unstructured discretizations, making them a promising paradigm for physics-informed machine learning in computational mechanics. Motivated by discrete physics losses and…
Feature-based image matching has extensive applications in computer vision. Keypoints detected in images can be naturally represented as graph structures, and Graph Neural Networks (GNNs) have been shown to outperform traditional deep…
Mesh-based simulations play a key role when modeling complex physical systems that, in many disciplines across science and engineering, require the solution of parametrized time-dependent nonlinear partial differential equations (PDEs). In…