Related papers: Automated dimensional analysis for PDEs
The finite element method is an indispensable tool in engineering, but its computational complexity prevents applications for control or at system-level. Model order reduction bridges this gap, creating highly efficient yet accurate…
Search-optimization problems are plentiful in scientific and engineering domains. Artificial intelligence has long contributed to the development of search algorithms and declarative programming languages geared towards solving and modeling…
Based upon two overlapped, body-unfitted meshes, a type of unified-field monolithic fictitious domain-finite element method (UFMFD-FEM) is developed in this paper for moving interface problems of dynamic fluid-structure interactions (FSI)…
Lumped parameter methods aim to simplify the evolution of spatially-extended or continuous physical systems to that of a "lumped" element representative of the physical scales of the modeled system. For systems where the definition of a…
We present a differentiable weak-form learning approach for accelerating finite element simulations. Rather than introducing black-box source terms in the strong form of the governing equations, we augment the momentum equation directly in…
Dimensional regularization of Euclidean momentum space integrals is a highly successful technique in renormalization of quantum field theories. While it yields a straightforward algorithmic method, with which to evaluate diagrams beyond…
This study is motivated by the need to develop an efficient numerical model to simulate non-uniform interfacial degradation of reinforcing steel in concrete in an accelerated corrosion setup. In this study, a multi physics finite element…
Partial differential equation (PDE) simulation holds extensive significance in scientific research. Currently, the integration of deep neural networks to learn solution operators of PDEs has introduced great potential. In this paper, we…
As a sequel to our previous work [C. Ma, Q. Zhang and W. Zheng, SIAM J. Numer. Anal., 60 (2022)], [C. Ma and W. Zheng, J. Comput. Phys. 469 (2022)], this paper presents a generic framework of arbitrary Lagrangian-Eulerian unfitted finite…
Nonlinear energy functionals appearing in the calculus of variations can be discretized by the finite element (FE) method and formulated as a sum of energy contributions from local elements. A fast evaluation of energy functionals…
Modeling complex spatiotemporal dynamics, particularly in far-from-equilibrium systems, remains a grand challenge in science. The governing partial differential equations (PDEs) for these systems are often intractable to derive from first…
This study presents the analytical formulation and the finite element solution of fractional order nonlocal plates under both Mindlin and Kirchoff formulations. By employing consistent definitions for fractional-order kinematic relations,…
In this article we consider the widely used immersed finite element method (IFEM), in both explicit and implicit form, and its relationship to our more recent one-field fictitious domain method (FDM). We review and extend the formulation of…
Mixed-dimensional partial differential equations (PDEs) are characterized by coupled operators defined on domains of varying dimensions and pose significant computational challenges due to their inherent ill-conditioning. Moreover, the…
Recent advances in scientific machine learning (SciML) have enabled neural operators (NOs) to serve as powerful surrogates for modeling the dynamic evolution of physical systems governed by partial differential equations (PDEs). While…
The finite cell method (FCM) belongs to the class of immersed boundary methods, and combines the fictitious domain approach with high-order approximation, adaptive integration and weak imposition of unfitted Dirichlet boundary conditions.…
We present a multiscale simulation framework that couples the Finite Element Method with molecular dynamics. Bypassing traditional equations of state (EOS) by using in-line atomistic simulations, the method offers the advantage of…
Irksome is a library based on the Unified Form Language (UFL) that enables automated generation of Runge--Kutta methods for time-stepping finite element spatial discretizations of partial differential equations (PDE). Allowing users to…
Federated learning (FL) is a privacy preserving machine learning paradigm designed to collaboratively learn a global model without data leakage. Specifically, in a typical FL system, the central server solely functions as an coordinator to…
Dimensionality reduction represents the process of generating a low dimensional representation of high dimensional data. Motivated by the formation control of mobile agents, we propose a nonlinear dynamical system for dimensionality…