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We propose a Pretrained Finite Element Method (PFEM),a physics driven framework that bridges the efficiency of neural operator learning with the accuracy and robustness of classical finite element methods (FEM). PFEM consists of a physics…

Thermal Energy Storage (TES) using Phase Change Materials (PCMs) represents a critical technology for sustainable energy management and grid stability. This study presents a novel Physics-Driven Deep Learning (PDDL) framework for modeling…

Mathematical Physics · Physics 2025-12-02 Meraj Hassanzadeh , Ehsan Ghaderi , Fatemeh Fatahi , Mohamad Ali Bijarchi

The eXtended Finite Element Method (XFEM) is used to solve interface problems with an unfitted mesh. We present an implementation of the XFEM in the FEM-library deal.II. The main parts of the implementation are (i) the appropriate…

Numerical Analysis · Mathematics 2015-07-16 Thomas Carraro , Sven Wetterauer

This paper proposes a finite element method for solving the periodic steady-state problem for the scalar-valued and vector-valued Poisson equations, a simple reduction model of the Maxwell equations under the Coulomb gauge. Introducing a…

Numerical Analysis · Mathematics 2022-01-13 Masaru Miyashita , Norikazu Saito

The boundary element method (BEM) is an efficient numerical method for simulating harmonic wave propagation. It uses boundary integral formulations of the Helmholtz equation at the interfaces of piecewise homogeneous domains. The…

Numerical Analysis · Mathematics 2022-11-01 Elwin van 't Wout , Seyyed R. Haqshenas , Pierre Gélat , Timo Betcke , Nader Saffari

This work focuses on a class of elliptic boundary value problems with diffusive, advective and reactive terms, motivated by the study of three-dimensional heterogeneous physical systems composed of two or more media separated by a selective…

Numerical Analysis · Mathematics 2018-04-20 Riccardo Sacco , Aurelio Giancarlo Mauri , Giovanna Guidoboni

This paper presents two approaches: the virtual element method (VEM) and the stabilization-free virtual element method (SFVEM) for analyzing thermomechanical behavior in electronic packaging structures with geometric multi-scale features.…

Numerical Analysis · Mathematics 2025-12-29 Yanpeng Gong , Sishuai Li , Fei Qin , Bingbing Xu

The Finite Element Method (FEM) is widely used to solve discrete Partial Differential Equations (PDEs) in engineering and graphics applications. The popularity of FEM led to the development of a large family of variants, most of which…

Numerical Analysis · Computer Science 2022-03-10 Teseo Schneider , Yixin Hu , Xifeng Gao , Jeremie Dumas , Denis Zorin , Daniele Panozzo

Partition of unity methods, such as the extended finite element method (XFEM) allow discontinuities to be simulated independently of the mesh [1]. This eliminates the need for the mesh to be aligned with the discontinuity or cumbersome…

Numerical Analysis · Mathematics 2011-07-26 Sundararajan Natarajan , D. Roy Mahapatra , Stephane PA Bordas

We propose an energy-stable parametric finite element method (ES-PFEM) to discretize the motion of a closed curve under surface diffusion with an anisotropic surface energy $\gamma(\theta)$ -- anisotropic surface diffusion -- in two…

Numerical Analysis · Mathematics 2021-10-26 Yifei Li , Weizhu Bao

We present a new discretization method for homogeneous convection-diffusion-reaction boundary value problems in 3D that is a non-standard finite element method with PDE-harmonic shape functions on polyhedral elements. The element stiffness…

Numerical Analysis · Mathematics 2017-08-29 Clemens Hofreither , Ulrich Langer , Steffen Weißer

A moving mesh finite element method is studied for the numerical solution of Bernoulli free boundary problems. The method is based on the pseudo-transient continuation with which a moving boundary problem is constructed and its steady-state…

Numerical Analysis · Mathematics 2025-03-05 Jinye Shen , Heng Dai , Weizhang Huang

In our recent work [H. Zhang, F.X. Trias, A. Oliva, D. Yang, Y. Tan, Y. Sheng. PIBM: Particulate immersed boundary method for fluid-particle interaction problems. Powder Technology. 272(2015), 1-13.], a particulate immersed boundary method…

Computational Physics · Physics 2015-02-05 Hao Zhang , Haizhuan Yuan , F. Xavier Trias , Aibing Yu , Yuanqiang Tan , Assensi Oliva

In this paper, we extend the recently proposed extended scaled boundary finite element method (xSBFEM)~\cite{natarajansong2013} to study fracture parameters of interfacial cracks and cracks terminating at the interface. The approach is also…

Numerical Analysis · Mathematics 2014-07-14 Sundararajan Natarajan , Chongmin Song , Salim Belouettar

We develop an efficient and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the regularization technique of Chen, Holst, and Xu; this technique made possible the first a…

Numerical Analysis · Mathematics 2010-10-01 Michael Holst , James Andrew McCammon , Zeyun Yu , Yongcheng Zhou , Yunrong Zhu

The finite element simulation of dynamic wetting phenomena, requiring the computation of flow in a domain confined by intersecting a liquid-fluid free surface and a liquid-solid interface, with the three-phase contact line moving across the…

Computational Physics · Physics 2012-02-20 J. E. Sprittles , Y. D. Shikhmurzaev

Understanding the flow of deformable particles such as liquid drops, synthetic capsules and vesicles, and biological cells confined in a small channel is essential to a wide range of potential chemical and biomedical engineering…

Fluid Dynamics · Physics 2020-11-18 J. M. Lyu , Paul G. Chen , G. Boedec , M. Leonetti , M. Jaeger

Several engineering applications involve complex materials with significant and discontinuous variations in thermophysical properties. These include materials for thermal storage, biological tissues with blood capillaries, etc. For such…

Numerical Analysis · Mathematics 2022-09-07 Naman Bartwal , Shantanu Shahane , Somnath Roy , Surya Pratap Vanka

In this paper, we introduce the Phantom Domain Finite Element Method (PDFEM), a novel computational approach tailored for the efficient analysis of heterogeneous and composite materials. Inspired by fictitious domain methods, this method…

Numerical Analysis · Mathematics 2025-05-06 Tianlong He , Philippe Karamian-Surville , Daniel Choï

This paper proposes a strategy to solve the problems of the conventional s-version of finite element method (SFEM) fundamentally. Because SFEM can reasonably model an analytical domain by superimposing meshes with different spatial…

Numerical Analysis · Mathematics 2023-10-09 Nozomi Magome , Naoki Morita , Shigeki Kaneko , Naoto Mitsume