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In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are designed for solving partial differential equations. The…

Numerical Analysis · Mathematics 2025-04-24 Yifan Wang , Zhongshuo Lin , Hehu Xie

Engineering simulations using boundary-value partial differential equations often implicitly assume that the uncertainty in the location of the boundary has a negligible impact on the output of the simulation. In this work, we develop a…

Tissues and Organs · Quantitative Biology 2024-06-11 S. Gerry Gralton , Farah Alkhatib , Ben Zwick , George Bourantas , Adam Wittek , Karol Miller

In the frame of isogeometric analysis, we consider a Galerkin boundary element discretization of the hyper-singular integral equation associated with the 2D Laplacian. We propose and analyze an adaptive algorithm which locally refines the…

Numerical Analysis · Mathematics 2020-03-03 Gregor Gantner , Dirk Praetorius , Stefan Schimanko

Wide variety of engineering design tasks can be formulated as constrained optimization problems where the shape and topology of the domain are optimized to reduce costs while satisfying certain constraints. Several mathematical approaches…

Optimization and Control · Mathematics 2015-03-10 Igor Ostanin , Denis Zorin , Ivan Oseledets

In this paper the numerical solution of potential problems defined on 3D unbounded domains is addressed with Boundary Element Methods (BEMs), since in this way the problem is studied only on the boundary, and thus any finite approximation…

Numerical Analysis · Mathematics 2022-05-06 Antonella Falini , Carlotta Giannelli , Tadej Kanduc , Maria Lucia Sampoli , Alessandra Sestini

We derive and discuss a posteriori error estimators for Galerkin and collocation IGA boundary element methods for weakly-singular integral equations of the first-kind in 2D. While recent own work considered the Faermann residual error…

Numerical Analysis · Mathematics 2016-11-24 Michael Feischl , Gregor Gantner , Alexander Haberl , Dirk Praetorius

The concept of isogeometric analysis, whereby the parametric func- tions that are used to describe CAD geometry are also used to approx- imate the unknown fields in a numerical discretisation, has progressed rapidly in recent years. This…

Numerical Analysis · Mathematics 2013-02-22 R. N. Simpson , S. P. A. Bordas , H. Lian , J. Trevelyan

The effect of higher order continuity in the solution field by using NURBS basis function in isogeometric analysis (IGA) is investigated for an efficient mixed finite element formulation for elastostatic beams. It is based on the Hu-Washizu…

Numerical Analysis · Mathematics 2023-09-08 Myung-Jin Choi , Roger A. Sauer , Sven Klinkel

The paper presents the Isogeometric Boundary Element Method (IGABEM) algorithm for solving the plane strain problem of an isotropic linearly elastic matrix containing an open material surface of arbitrary shape. Theoretical developments are…

Numerical Analysis · Mathematics 2025-11-06 Rohit Satish Patil , Zhilin Han , Sofia G. Mogilevskaya

This paper introduces a watertight technique to deal with the boundary representation of surface-surface intersection in CAD. Surfaces play an important role in today's geometric design. The mathematical model of non-uniform rational…

Graphics · Computer Science 2024-02-19 Hua Li , Lu Zhang , Ruoxi Guo , Zushang Xiao , Rui Guo

We consider the Galerkin boundary element method (BEM) for weakly-singular integral equations of the first-kind in 2D. We analyze some residual-type a posteriori error estimator which provides a lower as well as an upper bound for the…

Numerical Analysis · Mathematics 2015-04-24 Michael Feischl , Gregor Gantner , Dirk Praetorius

This paper is concerned with the Boundary Element simulation of elastic domains that contain thin inclusions that have elastic material properties, which are different to the domain. With thin inclusions we mean inclusions with extreme…

Numerical Analysis · Mathematics 2025-05-27 Vincenzo Mallardo , Christian Dunser , Gernot Beer

This paper presents a shape optimisation system to design the shape of an acoustically-hard object in the three-dimensional open space. Boundary element method (BEM) is suitable to analyse such an exterior field. However, the conventional…

Numerical Analysis · Mathematics 2021-05-11 Toru Takahashi , Daisuke Sato , Hiroshi Isakari , Toshiro Matsumoto

The boundary element method (BEM) provides an efficient numerical framework for solving multiple scattering problems in unbounded homogeneous domains, since it reduces the discretization to the domain boundaries, thereby condensing the…

Machine Learning · Computer Science 2025-12-03 Rémi Marsal , Stéphanie Chaillat

This work introduces a scaffolding framework to compactly parametrise solid structures with conforming NURBS elements for isogeometric analysis. A novel formulation introduces a topological, geometrical and parametric subdivision of the…

Computational Geometry · Computer Science 2022-06-10 Stefano Moriconi , Parashkev Nachev , Sebastien Ourselin , M. Jorge Cardoso

Boundary element methods (BEM) reduce a partial differential equation in a domain to an integral equation on the domain's boundary. They are particularly attractive for solving problems on unbounded domains, but handling the dense matrices…

Numerical Analysis · Mathematics 2020-06-30 Steffen Börm

Isogeometric cohesive elements are presented for modeling two and three dimensional delaminated composite structures. We exploit the knot insertion algorithm offered by NURBS (Non Uniform Rational B-splines) to generate cohesive elements…

Numerical Analysis · Mathematics 2014-10-07 Vinh Phu Nguyen , Pierre Kerfriden , Stephane Bordas

Implicitly described domains are a well established tool in the simulation of time dependent problems, e.g. using level-set methods. In order to solve partial differential equations on such domains, a range of numerical methods was…

Numerical Analysis · Computer Science 2016-01-16 Christian Engwer , Andreas Nüßing

In this work, we present a combination of a multigrid approach and the phi-FEM immersed boundary finite element method to reduce its computational cost while preserving its accuracy. To further reduce the numerical cost of the approach, we…

Numerical Analysis · Mathematics 2026-05-14 Raphaël Bulle , Michel Duprez , Vanessa Lleras , Killian Vuillemot

This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. In many applications, the spatial distribution of a field needs to be…

Machine Learning · Computer Science 2021-09-01 Roberto Ponciroli , Andrea Rovinelli , Lander Ibarra