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In this contribution we present a survey of concepts in localized model order reduction methods for parameterized partial differential equations. The key concept of localized model order reduction is to construct local reduced spaces that…

Numerical Analysis · Mathematics 2019-10-29 Andreas Buhr , Laura Iapichino , Mario Ohlberger , Stephan Rave , Felix Schindler , Kathrin Smetana

We develop a stabilized cut finite element method for the stationary convection diffusion problem on a surface embedded in ${\mathbb{R}}^d$. The cut finite element method is based on using an embedding of the surface into a three…

Numerical Analysis · Mathematics 2018-07-24 Erik Burman , Peter Hansbo , Mats G. Larson , Andre Massing , Sara Zahedi

We propose a projection-based model order reduction procedure for a general class of parametric quasi-static problems in nonlinear mechanics with internal variables. The methodology is integrated in the industrial finite element code code…

Numerical Analysis · Mathematics 2023-01-02 Eki Agouzal , Jean-Philippe Argaud , Michel Bergmann , Guilhem Ferté , Tommaso Taddei

Projection-based reduced order models are effective at approximating parameter-dependent differential equations that are parametrically separable. When parametric separability is not satisfied, which occurs in both linear and nonlinear…

Numerical Analysis · Mathematics 2021-10-22 Peter Sentz , Kristian Beckwith , Eric C. Cyr , Luke N. Olson , Ravi Patel

Reduced Order Models (ROMs) have been regarded as an efficient alternative to conventional high-fidelity Computational Fluid Dynamics (CFD) for accelerating the design and optimization processes in engineering applications. Many industrial…

Numerical Analysis · Mathematics 2026-01-15 Shenhui Ruan , Andreas G. Class , Gianluigi Rozza

In this contribution, we coupled the isogeometric analysis to a reduced order modelling technique in order to provide a computationally efficient solution in parametric domains. In details, we adopt the free-form deformation method to…

Numerical Analysis · Mathematics 2020-11-23 Fabrizio Garotta , Nicola Demo , Marco Tezzele , Massimo Carraturo , Alessandro Reali , Gianluigi Rozza

Computational analysis with the finite element method requires geometrically accurate meshes. It is well known that high-order meshes can accurately capture curved surfaces with fewer degrees of freedom in comparison to low-order meshes.…

Mathematical Software · Computer Science 2024-01-30 Ketan Mittal , Veselin A. Dobrev , Patrick Knupp , Tzanio Kolev , Franck Ledoux , Claire Roche , Vladimir Z. Tomov

This paper presents a novel approach for solving fourth-order phase-field models in brittle fracture mechanics using the Interior Penalty Finite Element Method (IP-FEM). The fourth-order model improves numerical stability and accuracy…

Numerical Analysis · Mathematics 2025-04-15 Tian Tian , Chen Chunyu , Wei Huayi

Parametric model order reduction using reduced basis methods can be an effective tool for obtaining quickly solvable reduced order models of parametrized partial differential equation problems. With speedups that can reach several orders of…

Numerical Analysis · Mathematics 2022-01-26 Mario Ohlberger , Stephan Rave

We present a machine-learning strategy for finite element analysis of solid mechanics wherein we replace complex portions of a computational domain with a data-driven surrogate. In the proposed strategy, we decompose a computational domain…

Numerical Analysis · Mathematics 2023-10-24 Eric Parish , Payton Lindsay , Timothy Shelton , John Mersch

We develop a high order cut finite element method for the Stokes problem based on general inf-sup stable finite element spaces. We focus in particular on composite meshes consisting of one mesh that overlaps another. The method is based on…

Numerical Analysis · Mathematics 2015-05-05 August Johansson , Mats G. Larson , Anders Logg

We propose a new reduced order modeling strategy for tackling parametrized Partial Differential Equations (PDEs) with linear constraints, in particular Darcy flow systems in which the constraint is given by mass conservation. Our approach…

Numerical Analysis · Mathematics 2023-11-27 Wietse M. Boon , Nicola R. Franco , Alessio Fumagalli , Paolo Zunino

Tactile perception is key to dexterous manipulation, yet simulating high-resolution elastomer deformation remains computationally prohibitive. Finite element methods (FEM) deliver high fidelity but demand costly remeshing, while Material…

Robotics · Computer Science 2026-05-07 Yuhu Guo , Zhikai Shen , Jiasheng Qu , Chenghao Qian , Yuming Huang , Bin Chen , Guoxing Fang

The scaled boundary finite element method (SBFEM) has recently been employed as an efficient means to model three-dimensional structures, in particular when the geometry is provided as a voxel-based image. To this end, an octree…

Numerical Analysis · Mathematics 2023-10-31 Hauke Gravenkamp , Albert A. Saputra , Sascha Eisenträger

We consider convection-diffusion problems in time-dependent domains and present a space-time finite element method based on quadrature in time which is simple to implement and avoids remeshing procedures as the domain is moving. The…

Numerical Analysis · Mathematics 2017-07-25 Sara Zahedi

In this work, we present an approach for the efficient treatment of parametrized geometries in the context of POD-Galerkin reduced order methods based on Finite Volume full order approximations. On the contrary to what is normally done in…

Numerical Analysis · Mathematics 2020-02-07 Giovanni Stabile , Matteo Zancanaro , Gianluigi Rozza

We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in $\mathbb{R}^d$ of arbitrary codimension. The method is based on using continuous…

Numerical Analysis · Mathematics 2016-10-07 Erik Burman , Peter Hansbo , Mats G. Larson , Andre Massing

We develop a new optimisation technique that combines multiresolution subdivision surfaces for boundary description with immersed finite elements for the discretisation of the primal and adjoint problems of optimisation. Similar to wavelets…

Numerical Analysis · Mathematics 2016-01-20 Kosala Bandara , Thomas Rüberg , Fehmi Cirak

The virtual element method (VEM) is a stabilized Galerkin method that is robust and accurate on general polygonal meshes. This feature makes it an appealing candidate for simulations involving meshes with embedded interfaces and evolving…

Numerical Analysis · Mathematics 2025-10-03 Ramsharan Rangarajan , N. Sukumar

Highly accurate simulations of complex phenomena governed by partial differential equations (PDEs) typically require intrusive methods and entail expensive computational costs, which might become prohibitive when approximating steady-state…

Machine Learning · Computer Science 2023-05-17 Paolo Conti , Giorgio Gobat , Stefania Fresca , Andrea Manzoni , Attilio Frangi