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Journal articles on the topic 'Unfitted mesh methods'

Author: Grafiati

Published: 5 October 2024

Last updated: 31 July 2025

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1

Fernández, Miguel A., and Mikel Landajuela. "Splitting schemes and unfitted-mesh methods for the coupling of an incompressible fluid with a thin-walled structure." IMA Journal of Numerical Analysis 40, no. 2 (2019): 1407–53. http://dx.doi.org/10.1093/imanum/dry098.

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Abstract Two unfitted-mesh methods for a linear incompressible fluid/thin-walled structure interaction problem are introduced and analyzed. The spatial discretization is based on different variants of Nitsche’s method with cut elements. The degree of fluid–solid splitting (semi-implicit or explicit) is given by the order in which the space and time discretizations are performed. The a priori stability and error analysis shows that strong coupling is avoided without compromising stability and accuracy. Numerical experiments with a benchmark illustrate the accuracy of the different methods propo

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2

Burman, Erik, Peter Hansbo, Mats G. Larson, and Sara Zahedi. "Cut finite element methods." Acta Numerica 34 (July 2025): 1–121. https://doi.org/10.1017/s0962492925000017.

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Cut finite element methods (CutFEM) extend the standard finite element method to unfitted meshes, enabling the accurate resolution of domain boundaries and interfaces without requiring the mesh to conform to them. This approach preserves the key properties and accuracy of the standard method while addressing challenges posed by complex geometries and moving interfaces.In recent years, CutFEM has gained significant attention for its ability to discretize partial differential equations in domains with intricate geometries. This paper provides a comprehensive review of the core concepts and key d

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3

Heimann, Fabian, Christoph Lehrenfeld, Paul Stocker, and Henry von Wahl. "Unfitted Trefftz discontinuous Galerkin methods for elliptic boundary value problems." ESAIM: Mathematical Modelling and Numerical Analysis, August 1, 2023. http://dx.doi.org/10.1051/m2an/2023064.

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We propose a new geometrically unfitted finite element method based on discontinuous Trefftz ansatz spaces. Trefftz methods allow for a reduction in the number of degrees of freedom in discontinuous Galerkin methods, thereby, the costs for solving arising linear systems significantly. This work shows that they are also an excellent way to reduce the number of degrees of freedom in an unfitted setting. We present a unified analysis of a class of geometrically unfitted discontinuous Galerkin methods with different stabilisation mechanisms to deal with small cuts between the geometry and the mesh

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4

Frachon, Thomas, Erik Nilsson, and Sara Zahedi. "Divergence-free cut finite element methods for Stokes flow." BIT Numerical Mathematics 64, no. 4 (2024). http://dx.doi.org/10.1007/s10543-024-01040-x.

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AbstractWe develop two unfitted finite element methods for the Stokes equations based on $$\textbf{H}^{{{\,\textrm{div}\,}}}$$ H div -conforming finite elements. Both cut finite element methods exhibit optimal convergence order for the velocity, pointwise divergence-free velocity fields, and well-posed linear systems, independently of the position of the boundary relative to the computational mesh. The first method is a cut finite element discretization of the Stokes equations based on the Brezzi–Douglas–Marini (BDM) elements and involves interior penalty terms to enforce tangential continuity

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5

Heimann, Fabian, and Christoph Lehrenfeld. "Geometry error analysis of a parametric mapping for higher order unfitted space–time methods." IMA Journal of Numerical Analysis, March 10, 2025. https://doi.org/10.1093/imanum/drae098.

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Abstract In Heimann, Lehrenfeld, and Preuß (2023, SIAM J. Sci. Comp., 45(2), B139–B165), new geometrically unfitted space–time Finite Element methods for partial differential equations posed on moving domains of higher-order accuracy in space and time have been introduced. For geometrically higher-order accuracy a parametric mapping on a background space–time tensor-product mesh has been used. In this paper, we concentrate on the geometrical accuracy of the approximation and derive rigorous bounds for the distance between the realized and an ideal mapping in different norms and derive results

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6

Jankuhn, Thomas, Maxim A. Olshanskii, Arnold Reusken, and Alexander Zhiliakov. "Error analysis of higher order trace finite element methods for the surface Stokes equation." Journal of Numerical Mathematics, October 4, 2020. http://dx.doi.org/10.1515/jnma-2020-0017.

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AbstractThe paper studies a higher order unfitted finite element method for the Stokes system posed on a surface in ℝ3. The method employs parametric Pk-Pk−1 finite element pairs on tetrahedral bulk mesh to discretize the Stokes system on embedded surface. Stability and optimal order convergence results are proved. The proofs include a complete quantification of geometric errors stemming from approximate parametric representation of the surface. Numerical experiments include formal convergence studies and an example of the Kelvin--Helmholtz instability problem on the unit sphere.

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7

Pretti, Giuliano, Robert E. Bird, Nathan D. Gavin, William M. Coombs, and Charles E. Augarde. "A Stable Poro‐Mechanical Formulation for Material Point Methods Leveraging Overlapping Meshes and Multi‐Field Ghost Penalisation." International Journal for Numerical Methods in Engineering 126, no. 5 (2025). https://doi.org/10.1002/nme.7630.

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ABSTRACTThe Material Point Method (MPM) is widely used to analyse coupled (solid‐water) problems under large deformations/displacements. However, if not addressed carefully, MPM u‐p formulations for poromechanics can be affected by two major sources of instability. Firstly, inf‐sup condition violation can arise when the spaces for the displacement and pressure fields are not chosen correctly, resulting in an unstable pressure field when the equations are monolithically solved. Secondly, the intrinsic nature of particle‐based discretisation makes the MPM an unfitted mesh‐based method, which can

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8

Kirchhart, Matthias. "On particles and splines in bounded domains." ESAIM: Mathematical Modelling and Numerical Analysis, May 6, 2020. http://dx.doi.org/10.1051/m2an/2020032.

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We propose numerical schemes that enable the application of particle methods for advection problems in general bounded domains. These schemes combine particle fields with Cartesian tensor product splines and a fictitious domain approach. Their implementation only requires a fitted mesh of the domain's boundary, and not the domain itself, where an unfitted Cartesian grid is used. We establish the stability and consistency of these schemes in $W^{s,p}$-norms, $s\in\mathbb{R}$, $1\leq p\leq\infty$.

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9

Yang, Fanyi. "The least squares finite element method for elasticity interface problem on unfitted mesh." ESAIM: Mathematical Modelling and Numerical Analysis, March 7, 2024. http://dx.doi.org/10.1051/m2an/2024015.

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In this paper, we propose and analyze the least squares finite element methods for the linear elasticity interface problem in the stress-displacement system on unfitted meshes. We consider the cases that the interface is $C^2$ or polygonal, and the exact solution $(\bsigma, \bu)$ belongs to $H^s(\div; \Omega_0 \cup \Omega_1) \times H^{1+s}(\Omega_0 \cup \Omega_1)$ with $s > 1/2$. Two types of least squares functionals are defined to seek the numerical solution. The first is defined by simply applying the $L^2$ norm least squares principle, and requires the condition $s \geq 1$. The second i

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10

Lehrenfeld, Christoph, Tim van Beeck, and Igor Voulis. "Analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem." Mathematics of Computation, October 30, 2024. http://dx.doi.org/10.1090/mcom/4027.

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In this paper we present a new H ( div ) H(\operatorname {div}) -conforming unfitted finite element method for the mixed Poisson problem which is robust in the cut configuration and preserves conservation properties of body-fitted finite element methods. The key is to formulate the divergence-constraint on the active mesh, instead of the physical domain, in order to obtain robustness with respect to cut configurations without the need for a stabilization that pollutes the mass balance. This change in the formulation results in a slight inconsistency, but does not affect the accuracy of the flu

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11

Corti, Daniele, Guillaume Delay, Miguel A. Fernández, Fabien Vergnet, and Marina Vidrascu. "Low-order fictitious domain method with enhanced mass conservation for an interface Stokes problem." ESAIM: Mathematical Modelling and Numerical Analysis, December 19, 2023. http://dx.doi.org/10.1051/m2an/2023103.

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One of the main difficulties that has to be faced with fictitious domain approximation of incompressible flows with immersed interfaces is related to the potential lack of mass conservation across the interface. In this paper, we propose and analyze a low order fictitious domain stabilized finite element method which mitigates this issue with the addition of a single velocity constraint. We provide a complete a priori numerical analysis of the method under minimal regularity assumptions. A comprehensive numerical study illustrates the capabilities of the proposed method, including comparisons

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12

Toprak, Teoman, Matthias Rieckmann, and Florian Kummer. "Cell Agglomeration Strategy for Cut Cells in eXtended Discontinuous Galerkin Methods." International Journal for Numerical Methods in Engineering 126, no. 9 (2025). https://doi.org/10.1002/nme.70013.

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ABSTRACTIn this work, a cell agglomeration strategy for the cut cells arising in the eXtended discontinuous Galerkin (XDG) method is presented. Cut cells are a fundamental aspect of unfitted mesh approaches, where complex geometries or interfaces separating subdomains are embedded into structured background grids to facilitate the mesh generation process. In such methods, arbitrary small cells occur due to the intersections of background cells with embedded geometries and lead to discretization difficulties due to their diminutive sizes. Furthermore, temporal evolutions of these geometries may

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13

Idesman, Alexander, and Bikash Dey. "3rd and 11th orders of accuracy of ‘linear’ and ‘quadratic’ elements for Poisson equation with irregular interfaces on Cartesian meshes." International Journal of Numerical Methods for Heat & Fluid Flow, December 31, 2021. http://dx.doi.org/10.1108/hff-09-2021-0596.

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Purpose The purpose of this paper is as follows: to significantly reduce the computation time (by a factor of 1,000 and more) compared to known numerical techniques for real-world problems with complex interfaces; and to simplify the solution by using trivial unfitted Cartesian meshes (no need in complicated mesh generators for complex geometry). Design/methodology/approach This study extends the recently developed optimal local truncation error method (OLTEM) for the Poisson equation with constant coefficients to a much more general case of discontinuous coefficients that can be applied to do

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14

Erdbrügger, Tim, Andreas Westhoff, Malte Höltershinken, et al. "CutFEM forward modeling for EEG source analysis." Frontiers in Human Neuroscience 17 (August 22, 2023). http://dx.doi.org/10.3389/fnhum.2023.1216758.

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IntroductionSource analysis of Electroencephalography (EEG) data requires the computation of the scalp potential induced by current sources in the brain. This so-called EEG forward problem is based on an accurate estimation of the volume conduction effects in the human head, represented by a partial differential equation which can be solved using the finite element method (FEM). FEM offers flexibility when modeling anisotropic tissue conductivities but requires a volumetric discretization, a mesh, of the head domain. Structured hexahedral meshes are easy to create in an automatic fashion, whil

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15

Petö, Márton, Fabian Duvigneau, Daniel Juhre, and Sascha Eisenträger. "Enhanced numerical integration scheme based on image compression techniques: Application to rational polygonal interpolants." Archive of Applied Mechanics, September 12, 2020. http://dx.doi.org/10.1007/s00419-020-01772-6.

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Abstract Polygonal finite elements offer an increased freedom in terms of mesh generation at the price of more complex, often rational, shape functions. Thus, the numerical integration of rational interpolants over polygonal domains is one of the challenges that needs to be solved. If, additionally, strong discontinuities are present in the integrand, e.g., when employing fictitious domain methods, special integration procedures must be developed. Therefore, we propose to extend the conventional quadtree-decomposition-based integration approach by image compression techniques. In this context,

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16

Marco, Martinolli, Biasetti Jacopo, Zonca Stefano, Polverelli Luc, and Vergara Christian. "Extended Finite Element Method for Fluid-Structure Interaction in Wave Membrane Blood Pumps." International Journal for Numerical Methods in Biomedical Engineering, May 27, 2021. https://doi.org/10.1002/cnm.3467.

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Numerical simulations of cardiac blood pump systems are integral to the optimization of device design, hydraulic performance and hemocompatibility. In wave membrane blood pumps, blood propulsion arises from the wave propagation along an oscillating immersed membrane, which generates small pockets of fluid that are pushed towards the outlet against an adverse pressure gradient.  We studied the Fluid-Structure Interaction between the oscillating membrane and the blood flow via three-dimensional simulations using the Extended Finite Element Method, an unfitted numerical technique that avoids

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