Unfitted Methods and Interface Problems

Erik Burman (Keynote)

University College London

Unfitted Space-Time FEM for the transport equation in a moving domain

We consider unfitted finite element methods for the transport equation posed on a moving domain. Building on the work of F. Heimann, we develop a space–time finite element formulation using elements that are continuous in space but discontinuous across time slabs. To address the inherent instability of hyperbolic problems, we incorporate an interior penalty stabilization, which also serves as a ghost penalty to handle small-cut elements. For the theoretical analysis, we establish a space–time inf-sup condition that ensures control of the material derivative and the L2(L2)-norm over the space–time domain. Combining stability with consistency, we derive error estimates that are optimal with respect to the material derivative and suboptimal by one-half order in the space–time L2(L2)-norm under suitable assumptions on the mesh parameters.

Christoph Lehrenfeld

Georg-August-Universität Göttingen

Time discretization in unfitted FEM on moving domains

Simulation problems with evolving geometries and topology changes challenge traditional finite element methods (FEM), which rely on fitted meshes that are difficult to generate and update. Geometrically unfitted FEM overcome this by decoupling the mesh from the geometry, using a simple background mesh that adapts to complex, time-dependent domains without costly remeshing. While this approach offers flexibility, it raises challenges in achieving stable and accurate time integration. This talk addresses these challenges, focusing on two strategies: an extension-based generalization of the method-of-lines approach and unfitted space-time FEM formulations. We outline the underlying ideas, key implementation aspects, and provide a priori error estimates alongside numerical results demonstrating the methods' effectiveness.

Xiaodong Wei

Shanghai Jiao Tong University

Stabilized Isogeometric Topology Optimization on Trimmed Geometries

In this work, we present a novel isogeometric topology optimization method tailored for complex design domains represented by boundary representations (B-reps). The topology optimization is reformulated as a generalized Cahn-Hilliard problem in which sensitivity analysis is not needed. The optimized structure is then found by finding the steady-state solution of the time-dependent, high-order nonlinear partial differential equation, which is solved on a cut Cartesian grid by the B-rep. To address the instability issue caused by small cut elements, we employ the minimal stabilization method in the nonlinear setting with the Nitsche's formulation. Numerical integration is handled by folded decomposition where negative Jacobian is allowed for the integration purpose only. Through several numerical examples, we demonstrate the efficacy and robustness of the proposed method in handling complex design domains.

Susanne Claus

ONERA

From Implicit Geometry to Differentiable Simulation: Linking Shape Modeling and CutFEM

This work introduces a consistent framework linking differentiable geometry representations with immersed finite element methods. It is composed of three software modules: (1) a differentiable shape library that expresses complex geometries through smooth implicit functions and supports automatic differentiation with respect to geometric parameters; (2) a lookup-table engine for robust and efficient intersection of level sets with a variety of cell shapes; and (3) a cut finite element library integrated with FEniCSx that enables the definition of differentiable weak forms on implicitly defined domains. Together, these developments create a seamless bridge between geometry and analysis, allowing gradients of physical quantities to be traced back to geometric design variables. This integrated approach opens new perspectives for shape optimization, sensitivity analysis, and multi-physics coupling within immersed finite element frameworks.

Andrea Bressan

IMATI Pavia

Sum factorization for trimmed domains

A fundamental integration technique is to recursively reduce multivariate integrals to a sequence of lower dimensional integrals by Fubini-Tonelli theorem. When applied to Cartesian domains the multivariate integral is the product of univariate integrals, and this leads the way to high performance numerical integration using the sum factorization algorithm. It seems thus worth exploring how much of this approach and algorithms works for unfitted domains, particularly in the context of IGA.

Alexander Düster (Keynote)

TU Hamburg

From CT scans to simulation: An FCM approach to material interfaces

In recent years, fictitious domain methods have attracted attention as they eliminate the need to generate body-conforming finite element meshes. The finite cell method (FCM) combines this fictitious domain approach with high-order finite elements. By employing simple Cartesian meshes, pre-processing is significantly simplified. However, because the mesh does not conform to the problem's geometry, the treatment of internal interfaces requires special consideration.

This talk provides an overview of the finite cell method applied to problems in solid mechanics. Particular emphasis is placed on the treatment of material interfaces, such as those found in microstructured and heterogeneous materials. We present an image-based approach using CT scans that allows for an efficient treatment of problems with material interfaces. Finally, we discuss the extension of this framework to fracture modeling using a phase-field approach.

Stefan Kollmannsberger

Bauhaus-Universität Weimar

Finding Interfaces

The task to find homogeneous Neumann Interfaces is the same as finding voids in a medium. An efficient corresponding method is using wave propagation and termed Full Waveform Inversion (FWI). Naturally, the interfaces are not known a-priory, which is why boundary conforming mesh generation is impossible. Unfitted methods are, therefore, a natural choice. The presentation will introduce FWI and discuss solution approaches using unfitted methods. It will then be demonstrated how deep learning can help to achieve better solutions.

Yusuf T. Elbadry

TU Darmstadt

Immersed IGA with boundary conformal method for magnetostatics: challenges and remedies

Numerical simulation of complex geometries can be computationally expensive and time-consuming, mainly due to the extensive effort required for geometry preparation and mesh generation. This challenge is particularly pronounced in magnetostatics, where multiple materials, including air gaps, must be modeled within the same physical domain. Classical FEM and multi-patch IGA approaches require boundary-fitted, typically conformal meshes, significantly increasing preprocessing effort. Immersed boundary methods provide an efficient alternative by automating the simulation process and reducing meshing complexity. In magnetostatics, these methods embed non-dominant material bodies within the dominant material represented by a Cartesian grid. In this work, we employ an immersed boundary–conformal method that immerses different bodies and surfaces into a background mesh, where interfaces can additionally enclosed by conformal boundary layers. Using spline-based immersed isogeometric analysis, we eliminate the need for body-conformal meshes. The 2D horseshoe magnet problem is used to benchmark and compare immersed approaches to the conformal IGA solution.

Alexei Lozinski

University of Franche-Comté

phi-FEM: a finite element method on unfitted meshes in space-time

phi-FEM is a fictitious domain type method suitable for PDEs on domains described by a level-set function, incorporating this function directly into the variational setting. The method employs the standard finite element spaces of any order and the straightforward numerical integration, while maintaining the optimal convergence orders. It was originally proposed for stationary elliptic problems. In this talk, we shall present an adaptation to time-dependent parabolic problems in moving domains, employing the space-time finite elements. This is ongoing joint work with Michel Duprez (Strasbourg) and Igor Voulis (Göttingen).

Guglielmo Scovazzi

Duke University

Immersed approximate domains: The Shifted Boundary Method

The Shifted Boundary Method (SBM) is an alternative to traditional unfitted FEM approaches that addresses the small cut-cell problem by eliminating cell cutting altogether.

The SBM introduces surrogate domains which contain only fully formed elements. The true boundary conditions are shifted onto the location of the surrogate domain boundary by extension operators constructed with Taylor expansions, preserving accuracy.

The Shifted Boundary Method is simple for Dirichlet boundary conditions, but more challenging for Neumann boundary conditions, since higher-order terms in the Taylor expansion are unavailable.

We propose an approach to Neumann boundary conditions that does not require a mixed formulation, maintaining the same data structure requirements as for Dirichlet boundary conditions. This approach yields optimal accuracy, achieved using approximate integration formulas of the weak form in the gap between the true and surrogate boundary. Note that no cut-cell integration is performed, but only integrals on the surrogate boundary.

John Evans (Keynote)

University of Colorado Boulder

Interpolation-Based Immersed Finite Element and Meshless Analysis: Foundations and Applications

Immersed finite element and meshless methods offer powerful tools for solving multi-material and multi-physics problems on complex domains, yet their implementation remains technically challenging. This talk introduces an interpolation-based approach that enables classical finite element codes to be transformed into immersed or meshless codes with minimal implementation effort. The central idea is to interpolate a background basis onto a Lagrange basis defined over a foreground mesh that conforms to geometric boundaries and material interfaces. However, the foreground mesh may be of poor quality, and this flexibility dramatically reduces the computational cost and human effort associated with the generation of a high-quality body-fitted mesh. I will present recent theoretical and algorithmic developments, demonstrate performance on benchmark multi-material and multi-physics problems, and illustrate how the same interpolation framework enables classical finite element codes such as FEniCS to be used for shape and topology optimization with little user intervention.

Maxim Olshanskiy (Keynote)

University of Houston

Analysis of a finite element method for PDEs in evolving domains with topological changes

The talk presents the first rigorous error analysis of an unfitted finite element method for linear parabolic problems posed on time-dependent domains that may undergo topological changes. The domain evolution is assumed to be smooth away from a critical time, at which the topology may change. To accommodate such transitions in the error analysis, we introduce several structural assumptions on the evolution of the domain in the vicinity of the critical time. These assumptions guarantee a specific control over the variation of a solution norm in time, even across singularities, and form the foundation for the numerical analysis. We demonstrate the applicability of our assumptions with examples of level-set domains undergoing topological transitions and discuss cases where analysis fails. The theoretical error estimate is supported by the results of a numerical experiment. Questions that remain open will be outlined.

Frits de Prenter

TU Delft

Stability and conditioning of immersed finite element methods: analysis and remedies

I will present an overview of the adverse effects of small cut elements and how these can be remedied, following the structure presented in our review paper (with same title). There will be special focus on the preconditioning technique we developed, highlighting the strengths and also the scenarios where it breaks down. If time permits, I will finish with a few slides on our recent work on higher-order boundary conditions for Lattice Boltzmann, based on unfitted finite element projections.

Robin Bouclier

INSA Toulouse

Immersed Isogeometric LaTIn Method for the Efficient Simulation of Composite Microstructures with Multiple Nonlinear Interfaces

Computational micromechanics, aimed at performing numerical simulations at the microscale, is of utmost importance, particularly in the current context of digital twins in the mechanics of materials. This talk presents the development of an immersed isogeometric LaTIn method designed for the efficient and accurate simulation of complex composite microstructures featuring multiple nonlinear interfaces. The first variant of the approach combines IsoGeometric Analysis, for its high accuracy, with an the immersed boundary-conformal strategy that eliminates complex meshing while maintaining conformal matrix/inclusion interfaces through the construction of conformal layers. Nonlinear behaviors are localized at these conformal interfaces and efficiently handled by the LaTIn algorithm, while the non-conformal coupling between subdomains remains in the linear elastic regime and is treated using Nitsche's method. In the second version of the approach, the conformal layers are removed, offering increased flexibility but requiring additional stabilization of the immersed isogeometric LaTIn formulation. The efficiency and robustness of the framework are demonstrated through several nonlinear 2D benchmark cases, including unilateral contact, frictional contact, and delamination in composite microstructures.

Stein Stoter

TU Eindhoven

Ghost mass for immersed explicit analysis

Explicit dynamics and unfitted discretization methods may seem like an unlikely marriage. Small-cut elements in the unfitted settings are detrimental to the stability of explicit time integration, manifesting as an impossibly small critical time-step size. To address this, we propose a ghost-penalty-based mass-scaling approach that enhances robustness and preserves the accuracy of explicit dynamics computations on unfitted geometries. We then investigates the influence of cut elements on the stability limits of explicit schemes for various weak enforcement techniques of Dirichlet boundary conditions, including penalty and Nitsche-type methods.

Vincenzo Gulizzi

University of Palermo

Embedded-boundary Discontinuous Galerkin Methods for solid, fluid and structure mechanics

This talk focuses on embedded-boundary Discontinuous Galerkin (EBDG) methods for solving partial differential equations that are of interest in science and engineering. Unlike continuous Galerkin approaches such as the Finite Element Method, DG methods are based on the use of discontinuous basis functions and suitably defined boundary integrals to enforce solution continuity and boundary conditions. The discontinuous nature of this formulation enables high-order accuracy via compact stencils, a simple way to handle hp refinement, block-structured mass matrices, and local conservation properties in combination with generally shaped elements, including those stemming from the embedded-boundary procedure. Numerical applications of EBDG methods are presented in a unified fashion for statics and elasto-dynamics of solids and structures, as well as for incompressible and compressible fluid dynamics. The talk also highlights specific aspects of the formulation and the implementation such as dynamic adaptive mesh refinement and computer parallelization performance.