Unfitted Finite Element Method Research Articles

An unfitted finite element method with direct extension stabilization for time-harmonic Maxwell problems on smooth domains

An unfitted finite element method with direct extension stabilization for time-harmonic Maxwell problems on smooth domains

A weighted shifted boundary method for immersed moving boundary simulations of Stokes' flow

The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods, and relies on reformulating the original boundary value problem over a surrogate (approximate) computational domain. The surrogate domain is constructed so as to avoid cut cells and the associated problematic implementation and numerical integration issues. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions: hence the name of the method, that shifts the location and values of the boundary conditions. In this article, we extend the SBM to the simulation of incompressible Stokes flow, by appropriately weighting its variational form with the elemental volume fraction of active fluid. This approach allows to drastically reduce spurious pressure oscillations in time, which are produced if the total volume of active fluid were to change abruptly over a time step. The proposed Weighted SBM (W-SBM) exactly preserves states of hydrostatic equilibrium, and induces small mass and momentum conservation errors, which converge as the grid is refined. This is in analogy to cutFEMs and related unfitted approaches, which rely on an affine representation of cut boundaries. We demonstrate the robustness and accuracy of the proposed method with an extensive suite of two-dimensional tests.

A CutFEM method for phase change problems with natural convection

In this paper, we present a Nitsche-based CutFEM approach for solving phase change problems while taking natural convection into account in the liquid phase. In the proposed algorithm, the interface is implicitly tracked through a level-set approach, where the level-set function is updated by solving a transport equation. To stabilize the finite element formulation of this equation, we use the continuous interior penalty method (CIP). We use the Backward Euler scheme for the time discretization and the of continuous piecewise linear polynomials (P1) for the spatial discretization of the temperature and the level-set function. For flow velocity and pressure in the liquid phase, the finite element formulation is based on the inf-sup stable pair of Hood-Taylor finite element function spaces P2−P1. As we are dealing with unfitted finite element methods, stability issues may arise due to geometry discretization. To address this issue, we incorporate ghost penalty terms into the finite element formulations of the heat and Navier–Stokes equations. According to Stefan’s condition, the normal velocity of the interface is proportional to the jump in the interfacial flux. For an efficient approximation of this jump, we use a ghost penalty-based domain integral approach which has shown to be highly accurate. To validate the proposed method, we first provide two manufactured solutions that confirm optimal convergence in both temporal and spatial discretizations. We next consider challenging problems such as the melting of n-octadecane in a differentially heated cavity, the multi-cellular melting of gallium, and the freezing of water to further validate our algorithm. We compare our results to existing numerical and experimental data.

An Eulerian finite element method for tangential Navier-Stokes equations on evolving surfaces

The paper introduces a geometrically unfitted finite element method for the numerical solution of the tangential Navier–Stokes equations posed on a passively evolving smooth closed surface embedded in R 3 \mathbb {R}^3 . The discrete formulation employs finite difference and finite elements methods to handle evolution in time and variation in space, respectively. A complete numerical analysis of the method is presented, including stability, optimal order convergence, and quantification of the geometric errors. Results of numerical experiments are also provided.

Unfitted Finite Element Methods for Axisymmetric Two-Phase Flow

Unfitted Finite Element Methods for Axisymmetric Two-Phase Flow

CutFEM forward modeling for EEG source analysis.

Source 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, while tetrahedral meshes are better suited to model curved geometries. Tetrahedral meshes, thus, offer better accuracy but are more difficult to create. We introduce CutFEM for EEG forward simulations to integrate the strengths of hexahedra and tetrahedra. It belongs to the family of unfitted finite element methods, decoupling mesh and geometry representation. Following a description of the method, we will employ CutFEM in both controlled spherical scenarios and the reconstruction of somatosensory-evoked potentials. CutFEM outperforms competing FEM approaches with regard to numerical accuracy, memory consumption, and computational speed while being able to mesh arbitrarily touching compartments. CutFEM balances numerical accuracy, computational efficiency, and a smooth approximation of complex geometries that has previously not been available in FEM-based EEG forward modeling.

Unfitted Trefftz discontinuous Galerkin methods for elliptic boundary value problems

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. We cover stability and derive a-priori error bounds, including errors arising from geometry approximation for the class of discretisations for a model Poisson problem in a unified manner. The analysis covers Trefftz and full polynomial ansatz spaces, alike. Numerical examples validate the theoretical findings and demonstrate the potential of the approach.

An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation

We consider the reliable implementation of high-order unfitted finite element methods on Cartesian meshes with hanging nodes for elliptic interface problems. We construct a reliable algorithm to merge small interface elements with their surrounding elements to automatically generate the finite element mesh whose elements are large with respect to both domains. We propose new basis functions for the interface elements to control the growth of the condition number of the stiffness matrix in terms of the finite element approximation order, the number of elements of the mesh, and the interface deviation which quantifies the mesh resolution of the geometry of the interface. Numerical examples are presented to illustrate the competitive performance of the method.

Unfitted finite element methods based on correction functions for Stokes flows with singular forces of low regularity

This paper presents an unfitted finite element method based on correction functions for solving stationary Stokes flows with singular forces acting on an immersed interface. It has been shown that the singular force is equivalent to a nonhomogeneous jump condition on the interface. In this paper, we consider the case that the jump has a low regularity so that it is impossible to use pointwise values on the interface to construct correction functions, as done in Guzmán et al. (2016). The natural way to deal with the problem is to use mean values of the jump on the parts of the interface cut by elements, instead of using pointwise values. However, we show that it may cause instability and the constant in the error estimate may depend on the interface location relative to the mesh. Inspired by Guo et al. (2019), we use a larger fictitious circle to overcome these issues. Associated with the correction functions, we consider two stable finite element pairs: the Mini element and the P2−P0 element, including the cases of continuous and discontinuous pressures. The optimal approximation capabilities of the correction functions and optimal error estimates of the finite element methods are both derived with a hidden constant independent of the interface location relative to the mesh. Numerical examples are provided to validate the theoretical results.

Construction of discontinuous enrichment functions for enriched or generalized FEM’s for interface elliptic problems in 1D

We introduce an enriched unfitted finite element method to solve 1D elliptic interface problems with discontinuous solutions, including those having implicit or Robin-type interface jump conditions. We present a novel approach to construct a one-parameter family of discontinuous enrichment functions by finding an optimal order interpolating function to the discontinuous solutions. In the literature, an enrichment function is usually given beforehand, not related to the construction step of an interpolation operator. Furthermore, we recover the well-known continuous enrichment function when the parameter is set to zero. To prove its efficiency, the enriched linear and quadratic elements are applied to a multi-layer wall model for drug-eluting stents in which zero-flux jump conditions and implicit concentration interface conditions are both present.

Error Analysis for a Parabolic PDE Model Problem on a Coupled Moving Domain in a Fully Eulerian Framework

We introduce an unfitted finite element method with Lagrange-multipliers to study an Eulerian time stepping scheme for moving domain problems applied to a model problem where the domain motion is implicit to the problem. We consider a parabolic partial differential equation (PDE) in the bulk domain, and the domain motion is described by an ordinary differential equation (ODE), coupled to the bulk partial differential equation through the transfer of forces at the moving interface. The discretisation is based on an unfitted finite element discretisation on a time-independent mesh. The method-of-lines time discretisation is enabled by an implicit extension of the bulk solution through additional stabilisation, as introduced by Lehrenfeld & Olshanskii (ESAIM: M2AN, 53:585-614, 2019). The analysis of the coupled problem relies on the Lagrange-multiplier formulation, the fact that the Lagrange-multiplier solution is equal to the normal stress at the interface and that the motion of the interface is given through rigid body motion. This paper covers the complete stability analysis of the method and an error estimate in the energy norm, under an assumption on the discrete interface velocity. This includes the dynamic error in the domain motion resulting from the discretised ODE and the forces from the discretised PDE. To the best of our knowledge this is the first error analysis of this type of coupled moving domain problem in a fully Eulerian framework. Numerical examples illustrate the theoretical results.

Adaptive geometric multigrid for the mixed finite cell formulation of Stokes and Navier–Stokes equations

AbstractThe unfitted finite element methods have emerged as a popular alternative to classical finite element methods for the solution of partial differential equations and allow modeling arbitrary geometries without the need for a boundary‐conforming mesh. On the other hand, the efficient solution of the resultant system is a challenging task because of the numerical ill‐conditioning that typically entails from the formulation of such methods. We use an adaptive geometric multigrid solver for the solution of the mixed finite cell formulation of saddle‐point problems and investigate its convergence in the context of the Stokes and Navier–Stokes equations. We present two smoothers for the treatment of cutcells in the finite cell method and analyze their effectiveness for the model problems using a numerical benchmark. Results indicate that the presented multigrid method is capable of solving the model problems independently of the problem size and is robust with respect to the depth of the grid hierarchy.

Space-time unfitted finite element methods for time-dependent problems on moving domains

We propose a space-time scheme that combines an unfitted finite element method in space with a discontinuous Galerkin time discretisation for the accurate numerical approximation of parabolic problems with moving domains or interfaces. We make use of an aggregated finite element space to attain robustness with respect to the cut locations. The aggregation is performed slab-wise to have a tensor product structure of the space-time discrete space, which is required in the numerical analysis. As an alternative, we also propose a space-time ghost penalty stabilisation term to attain robustness. We analyse the proposed algorithm, providing stability, condition number bounds and anisotropic a priori error estimates. A set of numerical experiments confirm the theoretical results for a parabolic problem on a moving domain. The method is applied for a mass transfer problem with changing topology.

A multigrid technique for Lagrange multiplier-based unfitted Finite Element contact issues.

Internal interfaces in a domain could exist as a material defect or they can appear due to propagations of cracks. Discretization of such geometries and solution of the contact problem on the internal interfaces can be computationally challenging. We employ an unfitted Finite Element (FE) framework for the discretization of the domains and develop a tailored, globally convergent, and efficient multigrid method for solving contact problems on the internal interfaces. In the unfitted FE methods, structured background meshes are used and only the underlying finite element spaces are modified to incorporate the discontinuities. The non-penetration conditions on the embedded interfaces of the domains are discretized using the method of Lagrange multipliers. We reformulate the arising variational inequality problem as a quadratic minimization problem with linear inequality constraints. Our multigrid method can solve such problems by employing a tailored multilevel hierarchy of the FE spaces and a novel approach for tackling the discretized non-penetration conditions. We employ pseudo-L2 projection- based transfer operators to construct a hierarchy of nested FE spaces from the hierarchy of non-nested meshes. The essential component of our multigrid method is a technique that decouples the linear constraints using an orthogonal transformation. The decoupled constraints are handled by a modified variant of the projected Gauss–Seidel method, which we employ as a smoother in the multigrid method. These components of the multigrid method allow us to enforce linear constraints locally and ensure the global convergence. We will demonstrate the robustness, efficiency, and level independent convergence property of the proposed method for Signorini’s problem and two-body contact problems

Tangential Navier–Stokes equations on evolving surfaces: Analysis and simulations

The paper considers a system of equations that models a lateral flow of a Boussinesq–Scriven fluid on a passively evolving surface embedded in [Formula: see text]. For the resulting Navier–Stokes type system, posed on a smooth closed time-dependent surface, we introduce a weak formulation in terms of functional spaces on a space-time manifold defined by the surface evolution. The weak formulation is shown to be well-posed for any finite final time and without smallness conditions on data. We further extend an unfitted finite element method, known as TraceFEM, to compute solutions to the fluid system. Convergence of the method is demonstrated numerically. In another series of experiments we visualize lateral flows induced by smooth deformations of a material surface.

Analysis of optimal preconditioners for CutFEM

AbstractIn this article, we consider a class of unfitted finite element methods for scalar elliptic problems. These so‐called CutFEM methods use standard finite element spaces on a fixed unfitted triangulation combined with the Nitsche technique and a ghost penalty stabilization. As a model problem we consider the application of such a method to the Poisson interface problem. We introduce and analyze a new class of preconditioners that is based on a subspace decomposition approach. The unfitted finite element space is split into two subspaces, where one subspace is the standard finite element space associated to the background mesh and the second subspace is spanned by all cut basis functions corresponding to nodes on the cut elements. We will show that this splitting is stable, uniformly in the discretization parameter and in the location of the interface in the triangulation. Based on this we introduce an efficient preconditioner that is uniformly spectrally equivalent to the stiffness matrix. Using a similar splitting, it is shown that the same preconditioning approach can also be applied to a fictitious domain CutFEM discretization of the Poisson equation. Results of numerical experiments are included that illustrate optimality of such preconditioners for the Poisson interface problem and the Poisson fictitious domain problem.

An Unfitted Finite Element Method by Direct Extension for Elliptic Problems on Domains with Curved Boundaries and Interfaces

We propose and analyze an unfitted finite element method of arbitrary order for solving elliptic problems on domains with curved boundaries and interfaces. The approximation space on the whole domain is obtained by the direct extension of the finite element space defined on interior elements, in the sense that there is no degree of freedom locating in boundary/interface elements. We apply a non-symmetric bilinear form and the boundary/jump conditions are imposed in a weak sense in the scheme. The method is shown to be stable without any mesh adjustment or any special stabilization. The optimal convergence rate under the energy norm is derived, and \(O(h^{-2})\)-upper bounds of the condition numbers are shown for the final linear systems. Numerical results in both two and three dimensions are presented to illustrate the accuracy and the robustness of the method.

A cut finite element method for two-phase flows with insoluble surfactants

We propose a new unfitted finite element method for simulation of two-phase flows in presence of insoluble surfactant. The key features of the method are 1) discrete conservation of surfactant mass; 2) the possibility of having meshes that do not conform to the evolving interface separating the immiscible fluids; 3) accurate approximation of quantities with weak or strong discontinuities across evolving geometries such as the velocity field and the pressure. The new discretization of the incompressible Navier-Stokes equations coupled to the convection-diffusion equation modeling the surfactant transport on evolving surfaces is based on a space-time cut finite element formulation with quadrature in time and a stabilization term in the weak formulation that provides function extension. The proposed strategy utilizes the same computational mesh for the discretization of the surface Partial Differential Equation (PDE) and the bulk PDEs and can be combined with different techniques for representing and evolving the interface, here the level set method is used. Numerical simulations in both two and three space dimensions are presented including simulations showing the role of surfactant in the interaction between two drops.

A multi-point constraint unfitted finite element method

In this work a multi-point constraint unfitted finite element method for the solution of the Poisson equation is presented. Key features of the approach are the strong enforcement of essential boundary, and interface conditions. This, along with the stability of the method, is achieved through the use of multi-point constraints that are applied to the so-called ghost nodes that lie outside of the physical domain. Another key benefit of the approach lies in the fact that, as the degrees of freedom associated with ghost nodes are constrained, they can be removed from the system of equations. This enables the method to capture both strong and weak discontinuities with no additional degrees of freedom. In addition, the method does not require penalty parameters and can capture discontinuities using only the standard finite element basis functions. Finally, numerical results show that the method converges optimally with mesh refinement and remains well conditioned.

A fourth-order unfitted characteristic finite element method for free-boundary problems

A fourth-order unfitted characteristic finite element method (UCFEM) is proposed to solve free-boundary problems of time-dependent partial differential equations (PDEs). The boundary of the domain is implicitly driven by the solution of the PDE, while the PDE is proposed on the time-varying unknown domain. In this way, they form a coupled nonlinear system. The key ingredient of the UCFEM is to trace the domain explicitly with a fourth-order forward flow map and discretize the PDE in time with a fourth-order backward flow map. The PDE is solved with a fourth-order unfitted finite element method on a fixed mesh. Since the boundary of the domain is expressed explicitly with cubic spline functions, quadrature on cut elements can be done accurately and efficiently even the domain undergoes severe deformations. The UCFEM provides a framework of designing high-order numerical methods for free-boundary problems. We apply the UCFEM to a two-dimensional convection-diffusion equation and Navier-Stokes equations, and obtain the overall fourth order of accuracy. With extensive numerical experiments, we show that the proposed method can achieve the fourth-order convergence even on severely deformed domains.