Unfitted Finite Element Method Research Articles

Analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem

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 flux variable. By applying post-processings for the scalar variable, in virtue of classical local post-processings in body-fitted methods, we retain optimal convergence rates for both variables and even the superconvergence after post-processing of the scalar variable. We present the method and perform a rigorous a priori error analysis of the method and discuss several variants and extensions. Numerical experiments confirm the theoretical results.

Divergence-free cut finite element methods for Stokes flow

We develop two unfitted finite element methods for the Stokes equations based on Hdiv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{H}^{{{\\,\ extrm{div}\\,}}}$$\\end{document}-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 of the velocity at interior edges in the mesh. The second method is a cut finite element discretization of a three-field formulation of the Stokes problem involving the vorticity, velocity, and pressure and uses the Raviart–Thomas (RT) space for the velocity. We present mixed ghost penalty stabilization terms for both methods so that the resulting discrete problems are stable and the divergence-free property of the Hdiv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{H}^{{{\\,\ extrm{div}\\,}}}$$\\end{document}-conforming elements is preserved also on unfitted meshes. In both methods boundary conditions are imposed weakly. We show that imposing Dirichlet boundary conditions weakly introduces additional challenges; (1) The divergence-free property of the RT and the BDM finite elements may be lost depending on how the normal component of the velocity field at the boundary is imposed. (2) Pressure robustness is affected by how well the boundary condition is satisfied and may not hold even if the incompressibility condition holds pointwise. We study two approaches of weakly imposing the normal component of the velocity at the boundary; we either use a penalty parameter and Nitsche’s method or a Lagrange multiplier method. We show that appropriate conditions on the velocity space has to be imposed when Nitsche’s method or penalty is used. Pressure robustness can hold with both approaches by reducing the error at the boundary but the price we pay is seen in the condition numbers of the resulting linear systems, independent of if the mesh is fitted or unfitted to the boundary.

A Shifted Boundary Method for the compressible Euler equations

The Shifted Boundary Method (SBM) is applied to compressible Euler flows, with and without shock discontinuities. The SBM belongs to the class of unfitted (or immersed, or embedded) finite element methods and avoids integration over cut cells (and the associated implementation/stability issues) by reformulating the original boundary value problem over a surrogate (approximate) computational domain. 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. We specifically discuss the advantages the proposed method offers in avoiding spurious numerical artifacts in two scenarios: (a) when curved boundaries are represented by body-fitted polygonal approximations and (b) when the Kutta condition needs to be imposed in immersed simulations of airfoils. An extensive suite of numerical tests is included.

An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation, Part II. Piecewise-smooth interfaces

We consider the reliable implementation of an adaptive high-order unfitted finite element method on Cartesian meshes for solving elliptic interface problems with geometrically curved singularities. We extend our previous work on the reliable cell merging algorithm for smooth interfaces to automatically generate the induced mesh for piecewise smooth interfaces. An hp a posteriori error estimate is derived for a new unfitted finite element method whose finite element functions are conforming in each subdomain. Numerical examples illustrate the competitive performance of the method.

An Unfitted Finite Element Poisson-Boltzmann Solver with Automatic Resolving of Curved Molecular Surface.

So far, the existing Poisson-Boltzmann (PB) solvers that accurately take into account the interface jump conditions need a pregenerated body-fitted mesh (molecular surface mesh). However, qualified biomolecular surface meshing and its implementation into numerical methods remains a challenging and laborious issue, which practically hinders the progress of further developments and applications of a bunch of numerical methods in this field. In addition, even with a molecular surface mesh, it is only a low-order approximation of the original curved surface. In this article, an interface-penalty finite element method (IPFEM), which is a typical unfitted finite element method, is proposed to solve the Poisson-Boltzmann equation (PBE) without requiring the user to generate a molecular surface mesh. The Gaussian molecular surface is used to represent the molecular surface and can be automatically resolved with a high-order approximation within our method. Theoretical convergence rates of the IPFEM for the linear PB equation have been provided and are well validated on a benchmark problem with an analytical solution (we also noticed from numerical examples that the IPFEM has similar convergence rates for the nonlinear PBE). Numerical results on a set of different-sized biomolecules demonstrate that the IPFEM is numerically stable and accurate in the calculation of biomolecular electrostatic solvation energy.

A High Order Unfitted Finite Element Method for Time-Harmonic Maxwell Interface Problems

A High Order Unfitted Finite Element Method for Time-Harmonic Maxwell Interface Problems

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