Cut Finite Element Research Articles

Shape optimization of a linearly elastic rolling structure under unilateral contact using Nitsche’s method and cut finite elements

The main motivation of this work is to develop a numerical strategy for the shape optimization of a rolling elastic structure under contact with respect to a uniform rolling criterion. A first objective is to highlight the influence on the treatment of the contact terms. To do so, we present a numerical comparison between a penalty-based approach and the use of Nitsche’s method which is known to have good consistency properties. A second task concerns the construction of an objective functional to force the uniform rolling criterion. Here, we present and compare two different strategies that will lead to quite similar results. All the numerical experiments proposed in this paper were performed using a fictitious domain approach coupled with a level set representation of the shape and the use of a cut finite element method to approximate the elastic equation.

Random geometries for optimal control PDE problems based on fictitious domain FEMs and cut elements

This work investigates an elliptic optimal control problem defined on uncertain domains and discretized by a fictitious domain finite element method and cut elements. Key ingredients of the study are to manage cases considering the usually computationally “forbidden” combination of poorly conditioned equation system matrices due to challenging geometries, optimal control searches with iterative methods, slow convergence to system solutions on deterministic and non-deterministic level, and expensive remeshing due to geometrical changes. We overcome all these difficulties, utilizing the advantages of proper preconditioners adapted to unfitted mesh methods, improved types of Monte Carlo methods, and mainly employing the advantages of embedded FEMs, based on a fixed background mesh computed once even if geometrical changes are taking place. The sensitivity of the control problem is introduced in terms of random domains, employing a Quasi Monte Carlo method and emphasizing on a deterministic target state. The variational discretization concept is adopted, optimal error estimates for the state, adjoint state and control are derived that confirm the efficiency of the cut finite element method in challenging geometries. The performance of a multigrid scheme especially developed for unfitted finite element discretizations adapted to the optimal control problem is also tested. Some fundamental preconditioners are applied to the arising sparse linear systems coming from the discretization of the state and adjoint state variational forms in the spatial domain. The corresponding convergence rates along with the quality of the prescribed preconditioners are verified by numerical examples.

Cut finite element methods and ghost stabilization techniques for space‐time discretizations of the Navier–Stokes equations

AbstractWe propose and analyze computationally a new fictitious domain method, based on higher order space‐time finite element discretizations, for the simulation of the nonstationary, incompressible Navier–Stokes equations on evolving domains. The physical domain is embedded into a fixed computational mesh such that arbitrary intersections of the moving domain's boundaries with the background mesh occur. The potential of such cut finite element techniques for higher order space‐time finite element methods has rarely been studied in the literature so far and deserves further elucidation. The key ingredients of the approach are the weak formulation of Dirichlet boundary conditions by Nitsche's method, the flexible and efficient integration over all types of intersections of cells by moving boundaries and the spatial extension of the discrete physical quantities to the entire computational background mesh including fictitious (ghost) subdomains of fluid flow. Thereby, an expensive remeshing and adaptation of the sparse matrix data structure are avoided and the computations are accelerated. To prevent spurious oscillations caused by irregular intersections of mesh cells, a penalization, defining also implicitly the extension to ghost domains, is added. These techniques are embedded in an arbitrary order, discontinuous Galerkin discretization of the time variable and an inf‐sup stable discretization of the spatial variables. The parallel implementation of the matrix assembly is described. The optimal order convergence properties of the algorithm are illustrated in a numerical experiment for an evolving domain. The well‐known 2d benchmark of flow around a cylinder as well as flow around moving obstacles with arising cut cells and fictitious domains are considered further.

High Order Discontinuous Cut Finite Element Methods for Linear Hyperbolic Conservation Laws with an Interface

We develop a family of cut finite element methods of different orders based on the discontinuous Galerkin framework, for hyperbolic conservation laws with stationary interfaces in both one and two space dimensions, and for moving interfaces in one space dimension. Interface conditions are imposed weakly and so that both conservation and stability are ensured. A CutFEM with discontinuous elements in space is developed and coupled to standard explicit time stepping schemes for linear advection problems and the acoustic wave problem with stationary interfaces. In the case of moving interfaces, we propose a space-time CutFEM based on discontinuous elements both in space and time for linear advection problems. We show that the proposed CutFEM are conservative and energy stable. For the stationary interface case an a priori error estimate is proven. Numerical computations in both one and two space dimensions support the analysis, and in addition demonstrate that the proposed methods have the expected accuracy.

A CutFEM based framework for numerical simulations of machine driven tunnels with arbitrary alignments

We describe a novel computational framework based on the cut finite element method (CutFEM) for process-oriented simulation in mechanized tunneling. The framework incorporates all relevant components required for the simulation of the tunnel advance process, namely the ground, the staged installation of the lining support, the tail void grouting and the tunnel boring machine. We demonstrate that CutFEM concepts can significantly facilitate the modeling, discretization and coupling of the different components, while maintaining the same accuracy as the standard boundary-fitted finite element method. The proposed CutFEM technology, which is being applied and investigated in the context of advancement simulations in mechanized tunneling for the first time, enables the seamless analysis of an arbitrary number of different tunnel alignment variants on the same structured background mesh without the need to set up a new model for each variant. This is a shift of paradigm in simulation-supported tunnel design, as the CutFEM based framework considerably facilitates a direct integration of geometric, building information and simulation models in early stages of a tunnel project. The simulation model allows for the damage assessment of the buildings during tunnel advancement with regards to different excavation scenarios, as shown in the numerical examples in this paper.

A Cut Finite Element Method for Two-Phase Flows with Insoluble Surfactants

A Cut Finite Element Method for Two-Phase Flows with Insoluble Surfactants

An interface penalty parameter free nonconforming cut finite element method for elliptic interface problems

In this paper, we explore a nonconforming cut finite element method based on the nonsymmetric version of Nitsche's method without interface penalty for elliptic interface problems. In contrast to symmetric Nitsche methods, this penalty parameter free scheme does not need “sufficiently large” parameters to ensure the stability. Our main results are that a special function is constructed and the stability of the bilinear form is proved by an inf-sup argument. The optimal convergence in the energy norm and the suboptimal convergence in L2-norm are derived. The suboptimality is due to the lack of adjoint consistency of our formulation. Numerical examples are provided to support the theoretical analysis.

Simple, accurate, and efficient embedded finite element methods for fluid–solid interaction

This work presents a new approach to implementing a recently proposed optimal order Cut Finite Element Method (CutFEM) for problems with moving embedded solid structures in viscous incompressible flows. This new approach uses the notion of equivalent polynomials, introduced previously in the context of the eXtended Finite Element Methods (XFEM), to implement exact integration for terms involving products of polynomials with Heaviside and Dirac distributions. Combining CutFEM and equivalent polynomials results in a method for fluid–structure interaction that (1) has the same number of degrees of freedom as the underlying conforming Galerkin method on the fixed background mesh, which is independent of the configuration of non-conforming interfaces, (2) has the same element assembly structure as classical FEM on the background mesh—with standard quadrature rules, and (3) retains the convergence properties, indeed the precise theoretical structure, of the original CutFEM method. The result is a method that is robust, accurate, and efficient for interacting particles, and which provides a convenient approach for upgrading legacy finite element codes to include embedded boundaries with CutFEM or similar formulations that can retain the same asymptotic accuracy as the underlying boundary conforming finite element method.

Numerical convergence of discrete extensions in a space‐time finite element, fictitious domain method for the Navier–Stokes equations

AbstractA key ingredient of our fictitious domain, higher order space‐time cut finite element (CutFEM) approach for solving the incompressible Navier–Stokes equations on evolving domains (cf. [1]) is the extension of the physical solution from the time‐dependent flow domain Ωtf to the entire, time‐independent computational domain Ω. The extension is defined implicitly and, simultaneously, aims at stabilizing the discrete solution in the case of unavoidable irregular small cuts. Here, the convergence properties of the scheme are studied numerically for variations of the combined extension and stabilization.

Flux recovery for Cut Finite Element Method and its application in a posteriori error estimation

In this article, we aim to recover locally conservative and H(div) conforming fluxes for the linear Cut Finite Element Solution with Nitsche’s method for Poisson problems with Dirichlet boundary condition. The computation of the conservative flux in the Raviart–Thomas space is completely local and does not require to solve any mixed problem. The L2-norm of the difference between the numerical flux and the recovered flux can then be used as a posteriori error estimator in the adaptive mesh refinement procedure. Theoretically we also prove the global reliability and local efficiency. The theoretical results are verified in the numerical results. Moreover, in the numerical results we also observe optimal convergence rate for the flux error.

A Reduced Order Model for a Stable Embedded Boundary Parametrized Cahn–Hilliard Phase-Field System Based on Cut Finite Elements

In the present work, we investigate a cut finite element method for the parameterized system of second-order equations stemming from the splitting approach of a fourth order nonlinear geometrical PDE, namely the Cahn–Hilliard system. We manage to tackle the instability issues of such methods whenever strong nonlinearities appear and to utilize their flexibility of the fixed background geometry—and mesh—characteristic, through which, one can avoid e.g. in parametrized geometries the remeshing on the full order level, as well as, transformations to reference geometries on the reduced level. As a final goal, we manage to find an efficient global, concerning the geometrical manifold, and independent of geometrical changes, reduced order basis. The POD-Galerkin approach exhibits its strength even with pseudo-random discontinuous initial data verified by numerical experiments.

A Reduced Order Cut Finite Element method for geometrically parametrized steady and unsteady Navier–Stokes problems

We focus on steady and unsteady Navier–Stokes flow systems in a reduced-order modeling framework based on Proper Orthogonal Decomposition within a levelset geometry description and discretized by an unfitted mesh Finite Element Method. This work extends the approaches of [1–3] to nonlinear CutFEM discretization. We construct and investigate a unified and geometry independent reduced basis which overcomes many barriers and complications of the past, that may occur whenever geometrical morphings are taking place. By employing a geometry independent reduced basis, we are able to avoid remeshing and transformation to reference configurations, and we are able to handle complex geometries. This combination of a fixed background mesh in a fixed extended background geometry with reduced order techniques appears beneficial and advantageous in many industrial and engineering applications, which could not be resolved efficiently in the past.

Comparison of Shape Derivatives Using CutFEM for Ill-posed Bernoulli Free Boundary Problem

In this paper we study and compare three types of shape derivatives for free boundary identification problems. The problem takes the form of a severely ill-posed Bernoulli problem where only the Dirichlet condition is given on the free (unknown) boundary, whereas both Dirichlet and Neumann conditions are available on the fixed (known) boundary. Our framework resembles the classical shape optimization method in which a shape dependent cost functional is minimized among the set of admissible domains. The position of the domain is defined implicitly by the level set function. The steepest descent method, based on the shape derivative, is applied for the level set evolution. For the numerical computation of the gradient, we apply the Cut Finite Element Method (CutFEM), that circumvents meshing and re-meshing, without loss of accuracy in the approximations of the involving partial differential models. We consider three different shape derivatives. The first one is the classical shape derivative based on the cost functional with pde constraints defined on the continuous level. The second shape derivative is similar but using a discretized cost functional that allows for the embedding of CutFEM formulations directly in the formulation. Different from the first two methods, the third shape derivative is based on a discrete formulation where perturbations of the domain are built into the variational formulation on the unperturbed domain. This is realized by using the so-called boundary value correction method that was originally introduced to allow for high order approximations to be realized using low order approximation of the domain. The theoretical discussion is illustrated with a series of numerical examples showing that all three approaches produce similar result on the proposed Bernoulli problem.

Contact modeling from images using cut finite element solvers

This paper presents a robust digital pipeline from CT images to the simulation of contact between multiple bodies. The proposed strategy relies on a recently developed immersed finite element algorithm that is capable of simulating unilateral contact between solids without meshing (Claus and Kerfriden in Int J Numer Methods Eng 113(6):938–966, 2018). It was shown that such an approach reduces the difficulties associated with the digital flow of information from analytically defined geometries to mechanical simulations. We now propose to extend our approach to include geometries, which are not defined mathematically but instead are obtained from images, and encoded in 3D arrays of voxels. This paper introduces two novel elements. Firstly, we reformulate our contact algorithm into an extension of an augmented Lagrangian CutFEM algorithm. Secondly, we develop an efficient algorithm to convert the surface data generated by standard segmentation tools used in medical imaging into level-set functions. These two elements give rise to a robust digital pipeline with minimum user intervention. We demonstrate the capabilities of our algorithm on a hip joint geometry with contact between the femur and the hip bone.

A consistent computational approach for general fluid-poroelasticity-structure-contact interaction problems

The focus of this contribution is the numerical treatment of interface-coupled problems concerned with the interaction of impermeable or permeable elastic bodies1 and the surrounding fluid including the contact interaction of deformable bodies. The presented formulation is based on the approach for fluid-structural-contact interaction in [1], and consistently takes into account fluid-filled poroelastic bodies including the conditions of solid-poroelastic contact, poroelastic-poroelastic contact, and viscous flow poroelasticity interaction. The interface conditions in normal and tangential orientation to formulate this type of coupled problem, based on the fundamental balances of mass and linear momentum, the no-slip condition, the Beavers Joseph condition, and the conditions for frictionless contact on the respective interfaces, are discussed. A continuous transition of the different types of tangential conditions is enabled by application of the general Navier condition with varying slip length. The fluid stress in the zone of closed contact, which is essential for the lift-off behavior of contacting bodies, is obtained by an extension approach augmented by the porous fluid state to ensure continuity and physical accuracy. To account for topological changes of the fluid domain, the numerical approach utilizes non-interface fitted computational meshes for the fluid domain enabled by the Cut Finite Element Method. All interface conditions are incorporated in a weak sense by Nitsche-based approaches. Different numerical examples analyze the proper contacting and lift-off behavior for contact between the different pairs of impermeable and permeable bodies and demonstrate the robustness for more challenging configurations, which include topological changes of the fluid domain, large contacting areas, and 3D configurations.

A cut finite element method for non-Newtonian free surface flows in 2D - application to glacier modelling

In ice sheet and glacier modelling, the Finite Element Method is rapidly gaining popularity. However, constructing and updating meshes for ice sheets and glaciers is a non-trivial and computationally demanding task due to their thin, irregular, and time dependent geometry. In this paper we introduce a novel approach to ice dynamics computations based on the unfitted Finite Element Method CutFEM, which lets the domain boundary cut through elements. By employing CutFEM, complex meshing and remeshing is avoided as the glacier can be immersed in a simple background mesh without loss of accuracy. The ice is modelled as a non-Newtonian, shear-thinning fluid obeying the p-Stokes (full Stokes) equations with the ice atmosphere interface as a moving free surface. A Navier slip boundary condition applies at the glacier base allowing both bedrock and subglacial lakes to be represented. Within the CutFEM framework we develop a strategy for handling non-linear viscosities and thin domains and show how glacier deformation can be modelled using a level set function. In numerical experiments we show that the expected order of accuracy is achieved and that the method is robust with respect to penalty parameters. As an application we compute the velocity field of the Swiss mountain glacier Haut Glacier d'Arolla in 2D with and without an underlying subglacial lake, and simulate the glacier deformation from year 1930 to 1932, with and without surface accumulation and basal melt.

A cut finite element method for spatially resolved energy metabolism models in complex neuro-cell morphologies with minimal remeshing

A thorough understanding of brain metabolism is essential to tackle neurodegenerative diseases. Astrocytes are glial cells which play an important metabolic role by supplying neurons with energy. In addition, astrocytes provide scaffolding and homeostatic functions to neighboring neurons and contribute to the blood–brain barrier. Recent investigations indicate that the complex morphology of astrocytes impacts upon their function and in particular the efficiency with which these cells metabolize nutrients and provide neurons with energy, but a systematic understanding is still elusive. Modelling and simulation represent an effective framework to address this challenge and to deepen our understanding of brain energy metabolism. This requires solving a set of metabolic partial differential equations on complex domains and remains a challenge. In this paper, we propose, test and verify a simple numerical method to solve a simplified model of metabolic pathways in astrocytes. The method can deal with arbitrarily complex cell morphologies and enables the rapid and simple modification of the model equations by users also without a deep knowledge in the numerical methods involved. The results obtained with the new method (CutFEM) are as accurate as the finite element method (FEM) whilst CutFEM disentangles the cell morphology from its discretisation, enabling us to deal with arbitrarily complex morphologies in two and three dimensions.

An unconditionally stable semi-implicit CutFEM for an interaction problem between an elastic membrane and an incompressible fluid

In this paper we introduce a finite element method for the Stokes equations with a massless immersed membrane. This membrane applies normal and tangential forces affecting the velocity and pressure of the fluid. Additionally, the points representing this membrane move with the local fluid velocity. We design and implement a high-accuracy cut finite element method (CutFEM) which enables the use of a structured mesh that is not aligned with the immersed membrane, and we formulate a time discretization that yields an unconditionally energy stable scheme. We prove that the stability is not restricted by the parameter choices that constrained previous finite element immersed boundary methods and illustrate the theoretical results with numerical simulations.

A consistent and versatile computational approach for general fluid‐structure‐contact interaction problems

AbstractWe present a consistent approach that allows to solve challenging general nonlinear fluid‐structure‐contact interaction (FSCI) problems. The underlying formulation includes both “no‐slip” fluid‐structure interaction as well as frictionless contact between multiple elastic bodies. The respective interface conditions in normal and tangential orientation and especially the role of the fluid stress within the region of closed contact are discussed for the general problem of FSCI. A continuous transition of tangential constraints from no‐slip to frictionless contact is enabled by using the general Navier condition with varying slip length. Moreover, the fluid stress in the contact zone is obtained by an extension approach as it plays a crucial role for the lift‐off behavior of contacting bodies. With the given continuity of the formulation, continuity of the discrete system of equations for any variation of the coupled system state (which is essential for the convergence of Newton's method) is reached naturally. As topological changes of the fluid domain are an inherent challenge in FSCI configurations, a noninterface fitted cut finite element method (CutFEM) is applied to discretize the fluid domain. All interface conditions, that is the “no‐slip” FSI, the general Navier condition, and frictionless contact are incorporated using Nitsche based methods, thus retaining the consistency of the model. To account for the strong interaction between the fluid and solid discretization, the overall coupled discrete system is solved monolithically. Numerical examples of varying complexity are presented to corroborate the developments. In a first example, the fundamental properties of the presented formulation such as the contacting and lift‐off behavior, the mass conservation, and the influence of the slip length for the general Navier interface condition are analyzed. Beyond that, two more general examples demonstrate challenging aspects such as topological changes of the fluid domain, large contacting areas, and underline the general applicability of the presented method.

A cut finite element method for a model of pressure in fractured media

We develop a robust cut finite element method for a model of diffusion in fractured media consisting of a bulk domain with embedded cracks. The crack has its own pressure field and can cut through the bulk mesh in a very general fashion. Starting from a common background bulk mesh, that covers the domain, finite element spaces are constructed for the interface and bulk subdomains leading to efficient computations of the coupling terms. The crack pressure field also uses the bulk mesh for its representation. The interface conditions are a generalized form of conditions of Robin type previously considered in the literature which allows the modeling of a range of flow regimes across the fracture. The method is robust in the following way: (1) Stability of the formulation in the full range of parameter choices; and (2) Not sensitive to the location of the interface in the background mesh. We derive an optimal order a priori error estimate and present illustrating numerical examples.