Nitsche's Method Research Articles

Nitsche-based isogeometric approach for free vibration analysis of laminated plate with multiple stiffeners and cutouts

On the basis of the first order shear deformation and Timoshenko beam theories, a general Nitsche-based isogeometric analysis (IGA) is developed for studying the free vibration behaviors of composite plates with/without stiffeners and cutouts. Firstly, the laminated plate with/without cutout is described by several non-overlapped NURBS patches. The stiffeners are placed on the common edges of adjacent patches and share the same control points with these sub-patch edges. Then, different patches are stuck together by the means of Nitsche method. Eventually, the proposed method is implemented for analyzing the free vibration of composite plates with/without stiffener and cutout. First of all, it avoids the discretization of the stiffener and the identification of plate elements where stiffener nodes are located at, that are required in finite element method and IGA. In addition, it can handle with laminated plate with multiple stiffeners and cutouts conveniently. At first, four meshing schemes with different element sizes are implemented to validate the good convergence and accuracy of the present method in case of laminated plate with stiffener. Then, rectangular, skew and elliptical plates without stiffener and cutout are considered and the obtained natural frequencies and mode shapes agree well with those reported in literature. Furthermore, results from composite plates with/without stiffener and cutout under different boundary conditions are analyzed by considering different lay-up numbers and orientations of fiber, shapes and numbers of cutout, orientation angles and numbers of stiffener.

A new mixed finite-element method for H2 elliptic problems

Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of the different boundary conditions that arise naturally in a variational formulation. Our formulation is based on introducing the gradient of the solution as an explicit variable, constrained using a Lagrange multiplier. The essential boundary conditions are enforced weakly, using Nitsche's method where required. As a result, the problem is rewritten as a saddle-point system, requiring analysis of the resulting finite-element discretization and the construction of optimal linear solvers. Here, we discuss the analysis of the well-posedness and accuracy of the finite-element formulation. Moreover, we develop monolithic multigrid solvers for the resulting linear systems. Two and three-dimensional numerical results are presented to demonstrate the accuracy of the discretization and efficiency of the multigrid solvers proposed.

Free vibration analysis of stiffened rectangular plate with cutouts using Nitsche based IGA method

The Nitsche based non-uniform rational B-splines (NURBS) isogeometric analysis (IGA) method is employed to study the free vibration of stiffened plate with holes. The plate with stiffeners is modeled by the Reissner–Mindlin theory and Timoshenko beam theory. Along the intersection lines between middle surface of plate and stiffeners, the plate is artificially divided into several non-overlapping NURBS patches, and stiffeners are located at the intersection lines. The holes are also constructed using several different non-overlapping NURBS patches along the common boundaries. Nitsche method is employed to glue different patches. Then, the free vibration analysis of stiffened plate with holes are conducted. One of the advantages of the presented method is that it is not necessary to identify the plate elements within which stiffener nodes located while the plate and stiffener need to be remeshed. The other advantage is that the presented method can be conveniently used to analyze the vibration of stiffened plate with multiple holes. Numerical examples are given to validate the correctness of presented method. The dimensionless frequencies and mode shapes of stiffened plate with holes or without holes are obtained by the present method and compared with the results that are available in the literature or those obtained using the ANSYS software. The results show that the effectiveness of the present method.

Nitsche method for contact with Coulomb friction: Existence results for the static and dynamic finite element formulations

We study the Nitsche-based finite element method for contact with Coulomb friction considering both static and dynamic situations. We provide existence and/or uniqueness results for the discretized problems under appropriate assumptions on physical and numerical parameters. In the dynamic case, existence and uniqueness of the space semi-discrete problem holds for every value of the friction coefficient and the Nitsche parameter. In the static case, if the Nitsche parameter is large enough, existence is ensured for any friction coefficient, and uniqueness can be obtained provided that the friction coefficient is below a bound that depends on the mesh size. These results are complemented by a numerical study.

BR2 discontinuous Galerkin methods for finite hyperelastic deformations

In this work we introduce a DG framework for nonlinear elasticity based on a Bassi-Rebay (BR2) formulation. The framework encompasses compressible and incompressible hyperelastic materials and is capable of dealing with large deformations. In order to achieve stability, we combine higher-order lifting operators for the BR2 stabilisation term with an adaptive stabilisation strategy which relies on the BR2 Laplace operator stabilisation and a penalty parameter based on the spectrum of the fourth-order elasticity tensor. Dirichlet boundary conditions for the displacement can be imposed by means of Lagrange multipliers and Nitsche method. Efficiency of the solution strategy is achieved by means of state-of-the-art agglomeration based h-multigrid preconditioners and the code implementation supports distributed memory execution on modern parallel architectures. Several benchmark test cases are proposed in order to investigate some relevant computational aspects, namely the performance of the h-multigrid iterative solver varying the stabilisation parameters and the influence of Dirichlet boundary conditions on Newton's method globalisation strategy.

A Nitsche Hybrid Multiscale Method with Non-matching Grids

We propose a Nitsche method for multiscale partial differential equations, which retrieves the macroscopic information and the local microscopic information at one stroke. We prove the convergence of the method for second order elliptic problem with bounded and measurable coefficients. The rate of convergence may be derived for coefficients with further structures such as periodicity and ergodicity. Extensive numerical results confirm the theoretical predictions.

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.

The finite cell method with least squares stabilized Nitsche boundary conditions

We apply the recently developed least squares stabilized symmetric Nitsche method for enforcement of Dirichlet boundary conditions to the finite cell method. The least squares stabilized Nitsche method in combination with finite cell stabilization leads to a symmetric positive definite stiffness matrix and relies only on elementwise stabilization, which does not lead to additional fill in. We prove a priori error estimates and bounds on the condition numbers.

A posteriori error estimates via equilibrated stress reconstructions for contact problems approximated by Nitsche's method

We present an a posteriori error estimate based on equilibrated stress reconstructions for the finite element approximation of a unilateral contact problem with weak enforcement of the contact conditions. We start by proving a guaranteed upper bound for the dual norm of the residual. This norm is shown to control the natural energy norm up to a boundary term, which can be removed under a saturation assumption. The basic estimate is then refined to distinguish the different components of the error, and is used as a starting point to design an algorithm including adaptive stopping criteria for the nonlinear solver and automatic tuning of a regularization parameter. We then discuss an actual way of computing the stress reconstruction based on the Arnold–Falk–Winther finite elements. Finally, after briefly discussing the efficiency of our estimators, we showcase their performance on a panel of numerical tests.

Deep Unfitted Nitsche Method for Elliptic Interface Problems

This paper proposes a deep unfitted Nitsche method for computing elliptic interface problems with high contrasts in high dimensions. To capture discontinuities of the solution caused by interfaces, we reformulate the problem as an energy minimization involving two weakly coupled components. This enables us to train two deep neural networks to represent two components of the solution in high-dimensional. The curse of dimensionality is alleviated by using the Monte-Carlo method to discretize the unfitted Nitsche energy function. We present several numerical examples to show the performance of the proposed method.

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.

A stabilized parametric level‐set XFEM topology optimization method for thermal‐fluid problem

AbstractThis article presents a stabilized level‐set topology optimization method for the design of a thermal‐fluid system. The steady‐state laminar Navier–Stokes equation and the convection–diffusion equation are used to model the thermal‐fluid problem. The compactly supported radial basis functions are employed to describe the fluid–solid interfaces. The extended finite element method is used to interpolate the material domain and the non‐slip interface conditions are weakly enforced by the Nitsche's method. To overcome the numerical instabilities during the optimization process, a distance‐regularized method based on adopting a modified energy potential functional is proposed and the signed distance property around the structural boundaries is obtained. In the numerical experiments, two‐dimensional numerical models considering minimizing the average or maximum temperature are used to reveal the validity and merits of the proposed method.

Weak enforcement of interface continuity and generalized periodicity in high‐order electromechanical problems

AbstractWe present a formulation for the weak enforcement of continuity conditions at material interfaces in high‐order problems by means of Nitsche's method, which is particularly suited for unfitted discretizations. This formulation is extended to impose generalized periodicity conditions at the unit cell boundaries of periodic structures. The formulation is derived for flexoelectricity, a high‐order electromechanical coupling between strain gradient and electric field, mathematically modeled as a coupled system of fourth‐order PDEs. The design of flexoelectric devices requires the solution of high‐order boundary value problems on complex material architectures, including general multimaterial arrangements. This can be efficiently achieved with an immersed boundary B‐splines approach. Furthermore, the design of flexoelectric metamaterials also involves the analysis of periodic unit cells with generalized periodicity conditions. Optimal high‐order convergence rates are obtained with an unfitted B‐spline approximation, confirming the reliability of the method. The numerical simulations illustrate the usefulness of the proposed approach toward the design of functional electromechanical multimaterial devices and metamaterials harnessing the flexoelectric effect.

Weak imposition of Dirichlet boundary conditions for analyses using Powell–Sabin B‐splines

AbstractPowell–Sabin B‐splines are enjoying an increased use in the analysis of solids and fluids, including fracture propagation. However, the Powell–Sabin B‐spline interpolation does not hold the Kronecker delta property and, therefore, the imposition of Dirichlet boundary conditions is not as straightforward as for the standard finite elements. Herein, we discuss the applicability of various approaches developed to date for the weak imposition of Dirichlet boundary conditions in analyses which employ Powell–Sabin B‐splines. We take elasticity and fracture propagation using phase‐field modeling as a benchmark problem. We first succinctly recapitulate the phase‐field model for propagation of brittle fracture, which encapsulates linear elasticity, and its discretization using Powell–Sabin B‐splines. As baseline solution we impose Dirichlet boundary conditions in a strong sense, and use this to benchmark the Lagrange multiplier, penalty, and Nitsche's methods, as well as methods based on the Hellinger‐Reissner principle, and the linked Lagrange multiplier method and its modified version.

Consistent immersed volumetric Nitsche methods for composite analysis

Generating quality body-fitting meshes for complex composite microstructures is a non-trivial task. In particular, micro-CT images of composites can contain numerous irregularly-shaped inclusions. Among the methods available, immersed boundary methods that discretize bodies independently provide potential for tackling these types of problems since a matching discretization is not needed. However, these techniques still entail the explicit parameterization of the interfaces, which may be considerable in number. In this work, immersed volumetric Nitsche methods are developed in order to avoid the difficulty of generating body fitting meshes for composite materials with complicated microstructures, and overcome the issues in the surface-type methods. These approaches are developed using Nitsche’s techniques to enforce volumetric continuity between the inclusion and background domains. It is shown that the proposed weak forms are fully consistent with the strong form of the composite problem. The present approach permits C0 approximations for the foreground discretization, and C1 approximations for the background. The effectiveness of these methods is demonstrated by solving homogeneous and inhomogeneous composite benchmark problems, where it is shown that the non-symmetric version of Nitsche’s approach is the most robust in all settings.

Falling balls in a viscous fluid with contact: Comparing numerical simulations with experimental data

We evaluate a number of different finite-element approaches for fluid–structure (contact) interaction problems against data from physical experiments. This consists of trajectories of single particles falling through a highly viscous fluid and rebounding off the bottom fluid tank wall. The resulting flow is in the transitional regime between creeping and turbulent flows. This type of configuration is particularly challenging for numerical methods due to the large change in the fluid domain and the contact between the wall and the particle. In the finite-element simulations, we consider both rigid body and linear elasticity models for the falling particles. In the first case, we compare the results obtained with the well-established Arbitrary Lagrangian–Eulerian (ALE) approach and an unfitted moving domain method together with a simple and common approach for contact avoidance. For the full fluid–structure interaction (FSI) problem with contact, we use a fully Eulerian approach in combination with a unified FSI-contact treatment using Nitsche's method. For higher computational efficiency, we use the geometrical symmetry of the experimental setup to reformulate the FSI system into two spatial dimensions. Finally, we show full three-dimensional ALE computations to study the effects of small perturbations in the initial state of the particle to investigate deviations from a perfectly vertical fall observed in the experiment. The methods are implemented in open-source finite element libraries, and the results are made freely available to aid reproducibility.

An analysis on the penalty and Nitsche's methods for the Stokes–Darcy system with a curved interface

We analyze the penalty and Nitsche's methods, in the continuous and discrete senses, for the Stokes–Darcy system with a curved interface. In the continuous sense, we prove the stability and optimal convergence for the penalty approach. In discretization, the curved interface is approximated by polygonal surface. We propose the discontinuous Galerkin (DG) penalty/Nitsche schemes, and establish the stability and error analysis taking the domain perturbation into account. To obtain sharp estimates on the pressure constant and the inf-sup condition involving the integration on the interface, we study the broken H1/2 norm of the DG element, and prove a DG version of the reversed trace operator.

A benchmark problem to evaluate implementational issues for three-dimensional flows of incompressible fluids subject to slip boundary conditions

We consider flows of an incompressible Navier–Stokes fluid in a tubular domain with Navier’s slip boundary condition imposed on the impermeable wall. We focus on several implementational issues associated with this type of boundary conditions within the framework of the standard Taylor-Hood mixed finite element method and present the computational results for flows in a tubular domain of finite length with one inlet and one outlet. In particular, we present the details regarding variants of the Nitsche method concerning the incorporation of the impermeability condition on the wall. We also find that the manner in which the normal to the boundary is numerically implemented influences the nature of the computational results. As a benchmark, we set up steady flows in a tube of finite length and compare the computational results with the analytical solutions. Finally, we identify various quantities of interest, such as the dissipation, wall shear stress, vorticity, pressure drop, and provide their precise mathematical definitions. We document how well these quantities are computationally approximated in the case of the benchmark.Although the geometry of the benchmark is simple, the correct computational results require careful selection of numerical methods and surprisingly non-trivial computational resources. Our goal is to test, using the setting with a known analytical solution, a robust computational tool that would be suitable for computations on real complex geometries that have relevance to problems in engineering and medicine. The model parameters in our computations are chosen based on flows in large arteries.

Nitsche's Method for Kirchhoff Plates

We introduce Nitsche's method for the numerical approximation of the Kirchhoff--Love plate equation under general Robin-type boundary conditions. We analyze the method by presenting a priori and a ...

A Next-Generation Mathematical Model for Drug-Eluting Stents

We propose a Navier--Stokes--Biot fluid-structure interaction (FSI) model to study the interaction between an incompressible, viscous Newtonian fluid and a poroelastic medium with permeability depending on the volumetric change of pore size. The FSI model is coupled to a set of advection-reaction-diffusion equations defined on moving domains so that we may study the interaction between the blood flow, a stented coronary artery, and time-dependent pharmacokinetics of drug absorption in drug-eluting stents. A monolithic approach is used to implement the proposed problem numerically within the context of finite element discretization. Nitsche's method is employed to enforce one of the coupling conditions at the moving fluid-poroelastic structure interface. Stability analysis is presented, providing conditions on Nitsche's penalty parameter under which the scheme is unconditionally stable. Using 3D simulations, five geometrically different metallic stent platforms are considered with two different pharmacokinetics to show how stent geometry and type of coating impact the biomechanical environment, the local hemodynamics, and the concentration of the pharmacological agents within the vascular wall and artery lumen. It is found that stent implantation changes the permeability properties of the arterial wall, as well as local hemodynamics, which may be responsible for the so-called edge effect, i.e., suboptimal reduction in restenosis rates near the edges of drug-eluting stents. To the best of our knowledge, this is the first study of drug-eluting stents that takes into account fluid-poroelastic structure interaction with permeability depending on the volumetric change of the pore size, coupled to an advection-reaction-diffusion model defined on moving domains.