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... • Méthode purement primale : dans code_aster, toutes les formulations disponibles pour le problème de contact sont sous la forme primale-duale, ce qui justifie notre intérêt pour l'utilisation de cette formulation. Ces dernières années, des formulations purement primales basées sur la méthode de Nitsche [85] ont reçu une attention croissante afin de formuler le problème de contact avec ou sans frottement en hypothèse de petites et grandes déformations. Contrairement aux méthodes sous forme primale-duale où l'on introduit une nouvelle variable, ces méthodes ne nécessitent d'introduire aucune inconnue supplémentaire. ...

... Comme exemple, on peut citer les méthodes de pénalisation [99]. Dans le Chapitre 4, nous nous intéressons à une approche primale basée sur la méthode de Nitsche [85]. Cette méthode a été originellement introduite pour la reformulation des conditions de Dirichlet dans les problèmes elliptiques. ...

... One example are penalty methods [99]. Here, we focus on another primal approach based on Nitsche's method [85]. This method was originally introduced for the reformulation of Dirichlet boundary conditions and extended in [27] to the frictionless contact problem in the framework of the finite element method. ...

In this thesis, we investigate model reduction for parameter-dependent variational inequalities. First, we consider the case where the problem is formulated by a mixed method, where the challenge is that the reduced model must satisfy an inf-sup stability condition. For this purpose, we have devised a new greedy algorithm, based on a theoretical result, which allows to efficiently build a stable reduced model. In addition, we have also devised a method to construct the dual basis leading to a larger aperture. In the case of friction, a new idea deals with the enforcement of the tangential constraints in the reduced model by means of a collocation method. Numerical results are presented for the problem of contact with and without friction. A first integration in the industrial software code_aster has been achieved in the frictionless case. In a second step, we study the reduced basis method applied to the contact problem formulated with Nitsche’s method. This method, which is purely primal, leads to an inefficient reduced model owing to the nonlinearity in the formulation. To overcome this problem, we propose a model reduction procedure based on the empirical interpolation method. Numerical tests confirm the robustness and efficiency of the approach.

... Over the past two decades, a theoretical foundation for the formulation of stabilized CutFEM has been developed by extending the ideas of Nitsche, presented in [1], to a general weak formulation of the interface conditions, thereby removing the need for domainfitted meshes. The foundations of CutFEM were presented in [2] and then extended to overlapping meshes in [3]. ...

... To obtain the finite element solution u h , we have used piecewise linear basis functions in space, and in time we have used the discontinuous Galerkin methods dG(0) and dG (1). In other words, the finite element method defined by (3.17) for p = 1 and q = 0, 1. ...

... The slope of the LLS of the error versus k and h for different values of µ for dG(1). ...

We present a cut finite element method for the heat equation on two overlapping meshes. By overlapping meshes we mean a mesh hierarchy with a stationary background mesh at the bottom and an overlapping mesh that is allowed to move around on top of the background mesh. Overlapping meshes can be used as an alternative to costly remeshing for problems with changing or evolving interior geometry. In this paper the overlapping mesh is prescribed a dG(0) movement, meaning that its location as a function of time is discontinuous and piecewise constant. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche's method. The dG(0) mesh movement results in a space-time discretization with a nice product structure between space and time which allows for existing analysis methodologies to be applied with only minor modifications. We follow the analysis methodology presented by Eriksson and Johnson in [12, 13], here referred to as an $L^2$-analysis because of the norm used in the error analysis. The greatest modification is the use of a shift operator that generalizes the Ritz projection operator. The shift operator is used to handle the shift in the overlapping mesh's location at discrete times. The $L^2$-analysis consists of the corresponding standard stability estimates and a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders.

... In this work, the objective is to evaluate the performance of an immersed boundary technique for 3D computational simulations of incompressible fluid flows governed by Navier-Stokes equations. In particular, we are interested in the Nitsche's method (Nitsche [3]) since, among other issues, it does not increase the system's degrees of freedom and has a rather straightforward implementation. A Newmark scheme (Newmark [4]) is used to time-integrate the governing equations. ...

... In the realm of fixed grid methods, the Nitsche's method (Nitsche [3]) has continually gained great attention in the context of implicit interfaces modeled by the XFEM (Dolbow and Harari [9]), also with imposition of constraints along non-matching surface grids (Bazilevs and Hughes [10]). In this work, the Nitsche's method is used to enforce the interface constraints (Dirichlet boundary conditions) for treating the mechanical interactions of overlapping finite element meshes, as depicted in Figure 1. ...

... Hm with Re 45 (Reynolds number). The fluid's density is3 1.0kg m and the kinematic viscosity is 32 10 ms . The coordinates of the center of the sphere are W and the boundary condition at outflow plane is zero traction. ...

This work investigates the outcomes of using an immersed boundary technique for the mixed finite element formulation of tridimensional incompressible fluid flows governed by the Navier-Stokes equations with internal fluid-body interfaces. A classical Eulerian approach is followed to describe the fluid. A Newton-Raphson scheme is devised to solve the resulting non-linear equations within a time step. The fluid-body interface is treated by the Nitsche's method, which is an immersed boundary technique whereby the fluid boundary conditions over the contact with the bodies are imposed weakly. In order to ascertain the accuracy and efficiency of the adopted method, numerical simulations of tridimensional flows of an incompressible fluid are analyzed and compared against reference solutions. This work refers to an intermediate stage of a PhD research that aims to model problems of fluid-particle interaction (FPI) and particle-laden fluids.

... They do not influence the weak convergence property sketched above (because τ is compactly supported). However, the value of the deformation and its gradient imposed on Γ D ∪ Γ M must be added separately for strong consistency [60]. To this end, we introduce the lifting of the boundary conditions (42) and (43) ...

... The discrete problem is thus to seek y h ∈ [V k h ] 3 such that (60) y h ∈ arg min ...

... We start with the unconstrained minimization problem (60) and recall that without external forces ...

We review different (reduced) models for thin structures using bending as principal mechanism to undergo large deformations. Each model consists in the minimization of a fourth order energy, potentially subject to a nonconvex constraint. Equilibrium deformations are approximated using local discontinuous Galerkin (LDG) finite elements. The design of the discrete energies relies on a discrete Hessian operator defined on discontinuous functions with better approximation properties than the piecewise Hessian. Discrete gradient flows are put in place to drive the minimization process. They are chosen for their robustness and ability to preserve the nonconvex constraint. Several numerical experiments are presented to showcase the large variety of shapes that can be achieved with these models.

... However, unlike the finite element formulation, the boundary terms in the weak form on the Dirichlet part of the physical boundary D do not vanish because the solution space of the finite cell method in general does not conform to the physical domain. The Nitsche's method [73] for the weak imposition of the boundary conditions is used, which leads to a consistent weak form and does not introduce additional degrees of freedom in the global system. In addition to the considerations for the imposition of essential boundary conditions, volume integrals are multiplied by a function˛.x/ ...

... 73 Influence of considering the HM interactions and the corresponding material parameters evolution on the model responses[58]. a ground surface settlement along the tunnel, b axial forces in the lining, c bending moments in the tunnel lining ...

The excavation process in mechanised tunnelling consists of various technical components whose interaction enables safe tunnel driving. In reference to the existing geological and hydrogeological conditions, different types of face support principles are applied. In case of fine-grained cohesive soils, the face support is provided by Earth Pressure Balanced (EPB) machines, while the Slurry Shield (SLS) technology is adapted in medium-grained to coarse grained non-cohesive soils even under high groundwater pressure. For both machine techniques, the support medium (the excavated and conditioned soil (EPB) or the bentonite suspension (SLS)) needs to be adapted for the specific application. Within this chapter, the theoretical, experimental and numerical developments and results are presented concerning the fundamentals of face support in EPB and SLS tunnelling including the rheology of the support medium, the material transport and mixing process of the excavated soil and the added conditioning agent in the excavation chamber of an EPB shield machine as well as the constitutive models for investigations of the near field interactions between surrounding soil and advancing shield machine.

... Nitsche's method was originally introduced for the treatment of the Dirichlet boundary conditions as a consistent method without additional Lagrangian multipliers [46]. In recent years, it has been introduced for contact problems. ...

... Initially introduced by Nitsche [46] (see also [59]), the method has been extended by [15,21] for the unilateral contact problem, then in elastodynamics [18,19,20]. We are going to define the space-discrete weak formulation of the Nitsche-FEM method: firstly, we introduce the linear discrete operator ...

This work focuses on the numerical performance of HHT-α and TR-BDF2 schemes for dynamic frictionless unilateral contact problems between an elastic body and a rigid obstacle. Nitsche’s method, the penalty method, and the augmented Lagrangian method are considered to handle unilateral contact conditions. Analysis of the convergence of an opposed value of the parameter α̃ for the HHT-α method is achieved. The mass redistribution method has also been tested and compared with the standard mass matrix. Numerical results for 1D and 3D benchmarks show the functionality of the combinations of schemes and methods used.

... on curved domains with control constraints. Nitsche's method for elliptic PDEs was proposed in 1971 by Nitsche [29] to weakly impose the Dirichlet boundary condition. The Dirichlet boundary control enters in the right hand side of the variational formulation and a penalty term is included in the bilinear form for the stability but does not introduce any consistency error. ...

... The discrete admissible control set is defined by . Recall that Nitsche's method [29] for the state equation (1.2) is defined as seeking a function such that (3.1) where the penalty parameter is taken sufficiently large to ensure the stability. We define the linear operators and by (3.2) and . ...

We consider Nitsche’s method for solving elliptic Dirichlet boundary control problems on curved domains with control constraints. By using Nitsche’s method for the treatment of inhomogeneous Dirichlet boundary conditions, the L² boundary control enters in the variational formulation in a natural sense. The idea was first used in Chang, et al. (Math. Anal. Appl.453, 529–557 2017) where the curved boundary was approximated by a broken line and a locally defined mapping was needed to obtain the numerical control on the curved boundary. In this paper, we develop a method defined on curved domains directly. We derive a priori estimates of quasi-optimal order for the control in the L² norm, and quasi-optimal order for the state and adjoint state in energy norms. Numerical examples are provided to show the performance of the proposed method.

... An early approach to weak boundary conditions for finite element methods was introduced orignally by Nitsche in [18], using a method that is related to ALM, but without any multiplier. Indeed here the multiplier has been replaced by its physical representation, the normal boundary flux. ...

... Recalling next that formally the Lagrange multiplier in (99) is given by = ∇ n v , which provides a direct way of computing the Lagrange multiplier from the primal solution, we obtain This is our stabilised ALM, the minimiser to which solves the problem of finding u h ∈ V h such that where We identify the classical method of Nitsche [18], stable if 0 is chosen so that 0 > C , where C is the constant in the inverse inequality Remark 1 As shown by Stenberg [19] (and discussed in Sec. 4.4), Nitsche's method can be viewed as a particular instance of the GLS stabilisation method of Barbosa-Hughes [99]; in this sense the ALM is a variant of GLS, with the multiplier eliminated. ...

In this paper we will present a review of recent advances in the application of the augmented Lagrange multiplier method as a general approach for generating multiplier-free stabilised methods. The augmented Lagrangian method consists of a standard Lagrange multiplier method augmented by a penalty term, penalising the constraint equations, and is well known as the basis for iterative algorithms for constrained optimisation problems. Its use as a stabilisation methods in computational mechanics has, however, only recently been appreciated. We first show how the method generates Galerkin/Least Squares type schemes for equality constraints and then how it can be extended to develop new stabilised methods for inequality constraints. Application to several different problems in computational mechanics is given.

... A generalized Heaviside enrichment strategy with multiple enrichment levels is used to ensure independent approximation on each connected fluid or solid subregion. Boundary and interface conditions are enforced weakly via Nitsche's method, see Nitsche (1971). Additionally, the face-oriented ghost stabilization is used to mitigate numerical instabilities resulting from the creation of small integration subdomains, see Burman (2010) and Burman and Hansbo (2014). ...

... Boundary conditions on the fluid velocity field are imposed weakly via Nitsche's formulation, see Nitsche (1971) and Bazilevs and Hughes (2007). To prescribe a velocity D on Γ f D , the following contributions are added to the velocity and pressure residuals: ...

Solving conjugate heat transfer design problems is relevant for various engineering applications requiring efficient thermal management. Heat exchange between fluid and solid can be enhanced by optimizing the system layout and the shape of the flow channels. As heat is transferred at fluid/solid interfaces, it is crucial to accurately resolve the geometry and the physics responses across these interfaces. To address this challenge, this work investigates for the first time the use of an eXtended Finite Element Method (XFEM) approach to predict the physical responses of conjugate heat transfer problems considering turbulent flow. This analysis approach is integrated into a level set-based optimization framework. The design domain is immersed into a background mesh and the geometry of fluid/solid interfaces is defined implicitly by one or multiple level set functions. The level set functions are discretized by higher-order B-splines. The flow is predicted by the Reynolds Averaged Navier–Stokes equations. Turbulence is described by the Spalart–Allmaras model and the thermal energy transport by an advection–diffusion model. Finite element approximations are augmented by a generalized Heaviside enrichment strategy with the state fields being approximated by linear basis functions. Boundary and interface conditions are enforced weakly with Nitsche’s method, and the face-oriented ghost stabilization is used to mitigate numerical instabilities associated with the emergence of small integration subdomains. The proposed XFEM approach for turbulent conjugate heat transfer is validated against benchmark problems. Optimization problems are solved by gradient-based algorithms and the required sensitivity analysis is performed by the adjoint method. The proposed framework is illustrated with the design of turbulent heat exchangers in two dimensions. The optimization results show that, by tuning the shape of the fluid/solid interface to generate turbulence within the heat exchanger, the transfer of thermal energy can be increased.

... Ern et JL. Guermond (voir [26]), pour les EDP elliptiques, les méthodes DG issues des travaux de Nitsche sur les méthodes de pénalité aux limites [37] et l'utilisation des pénalités intérieures (IP) pour appliquer faiblement les conditions de continuité imposées sur la solution ou ses dérivées à travers les interfaces entre les éléments adjacents ; voir, par exemple, Babuska [5], Babuska et Zlamal [41], Douglas et Dupont [13], Baker [7], Wheeler [14] et Arnold [1]. Les méthodes DG pour les problèmes elliptiques sous formes combinées ont été introduites plus récemment (initialement, une approximation discontinue a été utilisée uniquement pour la variable primale, le flux étant toujours discrétisé de manière conforme ; voir, par exemple, Dawson [10,11]). ...

... 37) où C est une constante positive.LaFigure 3.1 trace les solutions exacte et numérique pour N = 30 et m = 3. On remarque que ces deux figures sont similaires. ...

... Examples of these are direct elimination (also reduced-gradient algorithms), penalty and barrier methods, and Lagrange multiplier methods with the complementarity condition [4]. Stresses from the underlying discretization are often used to assist the normal condition with Nitsche's method [5]. 2. Frictional effects (and more generally, constitutive contact laws) and cone-complementarity forms [6,7]. ...

We introduce a PDE-based node-to-element contact formulation as an alternative to classical, purely geometrical formulations. It is challenging to devise solutions to nonsmooth contact problem with continuous gap using finite element discretizations. We herein achieve this objective by constructing an approximate distance function (ADF) to the boundaries of solid objects, and in doing so, also obtain universal uniqueness of contact detection. Unilateral constraints are implemented using a mixed model combining the screened Poisson equation and a force element, which has the topology of a continuum element containing an additional incident node. An ADF is obtained by solving the screened Poisson equation with constant essential boundary conditions and a variable transformation. The ADF does not explicitly depend on the number of objects and a single solution of the partial differential equation for this field uniquely defines the contact conditions for all incident points in the mesh. Having an ADF field to any obstacle circumvents the multiple target surfaces and avoids the specific data structures present in traditional contact-impact algorithms. We also relax the interpretation of the Lagrange multipliers as contact forces, and the Courant–Beltrami function is used with a mixed formulation producing the required differentiable result. We demonstrate the advantages of the new approach in two- and three-dimensional problems that are solved using Newton iterations. Simultaneous constraints for each incident point are considered.

... The Nitsche's method [27,16] is a classical method for imposing interface conditions weakly and can avoid introducing any Lagrange multiplier [8]. In this paper, we add Nitsche's penalty term in the variational formulation of the governing equations to overcome the difficulty of instability, which arises from the interface condition of the heat flux. ...

... Nitsche's method has been considered recently to discretize contact and friction conditions. The Nitsche method orginally proposed in [57] aims at treating the boundary or interface conditions in a weak sense, with appropriate consistent terms that involve only the primal variables. It differs in this aspect from standard penalization techniques which are generally non-consistent [49]. ...

We consider frictional contact problems in small strain elasticity discretized with finite elements and Nitsche method. Both bilateral and unilateral contact problems are taken into account, as well as both Tresca and Coulomb models for the friction. We derive residual a posteriori error estimates for each friction model, following [Chouly et al, IMA J. Numer. Anal. (38) 2018, pp. 921-954]. For the incomplete variant of Nitsche, we prove an upper bound for the dual norm of the residual, for Tresca and Coulomb friction, without any extra regularity and without a saturation assumption. Numerical experiments allow to assess the accuracy of the estimates and their interest for adaptive meshing in different situations.

... The Heaviside enriched XFEM formulation outlined above enables the modeling of C −1 intra-element discontinuities of state variables within a non-conforming background element. Essential boundary conditions can be enforced weakly by, for example, Nitsche's method [9,34] or the stabilized Lagrange multiplier method [22]. Immersing geometry into the XFEM background mesh can result in basis functions with small support within the geometric domain, leading to poorly conditioned systems of discretized governing equations. ...

This paper presents an immersed, isogeometric finite element framework to predict the response of multi-material, multi-physics problems with complex geometries using locally refined discretizations. To circumvent the need to generate conformal meshes, this work uses an extended finite element method (XFEM) to discretize the governing equations on non-conforming, embedding meshes. A flexible approach to create truncated hierarchical B-splines discretizations is presented. This approach enables the refinement of each state variable field individually to meet field-specific accuracy requirements. To obtain an immersed geometry representation that is consistent across all hierarchically refined B-spline discretizations, the geometry is immersed into a single mesh, the XFEM background mesh, which is constructed from the union of all hierarchical B-spline meshes. An extraction operator is introduced to represent the truncated hierarchical B-spline bases in terms of Lagrange shape functions on the XFEM background mesh without loss of accuracy. The truncated hierarchical B-spline bases are enriched using a generalized Heaviside enrichment strategy to accommodate small geometric features and multi-material problems. The governing equations are augmented by a formulation of the face-oriented ghost stabilization enhanced for locally refined B-spline bases. We present examples for two- and three-dimensional linear elastic and thermo-elastic problems. The numerical results validate the accuracy of our framework. The results also demonstrate the applicability of the proposed framework to large, geometrically complex problems.

... It follows the lines of [10], where continuous in time Galerkin methods are applied to (1.1), and [3,2,34,33], where discontinuous in time Galerkin methods are used to discretize the Navier-Stokes system. In contrast to [10], Dirichlet boundary conditions are implemented here by Nitsche's method [12,24,50]. This yields a strong link between two different families of inf-stable finite element pairs for the space discretization. ...

We present families of space-time finite element methods (STFEMs) for a coupled hyperbolic-parabolic system of poro- or thermoelasticity. Well-posedness of the discrete problems is proved. Higher order approximations inheriting most of the rich structure of solutions to the continuous problem on computationally feasible grids are naturally embedded. However, the block structure and solution of the algebraic systems become increasingly complex for these members of the families. We present and analyze a robust geometric multigrid (GMG) preconditioner for GMRES iterations. The GMG method uses a local Vanka-type smoother. Its action is defined in an exact mathematical way. Due to nonlocal coupling mechanisms of unknowns, the smoother is applied on patches of elements. This ensures the damping of error frequencies. In a sequence of numerical experiments, including a challenging three-dimensional benchmark of practical interest, the efficiency of the solver for STFEMs is illustrated and confirmed. Its parallel scalability is analyzed. Beyond this study of classical performance engineering, the solver's energy efficiency is investigated as an additional and emerging dimension in the design and tuning of algorithms and their implementation on the hardware.

... On se propose donc de reformuler le problème satisfait par le champ de transformation ψ ρ afin qu'il soit possible de lui appliquer la réduction de modèle. La méthode suivante, introduite en 1971 par Nitsche [65], est utilisée en mécanique pour des problèmes de contact [24,56]. Dans ce contexte, elle est considérée comme une alternativeà la méthode des multiplicateurs de Lagrange, ou aux méthodes de pénalisation. ...

Cette thèse propose de coupler deux outils préexistant pour la modélisation mathématique en mécanique : l’homogénéisation périodique et la réduction de modèle, afin de modéliser la corrosion des structures de béton armé exposées à la pollution atmosphérique et au sel marin. Cette dégradation est en effet difficile à simuler numériquement, eu égard la forte hétérogénéité des matériaux concernés, et la variabilité de leur microstructure. L’homogénéisation périodique fournit un modèle multi-échelle permettant de s’affranchir de la première de ces deux difficultés. Néanmoins, elle repose sur l’existence d’un volume élémentaire représentatif (VER) de la microstructure du matériau poreux modélisé. Afin de prendre en compte la variabilité de cette dernière, on est amenés à résoudre en temps réduit les équations issues du modèle multi-échelle pour un grand nombre VER. Ceci motive l’utilisation de la méthode POD de réduction de modèle. Cette thèse propose de recourir à des transformations géométriques pour transporter ces équations sur la phase fluide d’un VER de référence. La méthode POD ne peut, en effet, pas être utilisée directement sur un domaine spatial variable (ici le réseau de pores du matériau). Dans un deuxième temps, on adapte ce nouvel outil à l’équation de Poisson-Boltzmann, fortement non linéaire, qui régit la diffusion ionique à l’échelle de la longueur de Debye. Enfin, on combine ces nouvelles méthodes à des techniques existant en réduction de modèle (MPS, interpolation ITSGM), pour tenir compte du couplage micro-macroscopique entre les équations issues de l’homogénéisation périodique.

... Examples of these are direct elimination (also reducedgradient algorithms), penalty and barrier methods, and Lagrange multiplier methods with the complementarity condition [4]. Stresses from the underlying discretization are often used to assist the normal condition with Nitsche's method [5]. ...

We introduce a PDE-based node-to-element contact formulation as an alternative to classical, purely geometrical formulations. It is challenging to devise solutions to nonsmooth contact problem with continuous gap using finite element discretizations. We herein achieve this objective by constructing an approximate distance function (ADF) to the boundaries of solid objects, and in doing so, also obtain universal uniqueness of contact detection. Unilateral constraints are implemented using a mixed model combining the screened Poisson equation and a force element, which has the topology of a continuum element containing an additional incident node. An ADF is obtained by solving the screened Poisson equation with constant essential boundary conditions and a variable transformation. The ADF does not explicitly depend on the number of objects and a single solution of the partial differential equation for this field uniquely defines the contact conditions for all incident points in the mesh. Having an ADF field to any obstacle circumvents the multiple target surfaces and avoids the specific data structures present in traditional contact-impact algorithms. We also relax the interpretation of the Lagrange multipliers as contact forces, and the Courant--Beltrami function is used with a mixed formulation producing the required differentiable result. We demonstrate the advantages of the new approach in two- and three-dimensional problems that are solved using Newton iterations. Simultaneous constraints for each incident point are considered.

... The penalty method can be augmented by terms to make it variationally consistent, mitigating many of the just mentioned drawbacks. When this is done in a symmetric manner, this is called Nitsche's method [42]: ...

In this article, we study the effect of small-cut elements on the critical time-step size in an immersogeometric context. We analyze different formulations for second-order (membrane) and fourth-order (shell-type) equations, and derive scaling relations between the critical time-step size and the cut-element size for various types of cuts. In particular, we focus on different approaches for the weak imposition of Dirichlet conditions: by penalty enforcement and with Nitsche's method. The stability requirement for Nitsche's method necessitates either a cut-size dependent penalty parameter, or an additional ghost-penalty stabilization term is necessary. Our findings show that both techniques suffer from cut-size dependent critical time-step sizes, but the addition of a ghost-penalty term to the mass matrix serves to mitigate this issue. We confirm that this form of `mass-scaling' does not adversely affect error and convergence characteristics for a transient membrane example, and has the potential to increase the critical time-step size by orders of magnitude. Finally, for a prototypical simulation of a Kirchhoff-Love shell, our stabilized Nitsche formulation reduces the solution error by well over an order of magnitude compared to a penalty formulation at equal time-step size.

... where d is the spatial dimension and H 1 denotes the Sobolev space of degree one. Nitsche's method [66] is adopted to impose the Dirichlet boundary condition in Eq. (2e). The symmetric Nitsche-type weak form [30,32,67,68] is utilized, which reads: find u ∈ U such that ∫ ...

Fractures have attracted the attention of computational scientists for several decades. The modeling and simulation of fractures have been a major motivation for developing enriched finite element methods (FEMs), such as the numerical manifold method (NMM). However, ill-conditioning has always haunted NMM and other enriched FEMs when they are utilized for linear elastic fracture problems. Generally, ill-conditioning for a fracture problem is caused by two main issues: the arbitrary cut of the mesh by the fracture path and linear dependence related to the crack-tip enrichments. It is significantly challenging to overcome these two types of ill-conditioning using a single technique. In this study, we employ a preconditioner based on global normalization and local Gram-Schmidt orthogonalization of bases to eliminate these two ill-conditioning issues in NMM entirely and simultaneously. Various numerical examples have demonstrated that the proposed preconditioning strategy is highly effective in reducing the condition number and iteration counts of a iterative solver. It is highly robust, stable, and efficient and can be incorporated into enriched FEM programs to significantly facilitate the analyses of linear elastic fractures.

... Although the imposition of nonhomogeneous boundary conditions can be realized by embedding other media into the continuum body 24,25 , it is difficult to handle more complex problems involving various types of boundary conditions in this way. Since the challenge explained above is widely encountered in mesh-free methods 26,27,28 , several approaches, such as the penalty method, the method of Lagrange multiplier and Nitsche's method 29 , have been studied for imposing Dirichlet boundary conditions, and these approaches have been applied to MPM frameworks 30,31,32 . However, even though Dirichlet boundary conditions are properly handled, another problem arises. ...

An enhancement of the extended B-spline-based implicit material point method (EBS-MPM) is developed to avoid pressure oscillation and volumetric locking. The EBS-MPM is a stable implicit MPM that enables the imposition of arbitrary boundary conditions thanks to the higher-order EBS basis functions and the help of Nitsche's method. In particular, by means of the higher-order EBS basis functions, the EBS-MPM can suppress the cell-crossing errors caused by material points crossing the background grid boundaries and can avoid both the stress oscillations arising from inaccurate numerical integration and the ill-conditioning of the resulting tangent matrices. Although the higher-order EBS basis functions are known to avoid volumetric locking, the problem of pressure oscillation has not yet been resolved. Therefore, to suppress pressure oscillation due to quasi-incompressibility, we propose the incorporation of the F-bar projection method into the EBS-MPM, which is compatible with the higher-order EBS basis functions. Three representative numerical examples are presented to demonstrate the capability of the proposed method in suppressing both the pressure oscillation and volumetric locking. The results of the proposed method are compared to those of the finite element method with F-bar elements and those of isogeometric analysis with quadratic NURBS elements.

... Nitsche's method has been considered only recently to discretize contact and friction conditions, despite the fact that it has gained popularity for other boundary conditions. The Nitsche's method orginally proposed in [46] aims at treating the boundary or interface conditions in a weak sense, with appropriate consistent terms that involve only the primal variables. It differs in this aspect from standard penalization techniques which are generally non-consistent [43]. ...

This work deals with the discretization of single-phase Darcy flows in fractured and de-formable porous media, including frictional contact at the matrix-fracture interfaces. Fractures are described as a network of planar surfaces leading to so-called mixed-dimensional models. Small displacements and a linear poro-elastic behavior are considered in the matrix. One key difficulty to simulate such coupled poro-mechanical models is related to the formulation and discretization of the contact mechanical sub-problem. Our starting point is based on the mixed formulation using facewise constant Lagrange multipliers along the fractures representing normal and tangential stresses. This is a natural choice for the discretization of the contact dual cone in order to account for complex fracture networks with corners and intersections. It leads to local expressions of the contact conditions and to efficient semi-smooth nonlinear solvers. On the other hand, such a mixed formulation requires to satisfy a compatibility condition between the discrete spaces restricting the choice of the displacement space and potentially leading to sub-optimal accuracy. This motivates the investigation of two alternative formulations based either on a stabilized mixed formulation or on the Nitsche's method. These three types of formulations are first investigated theoritically in order to enhance their connections. Then, they are compared numerically in terms of accuracy and nonlinear convergence. The sensitivity to the choice of the formulation parameters is also investigated. Several 2D test cases are considered with various fracture networks using both P1 and P2 conforming Finite Element discretizations of the displacement field and an Hybrid Finite Volume discretization of the mixed-dimensional Darcy flow model.

... (8) with additional constraints. To this end, Nitsche's method [36] or Penalty method [37] can be employed, e.g. [38]. ...

The advancements in additive manufacturing (AM) technology have allowed for the production of geometrically complex parts with customizable designs. This versatility benefits large-scale space-frame structures, as the individual design of each structural node can be tailored to meet specific mechanical and other functional requirements. To this end, however, the design and analysis of such space-frames with distinct structural nodes needs to be highly automated. A critical aspect in this context is automated integration of the local 3D features into the 1D large-scale models. In the present work, a two-scale modeling approach is developed to improve the design and linear-elastic analysis of space frames with complex additively manufactured nodes. The mechanical characteristics of the 3D nodes are numerically reduced through an automated dimensional reduction process based on the Finite Cell Method (FCM) and substructuring. The reduced stiffness quantities are assembled in the large-scale 1D model which, in turn, enables efficient structural analysis. The response of the 1D model is passed on to the local model, enabling fully resolved 3D linear-elastic analysis. The proposed approach is numerically verified on a simplified beam example. Furthermore, the workflow is demonstrated on a tree canopy structure with additively manufactured nodes with bolted connections. The form of the large-scale structure is found based on the Combinatorial Equilibrium Modeling framework, and the different designs of the local structural nodes are based on generative exploration of the design space. It is demonstrated that the proposed methodology effectively automates the design and analysis of space-frame structures with complex, distinct structural nodes.

... Stenberg [31] has shown that Barbosa and Hughes [3] method is equivalent to Nitsche [26] method when Λ h = P 0 (Γ h ) and V h = P 1 (Ω h ) (when the mesh on Γ is the trace of the mesh in Ω): ...

The inf-sup condition, also called the Ladyzhenskaya--Babu\v ska--Brezzi (LBB) condition, ensures the existence, uniqueness and well-posedness of a saddle point problem, relative to a partial differential equation. Discretization by the finite element method gives the discrete problem which must satisfy the discrete inf-sup condition. But, depending on the choice of finite elements, the discrete condition may fail. This paper attempts to explain why it fails from an engineer's perspective, and reviews current methods to work around this failure. The last part recalls the mathematical bases.

... There has been some earlier related work. They include the bending-strip method for shell-shell [78] and shell-beam [79] structures, where the beam is actually a bending-stabilized cable, penalty formulation [80], and techniques based on Nitsche's method [81]. They also include using extra mesh refinement along the membrane edge [45] to attain C 0 continuity in both the edge direction and the other direction. ...

We present a T-splines computational method and its implementation where structures with different parametric dimensions are connected with continuity and smoothness. We derive the basis functions in the context of connecting structures with 2D and 1D parametric dimensions. Derivation of the basis functions with a desired smoothness involves proper selection of a scale factor for the knot vector of the 1D structure and results in new control-point locations. While the method description focuses on $$C^0$$ C 0 and $$C^1$$ C 1 continuity, paths to higher-order continuity are marked where needed. In presenting the method and its implementation, we refer to the 2D structure as “membrane” and the 1D structure as “cable.” It goes without saying that the method and its implementation are applicable also to other 2D–1D cases, such as shell–cable and shell–beam structures. We present test computations not only for membrane–cable structures but also for shell–cable structures. The computations demonstrate how the method performs.

... This is due to the fact that our shape functions φ i ϑ n i are neither interpolatory on the boundary ∂Ω nor do they vanish on it. A common way to overcome these issues and impose Dirichlet boundary conditions in meshfree methods is to use a variational approach due to Nitsche [89]. This approach has been shown to work in the setting of our flat-top PUM in [55]. ...

Even today, the treatment of industrial-grade geometries is a huge challenge in the field of numerical simulations. The geometries that are created by computer aided design (CAD) are often very complex and contain many flaws. Hence the discretization by mesh-based methods like the finite element method (FEM) is very time consuming and can take several months when human interaction is required. Therefore, a growing interest in so-called meshfree methods arose in the scientific community over the last few decades.
One such meshfree method is the partition of unity method (PUM), which is very promising because of its flexibility due to its very abstract formulations. But even though the PUM is meshfree in its core, the treatment of complex geometries is still lacking. In this thesis we develop methods to close that gap.
First we propose a post-processing step to the original cover construction algorithm employed in the PUM, that guarantees that stable approximation spaces can be constructed for arbitrary geometries in two and three space-dimensions. Then, we tackle the problem of efficient and robust integration in 2D, by proposing a monotone decomposition of the input geometry. By exploiting properties of the resulting decomposition, we can prove that all required intersection operations can be implemented reliably. By adding all inflection points of the domain's boundary when constructing local decompositions of the integration domains, we can prove that the resulting curved triangles always form a valid decomposition. In 3D, we propose to create a linear approximation of the input geometry. The linear representation allows all subsequent operations to be performed reliably and fast. Then, we develop a method to estimate the domain approximation error and relate that error to the approximation error of the PUM discretization. Refinement controlled by those error estimates then yields a method that can overall converge with optimal rates.
All methods proposed throughout that thesis are validated by numerical experiments. Thereby, we demonstrate the robustness on real-world industrial use cases. In 2D, we present results for a shell problem on the door of a car. In 3D, results for mechanical parts of the landing-gear of an Airbus A380 are presented.

... The DGFEM includes three common types: the symmetric interior penalty Galerkin method (SIPG) which comes from Nitsche method [35], the non-symmetric interior penalty Galerkin method (NIPG) (cf. see [14,36]), and the incomplete interior penalty Galerkin method (IIPG) (see, e.g., [37]). ...

In this paper, a discontinuous Galerkin finite element method of Nitsche's version for the Steklov eigenvalue problem in linear elasticity is presented. The a priori error estimates are analyzed under a low regularity condition, and the robustness with respect to nearly incompressible materials (locking-free) is proven. Furthermore, some numerical experiments are reported to show the effectiveness and robustness of the proposed method.

... Note that also inhomogeneous Dirichlet data g ∈ H 1 / 2 ( D ) can be handled by the usual superposition ansatz, i.e., u = u D + g , where g ∈ H 1 ( ) is a lifting of the inhomogeneous boundary data (e.g., by L 2 ( ) -projection or by nodal interpolation in the discrete case) and u D ∈ H 1 ( ) with u D = 0 on D solves an appropriate variant of (1) ; see, e.g., [8][9][10][11] . Due to the modular design of MooAFEM, also other methods for incorporating inhomogeneous Dirichlet data (such as penalty methods [12] or Nitsche's method [13] ) can be implemented analogously to Robin-type boundary conditions. MooAFEM is able to discretize problem (1) with conforming FEM spaces of arbitrary polynomial order and, by use of OOP, allows that the coefficients A , b, c, and α, as well as the data f , f , γ , and φ can be any function that depends on a spatial variable; in particular, FEM functions are also valid as coefficients. ...

We present an easily accessible, object oriented code (written exclusively in Matlab) for adaptive finite element simulations in 2D. It features various refinement routines for triangular meshes as well as fully vectorized FEM ansatz spaces of arbitrary polynomial order and allows for problems with very general coefficients. In particular, our code can handle problems typically arising from iterative linearization methods used to solve nonlinear PDEs. Due to the object oriented programming paradigm, the code can be used easily and is readily extensible. We explain the basic principles of our code and give numerical experiments that underline its flexibility as well as its efficiency.

... For Dirichlet's boundary conditions given by to zero-value on the discretized boundary, i.e. f ini | ∂Ω L p ≡ 0, Nitsche's method [9] applies, hence the weak the semi-discrete form reads ...

Arising from averaging and linearizing over the original Vlasov-Maxwell system for magnetized plasmas, the quasilinear theory describes the resonant interaction between particles and waves. Such a model reduction in weak turbulence regime results in a kinetic diffusion process in momentum space for the particle probability density function(pdf ), where the diffusion coefficients are determined by the wave spectral energy density(sed). Meanwhile, a reaction equation in spectral space governs the time dynamics of the wave sed, with growth rates linearly dependent on the particle pdf. We propose a conservative Galerkin scheme for the quasilinear diffusion model in three-dimensional momentum space and three-dimensional spectral space, with cylindrical symmetry. The conservation laws are preserved by adopting the conservative discrete integro-differential operators and a consistent quadrature rule. We introduce a semi-implicit time discretization, and the stability condition is discussed. Numerical examples with applications in the electron runaway problem are provided, they show that the particle-wave interaction results in a strong anisotropic diffusion effect on the particle pdf.

... The Heaviside enriched XFEM formulation outlined above enables the modeling of C −1 intra-element discontinuities of state variables within a non-conforming background element. Essential boundary conditions can be enforced weakly by, for example, Nitsche's method (Nitsche (1971); Burman (2012)) or the stabilized Lagrange multiplier method (Gerstenberger and Wall (2008)). ...

This paper presents an immersed, isogeometric finite element framework to predict the response of multi-material, multi-physics problems with complex geometries using locally refined discretizations. To circumvent the need to generate conformal meshes, this work uses an eXtended Finite Element Method (XFEM) to discretize the governing equations on non-conforming, embedding meshes. A flexible approach to create truncated hierarchical B-splines discretizations is presented. This approach enables the refinement of each state variable field individually to meet field-specific accuracy requirements. To obtain an immersed geometry representation that is consistent across all hierarchically refined B-spline discretizations, the geometry is immersed into a single mesh, the XFEM background mesh, which is constructed from the union of all hierarchical B-spline meshes. An extraction operator is introduced to represent the truncated hierarchical B-spline bases in terms of Lagrange shape functions on the XFEM background mesh without loss of accuracy. The truncated hierarchical B-spline bases are enriched using a generalized Heaviside enrichment strategy to accommodate small geometric features and multi-material problems. The governing equations are augmented by a formulation of the face-oriented ghost stabilization enhanced for locally refined B-spline bases. We present examples for two- and three-dimensional linear elastic and thermo-elastic problems. The numerical results validate the accuracy of our framework. The results also demonstrate the applicability of the proposed framework to large, geometrically complex problems.

... To incorporate (6) a Nitsche penalty term [23] is added, which weakly couples the domains and ensures coercivity [21]: ...

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. Methods: 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. Results: CutFEM outperforms competing FEM approaches with regard to numerical accuracy, memory consumption and computational speed while being able to mesh arbitrarily touching compartments. Conclusion: 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.

... On the other hand, for the penalty method, while it remains primal and is easy to implement, a carefully chosen penalty parameter is crucial to avoiding the loss of coercivity [10] and ill-conditioned systems with too many constraints [11]. The Nitschetype method, originally proposed to enforce element boundaries in a weak sense [12,13], is another method used to model contact and frictional conditions [2,10,14,15]. Different from the Penalty-type method, the Nitsche-type method is generally consistent without penetration. ...

In this paper, a thermomechanical coupled phase field method is developed to model cracks with frictional contact. Compared to discrete methods, the phase field method can represent arbitrary crack geometry without an explicit representation of the crack surface. The two distinguishable features of the proposed phase field method are: (1) for the mechanical phase, no specific algorithm is needed for imposing contact constraints on the fracture surfaces; (2) for the thermal phase, formulations are proposed for incorporating the phase field damage parameter so that different thermal conductance conditions are accommodated. While the stress is updated explicitly in the regularized interface regions under different contact conditions, the thermal conductivity is determined under different conductance conditions. In particular, we consider a pressure-dependent thermal conductance model (PDM) that is fully coupled with the mechanical phase, along with the other three thermal conductance models, i.e., the fully conductive model (FCM), the adiabatic model (ACM), and the uncoupled model (UCM). The potential of this formulation is showcased by several benchmark problems. We gain insights into the role of the temperature field affecting the mechanical field. Several 2D boundary value problems are addressed, demonstrating the model’s ability to capture cracking phenomena with the effect of the thermal field. We compare our results with the discrete methods as well as other phase field methods, and a very good agreement is achieved.

... Description of the coupling problem and prevalent methods Before presenting the framework we use for multi-patch analysis, let us shortly review the main numerical methods that have been investigated in the particular case of isogeometric analysis. Three methods stand out: the penalty coupling (Leidinger et al., 2019, Leonetti et al., 2020, Pasch et al., 2021, mortar coupling (Bernardi et al., 1993, Temizer et al., 2011, Hesch and Betsch, 2012, Brivadis et al., 2015, Dornisch et al., 2015, Dornisch and Müller, 2016, Matzen and Bischoff, 2016, Bouclier et al., 2017, Hirschler et al., 2019c, Buffa et al., 2020, Chasapi et al., 2020, and Nitsche coupling (Nitsche, 1971, Sanders et al., 2012, Apostolatos et al., 2014, Nguyen et al., 2014, Ruess et al., 2014, Du et al., 2015, Guo and Ruess, 2015, Schillinger et al., 2016, Hu et al., 2018, Antolin et al., 2019c, Elfverson et al., 2019, Du et al., 2020. Specifically, Apostolatos et al. (2014) discuss these domain decomposition methods for their application in isogeometric analysis. ...

Uniting the workflows of geometric design and numerical analysis is one of the challenging aims of IsoGeometric Analysis. Such a goal is addressed by using the same mathematical functions — namely, Non-Uniform Rational B-Splines — to describe the geometry and to serve as a support to solve the analysis. Amongst other advantages, NURBS functions benefit from a higher continuity with regards to Lagrange polynomials, and coarser meshes can be used, reducing the analysis time. When it comes to shape optimisation, IGA offers the advantage of providing a model that is compatible with Computer Aided Design software, without further processing. The aircraft engine design and manufacturing industry widely uses numerical methods, and hence can benefit from the advantageous features of IGA. Specific concerns arise in this industrial context, the volumetric definition of spinning parts such as blades being a prominent one. The purpose of this work is to propose a complete framework for the design, analysis and shape optimisation of aircraft engine blades using IGA. Using an actual industrial blade geometry, we propose a procedure to reconstruct a B-spline analysis-suitable volumetric model of the blade, ensuring its geometric accuracy and parametrisation regularity. Shape optimisation is performed using the spatial coordinates of control points as design variables. The mechanical response of the structure is computed using the open-source IGA code Yeti. The rest of the assembly, including the platform and tenon of the blade, is considered using a mortar approach for weak patch coupling. An embedded solid element formulation was developed during this study, enabling accurate modelling of the fillet linking the blade to its platform. In addition, it guarantees the geometric compatibility of the interfaces between adjacent patches during shape updates in the course of the shape optimisation process. The results demonstrate the efficiency of the method and its relevance for industrial aircraft engine blade design and shape optimisation.

... Although the imposition of nonhomogeneous boundary conditions can be realized by embedding other media into the continuum body 24,25 , it is difficult to handle more complex problems involving various types of boundary conditions in this way. Since the challenge explained above is widely encountered in mesh-free methods 26,27,28 , several approaches, such as the penalty method, the method of Lagrange multiplier and Nitsche's method 29 , have been studied for imposing Dirichlet boundary conditions, and these approaches have been applied to MPM frameworks 30,31,32 . However, even though Dirichlet boundary conditions are properly handled, another problem arises. ...

An enhancement of the extended B-spline-based implicit material point method (EBS MPM) is developed to avoid pressure oscillation and volumetric locking. The EBS-MPM is a stable implicit MPM that enables the imposition of arbitrary boundary conditions thanks to the higher-order EBS basis functions and the help of Nitsche's method. In particular, by means of the higher-order EBS basis functions, the EBS-MPM can suppress the cell-crossing errors caused by material points crossing the background grid boundaries and can avoid both the stress oscillations arising from inaccurate numerical integration and the ill-conditioning of the resulting tangent matrices. Although the higher order EBS basis functions are known to avoid volumetric locking, the problem of pressure oscillation has not yet been resolved. Therefore, to suppress pressure oscillation due to quasi-incompressibility, we propose the incorporation of the F-bar projection method into the EBS-MPM, which is compatible with the higher-order EBS basis functions. Three representative numerical examples are presented to demonstrate the capability of the proposed method in suppressing both the pressure oscillation and volumetric locking. The results of the proposed method are compared to those of the finite element method with F-bar elements and those of isogeometric analysis with quadratic NURBS elements.

... Fixed-grid immersed representations, such as web splines [42] or unstructured collections of radial basis functions [43], require careful adaptation of computations near the implicitly enforced boundaries. Penalty methods, e.g., in [44], can add smoothness constraints as part of the solution process at the cost of increasing the size of the problem. The approach requires a judicious choice of penalty parameters. ...

Unstructured hex meshes are partitions of three spaces into boxes that can include irregular edges, where n≠4 boxes meet along an edge, and irregular points, where the box arrangement is not consistent with a tensor-product grid. A new class of tri-cubic C1 splines is evaluated as a tool for solving elliptic higher-order partial differential equations over unstructured hex meshes. Convergence rates for four levels of refinement are computed for an implementation of the isogeometric Galerkin approach applied to Poisson’s equation and the biharmonic equation. The ratios of error are contrasted and superior to an implementation of Catmull-Clark solids. For the trivariate Poisson problem on irregularly partitioned domains, the reduction by 24 in the L2 norm is consistent with the optimal convergence on a regular grid, whereas the convergence rate for Catmull-Clark solids is measured as O(h3). The tri-cubic splines in the isogeometric framework correctly solve the trivariate biharmonic equation, but the convergence rate in the irregular case is lower than O(h4). An optimal reduction of 24 is observed when the functions on the C1 geometry are relaxed to be C0.

... This ensures that ϕ and all its derivatives exist and are bounded in Ω, although ϕ may be very small in regions close to many segments s i . M D : Using Nitsche's method [22]. The goal of this method is to variationally impose the Dirichlet boundary conditions. ...

In this paper, we present and compare four methods to enforce Dirichlet boundary conditions in Physics-Informed Neural Networks (PINNs) and Variational Physics-Informed Neural Networks (VPINNs). Such conditions are usually imposed by adding penalization terms in the loss function and properly choosing the corresponding scaling coefficients; however, in practice, this requires an expensive tuning phase. We show through several numerical tests that modifying the output of the neural network to exactly match the prescribed values leads to more efficient and accurate solvers. The best results are achieved by exactly enforcing the Dirichlet boundary conditions by means of an approximate distance function. We also show that variationally imposing the Dirichlet boundary conditions via Nitsche's method leads to suboptimal solvers.

... The immersogeometric flow analysis methodology consists of three main components: 1) The thermal fluid system is modeled using stabilized finite element methods for incompressible [37][38][39] and compressible [40][41][42] flows. 2) The Dirichlet boundary conditions imposed on the immersed objects are enforced weakly in the sense of Nitsche's method [2,16,43]. 3) To accurately capture the geometry of the flow domain, the concept of the Finite Cell Method (FCM) is employed in which the quadrature rules are adaptively refined [2,44,45]. These numerical ingredients are presented in this section. ...

Immersogeometric analysis (IMGA) is a geometrically flexible method that enables one to perform multiphysics analysis directly using complex computer-aided design (CAD) models. In this paper, we develop a novel IMGA approach for simulating incompressible and compressible flows around complex geometries represented by point clouds. The point cloud object's geometry is represented using a set of unstructured points in the Euclidean space with (possible) orientation information in the form of surface normals. Due to the absence of topological information in the point cloud model, there are no guarantees for the geometric representation to be watertight or 2-manifold or to have consistent normals. To perform IMGA directly using point cloud geometries, we first develop a method for estimating the inside-outside information and the surface normals directly from the point cloud. We also propose a method to compute the Jacobian determinant for the surface integration (over the point cloud) necessary for the weak enforcement of Dirichlet boundary conditions. We validate these geometric estimation methods by comparing the geometric quantities computed from the point cloud with those obtained from analytical geometry and tessellated CAD models. In this work, we also develop thermal IMGA to simulate heat transfer in the presence of flow over complex geometries. The proposed framework is tested for a wide range of Reynolds and Mach numbers on benchmark problems of geometries represented by point clouds, showing the robustness and accuracy of the method. Finally, we demonstrate the applicability of our approach by performing IMGA on large industrial-scale construction machinery represented using a point cloud of more than 12 million points.

Noise is nowadays omnipresent in our society, which encourages us to reduce its impact on health. Thanks to their design flexibility and lightness, sound absorbing packages made of porous materials might hold a pivotal position among noise reduction approaches. Our prime interest is in sound packages which have multiple layers with significant thickness disparity ranging from several meters to millimeters and potentiallycomplex geometries. We aim at elaborating on more efficient numerical methods to identify and predict the vibroacoustic behaviour of suchpackages compared to the classical Finite Element Method (FEM). Based on the eXtended Finite Element Method (X-FEM), enrichment and discretization strategies are developed to couple porous media involving mixed Biot’s equations. Stable and robust variational formulations are proposed to represent the acoustic effects of thin porous layers. Our approaches are demonstrated to be capable of reducing considerably the preprocessing and resolution times while maintaining the accuracy level in comparison with classical FEM.

We present a cut finite element method for the heat equation on two overlapping meshes. By overlapping meshes we mean a mesh hierarchy with a stationary background mesh at the bottom and an overlapping mesh that is allowed to move around on top of the background mesh. Overlapping meshes can be used as an alternative to costly remeshing for problems with changing or evolving interior geometry. In this paper the overlapping mesh is prescribed a cG(1) movement, meaning that its location as a function of time is continuous and piecewise linear. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche's method and also includes an integral term over the space-time boundary that mimics the standard discontinuous Galerkin time-jump term. The cG(1) mesh movement results in a space-time discretization for which existing analysis methodologies either fail or are unsuitable. We therefore propose, to the best of our knowledge, a new energy analysis framework that is general and robust enough to be applicable to the current setting$^*$. The energy analysis consists of a stability estimate that is slightly stronger than the standard basic one and an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders. $*$ UPDATE and CORRECTION: After this work was made public, it was discovered that the core components of the new energy analysis framework seemed to have been discovered independently by us and Cangiani, Dong, and Georgoulis in [1].

Finite element methods for electromagnetic problems modeled by Maxwell-type equations are highly sensitive to the conformity of approximation spaces, and non-conforming methods may cause loss of convergence. This fact leads to an essential obstacle for almost all the interface-unfitted mesh methods in the literature regarding the application to electromagnetic interface problems, as they are based on non-conforming spaces. In this work, a novel immersed virtual element method for solving a three-dimensional (3D) [Formula: see text] interface problem is developed, and the motivation is to combine the conformity of virtual element spaces and robust approximation capabilities of immersed finite element spaces. The proposed method is able to achieve optimal convergence. To develop a systematic framework, the [Formula: see text], [Formula: see text] and [Formula: see text] interface problems and their corresponding problem-orientated immersed virtual element spaces are considered all together. In addition, the de Rham complex will be established based on which the Hiptmair–Xu (HX) preconditioner can be used to develop a fast solver for the [Formula: see text] interface problem.

This work compares two Nitsche‐type approaches to treat non‐conforming triangulations for a high‐order discontinuous Galerkin (DG) solver for the acoustic conservation equations. The first approach (point‐to‐point interpolation) uses inconsistent integration with quadrature points prescribed by a primary element. The second approach uses consistent integration by choosing quadratures depending on the intersection between non‐conforming elements. In literature, some excellent properties regarding performance and ease of implementation are reported for point‐to‐point interpolation. However, we show that this approach can not safely be used for DG discretizations of the acoustic conservation equations since, in our setting, it yields spurious oscillations that lead to instabilities. This work presents a test case in that we can observe the instabilities and shows that consistent integration is required to maintain a stable method. Additionally, we provide a detailed analysis of the method with consistent integration. We show optimal spatial convergence rates globally and in each mesh region separately. The method is constructed such that it can natively treat overlaps between elements. Finally, we highlight the benefits of non‐conforming discretizations in acoustic computations by a numerical test case with different fluids.

We formulate a physics-informed compressed sensing (PICS) method for the reconstruction of velocity fields from noisy and sparse phase-contrast magnetic resonance signals. The method solves an inverse Navier–Stokes boundary value problem, which permits us to jointly reconstruct and segment the velocity field, and at the same time infer hidden quantities such as the hydrodynamic pressure and the wall shear stress. Using a Bayesian framework, we regularize the problem by introducing
a priori
information about the unknown parameters in the form of Gaussian random fields. This prior information is updated using the Navier–Stokes problem, an energy-based segmentation functional, and by requiring that the reconstruction is consistent with the
k
-space signals. We create an algorithm that solves this inverse problem, and test it for noisy and sparse
k
-space signals of the flow through a converging nozzle. We find that the method is capable of reconstructing and segmenting the velocity fields from sparsely-sampled (15%
k
-space coverage), low (~10) signal-to-noise ratio (SNR) signals, and that the reconstructed velocity field compares well with that derived from fully-sampled (100%
k
-space coverage) high (>40) SNR signals of the same flow.

We propose an adaptive mesh refinement strategy for immersed isogeometric analysis, with application to steady heat conduction and viscous flow problems. The proposed strategy is based on residual-based error estimation, which has been tailored to the immersed setting by the incorporation of appropriately scaled stabilization and boundary terms. Element-wise error indicators are elaborated for the Laplace and Stokes problems, and a THB-spline-based local mesh refinement strategy is proposed. The error estimation and adaptivity procedure are applied to a series of benchmark problems, demonstrating the suitability of the technique for a range of smooth and non-smooth problems. The adaptivity strategy is also integrated into a scan-based analysis workflow, capable of generating error-controlled results from scan data without the need for extensive user interactions or interventions.

Immersogeometric analysis (IMGA) is a geometrically flexible method that enables one to perform multiphysics analysis directly using complex computer-aided design (CAD) models. While the IMGA approach is well-studied and has a remarkable advantage over traditional CFD, IMGA still requires a well-defined B-rep model to represent the geometry. Obtaining such a model can sometimes be equally as challenging as creating a body-fitted mesh. To address this issue, we develop a novel IMGA approach for the simulation of incompressible and compressible flows around complex geometries represented by point clouds in this work. The point cloud representation of geometries is a direct method for digitally acquiring geometric information using LiDAR scanners, optical scanners, or other passive methods such as multi-view stereo images. The point cloud object’s geometry is represented using a set of unstructured points in the Euclidean space with (possible) orientation information in the form of surface normals. Due to the absence of topological information in the point cloud model, there are no guarantees for the geometric representation to be watertight or 2-manifold or to have consistent normals. To perform IMGA directly using point cloud geometries, we first develop a method for estimating the inside–outside information and the surface normals directly from the point cloud. We also propose a method to compute the Jacobian determinant for the surface integration (over the point cloud) necessary for the weak enforcement of Dirichlet boundary conditions. We validate these geometric estimation methods by comparing the geometric quantities computed from the point cloud with those obtained from analytical geometry and tessellated CAD models. In this work, we also develop thermal IMGA to simulate heat transfer in the presence of flow over complex geometries. The proposed framework is tested for a wide range of Reynolds and Mach numbers on benchmark problems of geometries represented by point clouds, showing the robustness and accuracy of the method. Finally, we demonstrate the applicability of our approach by performing IMGA on large industrial-scale construction machinery represented using a point cloud of more than 12 million points.

SWe present in this thesis our work on the control and optimization of high field magnets.The physics involved in the operation of the magnet are presented, and their discretizationis detailed. It con- sists of a non-linear thermoelectric problem, a magnetostatic problemand a linear elasticity problem. The Hybrid Discontinuous Galerkin (HDG) method is used inorder to better approximate the fields of interests, such as the current density, the magneticfield or the stress. We developed and implemented the Integral Boundary Condition (IBC) tobe able to impose the current intensity directly instead of using the difference of potential.To solve our problem in real time, we used the Reduce Basis method (RB), combined withthe Empirical Interpolation Method (EIM), its discrete version, the Simultaneous EIM andRB method and the Empirical Quadrature Method (EQM). Finally, we applied our methodsto two applications of interest for the LNCMI, the identification of cooling parameters basedon experimental data, and the optimization of the cuttings of the magnets to improve itshomogeneity.

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.

We consider the approximation of initial/boundary value problems involving, possibly high-dimensional, dissipative evolution partial differential equations (PDEs) using a deep neural network framework. More specifically, we first propose discrete gradient flow approximations based on non-standard Dirichlet energies for problems involving essential boundary conditions posed on bounded spatial domains. The imposition of the boundary conditions is realized weakly via non-standard functionals; the latter classically arise in the construction of Galerkin-type numerical methods and are often referred to as “Nitsche-type” methods. Moreover, inspired by the seminal work of Jordan, Kinderleher, and Otto (JKO) Jordan et al. (1998), we consider the second class of discrete gradient flows for special classes of dissipative evolution PDE problems with non-essential boundary conditions. These JKO-type gradient flows are solved via deep neural network approximations. A key, distinct aspect of the proposed methods is that the discretization is constructed via a sequence of residual-type deep neural networks (DNN) corresponding to implicit time-stepping. As a result, a DNN represents the PDE problem solution at each time node. This approach offers several advantages in the training of each DNN. We present a series of numerical experiments which showcase the good performance of Dirichlet-type energy approximations for lower space dimensions and the excellent performance of the JKO-type energies for higher spatial dimensions.

Typescript (photocopy). Includes appendices. Descriptors: Thesis (M.S.)--University of Nevada, Las Vegas, 2005. Includes bibliographical references.

Numerical Solution of Boundary Value Problems by the Perturbated Variational Principle

- I Babuska