Embedded solids of any dimension in the X-FEM. Part II - Imposing Dirichlet boundary conditions

peer reviewedThis paper focuses on the design of a stable Lagrange multiplier space dedicated to enforce Dirichlet boundary conditions on embedded boundaries of any dimension. It follows a previous paper in a series of two, on the topic of embedded solids of any dimension within the context of the extended finite element method. While the first paper is devoted to the design of a dedicated P1 function space to solve elliptic equations defined on manifolds of codimension one or two (curves in 2D and surfaces in 3D, or curves in 3D), the general treatment of Dirichlet boundary conditions, in such a setting, remains to be addressed. This is the aim of this second paper. A new algorithm is introduced to build a stable Lagrange multiplier space from the traces of the shape functions defined on the background mesh. It is general enough to cover: (i) boundary value problems investigated in the first paper (with, for instance, Dirichlet boundary conditions defined along a line in a 3D mismatching mesh); but also (ii) those posed on manifolds of codimension zero (a domain embedded in a mesh of the same dimension) and already considered in Béchet et al. 2009. In both cases, the compatibility between the Lagrange multiplier space and that of the bulk approximation (the dedicated P1 function space used in (i), or classical shape functions used in (ii)) — resulting in the inf–sup condition — is investigate through the numerical Chapelle-Bath test. Numerical validations are performed against analytical and finite element solutions on problems involving 1D or 2D boundaries embedded in a 2D or 3D background mesh. Comparisons with Nitsche’s method and the stable Lagrange multiplier space proposed in Hautefeuille et al. 2012, when they are feasible, highlight good performance of the approach

Embedded solids of any dimension in the X-FEM. Part I - Building a dedicated P1 function space

peer reviewedThis paper focuses on the design of a dedicated P1 function space to model elliptic boundary value problem on a manifold embedded in a space of higher dimension. Using the traces of the linear P1 shape functions, it introduces an algorithm to reduce the function space into an equivalent space having the same properties than a P1 Lagrange approximation. Convergence studies involving problems of codimension one or two embedded in 2D or 3D show good accuracy with regard to classical finite element and analytical solutions. The effects of the relative position of the domain with respect to the mesh are studied in a sensitivity analysis; it illustrates how the proposed solution allows to keep the condition number bounded. A comparative study is performed with the method introduced by Olshanskii et al. 2009 on a closed surface to validate our approach. The robustness of the proposed approach is investigated with regard to their method and that of Burman et al. 2016. This paper is the first in a series of two, on the topic of embedded solids of any dimension within the context of the extended finite element method. It investigates problems involving borderless domains or domains with boundary subject to Dirichlet constraint defined only on the boundaries of the bulk mesh, while the forthcoming paper overcomes this limitation by introducing a new stable Lagrange multiplier space for Dirichlet boundary condition (and more generally stiff condition), that is valid for every combination of the background mesh and manifold dimensions. The combination of both algorithms allows to handle any embedding i.e. 1D, 2D and 3D problems embedded in 2D or 3D background meshes

The extended finite element method for three-dimensional reinforced composites.

This paper deals with the use of eXtended Finite Element Method (XFEM) to perform local effects in three-dimensional reinforced composites. This method was first introduced to model cracks. It is based on the partition of unity concept and the description of discontinuities like the location of holes and material interfaces is often realized by the level-set method. The approach considered allows (i) to easily model the real geometry of reinforcing fibers (not idealized), (ii) to impose arbitrary Dirichlet and Neumann boundary conditions on the implicit defined boundaries and (iii) to introduce models of degradation. Numerical applications are presented on some academic tests

Numerical simulation on embedded solids, in the X-FEM context

peer reviewedCe travail traite des différents enjeux pour conserver toutes les capacités des calculs par éléments finis, en s’affranchissant des maillages conformes à la géométrie. La représentation de la géométrie, la construction d’un espace fonctionnel adapté et l’imposition des conditions aux limites sont abordées dans le contexte de la méthode des éléments finis étendus, tout particulièrement en présence de solides plongés dans un espace de dimension supérieure. L’approche proposée permet de traiter tout type de plongement, c.-à-d. des problèmes 1, 2, et/ou 3D plongés dans des maillages 2 ou 3D

Numerical simulations on embedded solids : integration of CAD and eXtended Finite Element Analysis

The aim of this communication is to propose a procedure in order to dissociate the geometric description of the field’s approximation within the extended finite element method (X-FEM) and with non-matching meshes. Implicit and explicit approaches are combined in order to represent with accuracy all the CAD entities regardless of their dimension. The choice of appropriate tools such as Level Sets technique allows to describe evolving interfaces with great flexibility. The design of a dedicated P1 functional space is achieved by decimating the traces of standard finite element (FE) shape functions, thanks to a new algorithm, especially when the problem domain is embedded in a space of a higher dimension. An analysis of the approximation properties of the P1 FE trace spaces on hyper-surfaces is available in the literature and applied for solving PDEs on closed surfaces without boundary. Dirichlet boundary conditions are applied using a convenient choice of stable Lagrange multiplier space, according to a new generalized algorithm. That extends the existing solutions to every combination of the space domain and boundary dimensions. In terms of solvers, the introduction of double Lagrange multipliers can be used to recover the positive definiteness of the bilinear form. This approach allows to treat any embedding, i.e. 1, 2, or 3D problems embedded in 2 or 3D background meshes. The possibility of applying the methodology to beams is investigated, with a potential application to through-thickness reinforced composites in a mixed-dimensional modelling framework

Embedded solids of any dimension in the X-FEM defined on higher-order approximations

Embedded interface methods bring a significant simplification of the modelling process before analysis. Complex geometries and moving boundaries may be described with great flexibility, reducing the meshing step to that of a simple bounding box. Following this idea – to dissociate the field approximation from the geometric description – manifolds of different dimensions may be embedded in the same bulk mesh. However, special attention should be given to the difference of dimensions between that of problem domain and that of bulk mesh (the codimension). Whereas the direct use of the shape functions of the bulk mesh is possible for a problem domain of codimension zero, this approach is no longer possible in other configurations, for instance a beam in a 3D mesh. Unlike approaches introducing independent overlapping meshes for each subdomain, function spaces may be built from the traces of higher dimensional spaces built upon the bulk mesh. For closed curves in 2D and closed surfaces in 3D, the resulting discrete method based on P1 FE have already been studied in the literature. To avoid badly conditioned linear systems, specific treatments are required, e.g. preconditioning approaches or stabilization techniques. Here, we propose to deplete wisely the trace space. We investigate higher-order function spaces to solve the diffusion equation in embedded solids of any codimension. A new space-reducer algorithm is introduced to design the dedicated spaces that avoids ill-conditionning while treating boundary conditions. We present the results of several numerical experiments with convergence analyses. To conclude, applications of this technique to embedded beams or shells is discussed

Simulations numériques sur des solides plongés, dans un contexte X-FEM

International audienceCe travail traite des différents enjeux pour conserver toutes les capacités des calculs par éléments finis, en s'affranchissant des maillages conformes à la géométrie. La représentation de la géométrie, la construction d'un espace fonctionnel adapté et l'imposition des conditions aux limites sont abordées dans le contexte de la méthode des éléments finis étendus, tout particulièrement en présence de solides plongés dans un espace de dimension supérieure. L'approche proposée permet de traiter tout type de plongement, c'est-à-dire des problèmes 1, 2, et/ou 3D plongés dans des maillages 2 ou 3D

Plongements de solides de dimensions arbitraires dans le cadre de la méthode des éléments finis étendus

First, we propose a new methodology to enforce Dirichlet boundary conditions in the X-FEM, available for every combination of the space and boundary dimensions. Second, we introduce a new algorithm to define a reduced function space over an embedded solid of arbitrary dimension. The combination of both algorithms allows to treat any embedding i.e. 1, 2 and 3D problems embedded in 2 or 3D background meshes

Interface element for delamination simulation. A good usage for accuracy and performances.

This paper deals with the use of interface element for the simulation of crack propagation. The questions: "how to choose mesh size, material properties and model parameters in order to get a correct result in a reasonable time" will be discussed. An industrial test case with skin-stringer separation will also be presented