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

Duboeuf, Frédéric; Bchet, E.

doi:10.1016/j.finel.2016.12.001

Cited by 3 publications

(12 citation statements)

References 59 publications

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“…As model problem, we consider an elliptic equation with the same settings than in the preliminary paper [14].…”

Section: Model Problemmentioning

confidence: 99%

“…Specific treatments of the linear algebraic system of equations has been proposed in [69]. We choose an approach taking advantage of the fixed topology of the mesh introduced in [14]. The full P1 trace space may be reduced thanks to an algorithm -called vertex space reducer -which identifies the shape functions that are in excess and defines linear combinations for any codimension of the problem.…”

Section: Definition Of the Discrete Function Spacementioning

confidence: 99%

“…The proposed solution deeply uses the terminology introduced in [14]. We briefly recall the main ideas.…”

Section: Terminologymentioning

confidence: 99%

“…In line with the previous paper [14], we consider the approach modifying the geometry to address conditioning issues of the resulting linear system, in a preliminary step to the Algorithm 1. In such a way, Ω h is locally modified into Ω h before constructing the linear extension Ω h .…”

Section: Conditioning Of the Linear Systemmentioning

confidence: 99%

“…Problems involving domains of a higher codimension will also be investigated, as surfaces embedded in a 3D mesh, thanks to dedicated P1 function spaces introduced in a preliminary paper [14]. A detailed review of such other techniques dealing with embedded problem is also presented in that paper.…”

Section: Introductionmentioning

confidence: 99%

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

Duboeuf

Bchet

2017

Finite Elements in Analysis and Design

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This 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 (

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“…As model problem, we consider an elliptic equation with the same settings than in the preliminary paper [14].…”

Section: Model Problemmentioning

confidence: 99%

Section: Definition Of the Discrete Function Spacementioning

confidence: 99%

“…The proposed solution deeply uses the terminology introduced in [14]. We briefly recall the main ideas.…”

Section: Terminologymentioning

confidence: 99%

Section: Conditioning Of the Linear Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Duboeuf

Bchet

2017

Finite Elements in Analysis and Design

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Combination of topology optimization and Lie derivative-based shape optimization for electro-mechanical design

Kuci

Henrotte

Duysinx

et al. 2018

Struct Multidisc Optim

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This paper presents a framework for the simultaneous application of shape and topology optimization in electro-mechanical design problems. Whereas the design variables of a shape optimization are the geometrical parameters of the CAD description, the design variables upon which densitybased topology optimization acts represent the presence or absence of material at each point of the region where it is applied. These topology optimization design variables, which are called densities, are by essence substantial quantities. This means that they are attached to matter while, on the other hand, shape optimization implies ongoing changes of the model geometry. An appropriate combination of the two representations is therefore necessary to ensure a consistent design space as the joint shape-topology optimization process unfolds. The optimization problems dealt with in this paper are furthermore constrained to verify the governing partial differential equations (PDEs) of a physical model, possibly nonlinear and discretized by means of, e.g., the finite element method (FEM). Theoretical formulas, based on the Lie derivative, to express the sensitivity of the performance functions of the optimization problem are derived and validated to be used in gradient-based algorithms. The method is applied to the torque ripple minimization in an interior permanent magnet synchronous machine (PMSM).

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A mixed-dimensional CutFEM methodology for the simulation of fibre-reinforced composites

Kerfriden

Claus

Mihai

2020

Adv. Model. and Simul. in Eng. Sci.

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We develop a novel unfitted finite element solver for composite materials with quasi-1D fibrous reinforcements. The method belongs to the class of mixed-dimensional non-conforming finite element solvers. The fibres are treated as 1D structural elements that may intersect the mesh of the embedding structure in an arbitrary manner. No meshing of the unidimensional elements is required. Instead, fibre solution fields are described using the trace of the background mesh. A regularised "cut" finite element formulation is carefully designed to ensure that analyses using such non-conforming finite element descriptions are stable. We also design a dedicated primal/dual operator splitting scheme to resolve the coupling between structure and fibrous reinforcements efficiently. The novel computational strategy is applied to the solution of stiff computational models whereby fibrous reinforcements may lose their bond to the embedding material above a certain level of stress. It is shown that the primal-dual 1D/3D CutFEM scheme is convergent and well-behaved in variety of scenarios involving such highly nonlinear structural computations.

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

Cited by 3 publications

References 59 publications

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

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

Combination of topology optimization and Lie derivative-based shape optimization for electro-mechanical design

A mixed-dimensional CutFEM methodology for the simulation of fibre-reinforced composites

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