A stable and optimally convergent LaTIn‐CutFEM algorithm for multiple unilateral contact problems

Claus, Susanne; Kerfriden, Pierre

doi:10.1002/nme.5694

Cited by 33 publications

(46 citation statements)

References 45 publications

(115 reference statements)

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“…Of key importance to CutFEM methods [7,12,18] is the introduction of regularisation terms to ensure that system matrices remain well-conditioned, notably by applying some form of ghost-penalty [6] stabilisation. To this end, we define the set of intersected element edges by…”

Section: Cut Finite Element Formulation Finite Element Spacesmentioning

confidence: 99%

“…L is a non-dimensionalising parameter that represents the width of the matrix region that reacts to the deformations of a particular fibre. As shown in [12], the first term helps control the spurious oscillations that may appear in the Lagrange multiplier field. The second term eliminates zero energy modes.…”

Section: Stabilisationmentioning

confidence: 99%

“…The second term eliminates zero energy modes. We showed in [12] that this type of stabilised primal/dual formulation could be interpreted as an alternative formulation for the stabilised augmented Lagrangian proposed in [8], in the spirit of the classical approach for unilateral contact problems first introduced in [2].…”

Section: Stabilisationmentioning

confidence: 99%

“…• Set F In our example, we use the quadrature point corresponding to the integral of the product of two piecewise linear fields over f . This introduces a quadrature error, whose effect was studied and discussed in [12].…”

Section: "Inf-sup-regularised" Local Stage: Algorithmic Solutionmentioning

confidence: 99%

“…Secondly, we propose an efficient primal-dual numerical algorithm based on the LaTIn algorithm [3,21,22] to solve the 1D/3D coupled problem iteratively. This primal-dual formulation is a natural extension of our previous work on primal-dual CutFEM technologies for unilateral contact problems [12] (see [1,8,19,26] for closely related pieces of work from other research groups), and is relatively new in the context of CutFEM approaches, where consistent penalty formulations are usually chosen as a first step to the development of a coupling strategy [7,11,13]. Finally, CutFEM formulations need to be carefully regularised for system matrices to be well conditioned and, in the context of primal-dual algorithms, for inf-sup stability conditions to be satisfied.…”

Section: Introductionmentioning

confidence: 98%

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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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Section: Cut Finite Element Formulation Finite Element Spacesmentioning

confidence: 99%

Section: Stabilisationmentioning

confidence: 99%

Section: Stabilisationmentioning

confidence: 99%

Section: "Inf-sup-regularised" Local Stage: Algorithmic Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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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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ϕ‐FEM: An Efficient Simulation Tool Using Simple Meshes for Problems in Structure Mechanics and Heat Transfer

Cotin,

Duprez,

Lleras

et al. 2023

Partition of Unity Methods

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One of the major issues in the computational mechanics is to take into account the geometrical complexity. To overcome this difficulty and to avoid the expensive mesh generation, geometrically unfitted methods, i.e. the numerical methods using the simple computational meshes that do not fit the boundary of the domain, and/or the internal interfaces, have been widely developed. In the present work, we investigate the performances of an unfitted method called φ-FEM that converges optimally and uses classical finite element spaces so that it can be easily implemented using general FEM libraries. The main idea is to take into account the geometry thanks to a level set function describing the boundary or the interface. Up to now, the φ-FEM approach has been proposed, tested and substantiated mathematically only in some simplest settings: Poisson equation with Dirichlet/Neumann/Robin boundary conditions. Our goal here is to demonstrate its applicability to some more sophisticated governing equations arising in the computational mechanics. We consider the linear elasticity equations accompanied by either pure Dirichlet boundary conditions or by the mixed ones (Dirichlet and Neumann boundary conditios co-existing on parts of the boundary), an interface problem (linear elasticity with material coefficients abruptly changing over an internal interface), a model of elastic structures with cracks, and finally the heat equation. In all these settings, we derive an appropriate variant of φ-FEM and then illustrate it by numerical tests on manufactured solutions. We also compare the accuracy and efficiency of φ-FEM with those of the standard fitted FEM on the meshes of similar size, revealing the substantial gains that can be achieved by φ-FEM in both the accuracy and the computational time.

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PGD in thermal transient problems with a moving heat source: A sensitivity study on factors affecting accuracy and efficiency

Strobl,

Unger,

Ghnatios

et al. 2024

Engineering Reports

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Thermal transient problems, essential for modeling applications like welding and additive metal manufacturing, are characterized by a dynamic evolution of temperature. Accurately simulating these phenomena is often computationally expensive, thus limiting their applications, for example for model parameter estimation or online process control. Model order reduction, a solution to preserve the accuracy while reducing the computation time, is explored. This article addresses challenges in developing reduced order models using the proper generalized decomposition (PGD) for transient thermal problems with a specific treatment of the moving heat source within the reduced model. Factors affecting accuracy, convergence, and computational cost, such as discretization methods (finite element and finite difference), a dimensionless formulation, the size of the heat source, and the inclusion of material parameters as additional PGD variables are examined across progressively complex examples. The results demonstrate the influence of these factors on the PGD model's performance and emphasize the importance of their consideration when implementing such models. For thermal example, it is demonstrated that a PGD model with a finite difference discretization in time, a dimensionless representation, a mapping for a moving heat source, and a spatial domain non‐separation yields the best approximation to the full order model.

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A stable and optimally convergent LaTIn‐CutFEM algorithm for multiple unilateral contact problems

Cited by 33 publications

References 45 publications

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

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

ϕ‐FEM: An Efficient Simulation Tool Using Simple Meshes for Problems in Structure Mechanics and Heat Transfer

PGD in thermal transient problems with a moving heat source: A sensitivity study on factors affecting accuracy and efficiency

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