Octree-based integration scheme with merged sub-cells for the finite cell method: Application to non-linear problems in 3D

Petö, Márton; Garhuom, Wadhah; Duvigneau, Fabian; Eisenträger, Sascha; Düster, Alexander; Juhre, Daniel

doi:10.1016/j.cma.2022.115565

Cited by 16 publications

(11 citation statements)

References 74 publications

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“…Note that the simulation is conducted in Cartesian coordinates, thus, when applying the prescribed displacements and body loads, an appropriate transformation of the coordinates is required. For the numerical integration of Equation ( 14) in the cut cells (red cells in Figure 3A), several approaches could be applied that were shown to yield reliable results even for non-linear analysis [2,3]. In this example, however, due to the cylindrical shape of the void region, we use the smart octree numerical integration scheme based on integration sub-cells, that are perfectly aligned to the curved boundary via blending functions [8].…”

Section: Resultsmentioning

confidence: 99%

“…Here, the framework of the MoMS allows for an arbitrary choice for 𝑢 r . For the current manufactured solution, we define 𝑢 r = Λ 𝑗 (𝑟 − 𝑅) 2 ⏟⎴⏟⎴⏟ ûr ,…”

Section: Problem Statementmentioning

confidence: 99%

“…F I G U R E 1 Simulation of large deformations using the finite cell method (reproduced from Ref. [2]).…”

Section: Introductionmentioning

confidence: 99%

“…The cut cells pose several challenges when it comes to accurate numerical integration, enforcement of boundary conditions, and conditioning of the system. These issues become even more significant in non-linear analyses, where stability and computation time play a crucial role [2][3][4].…”

Section: Introductionmentioning

confidence: 99%

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Code verification of non‐linear immersed boundary simulations using the method of manufactured solutions

Petö,

Juhre,

Eisenträger

2023

Proc Appl Math and Mech

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Non‐standard finite element technologies, such as immersed boundary approaches, are typically based on novel algorithms and advanced methods, which require reliable testing of the implemented code. For this purpose, the method of manufactured solutions (MoMS) offers a great framework, enabling an easy and straightforward derivation of closed‐form reference solutions. In this contribution, the focus is kept on non‐linear analysis via the finite cell method (FCM), which is typically based on an unfitted geometry discretization and higher‐order shape functions. The code verification via MoMS generally requires the application of boundary conditions to all boundaries of the simulation domain, which need to be enforced in a weak sense on the immersed boundaries. To avoid this, we propose a novel way of deriving manufactured solutions, for which the necessary constraints on the embedded boundaries are directly fulfilled. Thus, weak boundary conditions can be eliminated from the FCM simulation, and the simulation complexity is reduced when testing other relevant features of the immersed code. In particular, we focus on finite strain analysis of 3D structures with a Neo‐Hookean material model, and show that the proposed technique enables a reliable code verification approach for all load steps throughout the deformation process.

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Section: Resultsmentioning

confidence: 99%

“…Here, the framework of the MoMS allows for an arbitrary choice for 𝑢 r . For the current manufactured solution, we define 𝑢 r = Λ 𝑗 (𝑟 − 𝑅) 2 ⏟⎴⏟⎴⏟ ûr ,…”

Section: Problem Statementmentioning

confidence: 99%

“…F I G U R E 1 Simulation of large deformations using the finite cell method (reproduced from Ref. [2]).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Code verification of non‐linear immersed boundary simulations using the method of manufactured solutions

Petö,

Juhre,

Eisenträger

2023

Proc Appl Math and Mech

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“…A myriad of techniques to enhance octree-subdivision has been developed [27]. A few prominent techniques include error-estimate-based adaptive octreesubdivision [28], moment-fitting [29], equivalent polynomial methods [30], and the merged sub-cell technique [31].…”

Section: Integration Of Cut Elementsmentioning

confidence: 99%

Critical time-step size analysis and mass scaling by ghost-penalty for immersogeometric explicit dynamics

Stoter¹,

Divi²,

Brummelen³

et al. 2023

Preprint

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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 timestep size.

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Remeshing in the finite cell method for different types of geometry descriptions

Sartorti,

Düster

2024

Proc Appl Math and Mech

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The numerical structural analysis of problems with complex geometries can be challenging, especially if standard finite elements are used. In contrast, immersed methods, such as the finite cell method, relieve the mesh generation such that simply shaped elements/cells can be used. Then, the domain boundary is considered during the numerical integration. In finite strain analysis, the elements/cells face large distortions, especially the cells that are intersected by the boundary. When the solution fails, remeshing can be applied to continue the simulation. This process is currently limited to geometries described by a triangulated surface. Therefore, the present work shows an alternative way of describing the deformed geometry by interpolating the displacement field. In this work, the inverse distance approach, and radial basis functions (RBF) with and without a constant extension are applied. It turns out that RBF with constant extension leads to the most robust results compared to the other methods. Moreover, different geometry description types are tested, and the present approach leads to promising results.

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Octree-based integration scheme with merged sub-cells for the finite cell method: Application to non-linear problems in 3D

Cited by 16 publications

References 74 publications

Code verification of non‐linear immersed boundary simulations using the method of manufactured solutions

Code verification of non‐linear immersed boundary simulations using the method of manufactured solutions

Critical time-step size analysis and mass scaling by ghost-penalty for immersogeometric explicit dynamics

Remeshing in the finite cell method for different types of geometry descriptions

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