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 12 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

Self Cite

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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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Boolean finite cell method for multi-material problems including local enrichment of the Ansatz space

et al. 2023

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The Finite Cell Method (FCM) allows for an efficient and accurate simulation of complex geometries by utilizing an unfitted discretization based on rectangular elements equipped with higher-order shape functions. Since the mesh is not aligned to the geometric features, cut elements arise that are intersected by domain boundaries or internal material interfaces. Hence, for an accurate simulation of multi-material problems, several challenges have to be solved to handle cut elements. On the one hand, special integration schemes have to be used for computing the discontinuous integrands and on the other hand, the weak discontinuity of the displacement field along the material interfaces has to be captured accurately. While for the first issue, a space-tree decomposition is often employed, the latter issue can be solved by utilizing a local enrichment approach, adopted from the extended finite element method. In our contribution, a novel integration scheme for multi-material problems is introduced that, based on the B-FCM formulation for porous media, originally proposed by Abedian and Düster (Comput Mech 59(5):877–886, 2017), extends the standard space-tree decomposition by Boolean operations yielding a significantly reduced computational effort. The proposed multi-material B-FCM approach is combined with the local enrichment technique and tested for several problems involving material interfaces in 2D and 3D. The results show that the number of integration points and the computational time can be reduced by a significant amount, while maintaining the same accuracy as the standard FCM.

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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 12 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

Boolean finite cell method for multi-material problems including local enrichment of the Ansatz space

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