Non-negative moment fitting quadrature for cut finite elements and cells undergoing large deformations

Garhuom, Wadhah; Düster, Alexander

doi:10.1007/s00466-022-02203-9

Cited by 22 publications

(19 citation statements)

References 60 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%

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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%

“…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%

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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“…This is particularly important as ductile materials such as steel are commonly used in space-frame construction. Moreover, the computational time associated with the substructure analysis can be reduced by utilizing advanced numerical integration techniques [53,54].…”

Section: Discussionmentioning

confidence: 99%

“…A significant portion of the computational time is spent on numerical integration over the complex domain, and the use of advanced techniques can greatly improve the efficiency of this process, e.g. [53,54]. Note that the process of generating a node design and conducting a subsequent substructure analysis is automated, eliminating the need for manual engineering effort to perform the analysis on the complex node model.…”

Section: Complex Tree Canopy Structurementioning

confidence: 99%

Two-scale analysis and design of spaceframes with complex additive manufactured nodes

Oztoprak¹,

Paolini²,

D’Acunto³

et al. 2023

Preprint

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The advancements in additive manufacturing (AM) technology have allowed for the production of geometrically complex parts with customizable designs. This versatility benefits large-scale spaceframe structures, as the individual design of each structural node can be tailored to meet specific mechanical and other functional requirements. To this end, however, the design and analysis of such space-frames with distinct structural nodes needs to be highly automated. A critical aspect in this context is automated integration of the local 3D features into the 1D large-scale models. In the present work, a two-scale modeling approach is developed to improve the design and linear-elastic analysis of space frames with complex additively manufactured nodes. The mechanical characteristics of the 3D nodes are numerically reduced through an automated dimensional reduction process based on the Finite Cell Method (FCM) and substructuring. The reduced stiffness quantities are assembled in the large-scale 1D model which, in turn, enables efficient structural analysis. The response of the 1D model is passed on to the local model, enabling fully resolved 3D linear-elastic analysis. The proposed approach is numerically verified on a simplified beam example. Furthermore, the workflow is demonstrated on a tree canopy structure with additively manufactured nodes with bolted connections. The form of the large-scale structure is found based on the Combinatorial Equilibrium Modeling framework, and the different designs of the local structural nodes are based on generative exploration of the design space. It is demonstrated that the proposed methodology effectively automates the design and analysis of space-frame structures with complex, distinct structural nodes.

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“…It comes at the cost of complexity of numerical integration. Within this context, the most popular integration methods which are widely used by researchers are space-tree [1,2] (quadtree in 2D and octree in 3D) and moment fitting [3,4] approaches. Although the space-tree methods are very robust, they are computationally expensive due to a high number of integration points required for a certain level of accuracy.…”

Section: Introductionmentioning

confidence: 99%

Accurate integration of trimmed cells based on Bezier approximation

Hosseini

Gorji

Düster

2023

Proc Appl Math and Mech

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In this work, a new adaptive integration method for simulation of two-dimensional linear elasticity problems is presented. The main benefit of the proposed method is the reduction of the computational cost by lowering the number of integration points required to reach a certain level of accuracy. The main concept of the proposed method is to calculate new weights for trimmed cells employing the advantage of Bezier parametric curves. Within this concept, it is possible to map a square to a triangle with one curved edge where any curved edge is approximated by a parametric Bezier curve. In this way, a new set of Gaussian quadrature points is introduced for each trimmed cell in a fast and robust way. Besides main mapping cases, the integration method includes supplementary cases as well to increase the robustness and generality of the method. In the next step, the proposed method is implemented in a two-dimensional fictitious domain code in MATLAB to solve structural problems. The results will be compared to those obtained through the commercial finite element code ABAQUS. It is shown that the proposed method is accurate and robust.

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Non-negative moment fitting quadrature for cut finite elements and cells undergoing large deformations

Cited by 22 publications

References 60 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

Two-scale analysis and design of spaceframes with complex additive manufactured nodes

Accurate integration of trimmed cells based on Bezier approximation

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