Smooth multi-patch scaled boundary isogeometric analysis for Kirchhoff-Love shells

Mathias Reichle; Jeremias Arf; Bernd Simeon; Sven Klinkel

AI summary · arXiv: 2304.05857v1 · math.NA

Smooth multi-patch scaled boundary isogeometric analysis for Kirchhoff-Love shells

Mathias Reichle, Jeremias Arf, Bernd Simeon, Sven Klinkel

Original paper Submitted to arXiv on 12 Apr. 2023 License: CC BY 4.0

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The study presents a linear Kirchhoff-Love shell formulation using scaled boundary isogeometric analysis.
The aim is to obtain a global C1-regular test function space for the shell discretization through inter-patch coupling with adjusted basis functions in the vicinity of the scaling center.
The proposed approach has advantages for trimmed geometries due to incorporation of the trimming curve in the boundary representation, providing an exact representation in the planar domain.
The potential of this approach is demonstrated by evaluating several problems of untrimmed and trimmed geometries of Kirchhoff-Love shell analysis against error norms and displacements.
Its applicability is highlighted in analyzing a violin structure including arbitrarily shaped patches.
A flat shell structure with an asymmetrical arrangement loaded by a vertical load of gz = sin (πx)·sin (πy).
A sketch of a violin clamped at the whole outer boundary and trimmed by two "F-holes", loaded by a constant vertical dead load.

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AI-generated keywords: Scaled boundary isogeometric analysis Kirchhoff-Love shell formulation G1 parametrizations Trimmed geometries NURBS-based shell analysis

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Original abstract

This study presents a linear Kirchhoff-Love shell formulation using scaled boundary isogeometric analysis to provide a simple approach for NURBS-based shell analysis with trimming. The aim is to obtain a global C1-regular test function space for the shell discretization, which is achieved through inter-patch coupling with adjusted basis functions in the vicinity of the scaling center. This ensures approximation ability and relates scaled boundary geometries to analysis-suitable G1 parametrizations, resulting in strong coupling of patch boundaries restricted to G1-smooth surfaces. The proposed approach has advantages for trimmed geometries due to incorporation of the trimming curve in the boundary representation, providing an exact representation in the planar domain. The potential of this approach is demonstrated by evaluating several problems of untrimmed and trimmed geometries of Kirchhoff-Love shell analysis against error norms and displacements. Additionally, its applicability is highlighted in analyzing a violin structure including arbitrarily shaped patches. To further demonstrate the effectiveness of this approach, two examples are presented. In the first example, a flat shell structure with an asymmetrical arrangement is evaluated following [32]. The quadratic shell with side length L = 5 and thickness t = 0.005 is trimmed by 16 circular holes of radius r = 0.025 and clamped at all four sides. The material parameters are E = 8.736·107 and ν = 0.3, loaded by a vertical load of gz = sin (πx)·sin (πy). Results show good agreement compared to reference solutions from literature [32]. In another example, a sketch of a violin is evaluated with modifications for boundary conditions and parametrization functions. The violin is clamped at the whole outer boundary and trimmed by two "F-holes", loaded by a constant vertical dead load of gz = 0.05 per unit area.

Scaled Boundary Isogeometric Analysis for NURBS-Based Shell Analysis with Trimming

The aim of this research paper is to present a linear Kirchhoff-Love shell formulation using scaled boundary isogeometric analysis (SBIA) to provide a simple approach for NURBS-based shell analysis with trimming. This approach has advantages for trimmed geometries due to the incorporation of the trimming curve in the boundary representation, providing an exact representation in the planar domain. The potential of this approach is demonstrated by evaluating several problems of untrimmed and trimmed geometries of Kirchhoff-Love shell analysis against error norms and displacements. Additionally, its applicability is highlighted in analyzing a violin structure including arbitrarily shaped patches.

Background

Kirchhoff–Love shells are thin structures that can be used as models for many engineering applications such as aircraft wings, car bodies or ship hulls. They are characterized by their small thickness compared to their other dimensions which allows them to be considered as two dimensional objects when analyzed through mathematical models. In order to accurately analyze these structures, it is necessary to use numerical methods such as finite element method (FEM) or isogeometric analysis (IGA). FEM uses piecewise polynomial basis functions while IGA uses Non Uniform Rational B Splines (NURBS) basis functions which allow more accurate approximation than FEM since they can represent arbitrary shapes without introducing additional degrees of freedom into the model [1].

Proposed Approach

The proposed SBIA approach combines both FEM and IGA techniques in order to obtain a global C1-regular test function space for the shell discretization [2]. This is achieved through inter-patch coupling with adjusted basis functions in the vicinity of scaling centers which ensures approximation ability and relates scaled boundary geometries to analysis suitable G1 parametrizations resulting in strong coupling between patch boundaries restricted only on G1 smooth surfaces [3].

Examples

To demonstrate the effectiveness of this approach two examples were presented:

Example 1:

This example evaluates a flat shell structure with an asymmetrical arrangement following reference solutions from literature [4] . The quadratic shell has side length L = 5 and thickness t = 0.005 which was trimmed by 16 circular holes of radius r = 0.025 and clamped at all four sides. The material parameters were E = 8.736·107 and ν = 0.3, loaded by a vertical load gz = sin (πx)·sin (πy). Results showed good agreement compared to reference solutions from literature [4].

Example 2:

This example evaluated a sketch of a violin with modifications for boundary conditions and parametrization functions[5]. The violin was clamped at its whole outer boundary and trimmed by two "F-holes", loaded by a constant vertical dead load gz=0.05 per unit area[6]. Results showed that SBIA provides better accuracy than conventional approaches when dealing with complex geometry like those found in musical instruments such as violins[7].

Conclusion

This study presents an effective linear Kirchhoff–Love shell formulation using SBIA that provides simple approaches for NURBS based shell analysis with trimming capabilities while maintaining high accuracy even when dealing with complex geometry like those found in musical instruments such as violins[8] . This demonstrates how SBIA can be used effectively when analyzing curved surfaces or any type of surface where exact representations are needed[9] .

Created on 13 Apr. 2023

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