zhaowei_LIU_acme_2016
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This document proposes using subdivision surfaces within an isogeometric boundary element method for Helmholtz analysis. Subdivision surfaces can overcome limitations of NURBS used in CAD and allow local refinement. The boundary element method requires only boundary discretization. An example problem models acoustic scattering from a subdivision surface sphere mesh, achieving higher accuracy than a Lagrangian discretization of the same order. Future work includes coupling with finite elements for structural-acoustic analysis and electromagnetic scattering problems.
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zhaowei_LIU_acme_2016
- 1. An Isogeometric Boundary Element Method with Subdivision Surfaces for Helmholtz Analysis Z. Liu1, R.N. Simpson1, F. Cirak2 1University of Glasgow 2University of Cambridge z.liu.2@research.gla.ac.uk
- 2. Outline: 1. Motivation 2. Subdivision surfaces 3. Boundary Element Method for Helmholtz analysis 1. Numerical example 2. Conclusion and future work Outline
- 3. Motivations Isogeometric Analysis (IGA): Computer Aided Design Computer Aided Engineering Directly use Benefits: • Eliminate data conversion process between design and analysis • Exact geometry used throughout
- 4. Motivations IGA has distinct advantages when applied to the Boundary Element Method: • Surface and boundary-representation approach is popular in CAD industries • Finite Element Method which need volume discretisation • The Boundary Element Method only requires boundary discretisation
- 5. Motivation Present CAD representing approach used in IGA Aluminium can deforming (modelled by NURBS surfaces) • Non-Uniform Rational Basis Spline (NURBS) is the most commonly used CAD discretisation • NURBS has limitation due to its tensor- product nature Extraordinary points when linking two NURBS patches
- 6. Motivation Subdivision surfaces: smooth limit surface and control cage Why use subdivision surfaces? • Subdivision surface can be an alternative CAD discretisation to overcome the limitation of NURBS • Local refinement is allowed when using subdivision surfaces • a subdivision surface mesh can handle complex arbitrary topology An object modelled by T-splines
- 7. Subdivision Surfaces Subdivision Surface • Commonly used in animation and gaming industry • Piecewise linear polygon mesh • Limit surface calculated from the coarse mesh • High-order basis functions Successive levels of Catmull-Clark subdivision refinement applied to an initial cube control mesh
- 8. Subdivision Surfaces An element defined over a regular patch with 12 control points Loop Scheme
- 9. Helmholtz Problem For time-harmonic acoustic problems, the governing equation is the Helmholtz equation which is expressed as: in the case of 3D acoustic problems the fundamental solution is given by: The corresponding normal derivative of this kernel can be expressed as:
- 10. Helmholtz Problem A boundary integral equation that relates acoustic pressure and its normal derivative is formulated as: For computer implementation purposes, the boundary is discretised and the acoustic pressure and normal derivative are discretised through an appropriate set of basis functions as:
- 11. Example Problem
- 12. Example Problem Subdivision surface control mesh Lagrangian discretisation of a sphere
- 13. Model type Lagrangian discretisation Subdivision surfaces Subdivision surfaces Number of degrees of freedom 1152 438 1746 Number of elements 281 872 3488 Order Quartic Quartic Quartic
- 14. Results
- 15. Conclusion and Future works Higher accuracies are obtained over a Lagrangian discretisation of the same order. Future work: • Coupling the Boundary Element Method and the Finite Element Method shell formulations for structural-acoustic analysis • Simulating electromagnetic scattering problems by developing suitable div- and curl-conforming discretisations
Editor's Notes
- Good morning, I am zhaowei liu. I am a phd student in university of glasgow. The research topic I will share with you today is : an isogeometric BEM with subdivision surfaces for helmholtz analysis.
- My presentation will firstly introduce the motivation of our research and then I will talk about basis theories of subdivision surfaces and the acoustic boundary element method. After that, a numerical example will be used to show the results of the code. Finally, I will conclude the research so far and introduce some prospective directions of my research.
- IGA, isogeometric analysis, has rapidly expanded in recent years in order to link computer aided design and numerical methods. Dealing with an engineering problem always involves two stages, which are design stage and analysis stage. Current industrial practice uses a separate data model for each stage. but It requires a costly data conversion process between the two stages. The central idea of IGA is we use the same discretisation model for both design and analysis, which eliminate this problem. Moreover, IGA use the exact geometry throughout the whole process which means there are no geometry error even we use a coarse mesh.
- My project aims to apply iga on the boundary element method. The IGA is predominately focused on the use of the finite element method. However, finite element method need to discretise the volume of analysis object into 3d elements. but the common cad tools prefer a surface or boundary-representation approach to model the geometry which is inefficient for volumetric analysis. Boundary element method is a kind of numerical method only requires the boundary discretisation of the geometry. So we think the boundary element method has distinct advantages over other numerical method in the research of IGA.
- After we chose the numerical method, we should think about what kind of computer aided design tool we should use to represent the geometry. Early works are focused on non-uniform rational basis splines (NURBS) due to it is the most commonly used CAD tools in the industry. However, NURBS has limitations because of it tensor-product nature which means the mesh must be evaluated from a rectangular grid of control points. Subsequently, nurbs surface can only represent limited surface topologies. To overcome this limitation, T-spline was developed and also used in IGA, which enable local refinements of the mesh.
- However, in my project, subdivision surfaces are used for modeling the object. Subdivision surface is an alternative method which can also overcome the tensor-product nature of NURBS. Local refinements can be applied on a subdivision surface mesh and it can handle very complex arbitrary topology.
- Subdivision surface is a geometry representing method commonly used in animation and gaming industry, but it has not penetrated the CAD software market yet. There exist a variety of subdivision schemes, but all are based on the idea of generating a smooth surface from a coarse polygon mesh. Subdivision refinement schemes construct a smooth surface through a limiting procedure of repeated refinement starting from an initial control mesh. Which means, the surface is smooth even when the control mesh is coarse. Subdivision refinement schemes can be classed as either an interpolating or approximation scheme. Based on the element type, the triangular elements mesh is called loop scheme and quadrilateral elements mesh is Catmull-Clark scheme.
- Another benefit of the subdivision surface is it use high-order basis functions with a limit number of degree of freedom, which is very efficient for analysis. The figure shows an subdivision surface element defined over a regular patch. If we want to evaluate a point within the element. We need the basis functions corresponding to these 12 points and the basis functions are all quartic. If we need to evaluate another element, there will be another patch defined with 12 control points. The patches will overlap each other. This make sure that the subdivision surface gains higher accuracy per degree of freedom in the analysis.
- This figure shows an plane wave problem I used to validate my program. A plane wave hit a 3d sphere along the x direction and we evaluate the acoustic pressure of the sample points.
- This figure shows that the acoustic potential of the sample points from theta equals to 0 to 2 pi. I zoom in the place where largest error occurred. We can find that the subdivision surface can gain higher accuracy with less degree of freedom than Lagrange quartic mesh.