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The concept of isogeometric analysis, whereby the parametric functions that are used to describe CAD geometry are also used to approximate the unknown fields in a numerical discretisation, has progressed rapidly in recent years. This paper advances the field further by outlining an isogeometric Boundary Element Method (IGABEM) that only requires a representation of the geometry of the domain for analysis, fitting neatly with the boundary representation provided completely by CAD. The method circumvents the requirement to generate a boundary mesh representing a significant step in reducing the gap between engineering design and analysis. The current paper focuses on implementation details of 2D IGABEM for elastostatic analysis with particular attention paid towards the differences over conventional boundary element implementations. Examples of Matlab R code are given whenever possible to aid understanding of the techniques used.

SUMMARYThe present work addresses shape sensitivity analysis and optimization in two-dimensional elasticity with a regularised isogeometric boundary element method (IGABEM). NURBS are used both for the geometry and the basis functions to discretize the regularised boundary integral equations. With the advantage of tight integration of design and analysis, the application of IGABEM in shape optimziation reduces the mesh generation/regeneration burden greatly. The work is distinct from the previous literatures in IGABEM shape optimization mainly in two aspects: 1) the structural and sensitivity analysis takes advantage of the regularized form of the boundary integral equations, eliminating completely the need of evaluating strongly singular integrals and jump terms and their shape derivatives, which were the main implementation difficulty in IGABEM, and 2) although based on the same CAD model, the mesh for structural and shape sensitivity analysis is separated from the geometrical design mesh, thus achieving a balance between less design variables for efficiency and refined mesh for accuracy. This technique was initially used in isogeometric finite element method and was incorporated into the present IGABEM implementation.

We develop a T-spline isogeometric boundary element method (IGABEM) [1, 2, 3] to shape sensitivity analysis and gradient-based shape optimization in three dimensional linear elasticity. Contrary to finite element based isogeometric analysis (IGA) approaches, no parametrization of the volume is required. Hence, the iterative optimization algorithm can be implemented directly from CAD without any mesh generation nor postprocessing step for returning the resulting structure to CAD designers. T-splines also guarantee a watertight geometry without the manual geometrical-repair work as with non-uniform rational B-splines (NURBS). We demonstrate the worth of the method by analysing problems with and without analytical solutions, including engineering examples involving complex shapes. Additionally, we provide all the derivations of the required sensitivities and the details pertaining to the geometries examined in the benchmarking, to provide helpful reference problems for 3D shape optimization.

The boundary element method (BEM) is a powerful tool in computational acoustics, because the analysis is conducted only on structural surfaces, compared to the finite element method (FEM) which resorts to special techniques to truncate infinite domains. The isogeometric boundary element method (IGABEM) is a recent progress in the category of boundary element approaches, which is inspired by the concept of isogeometric analysis (IGA) and employs the spline functions of CAD as basis functions to discretize unknown physical fields. As a boundary representation approach, IGABEM is naturally compatible with CAD and thus can directly perform numerical analysis on CAD models, avoiding the cumbersome meshing procedure in conventional FEM/BEM and eliminating the difficulty of volume parameterization in isogeometric finite element methods. The advantage of tight integration of CAD and numerical analysis in IGABEM renders it particularly attractive in the application of structural shape optimization because (1) the geometry and the analysis can be interacted, (2) remeshing with shape morphing can be avoided, and (3) an optimized solution returns a CAD