IMAM Institute of Mechanics and Advanced Materials Sensitivity Analysis and Shape Optimisation with Isogeometric Boundary Element Method Haojie Lian, Robert Simpson, Stéphane P.A. Bordas Institute of Mechanics and Advanced materials, Theoretical and Computational mechanics, Cardiff university, Cardiff, CF24 3AA, Wales, UK 1

Outline Isogeometric Analysis (IGA) Isogeometric Boundary Element Method (IGABEM) Sensitivity analysis and shape optimisation with IGABEM Numerical examples

Why isogeometric analysis Key idea 1 The key idea of isogeometric analysis (IGA) ( Hughes et al. 2005 ) is to approximate the unknown fields with the same basis functions (NURBS, T-splines … ) as that used to generate the CAD model. Reduce the time No creation of analysis-suitable geometry; Without the need of mesh generation. Exact representation of geometry Suitable for the problems which are sensitive to geometric imperfections. High order continuous field More flexible hpk-refinement. 3

NURBS curve 1. Knot vector: a non-decreasing set of coordinates in the parametric space. Where n is the number of basis functions, i is the knot index and p is the curve degree, 2. Control points: 3. NURBS basis function:

Properties of NURBS basis functions Partition of Unity Non-negative p-1 continuous derivatives if no knot repeated No Kronecker delta property Tensor product property Surface:

Challenges Surface representation Domain representation domain parameterization,

IGABEM Key idea 2: Isogeometric Boundary Element Method (IGABEM) (Simpson, et al. 2011). The NURBS basis functions are used to discretise Boundary Integral Equation (BIE). Recently this work is extended to incorporate analysis-suitable T-splines. Reasons: 1. Representation of boundaries; 2. Easy to represent complex geometries.

IGABEM formulation Regularised form of boundary integral equation for 2D linear elasticity where and are field point and source point respectively, and are displacement and traction around the boundary, and are fundamental solutions. The geometry is discretised by: The field is discretised by:

IGABEM formulation In Parametric space Integration in parent element Matrix equation

Special techniques for IGABEM 1. Collocation point (Greville abscissae) 2. Boundary condition Collocate on the prescribed boundary 3. Integration High order Gauss integration

Nuclear reactor

Dam

Propeller

IGABEM shape optimisation The advantages of IGABEM for shape optimisation. More efficient An interaction with CAD: 1. Analysis can read the CAD data directly without any preprocessing. 2. Analysis can return the data to CAD without any postprocessing. More accurate Design velocity field is exactly obtained for gradient-based shape optimisation.

IGABEM sensitivity analysis Governing equations in parametric space, which can be viewed as material coordinate system Differentiate the equation w.r.t. design variables (implicit differentiation) Discretise the derivatives of displacement and traction using NURBS basis Finally

Sensitivity Propagation h-refinement algorithm is also suitable for shape derivatives refinement, but need to convert NURBS in to B-splines in

Pressure cylinder problem Design variable is large radius b

Infinite plate with a hole Design variable is radius R

Cantilever Beam Design curve is AB Minimise the area without violating von Mises stress criterion

Fillet Design curve is ED Minimise the area without violating von Mises stress criterion

Conclusions Conclusions Future work: An isogeometric boundary element method (IGABEM) has been introduced. IGABEM can suppress the mesh burden and interact with CAD. IGABEM has been applied to gradient-based shape optimisation. IGABEM is more efficient and accurate for analysis and optimisation Future work: Topology optimisation with IGABEM Easy to handle topology optimisation compared to IGAFEM Easy to implement with the help of topology derivatives T-spline based IGABEM for 3D shape optimisation Local refinement Analysis suitable and flexible to construct the complex geometry 21

References TJR Hughes, JA Cottrell, and Y Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194(39-41):4135-4195, 2005. T Greville. Numerical procedures for interpolation by spline functions. Journal of the society for Industrial and Applied mathematics: Series B, Numerical Analysis, 1964. R Johnson. Higher order B-spline collocation at the Greville abscissae. Applied Numerical Mathematics, 52:63-75, 2005. Les Piegl and W Tiller. The NURBS book. Springer, 1995. RN Simpson, SPA Bordas, J Trevelyan and T Rabczuk. An Isogeometric boundary element method for elastostatic analysis. Computer Methods in Applied Mechanics and Engineering. 209-212 (2012) 87–100.