Parametric Surfaces with Volume of Solid Control for Optimisation of Three Dimensional Aerodynamic Topologies

Abstract

This paper presents a topologically flexible parameterisation method suitable for the optimisation of 3 dimensional aerodynamics using traditional boundary fitted meshes. This parameterisation extends the restricted-snake volume of solid (RSVS) parameterisation previously developed by the authors. This is achieved by extending restricted snakes, a parametric active contour method, into a restricted surface capable of efficiently evolving arbitrary complex topologies handling collisions with no floating point arithmetic. This is integrated with a surface generation rule which allows smooth shapes with intuitive parameterisation of topology. The 3D-RSVS is presented along with results displaying its smooth behaviour, it's ability to produce shapes of aerodynamic interest and it's topological flexibility. A hierarchical approach to the volume of solid design variables is presented for the RSVS, allowing automatic design of the grids on which VOS are specified. An optimisation framework is proposed and some example 2 dimensional optimisation results are presented.

Gradient-Limiting Shape Control for Efficient Aerodynamic Optimization

“…B-spline surfaces [20] Bezier surfaces [21] Volume splines: Bezier [22], B-spline [23], NURBS [3], Radial basis function [24] Hicks-Henne [25] CSTs [26] Fictitious loads [27] Singular value decomposition [28] Partial differential equation [29] Volume of solid active contour [30,31] Multilevel optimization Successive optimizations are performed where shape control is refined after each stage. Subdivision [32,33] B-spline/Bezier knot insertion [34] Radial basis function [35] Sensitivity filtering Mesh vertices are used for shape control, and surface sensitivities are filtered to remove high-frequency components.…”

“…For example, the partial differential equation (PDE) approach by Bloor and Wilson [29] solves a fourth-order PDE where the boundary conditions are defined by the design variables. More recently, the restricted snake volume of solid method of Payot et al [30,31] defines shapes as the minimal surface enclosing specified volume fractions on a background grid. In this approach the volume fractions of the background grid cells are the design variables that, in combination with the restricted snakes active contour method, are able to produce shapes of arbitrary topology.…”

Gradient-Limiting Shape Control for Efficient Aerodynamic Optimisation

“…Continuous shape functions are used to describe geometry such that the number of design variables is much less than the number of mesh nodes B-Spline surfaces [20] Bezier surfaces [21] Volume splines: Bezier [22], B-Spline [23], NURBS [3], RBF[24] Hicks-Henne [25] CSTs [26] Fictitious loads [27] SVD [28] Partial differential equation [29] Volume of solid active contour [30,31] Multilevel optimisation Successive optimisations are performed where shape control is refined after each stage Subdivision [32,33] B-Spline/Bezier knot insertion [34] RBF [35] Sensitivity filtering Mesh vertices are used for shape control and surface sensitivities are filtered to remove high frequency components…”

“…For example, the partial differential equation (PDE) approach by Bloor and Wilson [29] solves a fourth-order PDE where the boundary conditions are defined by the design variables. More recently, the restricted snake volume of solid (RSVS) method of Payot et al [30,31] defines shapes as the minimal surface enclosing specified volume fractions on a background grid. In this approach the volume fractions of the background grid cells are the design variables which, in combination with the restricted snakes active contour method, are able to produce shapes of arbitrary topology.…”