Isogeometric Analysis and Shape Optimization in Fluid Mechanics - PowerPoint PPT Presentation

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Isogeometric Analysis and Shape Optimization in Fluid Mechanics Peter Nørtoft DTU Compute Joint work with Jens Gravesen, Allan R. Gersborg, Niels L. Pedersen, Morten Willatzen, Anton Evgrafov, Dang Manh Nguyen, and Tor Dokken Scientific Computing Section Seminar, September 17, 2013

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Goals and Outline The aim is to analyze and optimize flows using isogeometric analysis Shape Optimization drag

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Goals and Outline The aim is to analyze and optimize flows using isogeometric analysis Navier-Stokes Flow Model Shape Optimization drag

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Goals and Outline The aim is to analyze and optimize flows using isogeometric analysis Navier-Stokes Flow Model Shape Optimization drag Flow Acoustics Model +

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Goals and Outline The aim is to analyze and optimize flows using isogeometric analysis Navier-Stokes Flow Model Shape Optimization Isogeometric Analysis � ✒ � drag Flow Acoustics Model +

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Fluid Mechanics: Navier-Stokes Equations Flow problems are governed by a boundary value problem Γ D u velocity p pressure Ω density ρ viscosity µ force f Γ N

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Fluid Mechanics: Navier-Stokes Equations Flow problems are governed by a boundary value problem Γ D u velocity p pressure Ω density ρ viscosity µ force f Γ N 2D steady-state incompressible Navier-Stokes flow ρ ( u · ∇ ) u + ∇ p − µ ∇ 2 u = ρ f in Ω ∇ · u = 0 in Ω u = u ∗ on Γ D ( µ ∇ u i − p e i ) · n = 0 on Γ N

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Fluid Mechanics: Navier-Stokes Equations Flow problems are governed by a boundary value problem Γ D u velocity Re = ρ UL � 10 3 p pressure µ Ω density ρ ◮ viscous fluid viscosity µ ◮ slow flow force f ◮ small scale Γ N [M. Van Dyke] 2D steady-state incompressible Navier-Stokes flow ρ ( u · ∇ ) u + ∇ p − µ ∇ 2 u = ρ f in Ω ∇ · u = 0 in Ω u = u ∗ on Γ D ( µ ∇ u i − p e i ) · n = 0 on Γ N

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Fluid Mechanics: Navier-Stokes Equations Flow problems are governed by a boundary value problem Γ D u velocity Re = ρ UL � 10 3 p pressure µ Ω density ρ ◮ viscous fluid viscosity µ ◮ slow flow force f ◮ small scale Γ N [M. Van Dyke] 2D steady-state incompressible Navier-Stokes flow ρ ( u · ∇ ) u + ∇ p − µ ∇ 2 u = ρ f in Ω ∇ · u = 0 in Ω u = u ∗ on Γ D ( µ ∇ u i − p e i ) · n = 0 on Γ N Challenge: solve this using isogeometric analysis [Bazilevs et al., 2006b; Bazilevs & Hughes, 2008; Akkerman et al., 2010; . . . ]

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method Both geometry, velocity and pressure are parametrized by B-splines Ω

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method Both geometry, velocity and pressure are parametrized by B-splines N g � x i R g X X ( ξ, η ) = i ( ξ, η ) ¯ Ω i =1 [0 , 1] 2 η y ξ x R g Bivariate NURBS i x i control point ¯

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method Both geometry, velocity and pressure are parametrized by B-splines N g � x i R g X X ( ξ, η ) = i ( ξ, η ) ¯ Ω i =1 [0 , 1] 2 N u η y � u ( ξ, η ) = u i P u i ( ξ, η ) ¯ ξ x i , R p i , R g i =1 R u Bivariate NURBS/B-spline i N p u i , ¯ p i , ¯ x i control point/variable ¯ p i P p � p ( ξ, η ) = ¯ i ( ξ, η ) i =1

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method Both geometry, velocity and pressure are parametrized by B-splines N g � x i R g X X ( ξ, η ) = i ( ξ, η ) ¯ Ω i =1 [0 , 1] 2 N u η y � u ( ξ, η ) = u i P u i ( ξ, η ) ¯ ξ x i , R p i , R g i =1 R u Bivariate NURBS/B-spline i N p u i , ¯ p i , ¯ x i control point/variable ¯ p i P p � p ( ξ, η ) = ¯ i ( ξ, η ) i =1 Univariate B-spline: 1 • knot vector 0.8 • polynomial degree 0.6 0.4 0.2 0 0 0.5 1 ξ 1 Parameter domain [0 , 1] 2

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method Both geometry, velocity and pressure are parametrized by B-splines N g � x i R g X X ( ξ, η ) = i ( ξ, η ) ¯ Ω i =1 [0 , 1] 2 N u η y � u ( ξ, η ) = u i P u i ( ξ, η ) ¯ ξ x i , R p i , R g i =1 R u Bivariate NURBS/B-spline i N p u i , ¯ p i , ¯ x i control point/variable ¯ p i P p � p ( ξ, η ) = ¯ i ( ξ, η ) i =1 Univariate B-spline: N i ( ξ 1 ) 1 • knot vector 0.8 • polynomial degree 0.6 0.4 0.2 0 0 0.5 1 ξ 1 Parameter domain [0 , 1] 2

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method Both geometry, velocity and pressure are parametrized by B-splines N g � x i R g X X ( ξ, η ) = i ( ξ, η ) ¯ Ω i =1 [0 , 1] 2 N u η y � u ( ξ, η ) = u i P u i ( ξ, η ) ¯ ξ x i , R p i , R g i =1 R u Bivariate NURBS/B-spline i N p u i , ¯ p i , ¯ x i control point/variable ¯ p i P p � p ( ξ, η ) = ¯ i ( ξ, η ) i =1 Univariate B-spline: N i ( ξ 1 ) 1 • knot vector 0.8 • polynomial degree 0.6 0.4 0.2 0 0 0.5 1 ξ 1 Parameter domain [0 , 1] 2

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method Both geometry, velocity and pressure are parametrized by B-splines N g � x i R g X X ( ξ, η ) = i ( ξ, η ) ¯ Ω i =1 [0 , 1] 2 N u η y � u ( ξ, η ) = u i P u i ( ξ, η ) ¯ ξ x i , R p i , R g i =1 R u Bivariate NURBS/B-spline i N p u i , ¯ p i , ¯ x i control point/variable ¯ p i P p � p ( ξ, η ) = ¯ i ( ξ, η ) i =1 Univariate B-spline: N i ( ξ 1 ) • knot vector 1 • polynomial degree 0.5 0 1 1 0.5 0.5 0 0 ξ 2 ξ 1 Parameter domain [0 , 1] 2

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method Both geometry, velocity and pressure are parametrized by B-splines N g � x i R g X X ( ξ, η ) = i ( ξ, η ) ¯ Ω i =1 [0 , 1] 2 N u η y � u ( ξ, η ) = u i P u i ( ξ, η ) ¯ ξ x i , R p i , R g i =1 R u Bivariate NURBS/B-spline i N p u i , ¯ p i , ¯ x i control point/variable ¯ p i P p � p ( ξ, η ) = ¯ i ( ξ, η ) i =1 Univariate B-spline: N i ( ξ 1 ) , M j ( ξ 2 ) • knot vector 1 • polynomial degree 0.5 0 1 1 0.5 0.5 0 0 ξ 2 ξ 1 Parameter domain [0 , 1] 2

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method Both geometry, velocity and pressure are parametrized by B-splines N g � x i R g X X ( ξ, η ) = i ( ξ, η ) ¯ Ω i =1 [0 , 1] 2 N u η y � u ( ξ, η ) = u i P u i ( ξ, η ) ¯ ξ x i , R p i , R g i =1 R u Bivariate NURBS/B-spline i N p u i , ¯ p i , ¯ x i control point/variable ¯ p i P p � p ( ξ, η ) = ¯ i ( ξ, η ) i =1 Univariate B-spline: N i ( ξ 1 ) , M j ( ξ 2 ) • knot vector 1 • polynomial degree 0.5 0 1 1 0.5 0.5 0 0 ξ 2 ξ 1 Parameter domain [0 , 1] 2

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method Both geometry, velocity and pressure are parametrized by B-splines N g � x i R g X X ( ξ, η ) = i ( ξ, η ) ¯ Ω i =1 [0 , 1] 2 N u η y � u ( ξ, η ) = u i P u i ( ξ, η ) ¯ ξ x i , R p i , R g i =1 R u Bivariate NURBS/B-spline i N p u i , ¯ p i , ¯ x i control point/variable ¯ p i P p � p ( ξ, η ) = ¯ i ( ξ, η ) i =1 Univariate B-spline: N i ( ξ 1 ) , M j ( ξ 2 ) • knot vector • polynomial degree Bivariate Tensor Product B-spline: • 2 knot vectors • 2 polynomial degrees • P i,j ( ξ 1 , ξ 2 ) = N i ( ξ 1 ) M j ( ξ 2 ) Parameter domain [0 , 1] 2

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method Both geometry, velocity and pressure are parametrized by B-splines N g � x i R g X X ( ξ, η ) = i ( ξ, η ) ¯ Ω i =1 [0 , 1] 2 N u η y � u ( ξ, η ) = u i P u i ( ξ, η ) ¯ ξ x i , R p i , R g i =1 R u Bivariate NURBS/B-spline i N p u i , ¯ p i , ¯ x i control point/variable ¯ p i P p � p ( ξ, η ) = ¯ i ( ξ, η ) i =1 Univariate B-spline: N i ( ξ 1 ) , M j ( ξ 2 ) • knot vector • polynomial degree Bivariate Tensor Product B-spline: • 2 knot vectors • 2 polynomial degrees • P i,j ( ξ 1 , ξ 2 ) = N i ( ξ 1 ) M j ( ξ 2 ) Parameter domain [0 , 1] 2 Physical domain Ω