Kiendl - Isogeometric Analysis and Shape Optimal Design of Shell Structures

7/18/2019 Kiendl - Isogeometric Analysis and Shape Optimal Design of Shell Structures

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Technische Universität München


Isogeometric Analysis and Shape Optimal Design

of Shell Structures

Josef Kiendl





Outline

- NURBS and isogeometric analysis

- The isogeometric Kirchhoff-Love shell

- Integration into CAD

- Isogeometric shape optimization

- Application to FSI simulations of wind turbine blades





B-Splines

• Piecewise polynomial functions

• Smoothness over element boundaries

• Used for modeling of free form curves and surfaces

NURBS and Isogeometric Analysis

n

i

i pi u N u1

, )()( PC

p>0: )()()( 1,1

11

1

1,, u N uu

uuu N

uu

uuu N pi

i pi

pi

pi

i pi

i pi

p=0:

:0

:1

)(1

, otherwise

uuu

u N ii

pi

Cubic B-Spline functionsCubic B-Spline curve

u



Technische Universität MünchenTechnische Universität München

Tensor product of basis functions:

jiq j

m

j

pi

n

i

v M u N vu ,,

1

,

1

)()(),( PS

B-Spline Surfaces and Solids

k jir k q j

l

k

pi

m

j

n

i

w Lv M u N wvu ,,,,

1

,

1 1

)()()(),,( PB





Weighted control points Pi (xi,yi,zi,wi):

=> Exact representation of conic sections

NURBSNon-Uniform Rational B-Splines

jiq j

m

j

pi

n

i

ji jiq j

m

j

pi

n

i

wv M u N

wv M u N

vu

,,

1

,

1

,,,

1

,

1

)()(

)()(

),(

P

S


i

n

i

pi

ii

n

i

pi

wu N

wu N

u

1

,

1

,

)(

)(

)(

P

C




Weighted control points Pi (xi,yi,zi,wi):

=> Exact representation of conic sections

NURBSNon-Uniform Rational B-Splines

Circle Sphere

=> standard in CAD-programs !


jiq j

m

j

pi

n

i

ji jiq j

m

j

pi

n

i

wv M u N

wv M u N

vu

,,

1

,

1

,,,

1

,

1

)()(

)()(

),(

P

S

i

n

i

pi

ii

n

i

pi

wu N

wu N

u

1

,

1

,

)(

)(

)(

P

C




Isogeometric Analysis with NURBS

Idea:

Use functions from CAD (NURBS) as basis functions for FE -> common geometry description

+ No meshing - CAD model implies a NURBS mesh

+ Mesh refinement by NURBS refinement (order elevation and knot insertion)

+ Analysis on exact geometry -> advantage especially for thin shells

+ High-order elements with high continuities between elements





Outline









Kirchhoff-Love Shell Theory

• Cross sections remain straight

• Cross sections remain normal-> transverse shear strain is neglected

-> no decoupling of rotations and translations

• Valid for thin shells R/t > 20 (e.g. cooling tower R/t > 500)

x

xz x

w

R

t

z

x

Isogeometric Kirchhoff-Love shell




Governing Equations

Internal virtual work

S - Piola-Kirchhoff 2 stress tensor E - Green-Lagrange strain tensor (large deformations)

=> Separating strain into membrane and bending action

ε - membrane strain

κ - change of curvature


=> Problem for FE: curvature -> 2nd derivatives -> C1 continuity between elements




Isogeometric Analysis: NURBS -> C1 continuity

geometrically nonlinear kinematics:

base vectors

unit normal vector

metric tensor

curvature tensor

membrane strain

change of curvature

Degrees of freedom = Control point displacements

No rotations!





Scordelis-Lo roof – displacement plot





Large deformations and rotations

Benchmark: Straight plate bent to a circle

Moment modeled by a pair of forces perpendicular to the geometry





Outline









Isogeometric shell analysis integrated into CAD

CAD model of an oil barrel Applying boundary conditions

CAD models are surface-based.

-> For thin-walled structures, CAD and shell model use the same geometry description.

-> Design-through-analysis process inside CAD program (Rhino with IGA-plug-in):

geometric modeling, preprocessing, analysis, postprocessing

Refinement

Integration into CAD




Isogeometric shell analysis integrated into CAD

Deformation (scaled) Normal forces n1 (circumferential) Normal forces n2 (longitudinal)

Integration into CAD

CAD models are surface-based.

-> For thin-walled structures, CAD and shell model use the same geometry description.

-> Design-through-analysis process inside CAD program (Rhino with IGA-plug-in):

geometric modeling, preprocessing, analysis, postprocessing




Outline









min.dV.50:e.g. σε

0)( sg0sh )( def :masse.g. mm

max:stresse.g. σ σ

.min)( s f

Shape Optimization

• Shape Optimization with FE -> two different geometry descriptions

• Optimization Problem:

objective function:

equality constraints:inequality constraints:

design variables:

initial design objective

sensitivities

• Design Parametrization:

CAD-based: CAD parameters

FE-based: FE nodes

n,...,1s ii iii z y xl, r ,, e.g.variablesshape

meshing

CAD model Analysis model

Isogeometric Shape Optimization




min.dV.50:e.g. σε

0)( sg0sh )( def :masse.g. mm

max:stresse.g. σ σ

.min)( s f

Shape Optimization

• Shape Optimization with FE -> two different geometry descriptions

• Optimization Problem:

objective function:

equality constraints:inequality constraints:

design variables:

• Design Parametrization:

CAD-based: CAD parameters

FE-based: FE nodes

Isogeometric: Control Points

n,...,1s ii iii z y xl, r ,, e.g.variablesshape


initial design objective

sensitivities

meshing

CAD model Analysis model





Design variables: Control points

CAD model Isogeometric analysis model

Isogeometric optimization model






Example: Tube under constant internal pressure

Find optimal shape of the section in order to maximize the stiffness

constant pressure






12 design variables:

x, y, w

of the four edge control points

Design Variables





Animation: Optimization steps






Section shape after optimization: circular

Circular section exactly represented by NURBS

constant pressureconstant pressure





Example: Circular Tube under point loads






coarse optimization model fine optimization model

Different refinements for optimization model






Optimized design model in CAD program





Outline









Application: FSI simulation of a Wind Turbine Blade

Cooperation project with Y. Bazilevs, M.-C. Hsu, UCSD:

Fully coupled, 3D FSI simulation of a wind turbine blade rotating in the air flow

Fluid: Isogeometric fluid elements

Structure: Isogeometric K-L shell

-> matching discretization at the interface

Wind turbine blade Fluid mesh

FSI simulation of wind turbine blades




Application: FSI simulation of a Wind Turbine Blade

Cooperation project with Y. Bazilevs, M.-C. Hsu, UCSD:

Fully coupled, 3D FSI simulation of a wind turbine blade rotating in the air flow

Fluid: Isogeometric fluid elements

Structure: Isogeometric K-L shell

-> matching discretization at the interface

Structural modelFluid mesh





3D FSI Simulation

Normal forces





Thank you for your attention

Kiendl - Isogeometric Analysis and Shape Optimal Design of Shell Structures

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