Enhanced Discretization and Space-Time Refinement in - - PowerPoint PPT Presentation

Enhanced Discretization and Space-Time
 Refinement in Moving-Boundary Flow Simulation

Marek Behr Chair for Computational Analysis of Technical Systems 
 
 RICAM Special Semester
 November 8, 2016

Outline

• Why moving boundaries?
• melt flow processes
• solid mechanics
• Which frame of reference?
• Lagrangian, Eulerian, mixed
• How to follow the interface?
• tracking and capturing
• Enhanced surface discretization
• NEFEM, IGA
• Space-time refinement
• simplex space-time grids

2

• S. Elgeti and H. Sauerland


Deforming Fluid Domains Within the Finite Element Method: Five Mesh-Based Tracking Methods in Comparison
 Archives of Computational Methods in Engineering
 23 (2016) 323–361
 arXiv:1501.05878

Why Moving Boundaries?

• Fluid:
• filling
• swelling
• solidification
• Solid:
• shrinking
• deformation

3

XC Production TP A3 SFB 1120 TP B2 SFB 1120 TP B5

Which Frame of Reference?

• Lagrangian:
• frame attached to moving material
• no inflow or outflow of material
• Eulerian:
• frame fixed in space
• material flows relative to the frame
• Lagrangian-Eulerian:
• neither Lagrangian nor Eulerian
• Lagrangian and Eulerian as special cases

4

How to Follow the Interface?

5

Interface capturing

fixed Eulerian grid cut interface cell empty cell

Interface tracking

deforming ALE grid interface = grid boundary

Interface Tracking

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• Advantages:
• low discretization errors at the interface
• easy inclusion of interface effects
• Disadvantages:
• deformation limited if no remeshing
• projection errors if remeshing
• Arbitrary Lagrangian-Eulerian (ALE), space-time finite elements

Interface Capturing

7

• Advantages:
• easy handling of large deformations
• allows topology changes
• Disadvantages:
• high discretization errors at the interface
• load imbalance
• Particle methods, volume-of-fluid, level-set, phase field

Interface Capturing • Level Set

• Level set function
• fluid A domain
• fluid B domain
• interface
• Due to Osher and Sethian (1988)
• Advected with fluid velocity:
• Determines material properties


at integration point:

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φ(x, t) φ < 0 φ > 0 φ = 0 ∂φ ∂t + v · rφ = 0 ρ, µ(φ) = ( ρA, µA, φ(x, t) < 0 ρB, µB, φ(x, t) > 0

Interface Capturing • Level Set Reinitialization

• Initial value is signed distance function
• Signed-distance property degrades:
• How to reinitialize?
• search for closest interface node
• narrow band search
• iterative solution of Eikonal equation
• Mass conservation is not guaranteed

9

Interface Capturing • Level Set Sharpening

• XFEM-based integration:
• Adaptive refinement:

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level set linearization quadrature

Interface-Tracking Problem

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• Three governing equations
• kinematic condition
• interior mesh update
• incompressible Navier-Stokes
• Solved within nonlinear iteration loop

begin time step loop begin nonlinear iteration loop solve elevation equation solve deformation equation solve flow equation end loop end loop

Governing Equations: Flow

12

• Incompressible Navier-Stokes equations:
• Constitutive equation:
• Boundary conditions:
• For plastics flow:
• Navier-Stokes → Stokes
• Newtonian fluid → shear-thinning or viscoelastic (Giesekus, Oldroyd-B)

Galerkin Least-Squares Formulation: Flow

13

• Find such that :
• Notation:

Governing Equations: Deformation

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• Displacements governed by linear elasticity:
• Boundary conditions:
• Mesh update:

Galerkin Formulation: Deformation

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• Find such that :
• Coefficients and adjusted to stiffen small cells:
• Alternate approaches:

▶ geometric (Masud, 1997 & 2007), ▶ stress-based (Oñate et al., 2000), ▶ ...

Governing Equations: Elevation

16

• Kinematic condition at the free surface:
• One condition but two or three unknowns per node
• Whether explicitly stated or not, equivalent to:



 
 
 where is the generalized elevation along direction e

• Often applied point-wise; OK in tanks, but not in channels or jets:

Galerkin Least-Squares Formulation: Elevation

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• Find such that :
• Stabilization and DC parameters, using residual :

Example: Olmsted Dam

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• Spillway of Olmsted dam on Ohio River
• Experiments in a scale model at ERDC:
• Periodic spillway section
• Obstacles dissipate flow energy


and reduce erosion:

32ft 239ft 182ft 79ft 280ft 301.5ft

Example: Olmsted Dam

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• Computed using 418K tetrahedra and 1000 time steps:

Example: Olmsted Dam

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• Mesh adapts to free surface movement:

Example: Olmsted Dam

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• Quasi-steady state reached without remeshing
• Stream-wise component of velocity shown
• Hydraulic jump position accurately predicted

Example: Olmsted Dam

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• Comparison of space-time (left) and level-set with XFEM (right)

t = 3.5 s t = 6.1 s t = 8.5 s t = 11.6 s t = 27.2 s t = 36.1 s t = 3.5 s t = 6.1 s t = 8.5 s t = 11.6 s t = 27.2 s t = 36.1 s

Sauerland et al., Computers and Fluids, 87 (2013) 41–49

NURBS-Enhanced Finite Elements (Huerta et al.)

• Most elements are standard finite elements
• Elements on the NURBS boundary represent the exact geometry

Stavrev, Knechtges, Elgeti and Huerta, International Journal for Numerical Methods in Fluids, 81 (2016) 426–450

θ

• Transient Stokes equations; parabolic inflow; no slip on the walls
• FEM: linear space-time elements
• NEFEM: linear space-time approximation; geometry with cubic NURBS

NEFEM in Die Swell

Swell Factor NEFEM Swell Factor FEM Mesh I: 768 DOF 1.1262 1.1374 Mesh II: 4074 DOF 1.1257 1.1271 Mesh III: 6006 DOF 1.1220 1.1220

• Analogous to isoparametric concept
• IGA uses NURBS to represent both the geometry and the unknown:

Isogeometric Analysis (Hughes et al.) uh =

n

X

i=1

Ri,p(θ)ui

Hughes et al., Isogeometric analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement, Computer Methods in Applied Mechanics and Engineering, 194, (2005) 4135–4195

Spline-Based Coupling for Fluid-Structure-Interaction

Standard FE IGA + Standard FE IGA + NEFEM

Storage Tanks

• plant with industrial storage tanks
• stability under earthquakes


(seismic loading) is an important design requirement diamond-shaped buckling elephant footing pressure bumps

Storage Tanks

• free-surface flow
• deformable domain with

arbitrary wall shapes

• deformable structure
• interface tracking
• NURBS-enhanced


finite elements

• arbitrary boundary

description through NURBS

• Isogeometric Analysis

for elastodynamics Requirements Methods

• Liquid-filled storage tank subjected to seismic excitation
• Linear FEM for fluid, IGA shell for solid

3D Cylindrical Tank

thickness height Navier-slip BC diameter excitation

• Fluid velocity (left) and mesh deformation (right) in rigid tank:
• Notice mesh nodes sliding across the NURBS edge

3D Cylindrical Tank • Fluid Flow Field

• Structural deformation without excitation
• Excitation started after equilibrium reached

3D Cylindrical Tank • Structural Solution

maximal and minimal deformation equilibrium vertical displacement

3D Cylindrical Tank • FSI Solution

surface grid point height in rigid (red) and flexible (blue) tank

Mesh Generation for Space-Time

33

• Hughes and Hulbert (1988), Maubach (1991):
• 3D case presents obvious difficulties…

Prisms versus Simplices

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• Standard prismatic extensions of 3D elements:
• Simplex elements required for fully-unstructured meshes:

3d6n 4d8n 3d4n 4d5n

Prisms to Simplices

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• Start with prismatic 4D mesh
• Add temporal refinement nodes along the spines:
• Perturb temporal coordinates:
• Delaunay triangulation using, e.g., qhull package

in1 in2 kn1 kn2 jn1 jn2 in1 in4 kn1 kn2 jn1 jn3 jn2 in3 in2

Final Steps

36

• Sliver elimination:
• Temporally refined 3D space-time mesh for 2D cylinder:

d1 d2

Test Case: Gaussian Hill in 2D and 3D

37

• Initial profile:



 
 
 advected according to:
 
 
 
 with and

• Spatial meshes in 2D and 3D:
• Numerical solutions at compared to exact solution

x y

0.00 0.00 1.00 0.01 0.01 0.99

x z y

Test Case: Gaussian Hill in 2D

38

• GLS with C-N (left), prisms (middle) and tetra space-time with
• GLS with C-N (left), prisms (middle) and tetra space-time with

x y t

0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u 0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u 0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u 0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u 0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u 0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u 0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u 0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u 0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u

x y t x y t

0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u 0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u 0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u 0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u 0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u 0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u 0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u 0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u 0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u

x y t x y t x y t

Test Case: Gaussian Hill in 3D

39

• GLS with pentatope space-time with
• Space-time edge for also shown
• Flow examples in the next talk of Violeta Karyofylli

t x

0.2 0.4 0.6 0.8 1 x

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u

Summary

40

• Why moving boundaries?
• during melt flow: filling, swelling
• after melt flow: solidification, shrinkage
• Which frame of reference?
• Lagrangian
• Eulerian and mixed
• How to follow the interface?
• interface tracking: ALE, space-time
• interface capturing: MAC, VOF, level set, phase field
• Enhanced surface discretization
• Space-time refinement

Acknowledgments

41

• RWTH Aachen University
• Dr. Stefanie Elgeti, Violeta Karyofylli, Markus Frings, Philipp Knechtges, Roland

Siegbert, Atanas Stavrev (CATS)

• Prof. Christoph Hopmann, Dr. Christian Windeck (Plastics Processing)
• Rice University
• Prof. Matteo Pasquali (Chemical Engineering and Chemistry)
• Forschungszentrum Jülich
• Dr. Eric von Lieres (Biotechnology)
• TU Munich
• Prof. Wolfgang Wall (Computational Mechanics)
• UPC Barcelona
• Prof. Antonio Huerta (LaCàN)
• Old Dominion University
• Prof. Nikos Chrisochoides (Computer Science)
• German Research Foundation:


GSC 111, EXC 128, SFB 1120, BE 3689/7, BE 3689/10, EL 741/3

• Federal Ministry of Research BMBF and Environment BMU