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Adaptive Mesh Refinement in Filling Simulations Based on Level Set RICAM Special Semester | Space-Time Methods for PDEs PhD Student: M.Sc. Violeta Karyofylli Advisor: Prof. Marek Behr, Ph.D. This is a joint work with Markus Frings , Loic Wendling

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SLIDE 1

Adaptive Mesh Refinement in Filling Simulations Based on Level Set

RICAM Special Semester | Space-Time Methods for PDEs

Advisor: Prof. Marek Behr, Ph.D. PhD Student: M.Sc. Violeta Karyofylli This is a joint work with Markus Frings, Loic Wendling and Dr. Stefanie Elgeti at the Chair for Computational Analysis of Technical Systems

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SLIDE 2

Table of Contents Governing Equations Simplex Space-Time Meshes Static Bubble (2D) Rising Bubble (2D) Rising Droplet (3D) Step Cavity Benchmark Case (2D) Coat Hanger Distributer and Rectangular Cavity Benchmark Case (2D) Conclusion

2 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

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SLIDE 3

Governing Equations

  • Two-phase flow

– Incompressible, transient, isothermal – Two immiscible Newtonian phases

s p ac e t im e melt air phase boundary

Temporal and spatial refinement in the vicinity of moving interfaces.

  • Navier-Stokes equations:

πœπ‘— (πœ–π― πœ–π‘’ + 𝐯 β‹… βˆ‡π― βˆ’ 𝐠) βˆ’ βˆ‡ β‹… πœπ‘— = 0 in Ω𝑗(𝑒), βˆ€π‘’ ∈ [0, π‘ˆ] (1) βˆ‡ β‹… 𝐯 = 0 βˆ€π‘’ ∈ [0, π‘ˆ] (2)

  • Level-Set equation:

πœ–πœš πœ–π‘’ + 𝐯 β‹… βˆ‡πœš = 0

  • n

Ω𝑗(𝑒), βˆ€π‘’ ∈ [0, π‘ˆ] (3)

3 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

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SLIDE 4

Simplex Space-Time Meshes

3d6n 4d8n

(a) Prism-type space-time elements.

3d4n 4d5n

(b) Simplex-type space-time elements. Comparison of prism- and simplex-type space-time elements. Black nodes correspond to π‘’π‘œ and white nodes correspond to π‘’π‘œ+1. [1]

  • Advantages of Simplex Space-Time Meshes

– Different temporal refinement in different parts of the domain – Connection of disparate spatial meshes

1Behr, M. (2008). Simplex space-time meshes in finite element simulations. International Journal for Numerical Methods in Fluids, 57(9), 1421-1434

4 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

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SLIDE 5

Static Bubble (2D)

  • Description2

– A perfectly stationary circular bubble at equilibrium – Laplace-Young law: π‘žπ‘—π‘œ = π‘žπ‘π‘£π‘’ + 𝜏/𝑠 – Surface tension coefficient: 𝜏 = 𝟀 𝗅𝗁/𝗍πŸ₯ – Density of both phases: 𝜍1 = 𝜍2 – Viscosity of both phases: 𝜈1 = 𝜈2

  • Time Discretization

– Time slab size: Δ𝑒 = 𝟣.𝟣𝟀 𝗍 – Total number of time slabs: 𝟀πŸ₯𝟨

Ξ©1 Ξ©2

0.5 0.5 0.5 1.0 1.0

Static bubble in 2D: Computational domain.

  • Boundary Conditions

– No-slip boundary conditions at all boundaries – A zero reference pressure at one corner

2Hysing, S. (2006). A new implicit surface tension implementation for interfacial flows. International Journal for Numerical Methods in Fluids, 51(6), 659-672

5 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

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SLIDE 6

Static Bubble (2D)

0.2 0.4 0.6 0.8 1 1 2 3 4 5 𝑦 𝑄𝑠𝑓𝑑𝑑𝑣𝑠𝑓 prismatic simplex

(a) β„Ž = 1/80

0.2 0.4 0.6 0.8 1 1 2 3 4 5 𝑦 𝑄𝑠𝑓𝑑𝑑𝑣𝑠𝑓 prismatic simplex

(b) β„Ž = 1/160 Pressure cut-lines at 𝑧 = 𝟣.𝟨 after 𝟀πŸ₯𝟨 time slabs for two different mesh sizes.

6 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

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SLIDE 7

Rising Bubble (2D)

  • Description2

– A bubble rising in a heavier fluid – Surface tension coefficient: 𝜏 = πŸ₯𝟧.𝟨 𝗅𝗁/𝗍πŸ₯ – Density of both phases: 𝜍1 = 𝟀𝟣𝟣𝟣 𝗅𝗁/π—‡πŸ¦; 𝜍2 = 𝟀𝟣𝟣 𝗅𝗁/π—‡πŸ¦ – Viscosity of both phases: 𝜈1 = 𝟀𝟣 𝗅𝗁/𝗇/𝗍; 𝜈2 = 𝟀 𝗅𝗁/𝗇/𝗍 – Gravity: 𝑔𝑧 = βˆ’π‘• = βˆ’πŸ£.𝟬𝟫 𝗇/𝗍πŸ₯

  • Time Discretization

– Time slab size: Δ𝑒 = 𝟣.𝟣𝟀 𝗍 – Total number of time slabs: 𝟦𝟣𝟣

Ξ©1 Ξ©2

0.5 0.5 0.5 2.0 1.0

Rising bubble in 2D: Computational domain.

  • Boundary Conditions

– No-slip boundary conditions at the top and bottom boundary – Slip boundary conditions along the vertical walls – Zero pressure specified at the upper boundary

2Hysing, S. (2006). A new implicit surface tension implementation for interfacial flows. International Journal for Numerical Methods in Fluids, 51(6), 659-672

7 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

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SLIDE 8

Rising Bubble (2D)

(a) 𝑒 = 𝟀.𝟣 𝗍 (b) 𝑒 = πŸ₯.𝟣 𝗍 (c) 𝑒 = 𝟦.𝟣 𝗍 Bubble position at various time instances, obtained using a prism-type space-time discretization and a simplex-type space-time discretization. Light grey color corresponds to the prismatic space-time discretization (left half of the bubble) and dark grey color corresponds to simplex-based space-time discretization (right half of the bubble).

8 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

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SLIDE 9

Rising Bubble (2D)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 𝑦 𝑧 prismatic simplex TP2D

Comparison of the bubble at 𝑒 = 𝟦.𝟣 𝗍 with reference data published by [3].

3Hysing, S. R., Turek, S., Kuzmin, D., Parolini, N., Burman, E., Ganesan, S., & Tobiska, L. (2009). Quantitative benchmark computations of two-dimensional bubble

  • dynamics. International Journal for Numerical Methods in Fluids, 60(11), 1259-1288

9 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

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SLIDE 10

Rising Bubble (2D)

  • Comparison between the results of in-house solver and reference data3

0.5 1 1.5 2 2.5 3 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

𝑒 𝑧 βˆ’ π‘‘π‘“π‘œπ‘’π‘“π‘  𝑝𝑔 𝑛𝑏𝑑𝑑

prismatic simplex TP2D

(a) Over the whole simulation time.

2.4 2.5 2.6 2.7 2.8 2.9 3 0.95 1 1.05 1.1

𝑒 𝑧 βˆ’ π‘‘π‘“π‘œπ‘’π‘“π‘  𝑝𝑔 𝑛𝑏𝑑𝑑

prismatic simplex TP2D

(b) Between the time instances 𝑒1 = πŸ₯.𝟧 𝗍 and 𝑒2 = 𝟦.𝟣 𝗍. The position of the center of mass X𝑑 in 𝑧-direction of the rising bubble in 2D.

3Hysing, S. R., Turek, S., Kuzmin, D., Parolini, N., Burman, E., Ganesan, S., & Tobiska, L. (2009). Quantitative benchmark computations of two-dimensional bubble

  • dynamics. International Journal for Numerical Methods in Fluids, 60(11), 1259-1288

10 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

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SLIDE 11

Rising Droplet (3D)

  • Description4

– A droplet rising in a heavier fluid – Surface tension coefficient: 𝜏 = πŸ₯𝟧.𝟨 𝗅𝗁/𝗍πŸ₯ – Density of both phases: 𝜍1 = 𝟀𝟣𝟣𝟣 𝗅𝗁/π—‡πŸ¦; 𝜍2 = 𝟀𝟣𝟣 𝗅𝗁/π—‡πŸ¦ – Viscosity of both phases: 𝜈1 = 𝟀𝟣 𝗅𝗁/𝗇/𝗍; 𝜈2 = 𝟀 𝗅𝗁/𝗇/𝗍 – Gravity: 𝑔𝑧 = βˆ’π‘• = βˆ’πŸ£.𝟬𝟫 𝗇/𝗍πŸ₯

  • Time Discretization

– Time slab size: Δ𝑒 = 𝟣.𝟣𝟀 𝗍 – Total number of time slabs: 𝟦𝟣𝟣

Ξ©1 Ξ©2

0.5 0.5 0.5 0.5 2.0 1.0

Rising droplet in 3D: Computational domain.

  • Boundary Conditions

– No-slip boundary conditions at all the boundaries – Zero pressure specified at the upper boundary

4Adelsberger, J., Esser, P., Griebel, M., Groß, S., Klitz, M., & Rüttgers, A. (2014, March). 3D incompressible two-phase flow benchmark computations for rising droplets.

In Proceedings of the 11th World Congress on Computational Mechanics (WCCM XI), Barcelona, Spain, 2014 11 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

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SLIDE 12

Rising Droplet (3D)

(a) 𝑒 = 𝟀.𝟣 𝗍 (b) 𝑒 = πŸ₯.𝟣 𝗍 (c) 𝑒 = 𝟦.𝟣 𝗍 Droplet position at various time instances, obtained using a prism-type space-time discretization and a simplex-type space-time discretization. Light grey color corresponds to the prismatic space-time discretization (left half of the droplet) and dark grey color corresponds to simplex-based space-time discretization (right half of the droplet).

12 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

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SLIDE 13

Rising Droplet (3D)

  • Comparison between the results of in-house solver and reference data4

0.5 1 1.5 2 2.5 3 0.5 0.7 0.9 1.1 1.3 1.5

𝑒 𝑧 βˆ’ π‘‘π‘“π‘œπ‘’π‘“π‘  𝑝𝑔 𝑛𝑏𝑑𝑑

prismatic simplex DROPS

(a) Over the whole simulation time.

2.4 2.5 2.6 2.7 2.8 2.9 3 1.27 1.3 1.35 1.4 1.45 1.5

𝑒 𝑧 βˆ’ π‘‘π‘“π‘œπ‘’π‘“π‘  𝑝𝑔 𝑛𝑏𝑑𝑑

prismatic simplex DROPS

(b) Between the time instances 𝑒1 = πŸ₯.𝟧 𝗍 and 𝑒2 = 𝟦.𝟣 𝗍. The position of the center of mass X𝑑 in 𝑧-direction of the rising droplet in 3D.

4Adelsberger, J., Esser, P., Griebel, M., Groß, S., Klitz, M., & Rüttgers, A. (2014, March). 3D incompressible two-phase flow benchmark computations for rising droplets.

In Proceedings of the 11th World Congress on Computational Mechanics (WCCM XI), Barcelona, Spain, 2014 13 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

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SLIDE 14

Step Cavity Benchmark Case (2D)

Ξ©1 Ξ©2

0.3 0.3 0.8 0.6 0.6

Step Cavity in 2D: Computational domain.

  • Description5

– Metal casting – Spreading influence of gravity on the front – No surface tension effects – Density of both phases: 𝜍1 = 𝟣.𝟀 𝗅𝗁/π—‡πŸ¦; 𝜍2 = 𝟀𝟣𝟣 𝗅𝗁/π—‡πŸ¦ – Viscosity of both phases: 𝜈1 = 𝟣.𝟣πŸ₯ 𝗅𝗁/𝗇/𝗍; 𝜈2 = 𝟣.πŸ₯ 𝗅𝗁/𝗇/𝗍 – Gravity: 𝑔𝑧 = βˆ’π‘• = βˆ’πŸ¬.𝟫𝟣 𝗇/𝗍πŸ₯

  • Boundary Conditions

– Slip boundary conditions on fixed walls – Inflow boundary: small uniform velocity 𝑣 = 𝟣.𝟀 𝗇/𝗍 – Outflow boundary: traction-free boundary conditions

  • Time Discretization

– Time slab size: Δ𝑒 = 𝟣.𝟣𝟣𝟨 𝗍 – Total number of time slabs: 𝟦πŸ₯𝟣

5Cruchaga, M., Celentano, D., & Tezduyar, T. (2002). Computation of mould filling processes with a moving Lagrangian interface technique. Communications in

Numerical Methods in Engineering, 18(7), 483-493 14 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

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SLIDE 15

Step Cavity Benchmark Case (2D)

  • Time Refinement

– Hybrid mesh – Different temporal discretization at the interface – One to five elements in time direction

Nodes, laying only on top and bottom of the time slab Time refinement at the interface Hybrid space-time mesh corresponding to one of the 𝟦πŸ₯𝟣 (in total) time slabs of the simulation.

15 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

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SLIDE 16

Step Cavity Benchmark Case (2D)

(a) 𝑒 = 𝟣.𝟧 𝗍 (b) 𝑒 = 𝟣.𝟫 𝗍 (c) 𝑒 = 𝟀.𝟩 𝗍 Molten material position at various time instances, obtained with a prism-type space-time discretization (top row) and a simplex-type space-time discretization (bottom row) and compared with reference data (middle row) [5].

5Cruchaga, M., Celentano, D., & Tezduyar, T. (2002). Computation of mould filling processes with a moving Lagrangian interface technique. Communications in

Numerical Methods in Engineering, 18(7), 483-493 16 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

slide-17
SLIDE 17

Coat Hanger Distributer and Rectangular Cavity Benchmark Case (2D)

(a) Prism-type space-time discretization. (b) Simplex-type space-time discretization. Molten material position at the end of the simulation.

  • Time Discretization

– Time slab size: Δ𝑒 = 𝟣.𝟣𝟣πŸ₯𝟨 𝗍 – Total number of time slabs: 𝟀𝟣𝟣

  • Description6

– Mould filling – Isothermal conditions without gravity – Horizontal mould orientation – A 2D slice down the center of the distributor and mold – Surface tension coefficient: 𝜏 = 𝟧πŸ₯.𝟨 𝗁/𝗍πŸ₯ – Density of both phases: 𝜍1 = 𝟣.𝟣𝟣𝟀 𝗁/π–½π—‡πŸ¦; 𝜍2 = 𝟀𝟣𝟣𝟣 𝗁/π–½π—‡πŸ¦ – Viscosity of both phases: 𝜈1 = 𝟀 𝗁/𝖽𝗇/𝗍; 𝜈2 = 𝟧.𝟨 𝗁/𝖽𝗇/𝗍

  • Boundary Conditions

– Navier-slip boundary conditions on fixed walls – Inflow boundary: uniform velocity 𝑣 = 𝟨 𝖽𝗇/𝗍 – Outflow boundary: traction-free boundary conditions

6Rao, Rekha R., et al. (2006). Modeling Injection Molding of Net- Shape Active Ceramic Components. Sandia National Laboratories, Albuquerque, NM

17 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

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SLIDE 18

Conclusion

  • Initial validation of the unstructured space-time mesh solver

– For the Navier-Stokes equations – For the level-set equation

  • The use of a hybrid mesh in filling simulations (2D space-dimensions)
  • Future work

– Arbitrary temporal refinement during the filling simulation of complex moulds – The efficiency aspects of the unstructured space-time mesh solver

18 of 19

  • Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli

Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

slide-19
SLIDE 19

Thank you for your attention

Any questions?

  • Prof. Marek Behr, Ph.D.
  • Schinkelstr. 2 (Rogowski building)

2nd Floor, Room 227 52062 Aachen, Germany E-Mail: behr@cats.rwth-aachen.de

M.Sc. Violeta Karyofylli

  • Schinkelstr. 2 (Rogowski building)

2nd Floor, Room 222a 52062 Aachen, Germany E-Mail: karyofylli@cats.rwth-aachen.de