High-Order Adaptive ALE Calculations of Solidification B. T. - PowerPoint PPT Presentation

High-Order Adaptive ALE Calculations of Solidification B. T. Helenbrook Mech. & Aero. Eng. Dept. Clarkson University Potsdam, NY USA

Clarkson Motivation �� U N I V E R S I T Y �� Th Tc solid liquid u, Th u, dT/dx = 0 Th ◮ Materials manufacturing, ◮ Horizontal ribbon growth ◮ Spin-casting ◮ Aluminum smelters ◮ Welding

Clarkson Field Equations �� U N I V E R S I T Y �� ◮ Solid ∂ρ s c p , s T + ∂ρ s u j c p , s T + ∂ � ∂ T � − k s = 0 ∂ t ∂ x j ∂ x j ∂ x j ◮ Liquid ∂ρ l c p , l T + ∂ρ l u j c p , l T + ∂ � ∂ T � − k l = 0 ∂ t ∂ x j ∂ x j ∂ x j ∂ρ l u i + ∂ρ l u j u i = ∂ p + ∂τ i , j ∂ t ∂ x j ∂ x i ∂ x j

Clarkson Discrete Field Equations �� U N I V E R S I T Y �� x = x ( ξ, η, τ ) y = y ( ξ, η, τ ) t = τ � � � � ∂ J ρ c p T � − ∂ v ∂ξ E − ∂ v � � ∂η F d ξ d η + v ( E , F ) · n d Γ v ∂τ Ω Γ n el � � � ∂ v ∂ξ a E + ∂ v � � ∂ J ρ c p T + ∂ ∂ξ E + ∂ � � + ¯ τ ∂η F d Ω = 0 ∂η a F ∂τ Ω n n =1 y ,-x ξ ◮ E, F = contravariant fluxes in ξ η coordinates: � ρ c p T ( u − x τ ) − k ∂ T � � ρ c p T ( v − y τ ) − k ∂ T � F = y ξ − x ξ ∂ x ∂ y ◮ a E , a F convective velocities a F = ( u − x τ ) y ξ − ( v − y τ ) x ξ

Clarkson Continued... �� U N I V E R S I T Y �� ◮ Stabilization constant 4 A h = ( p + 1) 2 h max h max = maximum edge length of element τ = 1 h ¯ � J a 2 E + a 2 F + ν/ h + h / ∆ t ◮ Time derivatives: 3 rd order accurate A-Stable DIRK(Williams et. al 2002) ◮ Implicit time step ∆ t ◮ Implicit equations solved using either geometric multgrid or SuperLU/Petsc

High-Order Adaptive ALE Calculations of Solidification B. T. - PowerPoint PPT Presentation

High-Order Adaptive ALE Calculations of Solidification B. T. Helenbrook Mech. & Aero. Eng. Dept. Clarkson University Potsdam, NY USA Clarkson Motivation

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