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Simulation of 1-d Stefan Problem using XFEM V. S. S. Srinivas
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Stefan problem T=T 0 0 °C Initial Temperature = 0°C h(t) Governing equations Initial and boundary conditions
Analytical solution Solve the transcendental equation, get  Transcendental equation:  Temperature profile:  Interface position:
Conventional method ,[object Object],[object Object],T=T 0 T=T 0 T=T 0 t=0 t=t 1 t=t 2 0<t 1 <t 2 T=0
XFEM ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],T=T 0 T=T 0 T=T 0 t=0 t=t 1 t=t 2 0<t 1 <t 2
Enriched shape function Level set function in 1d: Interface position: x b ,  Element length: l e =1 X b =0.5 X b =0.3
Contd.. X b =0.5 X b =0.3
Implementation ,[object Object],[object Object],[object Object],[object Object],[object Object]
Initial values determination-A case study ,[object Object],[object Object],[object Object],T=T 0 t=t 1 T=0 Interface conditions: 1. T=T m 2. Discontinuity in the gradient of temperature T 5 T 6 T 1 T 2 T 4 T 3 ,[object Object],[object Object],[object Object],Consider 2D case…
Contd.. ,[object Object],[object Object],[object Object]
Important features of XFEM ,[object Object],[object Object],[object Object],[object Object],[object Object]
A peep into the matrices  Governing equation after applying weighted residuals Discretized equation: Unsymmetric Symmetric
Validation:  Numerical against Analytical
Convergence check
T versus x at different times
Issues-1D ,[object Object],[object Object],[object Object],[object Object]
Challenges XFEM-2D ,[object Object],[object Object],[object Object]
Pros and Cons ,[object Object],[object Object],[object Object]
Discussion ,[object Object],[object Object],[object Object],[object Object]
Future directions ,[object Object],[object Object],[object Object],[object Object]
Thank you

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xfem using 1D stefan problem