Simulation of 1-d Stefan Problem using XFEM V. S. S. Srinivas

Outline Stefan melting problem Physical description Analytical solution Conventional numerical modeling Associated issues Review of Xfem Implementation Results Unresolved issues Challenges in Xfem-2d Discussion and future directions

Stefan melting problem

Physical description

Analytical solution

Conventional numerical modeling

Associated issues

Review of Xfem

Implementation

Results

Unresolved issues

Challenges in Xfem-2d

Discussion and future directions

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 Always the interface has to coincide with a node Necessitates re-meshing T=T 0 T=T 0 T=T 0 t=0 t=t 1 t=t 2 0<t 1 <t 2 T=0

Always the interface has to coincide with a node

Necessitates re-meshing

XFEM Enrichment to elements whose support is intersected When interface moves across elements new nodes get enriched while the old ones shed enrichment No re-meshing required Quite useful when re-meshing is costly and the interface has to be tracked explicitly T=T 0 T=T 0 T=T 0 t=0 t=t 1 t=t 2 0<t 1 <t 2

Enrichment to elements whose support is intersected

When interface moves across elements

new nodes get enriched

while the old ones shed enrichment

No re-meshing required

Quite useful when re-meshing is costly and the interface has to be tracked explicitly

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 Considered 1D as opposed to a 2D scenario Reduced the complexities associated with level sets Performed analytical integration in computing the element matrices using Mathematica Initial values of the enriched degrees of freedom are computed by applying the interface conditions The element nodes are enriched when its support is intersected by the interface and vice versa

Considered 1D as opposed to a 2D scenario

Reduced the complexities associated with level sets

Performed analytical integration in computing the element matrices using Mathematica

Initial values of the enriched degrees of freedom are computed by applying the interface conditions

The element nodes are enriched when its support is intersected by the interface and vice versa

Initial values determination-A case study At t=0, the interface is located in the first element Its nodes are enriched Initial values? 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 Unique solution No solution Many solutions Consider 2D case…

At t=0, the interface is located in the first element

Its nodes are enriched

Initial values?

Unique solution

No solution

Many solutions

Contd.. 4 unknowns 6 conditions This is unlike 1D case where there are 2 unknowns and 2 conditions

4 unknowns

6 conditions

This is unlike 1D case where there are 2 unknowns and 2 conditions

Important features of XFEM Enriched dofs are determined automatically Don’t need explicit attention as in determining initial values Matrices that have to be inverted remain symmetric Positions of enriched dofs changing Number of enriched dofs is also changing

Enriched dofs are determined automatically

Don’t need explicit attention as in determining initial values

Matrices that have to be inverted remain symmetric

Positions of enriched dofs changing

Number of enriched dofs is also changing

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 Problems in determining the initial values of enriched dof for certain positions of the interface Nodes are a set of examples May be endemic to 1D During the interface propagation, if it falls on a node; in one instance it caused the matrix to become singular

Problems in determining the initial values of enriched dof for certain positions of the interface

Nodes are a set of examples

May be endemic to 1D

During the interface propagation, if it falls on a node; in one instance it caused the matrix to become singular

Challenges XFEM-2D Level set schemes for embedding the interface in an implicit function Requires stabilization schemes May cause rank deficient matrices

Level set schemes for embedding the interface in an implicit function

Requires stabilization schemes

May cause rank deficient matrices

Pros and Cons Don’t need re-meshing Interfaces can be tracked explicitly May cause rank deficient matrices

Don’t need re-meshing

Interfaces can be tracked explicitly

May cause rank deficient matrices

Discussion What is our objective? Will this idea serve to be useful? What are the various possibilities in modeling our problem. Will this method score over others?

What is our objective?

Will this idea serve to be useful?

What are the various possibilities in modeling our problem.

Will this method score over others?

Future directions Check if the singularity issues related to determining initial values persist in 2D If yes, ways to eliminate them Extend this model to 2D Explore the possibility of using Abaqus with XFEM capability

Check if the singularity issues related to determining initial values persist in 2D

If yes, ways to eliminate them

Extend this model to 2D

Explore the possibility of using Abaqus with XFEM capability

Thank you