M. Paszyński 1 , P. Matuszyk 2

1. Comparison of Different Mesh Management Strategiesfor Non-Stationary Self-Adaptive hp Finite Element Method 1Department of Computer Science, AGH University of Science and Technology, Kraków, Poland, email: paszynsk@agh.edu.pl 2Department of Applied Computer Science and Modelling, AGH University of Science and Technology, Kraków, Poland,email: pjm@agh.edu.pl M. Paszyński1, P. Matuszyk2 ASEM 2008, May 26-28, Jeju, South Korea

2. Outline • The self-adaptive hp-FEM algorithm • Hp finite element • Self-adaptive hp-Finite Element Method (hp-FEM) • Motivation – phase-transition simulations • Extension of the self-adaptive hp-FEM algorithm to non-stationary problems • Three algorithms for mesh managements • Numerical experiments • Conclusions

3. hp Finite Element Method The Finite Element Method (FEM) consists in construction of a finite dimensional sub-space We seek the approximate solution as a linear combination of the basis functions

4. Self-adaptive hp Finite Element Method Goal: increase the number of the basis functions in order to increase the accuracy of the approximate solution hp adaptation generates a sequence of meshes delivering exponential convergence of the numberical error with respect to the mesh size It is done breaking selected finite elements into smaller elements and increasing polynomial order of approximation on selected finite elements.

5. Self-adaptive hp Finite Element Method The self-adaptive hp-FEM provides exponential convergence of the numerical error with respect to the mesh size

6. Motivation The simulations of the phase transition for solidification of Fe-C alloy Withthe non-stationary self-adaptive hpFinite Element Method (hp-FEM) Material data provided by Cellular Automata (CA) time step 100 time step 1 time step 50 concentration scalar field hp refined finite element mesh

7. Strong form of the non-stationary heat and mass transfer problems • Find the temperature scalar field such that: where heat transfer coefficient density specific heat TNambient temperature Provided by Cellular Automata simulation

8. Strong form of the non-stationary heat and mass transfer problems • Find the concentration scalar field such that: where D diffusion coefficient cNambient concentration Provided by Cellular Automata simulation

9. Weak (variatonal) form of the non-stationary heat and mass transfer problems • Find the temperature scalar field such that: where heat transfer coefficient density specific heat TNambient temperature Provided by Cellular Automata simulation

10. Weak (variatonal) form of the non-stationary heat and mass transfer problems • Find the concetration scalar field such that : where D diffusion coefficient cNambient concentration Provided by Cellular Automata simulation

11. Solution from the previous time step (denoted by k) Solution from the actual time step (denoted by k+1) Time discretization • For both heat and mass transport problems the following discretization with respect to time with trapezoid rule is applied: • Time disretization requires utilization of two computational meshes, from the actual and previous time step: M – mass matrix,  - time step,   [0,1] – time discretization scheme parameter.

12. Data structure for storing refined meshes • Actual computational mesh consists in finite elements represented by leaves of the refinement tree (the leaves representing active elements are denoted by blue color in the figure) • The refinement tree stores the history of mesh refinments executed on regular initial mesh • The self-adaptive hp-FEM code generates a sequence of nested meshes with corresponding approximation spaces

13. Incompatibility of meshes • The optimal meshes generated by the self-adaptive hp-FEM for the actual and previous time steps may be not compatible • If the meshes are not compatible, there is a need to project previous time step solution into the actual time step solution

14. General non-stationary algorithm generate initial_mesh with initial temperature and concentration fields distributions do time_step= 1, nr_time_steps generate coarse_mesh for actual time_step do iteration of self-adaptive hp­FEM for time_step project previous time_step solution into coarse_mesh solve the coarse_mesh problem generate the fine_mesh project previous time_step solution into fine_mesh solve the fine_mesh problem make decision about optimal refinements generate optimal_mesh coarse_mesh = optimal_mesh enddo • enddo

15. Initial mesh generation strategy I • The optimal mesh for the next time step is obtained from the same regular initial mesh as for the previous time step solution. • Advantages / disadvantages: • The optimal mesh for the new time step is not restricted by the structure of the previous time step • The whole process of generating the new optimal mesh is computationally expensive. • There is a need to store two computational meshes, from the actual and the previous time steps

16. Initial mesh generation strategy II • The optimal mesh from the previous time step is utilized as the initial mesh for the next time step. • Advantages / disadvantages: • The solution from the previous time step can be stored at the same data structure, since both refinement trees are compatible. • The optimal mesh from the previous time step may be not optimalfor the next time step, and multiple new refinements may be needed. • The size of the problem is growing with furthcoming time steps, since each time step requires some additional new refinements,

17. Initial mesh generation strategy III • The optimal mesh from the previous time step is utilized, but some partial unrefinement are executed, to step back before executing new refinements. • Advantages / disadvantages: • The optimal mesh for the new time step may require less refinements, and the process of the new optimal mesh generation may be less expensive. • The quality of the previous time step solution stored on the previous mesh can be decreased after performing proposed unrefinement. This can be overcome by storing a copy of the previous time mesh and projecting the previous time step solution into the actual mesh for the current time step.

18. Numerical experiments

19. Numerical experiments

20. Numerical experiments

21. Computational complexity Coarse mesh element Fine mesh element Number of degrees of freedom Thus, the fine mesh problem size where Computational complexity of a single iteration under the most efficient iterative solver computational complexity

22. Comparison of three strategies Comparison of the total computational costs for three different mesh management strategies

23. Numerical results - concentration field

24. Numerical results - temperature filed

25. Conclusions • We compared three different mesh managing strategies for the non-stationary self-adaptive hp­FEM code utilized to simulate the austenite-ferrite phase transformation. Each strategy concerns the utilization of the computational mesh and the solution from the previous time step during computations of an actual time step. • The first strategy generated a next time step mesh starting from the regular initial mesh. • The second strategy utilized previous time step mesh as a starting point for the next time step iteration. • The third strategy performed global unrefinement on the previous time step mesh before utilizing the mesh for the actual time step solution. • From the numerical simulations it follows that the second strategy is the most efficient strategy. This is because the number of time step iterations was small, and the optimal mesh sizes generated by other strategies are of the similar order. For the larger simulations, with larger number of time steps, the third strategy would be the most optimal.

26. Thank you The presented work has been supported by the Polish Ministry of Science grant no. 3 T08 B 055 29