Advanced Scaling Techniques for the Modeling of Materials Processing

1. Advanced Scaling Techniques for the Modeling of Materials Processing Patricio F. Mendez Colorado School of Mines

2. Goals • For people less familiar with scaling • will show how scaling is especially helpful for materials processes • For people familiar with scaling • will show a new relationship that permits to automate part of the scaling process • The reasoning applies to almost all materials processes • For clarity, I’ll use a particular welding problem as an example, but the approach is valid beyond welding

3. Materials Processes are “Multiphysics” and Coupled • Welding example: free surface depression of weld pool. Can induce defects and lower productivity

4. Materials Processes are “Multiphysics” and Coupled • Multiphysics in the weld pool (12) electrode arc solidified metal weld pool substrate

5. Materials Processes are “Multiphysics” and Coupled • Multiphysics in the weld pool • Inertial forces electrode arc solidified metal weld pool substrate

6. Materials Processes are “Multiphysics” and Coupled • Multiphysics in the weld pool • Inertial forces • Viscous forces electrode arc solidified metal weld pool substrate

7. Materials Processes are “Multiphysics” and Coupled • Multiphysics in the weld pool • Inertial forces • Viscous forces • Hydrostatic electrode arc rgh solidified metal weld pool substrate

8. Materials Processes are “Multiphysics” and Coupled • Multiphysics in the weld pool • Inertial forces • Viscous forces • Hydrostatic • Buoyancy electrode arc brghDT solidified metal weld pool substrate

9. Materials Processes are “Multiphysics” and Coupled • Multiphysics in the weld pool • Inertial forces • Viscous forces • Hydrostatic • Buoyancy • Conduction electrode arc solidified metal weld pool substrate

10. Materials Processes are “Multiphysics” and Coupled • Multiphysics in the weld pool • Inertial forces • Viscous forces • Hydrostatic • Buoyancy • Conduction • Convection electrode arc solidified metal weld pool substrate

11. Materials Processes are “Multiphysics” and Coupled • Multiphysics in the weld pool • Inertial forces • Viscous forces • Hydrostatic • Buoyancy • Conduction • Convection • Electromagnetic electrode arc J B B J×B solidified metal weld pool substrate

12. Materials Processes are “Multiphysics” and Coupled • Multiphysics in the weld pool • Inertial forces • Viscous forces • Hydrostatic • Buoyancy • Conduction • Convection • Electromagnetic • Free surface electrode arc solidified metal weld pool substrate

13. Materials Processes are “Multiphysics” and Coupled • Multiphysics in the weld pool • Inertial forces • Viscous forces • Hydrostatic • Buoyancy • Conduction • Convection • Electromagnetic • Free surface • Gas shear electrode arc t solidified metal weld pool substrate

14. Materials Processes are “Multiphysics” and Coupled • Multiphysics in the weld pool • Inertial forces • Viscous forces • Hydrostatic • Buoyancy • Conduction • Convection • Electromagnetic • Free surface • Gas shear • Arc pressure electrode arc solidified metal weld pool substrate

15. Materials Processes are “Multiphysics” and Coupled • Multiphysics in the weld pool • Inertial forces • Viscous forces • Hydrostatic • Buoyancy • Conduction • Convection • Electromagnetic • Free surface • Gas shear • Arc pressure • Marangoni electrode arc t solidified metal weld pool substrate

16. Materials Processes are “Multiphysics” and Coupled • Multiphysics in the weld pool (12) • Inertial forces • Viscous forces • Hydrostatic • Buoyancy • Conduction • Convection • Electromagnetic • Free surface • Gas shear • Arc pressure • Marangoni • Capillary electrode arc solidified metal weld pool substrate

17. Materials Processes are “Multiphysics” and Coupled • Inertial forces • Viscous forces Capillary Hydrostatic Buoyancy Marangoni • Conduction • Convection Arc pressure Gas shear Electromagnetic Free surface

18. Materials Processes are “Multiphysics” and Coupled • Inertial forces • Viscous forces Capillary Hydrostatic Buoyancy Marangoni • Conduction • Convection Arc pressure Gas shear Electromagnetic Free surface

19. Materials Processes are “Multiphysics” and Coupled • Inertial forces • Viscous forces Capillary Hydrostatic Buoyancy Marangoni • Conduction • Convection Arc pressure Gas shear Electromagnetic Free surface

20. Materials Processes are “Multiphysics” and Coupled • Inertial forces • Viscous forces Capillary Hydrostatic Buoyancy Marangoni • Conduction • Convection Arc pressure Gas shear Electromagnetic Free surface

21. Materials Processes are “Multiphysics” and Coupled • Inertial forces • Viscous forces Capillary Hydrostatic Buoyancy Marangoni • Conduction • Convection Arc pressure Gas shear Electromagnetic Free surface

22. Materials Processes are “Multiphysics” and Coupled • Inertial forces • Viscous forces Capillary Hydrostatic Buoyancy Marangoni • Conduction • Convection Arc pressure Gas shear Electromagnetic Free surface

23. Materials Processes are “Multiphysics” and Coupled • Inertial forces • Viscous forces Capillary Hydrostatic Buoyancy Marangoni • Conduction • Convection Arc pressure Gas shear Electromagnetic Free surface

24. Disagreement about dominant mechanism • Experiments cannot show under the surface • Numerical simulations have convergence problems with a very deformed free surface • Proposed explanations for very deformed weld pool • Ishizaki (1980): gas shear, experimental • Oreper (1983): Marangoni, numerical • Lin (1985): vortex, analytical • Choo (1991): Arc pressure, gas shear, numerical • Rokhlin (1993): electromagnetic, hydrodynamic, experimental • Weiss (1996): arc pressure, numerical

25. Scaling of a high current weld pool • Goals: • Identify dominant phenomena: • gas shear? Marangoni? electromagnetic? arc pressure? • Relate results to process parameters • materials properties, welding velocity, weld current • Estimate characteristic values: • velocity, thickness, temperature velocity thickness

26. Scaling of a high current weld pool • Governing equations (9): z’ x z w U

27. Scaling of a high current weld pool at free surface at solid-melt interface • Boundary Conditions: far from weld free surface solid-melt interface far from weld

28. Scaling of a high current weld pool • Variables and Parameters • independent variables (2) • dependent variables (9) • parameters (18) with so many parameters Dimensional Analysis is not effective from other models, experiments

29. Classical Scaling Approach • Scale variables and differential expressions • Assume a set of dominant driving forces • Normalize equations • Solve for the unknown terms • Verify self-consistency • If not self-consistent, return to 3. Roughly, this is the approach suggested by Dantzig and Tucker, Bejan, Kline, Denn, Deen, Sides, Chen, Astarita, and more

30. Classical Scaling Approach unknown characteristic values (9):

31. Classical Scaling Approach governing equation

32. Classical Scaling Approach governing equation scaled variables OM(1)

33. output input input Classical Scaling Approach governing equation scaled variables OM(1) normalized equation

34. output input input Classical Scaling Approach two possible balances B1

35. output input input Classical Scaling Approach two possible balances B1 B2

36. output input input Classical Scaling Approach two possible balances balance B1 generates one algebraic equation: B1 B2

37. output input input Classical Scaling Approach two possible balances balance B1 generates one algebraic equation: balance B2 generates a different equation: B1 B2

38. output input input Classical Scaling Approach two possible balances balance B1 generates one algebraic equation: balance B2 generates a different equation: self-consistency: choose the balance that makes the neglected term less than 1 B1 B2

39. Classical Scaling Approach two possible balances balance B1 generates one algebraic equation: balance B2 generates a different equation: self-consistency: choose the balance that makes the neglected term less than 1 TWO BIG PROBLEMS FOR MATERIALS PROCESSES!

40. Classical Scaling Approach ? two possible balances 1 equation 2 unknowns balance B1 generates one algebraic equation: ? ? ? 1 equation 3 unknowns balance B2 generates a different equation: ? self-consistency: choose the balance that makes the neglected term less than 1 TWO BIG PROBLEMS FOR MATERIALS PROCESSES! • Each balance equation involves more than one unknown

41. Classical Scaling Approach two possible balances • Each balance equation involves more than one unknown • A system of equations involves many thousands of possible balances balance B1 generates one algebraic equation: balance B2 generates a different equation: self-consistency: choose the balance that makes the neglected term less than 1 TWO BIG PROBLEMS FOR MATERIALS PROCESSES!

42. Scaled equations (9) all coefficients are power laws all terms in parenthesis expected to be OM(1)

43. Scaled BCs (6) Boundary conditions

44. Iterative process • Simple scaling approach involves 334098 possible combinations • There are 116 self-consistent solutions • there is no unicity of solution • we cannot stop at first self-consistent solution • self-consistent solutions are grouped into 55 classes (1- 6 solutions per class)

45. Automating iterative process • Power-law coefficients can be transformed into linear expressions using logarithms • Several power law equations can then be transformed into a linear system of equations • Normalizing an equation consists of subtracting rows

46. Matrix of Coefficients one row for each term of the equation 9 equations 6 BCs

47. 9 unknown charact. values 18 parameters one row for each term of the equation 9 equations 6 BCs

48. Calculation of a Balance • select 9 equations • select dom. input

49. Calculation of a Balance • select 9 equations • select dom. input • select dom. output

50. Calculation of a Balance • select 9 equations • select dom. input • select dom. output • build submatrix of selected normalized outputs 18 parameters 9 unknown charact. values [No]S 9x9 [No]P’