1. Advanced Scaling Techniques for the Modeling of Materials Processing Karem E. Tello Colorado School of Mines Ustun Duman Novelis Patricio F. Mendez Director, Canadian Centre for Welding and Joining University of Alberta

2. Phenomena in Materials Processing Transport processes play a central role – Heat transfer – Fluid Flow – Diffusion – Complex boundary conditions and volumetric factors: Free surfaces Marangoni Vaporization Electromagnetics Chemical reactions Phase transformations Multiple phenomena are coupled 2

3. Example: Weld Pool at High Currents 3 gouging region trailing region rim

4. Multiphysics in the Weld Pool 4 Driving forces in the weld pool (12) weld pool substrate solidified metal arc electrode

5. Multiphysics in the Weld Pool 5 Driving forces in the weld pool (12) – Inertial forces weld pool substrate solidified metal arc electrode

6. Multiphysics in the Weld Pool 6 Driving forces in the weld pool (12) – Inertial forces – Viscous forces weld pool substrate solidified metal arc electrode

7. Multiphysics in the Weld Pool 7 Driving forces in the weld pool (12) – Inertial forces – Viscous forces – Hydrostatic weld pool substrate solidified metal arc electrode  gh

8. Multiphysics in the Weld Pool 8 Driving forces in the weld pool (12) – Inertial forces – Viscous forces – Hydrostatic – Buoyancy weld pool substrate solidified metal arc electrode  gh  T

9. Multiphysics in the Weld Pool 9 Driving forces in the weld pool (12) – Inertial forces – Viscous forces – Hydrostatic – Buoyancy – Conduction weld pool substrate solidified metal arc electrode

10. Multiphysics in the Weld Pool 10 Driving forces in the weld pool (12) – Inertial forces – Viscous forces – Hydrostatic – Buoyancy – Conduction – Convection weld pool substrate solidified metal arc electrode

11. Multiphysics in the Weld Pool 11 Driving forces in the weld pool (12) – Inertial forces – Viscous forces – Hydrostatic – Buoyancy – Conduction – Convection – Electromagnetic weld pool substrate solidified metal arc electrode J B B J×BJ×B

12. Multiphysics in the Weld Pool 12 Driving forces in the weld pool (12) – Inertial forces – Viscous forces – Hydrostatic – Buoyancy – Conduction – Convection – Electromagnetic – Free surface weld pool substrate solidified metal arc electrode

13. Multiphysics in the Weld Pool 13 Driving forces in the weld pool (12) – Inertial forces – Viscous forces – Hydrostatic – Buoyancy – Conduction – Convection – Electromagnetic – Free surface – Gas shear weld pool substrate solidified metal arc electrode 

14. Multiphysics in the Weld Pool 14 Driving forces in the weld pool (12) – Inertial forces – Viscous forces – Hydrostatic – Buoyancy – Conduction – Convection – Electromagnetic – Free surface – Gas shear – Arc pressure weld pool substrate solidified metal arc electrode

15. Multiphysics in the Weld Pool 15 Driving forces in the weld pool (12) – Inertial forces – Viscous forces – Hydrostatic – Buoyancy – Conduction – Convection – Electromagnetic – Free surface – Gas shear – Arc pressure – Marangoni weld pool substrate solidified metal arc electrode 

16. Multiphysics in the Weld Pool 16 Driving forces in the weld pool (12) – Inertial forces – Viscous forces – Hydrostatic – Buoyancy – Conduction – Convection – Electromagnetic – Free surface – Gas shear – Arc pressure – Marangoni – Capillary weld pool substrate solidified metal arc electrode

17. Multicoupling in the Weld Pool 17 Hydrostatic Buoyancy Electromagnetic Free surface Capillary Gas shear Arc pressure Marangoni Inertial forces Viscous forces Conduction Convection

18. Multicoupling in the Weld Pool 18 Hydrostatic Buoyancy Electromagnetic Free surface Capillary Gas shear Arc pressure Marangoni Inertial forces Viscous forces Conduction Convection

19. Multicoupling in the Weld Pool 19 Hydrostatic Buoyancy Electromagnetic Free surface Capillary Gas shear Arc pressure Marangoni Inertial forces Viscous forces Conduction Convection

20. Multicoupling in the Weld Pool 20 Hydrostatic Buoyancy Electromagnetic Free surface Capillary Gas shear Arc pressure Marangoni Inertial forces Viscous forces Conduction Convection

21. Multicoupling in the Weld Pool 21 Hydrostatic Buoyancy Electromagnetic Free surface Capillary Gas shear Arc pressure Marangoni Inertial forces Viscous forces Conduction Convection

22. Multicoupling in the Weld Pool 22 Hydrostatic Buoyancy Electromagnetic Free surface Capillary Gas shear Arc pressure Marangoni Inertial forces Viscous forces Conduction Convection

23. Multicoupling in the Weld Pool 23 Hydrostatic Buoyancy Electromagnetic Free surface Capillary Gas shear Arc pressure Marangoni Inertial forces Viscous forces Conduction Convection

24. Disagreement about dominant mechanism 24 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. State of the Art in Understanding of Welding (and Materials) Processes Questions that can be “easily” answered – For a given current, gas, and geometry, what is the maximum velocity of the molten metal? – For a given set of parameters, what are the temperatures, displacements, velocities, etc? Questions more difficult to answer: – What mechanism is dominant in determining metal velocity? – If I am designing a weld, what current should I use to achieve a given penetration? – Can I alter one parameter and compensate with other parameters to keep the same result? 25

26. Scaling can help answer the “difficult” questions Dimensional Analysis – Buckingham’s “Pi” theorem “Informed” Dimensional Analysis – dimensionless groups based on knowledge about system Inspectional Analysis – dimensionless groups from normalized equations Ordering – Scaling laws from dominant terms in governing equations (e.g. Bejan, M M Chen, Dantzig and Tucker, Kline, Denn, Deen, Sides, Astarita, and more) 26

27. Typical ordering procedure 1.Write governing equations 2.Normalize the variables using their characteristic values. Some characteristic values might be unknown. This step results in differential expressions based on the normalized variables. 3.Replace normalized expressions into governing equations. 4.Normalize equations using the dominant coefficient 5.Solve for the unknown characteristic values – choose terms where they are present – make their coefficients equal to 1. 6.Verify that the terms not chosen are not larger than one. 7.If any term is larger than one, normalize equations again assuming different dominant terms. 27

28. Typical ordering procedure Limitations 1.Approximation of differential expressions can be grossly inaccurate not true in important practical cases! – Higher order derivatives – Functions with high curvature 28

29. Typical ordering procedure Limitations 2.Cannot perform manually balances for coupled problems with many equations when making coefficients equal to 1, there maybe more than one unknown impractical to check manually for all balances (there is no guaranteed unicity in ordering) 29

30. Order of Magnitude Scaling (OMS) Addresses the drawbacks 1.Table of improved characteristic values 2.Linear algebra treatment Mendez, P.F. Advanced Scaling Techniques for the Modeling of Materials Processing. Keynote paper in Sohn Symposium. August 27-31, 2006. San Diego, CA. p. 393-404. 30

31. OMS of a high current weld pool Goals: – Estimate characteristic values: velocity, thickness, temperature – Relate results to process parameters materials properties, welding velocity, weld current – Capture all physics, simplifications in the math – Identify dominant phenomena: gas shear? Marangoni? electromagnetic? arc pressure? 31 thickness velocity

32. 1. Governing Equations 32 U  z’ x z

33. 1. Governing Equations 33 Boundary Conditions: at free surface at solid-melt interface far from weld free surface solid-melt interface far from weld

34. 1. Governing Equations 34 Variables and Parameters – independent variables (2) – dependent variables (9) – parameters (18) from other models, experiments with so many parameters Dimensional Analysis is not effective

35. 2. Normalization of variables 35 unknown characteristic values (9):

36. 3. Replace into governing equations 36 governing equation

37. 3. Replace into governing equations 37 governing equation scaled variables OM(1)

38. 4. Normalize equations 38 governing equation scaled variables OM(1) normalized equation outputinput

39. 5. Solve for unknowns 39 outputinput two possible balances B1

40. 5. Solve for unknowns 40 outputinput two possible balances B1 B2

41. 5. Solve for unknowns 41 outputinput two possible balances B1 B2 balance B1 generates one algebraic equation:

42. 5. Solve for unknowns 42 outputinput two possible balances B1 B2 balance B1 generates one algebraic equation: balance B2 generates a different equation:

43. 6. Check for self-consistency 43 outputinput two possible balances B1 B2 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

44. Shortcomings of manual approach 44 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!

45. Shortcomings of manual approach 45 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! ? ? ? ? ? 1 equation 2 unknowns 1 equation 3 unknowns 1.Each balance equation involves more than one unknown

46. Shortcomings of manual approach 46 1.Each balance equation involves more than one unknown 2.A system of equations involves many thousands of possible balances 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!

47. Shortcomings of manual approach 47 all coefficients are power laws all terms in parenthesis expected to be OM(1)

48. Shortcomings of manual approach 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) 48

49. 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 49

50. Matrix of Coefficients 50 9 equations 6 BCs one row for each term of the equation

51. 51 9 equations 6 BCs one row for each term of the equation 18 parameters 9 unknown charact. values

52. Solve for unknowns using matrices 52 18 parameters 9 unknown charact. values [No]P’[No]P’ [N o ] S 9x9

53. 9 unknowns 18 parameters Matrix [S] Solve for unknowns using matrices

54. Check for self-consistency can be checked using matrix approach checking the 334098 combinations took 72 seconds using Matlab on a Pentium M 1.4 GHz 54 secondary terms submatrices of normalized secondary terms

55. Scaling results 55  c =36  m UcUc cc cc

56. Scaling results 56 arc pressure / viscouselectromagnetic / viscoushydrostatic / viscouscapillary / viscousMarangoni / gas shearbuoyancy / viscous gas shear / viscous convection / conduction inertial / viscous diff. = /diff.  plasma shear causes crater

57. Summary 57 Materials processes are “Multiphysics” and “Multicoupled” Scaling helps understand the dominant forces in materials processes Several thousand iterations are necessary for scaling The “Matrix of Coefficients” and associate matrix relationships help automate scaling

58. 58

59. Properties of Scaling Laws Simple closed-form expressions – Typically are exact solution of asymptotic cases – Display explicitly the trends in a problem insightful (explicit variable dependences) – generalize data, rules of thumb – Power Laws Only way to combine units “Everything plotted in log-log axes becomes a straight line” Are valid for a family of problems (which can be reduced to a “canonical” problem) – useful to interpolate / extrapolate, detect outliers – Range of validity can be determined (Process maps) Provide accurate approximations – can be used as benchmark for numerical models Useful for fast calculations – massive amounts of data (materials informatics) – meta-models, early stages of design – control systems Reductionist (system answers can be build by understanding the elements individually) 59 Simple, Accurate, General, Fast

60. 60 Calculation of a Balance 1.select 9 equations 2.select dom. input

61. 61 Calculation of a Balance 1.select 9 equations 2.select dom. input 3.select dom. output

62. 62 Calculation of a Balance 1.select 9 equations 2.select dom. input 3.select dom. output 4.build submatrix of selected normalized outputs 18 parameters 9 unknown charact. values [No]P’[No]P’ [N o ] S 9x9

63. 63 Scaling of FSW Crawford et al. STWJ 06 maximum temp? shear rate? thickness?

64. 64 FSW: Scaling laws

65. 65 FSW: Limits of validity “Slow moving heat source” – isotherms near the pin ≈ circular “Slow mass input” – deformation around tool has radial symmetry concentric with the tool “Thin shear layer” – the shear layer sees a flat (not cylindrical) tool Va/  << 1 Va  <<  a   << a (<0.3) (0.01-.3) (~0.1-0.3)

66. 66 FSW: Comparison with literature Stainless 304 Steel 1018 ~1 flat trend within limits

67. 67 FSW: Comparison with literature Stainless 304 Steel 1018 Ti-6Al-4V

68. 68 FSW: Comparison with literature Stainless 304 Steel 1018 C 1 = 0.76 C 2 = 0.33 C 3 = -0.89

69. 69 FSW: Comparison with literature Aluminum alloys ferrous alloys Ti-6Al-4V Corrected using trend based on shear layer thickness Good for aluminum, steel and Ti Good beyond hypotheses

70. 70 Other problems scaled Weld pool recirculating flows Arc – P.F. Mendez, M.A. Ramirez, G. Trapaga, and T.W. Eagar, Order of Magnitude Scaling of the Cathode Region in an Axisymmetric Transferred Electric Arc, Metallurgical Transactions B, 32B (2001) 547-554 Ceramic to metal bonding – J.-W. Park, P.F. Mendez, and T.W. Eagar, Strain Energy Distribution in Ceramic to Metal Joints, Acta Materialia, 50 (2002) 883-899 – J.-W. Park, P.F. Mendez, and T.W. Eagar, Residual Stress Release in Ceramic- to-Metal Joints by Ductile Metal Interlayers, Scripta Materialia, 53 (2005) 857-861 Penetration at high currents Electrode melting RSW

71. 71 Canadian Centre for Welding and Joining Vision and Mission: – Ensure that Canada is a leader of welding and joining technologies through research and development education application – The main focus of the Centre is meeting the needs of Canadian resource- based industries. Structure -Weldco/Industry Chair in Welding and Joining $4M -Metal products fabrication industry in Alberta: $4.8 billion in revenue in 2005, projected to $7.5 billion by 2009. -In oil sands, investment in major projects for the next 25 years exceed $200 billion with $86 billion already committed for starts by 2011

72. Shortcomings of manual approach 72 Boundary conditions

73. Promising approaches to answer the “difficult”questions closed form solutions – exact solutions – asymptotics / perturbation – dimensional analysis – regressions not considered “state of the art” – hold great promise – numerical, experiments are “state of the art” 73 Applied mathematics Engineering Scaling