1. Performing a Computer Simulation using C++

2. Structure of Presentation
Description of Simulation

3. Structure of Presentation
Description of Simulation
Results of Simulation

4. Structure of Presentation
Description of Simulation
Results of Simulation
Overview of C++

5. Structure of Presentation
Description of Simulation
What is being simulated
Results of Simulation
Overview of C++

6. Structure of Presentation
Description of Simulation
What is being simulated
What is the algorithm
Results of Simulation
Overview of C++

7. Description of Simulation

8. Description of Simulation
What is being simulated?

9. Description of Simulation
What is being simulated?
One-dimensional Lattice of masses on springs.

10. Description of Simulation
What is being simulated?
One-dimensional Lattice of masses on springs.

11. Description of Simulation
What is being simulated?
One-dimensional Lattice of masses on springs.

21. Comparison of Quadratic Potential
And Toda Potential

22. Quadratic Potential
We expect the energy to stay in the initial modes.

23. Quadratic Potential
We expect the energy to stay in the initial modes.
Toda Potential
We expect the energy to migrate into all modes.

24. Surprising result discovered by FPU:
Energy periodically migrates BACK into the initial modes.
Toda Potential
We expect the energy to migrate into all modes.

25. Surprising result discovered by FPU:
Energy periodically migrates BACK into the initial modes.
Surprising result #2: Solitons, particle-like waves that maintain their shape forever without dispersing or breaking up.
Toda Potential
We expect the energy to migrate into all modes.

26. Surprising result #2: Solitons, particle-like waves that maintain their shape forever without dispersing or breaking up.
Periodic boundary conditions required.
I won’t demonstrate these.

27. Comparison of Quadratic Potential
And Toda Potential

28. Quadratic Potential
Good to test that our algorithm works.

29. Quadratic Potential
Good to test that our algorithm works.
Toda Potential
Try to reproduce the periodic energy migration.

30. Description of Simulation
What is the algorithm?

31. Description of Simulation
What is the algorithm?
BAD approach: Euler’s method

32. BAD approach: Euler’s method

33. BAD approach: Euler’s method
•Energy diverges

34. BAD approach: Euler’s method
•Energy diverges
•Takes a long time to run

35. GOOD approach: Hessian matrix method

36. GOOD approach: Hessian matrix method

37. GOOD approach: Hessian matrix method

38. GOOD approach: Hessian matrix method

39. GOOD approach: Hessian matrix method

40. GOOD approach: Hessian matrix method

41. GOOD approach: Hessian matrix method
•Energy doesn’t diverge

42. GOOD approach: Hessian matrix method
•Energy doesn’t diverge
•Much faster to run

43. GOOD approach: Hessian matrix method

44. GOOD approach: Hessian matrix method

45. Review of Algorithm

46. Results of Simulation

47. Results of Simulation Quadratic Potential (Linear Force)
Algorithm is fast and stable

48. Results of Simulation Quadratic Potential (Linear Force)
Algorithm is fast and stable
Energy remains in initial modes

49. Results of Simulation Quadratic Potential (Linear Force)
Algorithm is fast and stable
Energy remains in initial modes
Toda Potential (Nonlinear Force)
Algorithm is stable if the nonlinear constant is not too large

50. Results of Simulation Quadratic Potential (Linear Force)
Algorithm is fast and stable
Energy remains in initial modes
Toda Potential (Nonlinear Force)
Algorithm is stable if the nonlinear constant is not too large
Takes too long to see energy migrate to other modes

51. Results of Simulation Quadratic Potential (Linear Force)
Algorithm is fast and stable
Energy remains in initial modes
Toda Potential (Nonlinear Force)
Algorithm is stable if the nonlinear constant is not too large
Takes too long to see energy migrate to other modes
For just the right combination, we see the energy migrate back to the first mode (m=k=b=1, dt=.01, 8 particles, 5 second simulation)

52. Quick overview of C++

53. Quick overview of C++
Object-oriented: Severe encapsulation

54. Quick overview of C++ Object-oriented: Severe encapsulation
Heavily type-safe

55. Quick overview of C++ Object-oriented: Severe encapsulation
Heavily type-safe
Most errors caught by compiler… once it compiles, it will probably run “fairly” well

56. Quick overview of C++ Object-oriented: Severe encapsulation
Heavily type-safe: what does it mean?

57. Quick overview of C++ Object-oriented: Severe encapsulation
Heavily type-safe: what does it mean?
•Every token is of a certain “type”

58. Quick overview of C++ Object-oriented: Severe encapsulation
Heavily type-safe: what does it mean?
• Easy debugging: debug just the type itself, not the program.

59. Quick overview of C++ Object-oriented: Severe encapsulation
Inheritance: derive new types from old types

60. Quick overview of C++ Object-oriented: Severe encapsulation
Inheritance: derive new types from old types
Almost no risk of introducing bugs into old functionality

61. Quick overview of C++ Object-oriented: Severe encapsulation
Inheritance: derive new types from old types
Almost no risk of introducing bugs into base type’s functionality
Derived type can be very small and easy to understand

62. Quick overview of C++ Object-oriented: Severe encapsulation
Inheritance: derive new types from old types
Almost no risk of introducing bugs into base type’s functionality
Derived type can be very small and easy to understand
Polymorphism: derived type can “override” key behavior of base type

63. Example: Shape Base type: SHAPE Derived Type: Circle
Circle overrides function “draw”
Derived Type: Polygon
Derived Type: Square
Square overrides function “draw”
Derived Type: Triangle
Triangle overrides function “draw”

64. Quick overview of C++ Object-oriented: Severe encapsulation
Heavily type-safe:

65. Quick overview of C++ Object-oriented: Severe encapsulation
Heavily type-safe:
Programmer is careful to use exactly the right types for the job

66. Quick overview of C++ Object-oriented: Severe encapsulation
Heavily type-safe:
Programmer is careful to use exactly the right types for the job
Programming consists of creating types. Once they are created, they do their job automatically.

67. Quick overview of C++ Object-oriented: Severe encapsulation
Programming consists of creating types. Once they are created, they do their job automatically.

68. Quick overview of C++ Object-oriented: Severe encapsulation
Programming consists of creating types. Once they are created, they do their job automatically.
This allows for highly extensible, error-free code.

69. Quick overview of C++
The “main” loop usually consists of creating one or two objects, that’s all.

70. Quick overview of C++
The “main” loop usually consists of creating one or two objects, that’s all.
It is a completely different way of programming than “structured” programming.

71. Quick overview of C++
The “main” loop usually consists of creating one or two objects, that’s all.
It is a completely different way of programming than “structured” programming.
The concept of “flow of control” is strictly avoided except at the lowest levels.

72. Quick overview of C++
Disadvantages: Takes a long time to program.

73. Quick overview of C++
Using the MatLab C++ library:

74. Quick overview of C++ Using the MatLab C++ library:
Matlab is a “structured” programming environment.

75. Quick overview of C++ Using the MatLab C++ library:
Matlab is a “structured” programming environment.
C++ is an “object-oriented” programming environment.

76. Quick overview of C++ Using the MatLab C++ library:
Matlab is a “structured” programming environment.
C++ is an “object-oriented” programming environment.
It is hard to make a library that works well for both MatLab and for C++ programmers.

77. Quick overview of C++ Using the MatLab C++ library:
It is hard to make a library that works well for both MatLab and for C++ programmers.

78. Quick overview of C++ Using the MatLab C++ library:
It is hard to make a library that works well for both MatLab and for C++ programmers.
MatLab chose to create a simple interface, without any derived objects.

79. Quick overview of C++ Using the MatLab C++ library:
It is hard to make a library that works well for both MatLab and for C++ programmers.
MatLab chose to create a simple interface, without any derived objects.
This allows MatLab programmers to use “structured” programming in C++, without utilizing object-oriented programming.

80. Quick overview of C++ Using the MatLab C++ library:
It is hard to make a library that works well for both MatLab and for C++ programmers.
MatLab chose to create a simple interface, without any derived objects.
This allows C++ programmers to create their own object-oriented libraries to fully utilize object-oriented programming.

81. Quick overview of C++ Using the MatLab C++ library:
My code is basically structural.
It would take a long time to develop an effective object-oriented library using the simple base classes provided by MatLab.

82. Further Work to be Done:
Add periodic boundary conditions
Observe solitons