1. Katsuyo Thornton*, R. Edwin García✝, Larry Aagesen*
Summer School for Integrated Computational Materials Education Computational Mechanics Pre-Module Lecture
Katsuyo Thornton*, R. Edwin García✝, Larry Aagesen*
*Materials Science & Engineering
University of Michigan
✝: Materials Engineering
Purdue University 1

2. Purposes of the Module
Develop understanding of and intuition for mechanics of materials through hands-on exercises
Learn how computational tools can be used to determine stress distributions in material microstructures
Demonstrate the importance of continuum mechanics via an application to a polycrystal. 2

3. How Do We Solve the Equations
Once you obtain a PDE, there are many ways to solve the problem.
Finite Element Analysis or Finite Element Method has been the dominant approach in computational solid mechanics.
Relatively good convergence (higher accuracy with fewer mesh points).
Internal boundary conditions.
There are other methods that allow solutions, including a reformulation of the original equation, which can easily be solved using the finite difference method. 17

4. What is FEM?
The finite element method is a numerical method to solve problems of engineering and physics.
Useful for problems with complicated geometries, loadings, and material properties where analytical solutions cannot be obtained.
Mathematically, the PDE is converted to its variational (integral) form. An approximate solution is given by a linear combination of trial functions. The solution is given by error reduction.
Physically, it is equivalent to dividing up a system into smaller pieces (elements) where each piece follows the law of nature. 18

5. Discretizations
Model a physical body by dividing it into an equivalent system of smaller bodies or units (finite elements) interconnected at points common to two or more elements (nodes or nodal points) and/or boundary lines and/or surfaces. 19

7. Steps in FEM (numerical)
Discretize and Select Element Type
Select a Displacement Function
Define Strain/Displacement and Stress/Strain Relationships
Derive Element Stiffness Matrix & Eqs.
Assemble Equations and Introduce B.C.s
Solve for the Unknown Displacements
Calculate Element Stresses and Strains
Interpret the Results 23

9. g13 g8
g14 g4
g12 g3 g9
g10 g6
g11 g2 g1 g7 g5

10. Finite Elements of Microstructures
1. Start with a micrograph
2. Assign material properties
3. Perform virtual experiments
4. Visualize
and quantify

11. Modeling Material Properties

12. Collection of properties
Material
Collection of properties
Microstructure
Associates materials to phases (pixels)
Image
Associates colors to pixels
Skeleton
Geometry of mesh
Mesh
Equations, fields, and boundary conditions
Solution
Macroscopic properties, material evolution

13. Running OOF2 in the nanoHUB

14. The Problem of Meshing Images

15. Fitting a Mesh to Pixel Boundaries
Mesh operations work to minimize an “energy” functional E of the mesh
ai= fractional area of pixel type i
N = total number of pixel types

16. homogeneity E
shape

17. Fitting a Mesh to Pixel Boundaries
Initial Coarse Mesh
Refining Elements
Monte Carlo Node Motion
and Edge Swapping
Operations to
reduce E
Final Mesh

18. Adapting your Mesh is Like Washing your Laundry
Not one single operation will do the job
Sometimes you need to apply the same set of operations several times to get the effect you want
The same script typically applies to similar looking microstructures
Dirty skeletons go here

19. The Final Skeleton?

20. What Defines a Good Mesh?
Mesh reproduces the geometrical features of material or part
Elements must have a “good” shape
The fields of mesh can be solved for
The solution of the fields are very close to the analytic solution

21. Badly Shaped Elements
Good Shapes
Bad Shapes

22. Badly Shaped Meshes

23. Coarse Meshes

24. +Q -Q

25. Mesh Size Effects

26. Mesh Size Effects

27. Mesh Size Effects

28. Unsolvable Meshes ; ; Zeroth value of properties
Large difference in properties

29. A Good Mesh?

30. Properties of a Good Mesh
Reproduces accurately the geometry of the system
Is coarse in those regions where the fields are not varying rapidly
Is refined wherever the fields are varying rapidly
Takes a reasonable time to solve
Has “just enough” degrees of freedom

32. Extras and Deleted Scenes

33. The Error of the Solution
Ideally, we want to solve exactly for
In practice, however, we get
The relative error is: *

34. Number of fields, f, per node
Error will Depend on:
Number of bits, x, to represent a double
(smallest number you can represent, S)
S =1/2x ×
Number of fields, f, per node f
Number of nodes, N N
smaller error will be:
emin > S*f*N

35. Something you will Hear About: The Condition Number
define the magnitude of a matrix as:
:eigenvalue
Define the condition number as:

36. Why you Should Care about the Condition Number
In an ideal world, A=1
However, in real-life materials: if if if

37. How it Relates to the Error of the Calculation
relative
error
relative
accuracy
of the solution
the bigger the A
the harder to solve

38. Assigning Fields and Equations
displacement:
making something
active:
to solve for it

39. Solving the Fields
tolerance: how much error are you willing to live with?
linear solvers: numerics
to invert the FEM
matrices
preconditioners: black magic to improve on solvability

41. Lots of Options in the Skeleton Page
Methods to
improve mesh

42. The Anneal Method randomly displace a node by an amount delta
if move decreases the energy of element, accept it
go to 1

43. The Refine Method
Refine those elements whose energy is greater than specified threshold

44. The Swap Edges Method

45. Snap Nodes