1. Materials Science & Engineering University of Michigan
Summer School for Integrated Computational Materials Education Computational Mechanics and Finite Element Method
Katsuyo Thornton
Materials Science & Engineering
University of Michigan
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

2. Topics What is FEM? What is FEM good for?
Stiffness Method for Mechanical Equilibrium
What goes into the method
How it’s solved
More General Approach
Acknowledgement: a few slides were adopted from Edwin Garcia’s lecture in 2017.
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

3. Purpose: Learn what Finite Element Method is and what mathematics & physics it is based on.
Motivation: It’s the method behind most widely used tools (such as ANSYS/Abaqus/Fluent, COMSOL, etc.). Understanding the method can guide you to use the tools correctly .
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

4. What is FEM?
The finite element method is a numerical method for solving 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 interpolation functions. The solution is given by error reduction.
Physically, it is equivalent to dividing up a system into smaller pieces (elements) where each piece follow the law of nature.

6. 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.
BIG PICTURE: Relate the nodal values of the solutions to each other and known values (boundary conditions). A PDE turns into a matrix equation, which can then be numerically solved.
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

7. Element Types Felippa C., FEM Modeling: Introduction
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

8. Typical Application of FEM
Structural/Stress Analysis
Fluid Flow
Heat Transfer
Electro-Magnetic Fields
Soil Mechanics
Acoustics
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

9. Advantages Irregular Boundaries General Loads Different Materials
Boundary Conditions
Variable Element Size
Easy Modification
Dynamics
Nonlinear Problems (Geometric or Material)
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

10. Steps in FEM 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
Solve for Element Stresses and Strains
Interpret the Results
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

11. Stiffness Method A physically based FEM
Divide up a system into smaller pieces (elements) where each piece follow the law of nature
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

12. Definitions for this section
For an element, a stiffness matrix
is a matrix such that
where relates local coordinates
and nodal displacements
to local forces of a single element.
Bold denotes vector/matrices.
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

13. Spring Element k 1 2 L
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

14. Definitions node node k - spring constant These are scalar values
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

15. Stiffness Relationship for a Spring
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

16. Steps in Process 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 Unknowns (Displacements)
Solve for Element Stresses and Strains
Interpret the Results
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

17. Step 1 - Select the Element Typek 1 2 T T L
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

18. Step 2 - Select a Displacement Function
This is a function, not a nodal value.
Assume a displacement function
Assume a linear function.
Number of coefficients = number of d-o-f
Write in matrix form.
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

19. Express as function of and
Solve for a2 :
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

20. Substituting back into:
Yields:
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

21. In matrix form:
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

22. Here, the bases are linear functions!
Shape Functions
N1 and N2 are called Shape Functions or
Interpolation Functions. They express the
shape of the assumed displacements.
N1 =1 N2 =0 at node 1
N1 =0 N2 =1 at node 2
N1 + N2 =1
Recall Fourier Transform, in which the basis functions are sinusoidal functions.
Here, the bases are linear functions!
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

23. N1 1 2 L
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

24. N2 1 2 L
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

25. N1 N2 1 2 L
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

26. Step 3 - Define Strain/Displacement and Stress/Strain Relationships
Here is where physics comes into play!
T - tensile force  - total elongation
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

27. Step 4 - Derive the Element Stiffness Matrix and Equations
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

28. Stiffness Matrix
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

29. Note: not simple addition! An example later.
Step 5 - Assemble the Element Equations to Obtain the Global Equations and Introduce the B.C.
(e) indicates “element” index
Note: not simple addition!
An example later.
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

30. Step 6 - Solve for Nodal Displacements
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

31. Step 7 - Solve for Element Forces
Once displacements at each
node are known, then substitute
back into element stiffness equations
to obtain element nodal forces.
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

32. Two Spring Assembly 2 1 3 k1 k2 2 1 x F3x F2x
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

33. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

34. Continuity/Compatibility Condition
Elements 1 and 2 remain connected
at node 3. This is called the continuity
or compatibility requirement.
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

35. This is just adding.
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

36. Substitution will give you these equations
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

37. But, this way is really cumbersome! Computers can’t do this either!
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

38. Assembly of [K] - An Alternative Method2 1 2 1 3 x
F3x
F2x k1 k2
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

39. Recall that
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

40. Assembly of [K] - An Alternative Method1 3
Insert row and column 2 with zeros
Flip row, flip columns, and insert row 1 with zeros 3 2
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

41. Expand Local [k] matrices to Global Size
Computers can do this!
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

42. Net Force
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

43. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

44. Compatibility Displacements of the shared nodes are equal.
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

45. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

46. Boundary Conditions Must Specify B.C.’s to prohibit rigid body motion.
Two type of B.C.’s
Homogeneous - displacements = 0
Nonhomogeneous - displacements = nonzero value
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

47. Delete row and column corresponding to B.C.
Homogeneous B.C.’s
Delete row and column corresponding to B.C.
Solve for unknown displacements.
Compute unknown forces (reactions) from original (unmodified) stiffness matrix.
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

48. 2 1 2 1 3 x
F3x
F2x k1 k2
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

49. Example: Homogeneous BC, d1x=0
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

50. Turn the Continuum Equations into Algebraic Equations
More General Approach
Turn the Continuum Equations into Algebraic Equations
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

51. Spatial Discretization of an Elastic Solid

52. Another Example: Heat Equation
Consider steady-state heat equation with source term and spatially dependent material properties:
COMSOL solution for steady-state heat transfer

53. General Approaches of Finite Element Method
Define a set of piecewise polynomial functions (related to the element types and shape functions).
The solution can be represented by this piecewise polynomial function.
The differential equation can then be integrated over each element with the approximate solution, leading to a set of algebraic equations (key!).
The computational mesh no longer needs to be Cartesian or regularly ordered, a primary advantage of the FEM.
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

54. Solution is Represented as a Sum of Basis Functions
Solution for u is represented by sum of the basis functions times constant coefficients.
In FEM, the basis function is called the shape function, and nonzero only on one element.
At each point, the coefficients of the polynomial are stored
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

55. In 1D Linear elements use linear basis functions
This is equivalent to piecewise linear approximation.
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

56. In 2D 7 8 9 6 1 2 3 4 5

57. For Mechanics: Energy Balance
stored energy = applied external loads
stored
elastic
energy
work performed by
gravity
externally applied
traction

58. Application to Heat Equation
Consider steady-state heat equation with source term and spatially dependent material properties:
The solution is represented by
The residual (how different the approximated solution is from the true solution) is given by:
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

59. The Method of Weighted Residuals (Specifically, Galerkin)
The method of weighted residuals requires the weighted integrals of the residuals to be zero.
Weight function, wi(x)=φi (x) (Galerkin)
Residual,
Weighted residual (WR):
L1 and L2 are the location of the domain boundaries. i i
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

60. MWR, continued Consider weight functions, For each ϕi(x), set WR=0:
This provides N equations that can be used to solve for in
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

61. Summary & Conclusion
We covered a high-level overview of FEM, focusing on the physical model (the stiffness method), advantages & applications.
FEM is a deep, complex approach. Some textbooks are in three volumes!
It can take an entire semester to cover introduction to FEM!
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

62. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

63. Summary & Conclusion
We covered a high-level overview of FEM, focusing on the physical model (the stiffness method), advantages & applications.
FEM is a deep, complex approach. Some textbooks are in three volumes!
It can take an entire semester to cover introduction to FEM!
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

64. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

65. Example of Software Finite Element Software (focused applications)
Fluent (fluid dynamics)
ANSYS Mechanical, Structural, ABAQUS, etc.
Finite Element Software (multiphysics)
With various modules: heat transfer, mechanics, etc.
COMSOL, ANSYS, SIMULIA, etc.
Free: OOF, FreeFEM++, FEniCS, etc.
See

66. Questions?
Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018