1. Materials Science & Engineering University of Michigan
Summer School for Integrated Computational Materials Education Computational Mechanics: Basic Concepts and Finite Element Method
Katsuyo Thornton
Materials Science & Engineering
University of Michigan

2. This lecture will...
Provide you with the general background related to the Computational Mechanics Module.
Topics
Brief review of continuum mechanics of an elastic solid
Finite element method for mechanics problems
(Stiffness method)
Finite element method as a general partial differential equation solver (brief, if time allows)
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

3. Continuum Mechanics The study of the physics of continuous materials
From Wikipedia
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

4. Continuum Mechanics The study of the physics of continuous materials
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

5. Basic Concepts of (Static) Elasticity
An elastically deformed material returns to its original shape upon the release of applied force – reversible.
Compare to plasticity – irreversible changes to materials.
Basic equation governing elasticity considers:
Mechanical equilibrium (Force must balance)
Constitutive equation (What reaction does material exhibit in response to strain?)
Will first examine one dimensional system.
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

6. 1D Example: Spring-Mass
State is described by force and displacement of each mass.
Mechanical equilibrium: Net force on each mass is zero.
Constitutive equation relates force with material properties and displacement. k 1 2 L
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

7. 1D Example: Spring-Mass
Mechanical equilibrium: net force at mass i
Constitutive equation of linear elasticity (Hooke’s Law) k 1 2 L
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

8. Solid Mechanics: 3D Consider a volume element inside a body.
From Wikipedia
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

9. Solid Mechanics: 3D
In multi-dimension, deformation along one direction leads to deformation along another direction.
Green: undeformed body
Red: after tensile strain
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

10. Tensors
Multidimensional array of numbers with respect to a basis (including scalar)
Orders of tensors
Scalar, 0th-order tensor, f
Vector, 1st-order tensor, v = [f1, f2, f3, ...]
Matrix, 2nd-order tensor
i.e., the order of a tensor can be understood as the dimension
The number of elements in each dimension is usually determined by the spatial dimension associated with the problem.
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

11. Stress Tensor Stress is a second-order tensor.
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

12. Strain Strain tensor Strain results from deformation of a body.
Strain is a gradient of displacement.
Constant displacement DOES NOT lead to deformation.
Constant strain is a uniform stretch/compression of the body.
In a spring-mass system, the displacement of the mass is measured away from the equilibrium position. The spring can be viewed as having to experience uniform strain.
Strain tensor
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

13. Stress & Strain
Stress is equivalent to the force in the spring-mass system.
Stress has a unit of force per unit area.
Strain is related to the displacement of the mass.
Strain is dimensionless, as it is the gradient of displacement (unit of length) with respect to the position (unit of length).
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

14. Linear Elasticity Equivalent to Hooke’s Law for springs.
In the most general form
Repeated indices imply summation.
For isotropic materials, the elastic constants can be reduced to
K and m are the bulk modulus and shear modulus.
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

15. Elasticity vs. Plasticity
When a material experiences a large deformation, its atomic constituency arranges itself in such a way that it will not recover to the original state.
Bond breaking
Dislocation motion
Plastic deformation is a challenging multiscale problem!
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

16. Force Balance
Change in stress with respect to position is the unbalanced force.
Force balance in 3D or
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

17. 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.
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

18. 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.
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

19. Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 201419

20. 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.
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

21. Element Types Felippa C., FEM Modeling: Introduction
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

22. 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 Santa Barbara, CA, July 14-25, 2014

23. Typical Applications of FEM
Structural/Stress Analysis
Fluid Flow
Heat Transfer
Electro-Magnetic Fields
Soil Mechanics
Acoustics
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

24. 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
Calculate Element Stresses and Strains
Interpret the Results
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

25. Stiffness Method A physically based FEM
Divide up a system into smaller pieces (elements) where each piece follow the law of nature

26. 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 Santa Barbara, CA, July 14-25, 2014

27. Spring Element k 1 2 L
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

28. Definitions node node k - spring constant
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

29. Stiffness Relationship for a Spring
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

30. 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)
Calculate Element Stresses and Strains
Interpret the Results
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

31. Step 1 - Select the Element Typek 1 2 T T L
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

32. Step 2 - Select a Displacement Function
Assume a displacement function
Assume a linear function.
Number of coefficients = number of local d-o-f (degree of freedom)
Write in matrix form.
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

33. Express as function of and
Solve for a2 :
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

34. Substituting back into:
Yields:
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

35. In matrix form: or
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

36. 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
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

37. N1 1 2 L
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

38. N2 1 2 L
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

39. N1 N2 1 2 L
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

40. Step 3 - Define Strain/Displacement and Stress/Strain Relationships
where
T - tensile force  - total elongation
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

41. Step 4 - Derive the Element Stiffness Matrix and Equations
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

42. This describes the interactions between two nodes (1 & 2)
Stiffness Matrix
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

43. Note: not simple addition!
Step 5 - Assemble the Element Equations to Obtain the Global Equations and Introduce the B.C.
Note: not simple addition!
An example later.
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

44. Step 6 - Solve for Nodal Displacements
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

45. 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 Santa Barbara, CA, July 14-25, 2014

46. Two Spring Assembly 2 1 3 k1 k2 2 1 x F3x F2x
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

47. Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

48. Elements 1 and 2 remain connected
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 Santa Barbara, CA, July 14-25, 2014

49. (Includes only those from springs)
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

50. Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

51. Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

52. Assembly of [K] - An Alternative Method2 1 2 1 3 x
F3x
F2x k1 k2
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

53. Assembly of [K] - An Alternative Method
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

54. Expand Local [k] matrices to Global Size
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

55. Force Equilibrium
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

56. Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

57. Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

58. Compatibility
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

59. Boundary Conditions (B.C.s)
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 Santa Barbara, CA, July 14-25, 2014

60. 2 1 2 1 3 x
F3x
F2x k1 k2
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

61. 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 Santa Barbara, CA, July 14-25, 2014

62. Example: Homogeneous BC, d1x=0
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

63. Beyond the Stiffness Method
The stiffness method provides a good model for solid mechanics problems.
However, it is unclear how the method could be applied to a diverse range of problems important in MSE.
Now, we will briefly learn about using FEM to solve a partial differential equation (diffusion equation).
The method is called the Method of Minimal Weighted Residual (MWR).
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

64. Model Equation: Steady-State Diffusion Equation
Consider dimensional steady-state diffusion equation with source term q(x):
For illustration, we restrict ourselves to 1D:
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

65. Approximate Solution
we express the approximate solution as a linear combination of basis functions:
where ai are constants.
For accuracy, the best bases are those that behave similarly to the solution. However, for computational efficiency, simpler bases (such as a linear function) are often a better fit.
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

66. Basis function = Shape function
FEM uses shape functions to approximate the solution.
shape function is another name for the basis function for the FEM.
For the FEM with linear basis, we use a similar form as the stiffness method: xi i
xi-1
i-1
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

67. Piecewise Linear Basis Function
xi-1 xi i 1
xi+1
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

68. Derivatives of the Piecewise Linear Basis Function
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

69. Residual
Take the model diffusion equation, and move the source term to the left hand side:
Let be the estimate of the solution for u. Then the residual is given by
For an exact solution, = 0 for all x.
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

70. Method of Minimal Weighted Residual
How do we evaluate how close the approximate solution is to the true solution?
Let the ith weight function, wi(x), to be nonzero only on two consecutive elements around the node i. (This is often the same as the basis function.)
Weighted residual to be minimized:
The weighted residual becomes algebraic once the integral is performed.
xi-1 xi i 1
xi+1
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

71. General Summary (1) Solution is approximated by
We need to determine ai
The condition for determining ai is to reduce the error.
The error (residual) is determined by considering how the value of a different diff. eq. is when the approximate solution is substituted for the solution.
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

72. General Summary (2)
In FEM, the basis functions i are the shape functions.
The residual can be calculated in various ways. We will focus on the method of weighted residual.
In particular, when the weight functions are identical to the basis functions, the method is called the Galerkin method.
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014

73. Steps in MWR FEM Discretize and Select Element Type
Select a Solution Function
Select a Set of Basis Functions
Derive Local “Stiffness” Matrix and Equations
Assemble Equations and Introduce B.C.’s
Solve for the Unknown Coefficients
Rebuild the Solution from the Coefficients
Interpret the Results
Summer School for Integrated Computational Materials Education Santa Barbara, CA, July 14-25, 2014