Summer School for Integrated Computational Materials Education 2015 Computational Mechanics: Basic Concepts and Finite Element Method Katsuyo Thornton

Summer School for Integrated Computational Materials Education 2015

Computational Mechanics:Basic Concepts and Finite

Element Method

Katsuyo Thornton

Materials Science & Engineering

University of Michigan

This lecture will...• Provide you with the general background related to

the Computational Mechanics Module.• Topics

o Brief review of continuum mechanics of an elastic solid

o Finite element method for mechanics problems

(Stiffness method)o Finite element method as a general partial

differential equation solver (brief, if time allows)

Summer School for Integrated Computational Materials Education Ann Arbor, Michigan, June 15-26, 2015

Continuum Mechanics

• The study of the physics of continuous materials

From Wikipedia


Continuum Mechanics

• The study of the physics of continuous materials


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.


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.

1 2k

LSummer School for Integrated Computational Materials Education

Ann Arbor, Michigan, June 15-26, 2015

1D Example: Spring-Mass

• Mechanical equilibrium: net force at mass i

• Constitutive equation of linear elasticity (Hooke’s

Law)

1 2k

LSummer School for Integrated Computational Materials Education

Ann Arbor, Michigan, June 15-26, 2015

Solid Mechanics: 3D

• Consider a volume element inside a body.

From Wikipedia


Solid Mechanics: 3D

• In multi-dimension, deformation along one direction leads to deformation along another direction.

• Green: undeformed body• Red: after tensile strain


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.


Stress Tensor

• Stress is a second-order tensor.


Strain• 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


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).


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.


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!


Force Balance

• Change in stress with respect to position is the unbalanced force.

• Force balance in 3D

or


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.


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.



19

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.


Element Types

Felippa C., FEM Modeling: Introductionhttp://caswww.colorado.edu/courses.d/IFEM.d/IFEM.Ch06.d/IFEM.Ch06.pdf


Advantages

• Irregular Boundaries• General Loads• Different Materials• Boundary Conditions• Variable Element Size• Easy Modification• Dynamics• Nonlinear Problems (Geometric or Material)


Typical Applications of FEM

• Structural/Stress Analysis• Fluid Flow• Heat Transfer• Electro-Magnetic Fields• Soil Mechanics• Acoustics


Steps in FEM

1. Discretize and Select Element Type

2. Select a Displacement Function

3. Define Strain/Displacement and Stress/Strain Relationships

4. Derive Element Stiffness Matrix & Eqs.

5. Assemble Equations and Introduce B.C.s

6. Solve for the Unknown Displacements

7. Calculate Element Stresses and Strains

8. Interpret the Results


Stiffness Method

A physically based FEM

Divide up a system into smaller pieces (elements) where each piece

follow the law of nature

Definitions for this section

For an element, a stiffness matrix is a matrix such thatwhere relates local coordinates and nodal displacementsto local forces of a single element.

Bold denotes vector/matrices.


Spring Element

1 2

k

L


Definitions

node

k - spring constant

node


Stiffness Relationship for a Spring


Steps in Process

1) Discretize and Select Element Type

2) Select a Displacement Function

3) Define Strain/Displacement and Stress/Strain Relationships

4) Derive Element Stiffness Matrix & Eqs.

5) Assemble Equations and Introduce B.C.s

6) Solve for the Unknowns (Displacements)

7) Calculate Element Stresses and Strains

8) Interpret the Results


Step 1 - Select the Element Type

1 2

k

L

T T


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.


Express as function of and

Solve for a2 :


Substituting back into:

Yields:


In matrix form:

or


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 1N1 =0 N2 =1 at node 2N1 + N2 =1


1 2

N1

L


1 2

N2

L


1 2

N1 N2

L


Step 3 - Define Strain/Displacement and Stress/Strain Relationships

T - tensile force - total elongation

where


Step 4 - Derive the Element Stiffness Matrix and Equations


Stiffness Matrix

This describes the interactions between two nodes (1 & 2)


Step 5 - Assemble the Element Equations to Obtain the Global Equations and Introduce the B.C.

Note: not simple addition!An example later.


Step 6 - Solve for Nodal Displacements


Step 7 - Solve for Element Forces

Once displacements at eachnode are known, then substitute back into element stiffness equationsto obtain element nodal forces.


Two Spring Assembly

k1

1 2

k2

1 2

3x

F3x F2x



Elements 1 and 2 remain connectedat node 3. This is called the continuity or compatibility requirement.

Continuity/Compatibility Condition


(Includes only those from springs)




Assembly of [K] - An Alternative Method

k1

1 2

k2

1 2

3 x

F3x F2x


Assembly of [K] - An Alternative Method


Expand Local [k] matrices to Global Size


Force Equilibrium




Compatibility


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


k1

1 2

k2

1 2

3 x

F3x F2x


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.


Example: Homogeneous BC, d1x=0


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).


Model Equation:Steady-State Diffusion Equation

• Consider dimensional steady-state diffusion equation with source term q(x):

• For illustration, we restrict ourselves to 1D:


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.



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

Piecewise Linear Basis Function

xi-1 xi

i

1

xi+1


Derivatives of the Piecewise Linear Basis Function


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.


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


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.


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.


Steps in MWR FEM

1. Discretize and Select Element Type

2. Select a Solution Function

3. Select a Set of Basis Functions

4. Derive Local “Stiffness” Matrix and Equations

5. Assemble Equations and Introduce B.C.’s

6. Solve for the Unknown Coefficients

7. Rebuild the Solution from the Coefficients

8. Interpret the Results


Documents

Summer School for Integrated Computational Materials Education 2015 Computational Mechanics: Basic Concepts and Finite Element Method Katsuyo Thornton