ME 405 Applications of Finite Element Analysis
Introduction to Finite Element Method
Introduction
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Finite Element Method (FEM) is a numerical method for solving differential equations
FEM
PDEs
for fluid or structural
mechanics
Algebraic
equations
[A]{x} = {B}
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Why? - because we generally cannot solve most of the equations analytically,
however we have several direct and iterative methods for algebraic system of
equations.
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Typical engineering applications
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Structural analysis - Computational Structural Dynamics (CSD)
Fluid flow analysis – Computational Fluid Dynamics
Heat transfer
Acoustics
Mathematical Modeling
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Introduction
The FEM basically includes the following steps
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Partition the domain with elements.
 Elements can be of arbitrary shape
 Unstructured elements are possible
 Triangular, quad; tetrahedral hexahedral or hybrid…
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Define Interpolating or shape functions
 In each element a parametric representation of the unknown variables is defined.
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Define an integral formulation (weak form) of the equations
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The integral formulation is discretized based on the variable definitions in each element and
the element geometry.
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Assemble the global system using the element stiffness matrices and nodal connectivity data.
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Apply the initial (for time dependent problems) and boundary conditions and solve the linear
system of equations.
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Basic concepts
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A simple element that can be used to examine the stiffness concept is the spring element.
Spring force-displacement relation
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Basic concepts
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Force equilibrium for spring element
Node i
Node j
Matrix form
Note that k is symmetric.
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Basic concepts
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Consider a system of two springs
Element 1
Element 2
Nodal forces
In matrix form
1
2
3
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Basic concepts
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Observe that the global system is a superposition of local element matrices
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Element 1
1
2
2
2
Element 2
3
2
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This matrix assembly procedure is very
simple and applies to finite element
modeling of any complex structure.
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Basic concepts
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Boundary conditions
 For the global system we need to apply boundary conditions otherwise the global stiffness
matrix will be singular.
 We will typically have displacement and traction BCs for structural analysis.
 Consider for example,
Applied loads.
Fixed at node 1.
 We want to solve for the unknown displacements and the reaction force 𝐹1 .
 Applying the BCs the system becomes
and
Solve for the
displacements
and
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Basic concepts
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Example
 For the following spring system find the global stiffness matrix.
assemble
symmetric, banded, but singular.
BCs need to be applied.
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Basic concepts
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Example
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Basic concepts
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Example
Apply BCs
The finite element equation system is
The 1st and 4th rows
and columns are deleted
The reactions are obtained from the 1st ans 4th equations
For spring 2
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Bar Element
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Linear-elastic material
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Small deformations
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Constant cross-sectional area (prizmatic)
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Directly applicable to the solution of pin-connected trusses (Plane or 3D trusses.)
• In 2D – x and y are global – can be used for
plane truss analysis for example.
• The local direction is 𝑥
• We will first consider the stiffnes matrix in
local coordinates with direct stiffnes
method.
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Bar Element
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Bar element in local coordinates
Hooke’s law:
Strain-displacement relation:
Force equilibrium
Select a displacement function
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Bar Element
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Shape functions
Displacement function
Shape
functions for
bar element
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Bar Element
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Stifness matrix – direct method
The bar act as a sprink with
element stiffness matrix
Element equilibrium equation is
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Bar Element
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Example
Element 1
Element 2
Find the stresses in the two bar assembly
which is loaded with force P, and
constrained at the two ends.
Assembled system
Now apply BCs.
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Bar Element
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Example
Boundary conditions
• Exact stress values within the
linear theory for 1-D bar structures
• No need to further divide the
elements.
• Averaged cross-sectional areas
should be used for elements if the
bars are tapered.
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Basic concepts review
FEM for structural analysis
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Divide structure into pieces (finite elements)
 The elements are connected at the nodes to model the complex structure.
 The unknows are defined at the nodes.
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Difine how the variables behave in each element
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Based on this definition each element will have a stiffness matrix which relates the nodal
displacement to element forces. For example,
𝑘𝑒 𝑥𝑒 = 𝑓 𝑒
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The size of the stiffness matrix depends on the degrees of freedom (number of element nodes and
number of unknowns at each node.)
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Each element stiffness matrix will be assembled based on nodal connectivity data to form a
global stiffness matrix for the whole structure, for example in the form of
𝐾 𝑥 = 𝐹
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When the BCs are applied this linear system can be solved for the nodal displacements, then the
stresses and strains can be calculated for each element.
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References
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Y. Liu, 1998. Introduction to the Finite Element Method, lecture notes, University of
Cincinnati.
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Daryl L. Logan, A First Course in the Finite Element Method, 2007.
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