Transcript FEA Short Course - University of Rhode Island
MCE 561 Computational Methods in Solid Mechanics
Introduction
Need for Computational Methods
•
Stress Analysis Solutions Using Either Strength of Materials or Theory of Elasticity Are Normally Accomplished for Regions and Loadings With Relatively Simple Geometry
•
Most Real World Problems Involve Cases with Complex Shape, Boundary Conditions and Material Behavior
•
Therefore a Gap Exists Between What Is Needed in Applications and What Can Be Solved by Analytical Closed form Methods
•
This Has Lead to the Development of Several Numerical/Computational Schemes Including: Finite Difference, Finite Element, Boundary Element, Discrete Element Methods
Finite Difference Method
Finite Difference Method (FDM)
The finite difference method replaces the derivatives in governing field equations by
difference quotients
which involve values of the solution at discrete mesh points in the domain under study. For example, the first derivative with respect to
x
may be represented by
(i-1) (i) (i+1) x
du
dx
u i u i
u i
1
x
1
u i u i
x
1
u i
2
x
1 ...
...
...
backward difference forward difference central difference
x
x
Repeated applications of this representation set up algebraic systems of equations in terms of the unknown nodal values
u i .
The major difficulty with this method lies in the inaccuracies in dealing with regions of complex shape, although this problem can be eliminated through the use of coordinate transformation techniques.
Finite Element Method (FEM)
Finite Element Method
Finite Element Method (FEM)
The fundamental concept of the finite element method lies in dividing the body under study into a finite number of pieces (subdomains) called
elements
. Particular assumptions are then made on the variation of the unknown dependent variable(s) across each element using so-called
interpolation or shape functions
. This approximated variation is quantified in terms of solution values at special locations within the element called the
nodes
. Through this discretization process, the method sets up an algebraic system of equations for unknown nodal values which approximate the continuous solution. Because element size and shape are variable, the finite element method can handle problem domains of quite complicated shape, and thus it has become a very useful and practical tool.
Boundary Element Method
Bounday Element Method (BEM)
The boundary element method is based upon an
integral statement
of the governing equations of the problem under study. The integral statement may be cast into a form which contains unknowns
only over the boundary
of the body domain. This boundary integral equation may then be solved using concepts from the finite element method; i.e. the boundary may be discretized into a number of elements and the interpolation approximation concept may then be applied. This method again produces an algebraic system of equations to solve for the unknown boundary nodal values, but the system is generally much smaller than that generated by internal discretization techniques such as finite element method. By avoiding interior discretization, the boundary element method can have significant advantages over finite element schemes for infinite or very large domains, and for cases where only boundary information is sought.
Boundary Element Node
Discrete Element Method
Discrete Element Method (DEM)
When the continuum under study is discrete in nature such as that found in particulate and granular materials, it is convenient to allow the element sub-domains to move freely subject to the mechanics of the problem. For such cases a technique called the discrete element method allows continuity of the element assembly to be relaxed and
contact conditions
between elements determine the deformations and motions of the material.
Individual elements can therefore have finite motions (displacements and rotations), and simplifying constitutive assumptions (rigid or linear elastic) are normally made for each element. Other similar techniques include
block element methods
or
discontinuous deformation analysis
.
Discrete Elements
Introduction to Finite Element Analysis
The finite element method is a computational scheme to solve field problems in engineering and science. The technique has very wide application, and has been used on problems involving stress analysis, fluid mechanics, heat transfer, diffusion, vibrations, electrical and magnetic fields, etc. The fundamental concept involves dividing the body under study into a finite number of pieces (subdomains) called elements (see Figure). Particular assumptions are then made on the variation of the unknown dependent variable(s) across each element using so-called interpolation or approximation functions. This approximated variation is quantified in terms of solution values at special element locations called nodes. Through this discretization process, the method sets up an algebraic system of equations for unknown nodal values which approximate the continuous solution. Because element size, shape and approximating scheme can be varied to suit the problem, the method can accurately simulate solutions to problems of complex geometry and loading and thus this technique has become a very useful and practical tool.
Advantages of Finite Element Analysis
Models Bodies of Complex Shape - Can Handle General Loading/Boundary Conditions - Models Bodies Composed of Composite and Multiphase Materials - Model is Easily Refined for Improved Accuracy by Varying Element Size and Type (Approximation Scheme) - Time Dependent and Dynamic Effects Can Be Included - Can Handle a Variety Nonlinear Effects Including Material Behavior, Large Deformations, Boundary Conditions, Etc.
Finite Element Codes
Based on the success and usefulness of the finite element method, numerous computer codes have been developed that implement the numerical scheme. Some codes are special purpose in-house while others are general purpose commercial codes. Some of the more popular commercial codes include: -
ABAQUS
-
ADINA
-
ALGOR
-
ANSYS
-
COSMOS
-
NASTRAN
Basic Concept of Finite Element Method
T
Any continuous solution field such as stress, displacement, temperature, pressure, etc. can be approximated by a discrete model composed of a set of piecewise continuous functions defined over a finite number of subdomains.
One-Dimensional Temperature Distribution
T
Exact Analytical Solution Approximate Piecewise Linear Solution
x x
Two-Dimensional Discretization
0 u(x,y) -1 -2 2 1 -3 4 3.5
Approximate Piecewise Linear Representation 3 2.5
y 2 1.5
1 -1 -0.5
0 0.5
1 x 1.5
2 2.5
3
Discretization Concepts
T Exact Temperature Distribution, T(x) x T
1
Finite Element Discretization Linear Interpolation Model (Four Elements) T 2 T
2
T 3 T 3 T 4 T 4 T 5 Quadratic Interpolation Model
1
(Two Elements) T 2 T 3 T 3 T 4 T 5 T T
1
T 2 T 3 T 4 T 5 Piecewise Linear Approximation
Temperature Continuous but with Discontinuous Temperature Gradients
x T T
1
T 2 T 3 T 4 T 5 x Piecewise Quadratic Approximation
Temperature and Temperature Gradients Continuous
Common Types of Elements
One-Dimensional Elements Line Rods, Beams, Trusses, Frames Two-Dimensional Elements Triangular, Quadrilateral Plates, Shells, 2-D Continua Three-Dimensional Elements Tetrahedral, Rectangular Prism (Brick) 3-D Continua
Discretization Examples
One-Dimensional Frame Elements Two-Dimensional Triangular Elements Three-Dimensional Brick Elements
Multi-Phase Material Problems
Material 1 Material 2 Material 3 Interface Behavior
Basic Steps in Finite Element Method
- Domain Discretization - Select Element Type (Shape and Approximation) - Derive Element Equations (Variational and Energy Methods) - Assemble Element Equations to Form Global System [K]{U} = {F} [K] = Stiffness or Property Matrix {U} = Nodal Displacement Vector {F} = Nodal Force Vector - Incorporate Boundary and Initial Conditions - Solve Assembled System of Equations for Unknown Nodal Displacements and Secondary Unknowns of Stress and Strain Values
Common Sources of Error in FEA
• • • •
Domain Approximation Element Interpolation/Approximation Numerical Integration Errors Computer Errors (Round-Off, Etc., )
Measures of Accuracy in FEA
Accuracy Error = |(Exact Solution)-(FEM Solution)| Convergence Limit of Error as: Number of Elements (h-convergence) or Approximation Order (p-convergence) Increases Ideally, Error
0 as Number of Elements or Approximation Order
Two-Dimensional Discretization Refinement Plate With a Circular Hole Example
(Discretization with 228 Elements) (Triangular Element) (Discretization with 912 Elements)
One Dimensional Examples
Bar Element
Uniaxial Deformation of Bars Using Strength of Materials Theory
u 1 u 2
1 2 Differenti al Equation :
d dx
(
au
)
cu
q
0 Boundary Conditions Specificat ion :
u
,
a du dx
q
1 Beam Element
Deflection of Elastic Beams Using Euler-Bernouli Theory
w 1 w 2
q
2
1 2 Differenti al Equation :
d
2
dx
2 (
b d
2
w
)
dx
2
f
(
x
) Boundary Conditions Specificat ion :
w
,
dw dx
,
b d
2
w dx
2 ,
d dx
(
b d
2
w
)
dx
2
Two Dimensional Scalar-Valued Problems
Triangular Element
f 3 3 2 f 2 1 f 1
One Degree of Freedom per Node
Example Differenti al Equation 2 f
x
2 2 f
y
2
f
(
x
,
y
) : Boundary Condtions Specificat ion f ,
d
f
dn
f
x n x
f
y n y
:
Two-Dimensional Vector/Tensor-Valued Problems
3
Triangular Element v 3 u 3 v 2 u 2
2
v 1
1
u 1
Two Degrees of Freedom per Node
Elasticity Field Equations in Terms of Displaceme nts 2
u
E
2 ( 1 )
x
u
x
v
y
F x
0 2
v
E
2 ( 1 )
y
u
x
v
y
F y
0 Boundary Conditons
u
,
v T T y x
C
11
u
x
C
12
C
66
u y
v y
n x
v
x
n x
C
12
C
66
u x
u
y C
v
x
n y
22
v y
n y
Development of Finite Element Equation
•
The Finite Element Equation Must Incorporate the Appropriate Physics of the Problem
•
For Problems in Structural Solid Mechanics, the Appropriate Physics Comes from Either Strength of Materials or Theory of Elasticity
•
FEM Equations are Commonly Developed Using Direct, Variational- Virtual Work or Weighted Residual Methods Direct Method Based on physical reasoning and limited to simple cases, this method is worth studying because it enhances physical understanding of the process Variational-Virtual Work Method Based on the concept of virtual displacements, leads to relations between internal and external virtual work and to minimization of system potential energy for equilibrium Weighted Residual Method Starting with the governing differential equation, special mathematical operations develop the “weak form” that can be incorporated into a FEM equation. This method is particularly suited for problems that have no variational statement. For stress analysis problems, a Ritz-Galerkin WRM will yield a result identical to that found by variational methods.
Simple Element Equation Example Direct Stiffness Derivation
u 1 u 2 F 1 F 2
1 2 Equilibriu Equilibriu
k
m at Node 1 m at Node 2
F
1
F
2
ku
1
ku
1
ku
2
ku
2 or in Matrix Fo rm
Stiffness Matrix
k k
k k
u u
2 1
F F
2 1 [
K
]{
u
} {
F
}
Nodal Force Vector
Common Approximation Schemes One-Dimensional Examples
Polynomial Approximation Most often polynomials are used to construct approximation functions for each element. Depending on the order of approximation, different numbers of element parameters are needed to construct the appropriate function. Linear Quadratic Cubic Special Approximation For some cases (e.g. infinite elements, crack or other singular elements) the approximation function is chosen to have special properties as determined from theoretical considerations