Intro to FEM in 1D 2D
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Transcript of Intro to FEM in 1D 2D
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8/19/2019 Intro to FEM in 1D 2D
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Gaurav
Indian Institute of Technology Gandhinagar
Short course on
Soil-Structure Interaction
Computer Applications and Material Models
19-23 January, 2015
Beam on elastic foundation
Seepage through a porous medium
q x
y, v
k
2
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Analytical Solution ‘exact’ solution
Difficult for nonlinear problems defined in non-regulargeometries
Finite Difference Method Easy computer implementation
Difficult for problems defined in non-regular geometries
Finite Element Method (FEM)
Applicable in most scenarios, nice mathematicalstructure
Cumbersome computer implementation
Other methods 3
Express the problem in energy/weak form We don’t work with PDEs directly unlike the finite difference
method
Discretize domain into multiple finite sub-domains
Compute quantities in all elements individually
Gather (assemble) contributions of all elements in a centrallocation
Enforce boundary conditions
Solve resulting system of algebraic equations
Compute quantities of interest (stress, strain, etc.)
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Element-level contribution
Assembly
It can be shown that
This is one of the desiredcharacteristics of numerical methods
that use discretization to solve governing equations
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Linear elementsQuadratic convergence
• This is a typically observed errorconvergence behavior in FEA.
n
p
–
P n
2
1
3
1
Quadratic elementsCubic convergence
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Approximating area of circle Approximating center of mass of an arbitrarily shaped
object
Estimation of the value of
7 x
y
Governing differential equation (strong form)
Analytical solution
q( x) x, u( x) P
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Equivalent governing equation (energy form)
Discretize geometry (say, we use three nodes)
Energy form can be distributed over individual elements
x = 0 x = L/2 x = L
n = 1 n = 2 n = 3e = 1 e = 2
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Represent u( x) using piecewise linear functions
u(0) = u1
12(0.5 – x/L) 2( x/L – 0.5)2 x/L 2(1 – x/L)
u( x) = N 1( x) u1 + N 2( x) u2 + N 3( x) u3
u( L/2) = u2 u(L) = u3
Unknown nodaldisplacements
Known shape functions
N 1( x) + N 2( x) + N 3( x) = 1 Partition of unity 10
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Represent u( x) using piecewise linear functions
Want u( x) as close to uexact as possible
How? Principle of minimum potential energy (MPE)
u1
u( x)
u2
u3
uexact
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We have
minimize wrt {u}
Stiffness matrix Force vector
Strain energy External work
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u(0) = 0, AE = 1, L = 1, P = 1/3, q( x) = – x2.
Can we solve for {u} in its present form? No! Stiffness matrix is singular
Need to apply boundary conditions
Know: u1 = 0Cancelled equation
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u(0) = 0, AE = 1, L = 1, P = 1/3, q( x) = – x2.
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Can we further improve/streamline the process?
Shape functions defined in x – different definitions indifferent ranges of x
Integrations over individual elements – limits depend onelements
Makes it less amenable to computer implementation
Solution?: Define standard reference element
Real Element Reference Element
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This allows formulation of standard, linear bar element
Reference Element
u1 u2
u( x) = N 1(r ) u1 + N 2(r ) u2
N 1(r ) N 2(r )
What does K ij physically represent? 16
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Derivation of stiffness matrix using ‘unit deflectionmethod’ Properties: A, E , L
– AE/L 1 AE/L
AE/L 1 – AE/L
a a’ b
b b’ a
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Reference element formulation also
allows use of numerical quadrature Gauss-Legendre quadrature
r i
w i
N GP = 1
0.000 000 000 000 000 2.000 000 000 000 000
N GP = 2
±0.677 350 269 189 626 1.000 000 000 000 000
N GP = 3
±0.744 596 669 241 483 0.555 555 555 555 556
0.000 000 000 000 000 0.888 888 888 888 889
If 2 N GP ≥ n+1,quadrature is exact!
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q x y, v
Strain energy
Element Stiffness
Shape functions
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q = 1
x
y, v
EI = 1
L = 1
q = 1
What can we say about errors?
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Timoshenko beam element Considers shear deformation, separate approximations
for v and θ .
Beam-column element Considers axial deformations as well
Essentially, combination of beam and bar elements
These were 2D1D idealizations
Also possible to do 3D1D idealizations Three displacements (u, v, w) and three rotations (θ x, θ y,
θ z ) at each node
• A.J.M. Ferreira, MATLAB Codes for Finite Element Analysis: Solids and Structures,Springer, 2009.
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Parallels 1D FE formulation via the ‘reference element’
Real Element Reference Element
x
y
r
s
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Shape functions
r
sBilinear shape functions
N 1(r )
r
s
1
N 1(r , s)
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Shape functions
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Based on Lagrange polynomials
Linear 1D case
Linear 2D case:
order of polynomial
node
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2nd –order 1D element
N 3(r )
N 2(r )
N 1(r )
Reference Element
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2nd –order 2D element
r
s
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Other possibilities Combine Lagrange polynomials of different order for r
and s
e.g.: N (r , s) = l 22(r ) l 1
1( s)
Serendipity elements
Elements with no interior nodes
r
s
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Gaurav
Indian Institute of Technology Gandhinagar