Intro to FEM in 1D 2D

8/19/2019 Intro to FEM in 1D 2D

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Gaurav

Indian Institute of Technology Gandhinagar

[email protected]

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

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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

[email protected]

Intro to FEM in 1D 2D

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