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FEM  Method  

We  will  explore:  

1-­‐D  linear  &  higher  order  elements  2-­‐D    triangular  &  rectangular  elements  

Powerful  method  developed  originally  to  solve  structural  mechanics  problems  (e.g.  bridges,  buildings,  etc..)  

Can  be  applied  to  flow,  solute,  and  heat  transport  problems  

FEM  Method  •  Has  underlying  theoreJcal  basis  in    branch  of  mathemaJcs  called  calculus  of  variaJon  

•  Within  each  element,  the  unknown  variable  is  approximated  using  polynomials  that  depend  only  on  the  spaJal  coordinates  (i.e.  h  =  a  +  bx  +  cy  +...).    The  verJces  of  the  elements  are  referred  to  as  nodes.    

•  Originally  developed  for  solid  mechanics  problems     Tri  

FEM Mesh of Baseball and Bat

FEM Mesh of Dam

Triangular Element

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Finite  Element  Method  &  Calculus  of  VariaJons  

•  Involves  minimizing  residual  errors  using  integraJon  •  Conceptually  similar  to  the  cannon  ball  trajectory  problem  solved  using  the  calculus  of  

variaJons  method  (see  Feynman’s  lecture  notes)  

x(t)

η(t) x(t)

η(t)

-  one possible path -  true path -  deviation

feynman_vc.pdf True path is the path of minimum energy!

•  FD  method  using  Taylor’s  Series  Expansion  

•  FE  method  uses  IntegraJon  by  Parts    

Example:

Solve the following Integral. The integrand is the product of two functions

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Steps  in  FEM  Method,  1D  Example  

•  Formulate  Diff.  Eq.  

T  –  transmisivity  (m2/day)  H  –  head  (m)  Q  –  recharge  (m/day)  

h(0,t)  =  specified  head  q(L,t)  =  specifed  flux  

•  DiscreJze  SoluJon  Domain  

•  Define  Local  Coordinate  System  

L

Steps  in  FEM  Method,  1D  Example  

•  Construct  (linear)  Trial  SoluJon  for  each  element  

φ  -­‐  Shape  funcJons    

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Steps  in  FEM  Method,  1D  Example  

•  Construct  (linear)  Trial  SoluJon  for  each  element  

φ  -­‐  Shape  funcJons    

Shape  FuncJon  ProperJes  

•  Linear  variaJon  across  element  from  0  to  1  

•  DerivaJves  of  linear  shape  funcJons    are  constant  

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1D-­‐FEM  

•  SubsJtute  Trial  SoluJon  into  Diff.  Eq.  

•  MulJply  trial  soluJon  by  a  weighJng  funcJon  and  require  the  weighted  residual  errors  integrate  to  zero  across  the  element  

v  –  weighJng  FuncJon  

1D-­‐FEM  Steps  

•  Use  IntegraJon  by  parts  to  “lower  the  order  of    the  PDE”  

•  SubsJtute  Trial  SoluJon  and  weighJng  funcJon  into  “weak  form  of  PDE”  

v  –  weighJng  FuncJon  

0

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1D-­‐FEM  Steps  

•  This  results  in  a  system  of  algebraic  EquaJons  

1D-­‐FEM  Steps  

•  Evaluate  Integrals:  m=1,  n=1  

•  Evaluate  Integrals:  m=2,  n=1  

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ProperJes  of  Shape  FuncJons:  

This is true for any point along the element (i.e. )

ProperJes  of  Shape  FuncJons:  

1D-FEM

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FEM-­‐1D  Steps  

Assemble Global Matrix:

Steps  1D  FEM  

•  Impose  specified  Head  Boundary  CondiJon  @  x=0  

•  Impose  specified  Flux  Boundary  CondiJon  @  x=L  

Generalized Tridiagonal Matrix….look familiar??

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1D  FEM  Steps  

•  Let:  

•  Global  Matrix  Becomes:    

•  SoluJon  (Gaussian  EliminaJon):  

AnalyJcal  SoluJon  Comparison  

Unlike the finite difference method, the finite element method provides us with an estimate of the hydraulic heads everywhere within the solution domain.

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DeterminaJon  of  Elemental  Fluxes  

FEM:

Analytical:

Higher  Order  Elements  

x=0 x=Δx/2 x=Δx

h1 h2 h3

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Lagrange  Family  of  Shape  FuncJons  

Example 1: N=2 (linear element)

Example 2: N=3 (quadatric element)

Lagrange Family Shape Functions: Quadratic Elements

Properties of Lagrange Shape Functions

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DerivaJves  of  Shape  FuncJons  

1D-­‐FEM-­‐QuadraJc  

•  SubsJtute  Trial  SoluJon  into  Diff.  Eq.  

•  MulJply  trial  soluJon  by  a  weighJng  funcJon  and  require  the  weighted  residual  errors  integrate  to  zero  across  the  element  

v  –  weighJng  FuncJon  

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1D-­‐FEM  QuadraJc  Steps  

•  This  results  in  a  system  of  algebraic  EquaJons  

Evaluate  QuadraJc  Amn  •  Evaluate  Integrals:  m=1,  n=1  

•  Evaluate  Integrals:  m=2,  n=1  

You can do the rest….

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Load  Vector  

You can do B3

Final  Amn  Matrix  

Final  Bm  Vector  

Quad. Element Linear Element

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Assemble  Global  Matrix  

Note Amn Matrix is no longer tridiagonal

Solve  System  of  EquaJons  Let: T(element-1) = 3 T(element-2) = 3 Δx = 1 qo = 1 ho = 0 Q(element-1) = 6 Q(element-2) = 6

h1 = 0 h2 = 1.91 h3 = 3.333 h4 = 4.25 h5 = 4.667

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Solve  System  of  EquaJons  

(use local coordinate system For elemental flux calc. (0<x<Δx)

Transient  FEM  Problems  

The finite element solution to this time dependent problem is very similar to the approach taken to solve Poisson's Equation. We use the same trial solutions:

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Transient  FEM  Problems  

Since hn is not a function of “t”, we get the following integrand For the time dependent derivative:

Transient  FEM  Problems  

So our governing equation can be recast in matrix form as:

Recall we already know what Amn is

So now we need to solve for Pmn:

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Transient  FEM  Problems  Evaluate the matrix:

Transient  FEM  Problems  Evaluate the matrix:

Recall:

Putting it all Together:

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Fully  Implicit  FormulaJon  

Writing out the equations for element 1, nodes “1” and “2”: