Finite Element Method in 1-D*cem01.ucsd.edu/~vitaliy/courses/ece222c/ece222c_set5_FEM.pdf · 1...
Transcript of Finite Element Method in 1-D*cem01.ucsd.edu/~vitaliy/courses/ece222c/ece222c_set5_FEM.pdf · 1...
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Finite Element Method in 1-D*ECE 222c, UCSD
Instructor: Vitaliy Lomakin
*Used material: Notes by J. Jin, ECE, UIUCand J. Jin, The Finite Element Method in EM, Wiley 2002
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Finite Element Method in 2-D*ECE 222c, UCSD
Instructor: Vitaliy Lomakin
*Based on the notes by J. Jin, ECE, UIUCand J. Jin, The Finite Element Method in EM, Wiley 2002
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Boundary-Value Problem
ΩΓ
n̂
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Equivalent Variational Problem
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Equivalent Variational Problem
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Equivalent Variational Problem
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Equivalent Variational Problem
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Basic FEM Steps
1. Discretization/subdivision of solution domain
2. Selection of interpolation schemes
3. Formulation of the system of equations
4. Solution of the system of equations
1-D: 2-D: 3-D:
Linear or higher-order polynomials
Using either the Ritz or Galerkin method:Formulate elemental equations and assemble
Using either a direct or iterative method
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FEM Analysis – Domain subdivision
Step 1: Domain Discretization
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FEM Analysis – Domain subdivision
Step 1: Domain Discretization
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FEM Analysis – Domain subdivision
Step 1: Domain Discretization
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FEM Analysis – Domain subdivision
Step 1: Domain Discretization
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FEM Analysis – Element interpolation
Step 2: Element Interpolation
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e
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FEM Analysis – Element interpolation
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FEM Analysis – Element formulation
Step 3: Formulation of the System of EquationsA. Elemental equations
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FEM Analysis – Element formulation
Elemental functional:
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FEM Analysis – Element formulation
Integration formula:
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FEM Analysis – Element formulation
Use matrix notation:
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FEM Analysis – Assembly
B. Assembly
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FEM Analysis – Assembly
Apply the stationarity condition:
Carry out the summation:
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FEM Analysis – Assembly
How to carry out the summation?
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FEM Analysis – Assembly
Example: 1
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4
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FEM Analysis
1. Start from a null matrix and add in the first element:
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FEM Analysis
2. Add in the second element:
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FEM Analysis
3. Add in the third element:
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FEM Analysis
4. Add in the fourth element:
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FEM Analysis
5. Follow a similar procedure:
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FEM Analysis – Apply BC
C. Impose the Dirichlet Boundary Condition:
To impose , simply set:
To maintain symmetry, set:
Approach #1:
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FEM Analysis – Apply BC
After imposing , , :
RemainsSymmetric!
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FEM Analysis – Apply BC
Can be made smaller:
Worthwhile when there are many prescribed nodes.
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FEM Analysis – Apply BC
Approach #2 (Simple one):
To impose , simply set:
After imposing , , :
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FEM Analysis – Sample Program
c.....Input data descriptionc c nn total number of nodes c For i = 1 to nn, input:c x(i) x-coordinate c y(i) y-coordinate c end forc c ne total number of elements c For e = 1 to ne, input:c alpha(e) value of alphac beta(e) value of beta c f(e) value of f c For i = 1 to 3, input:c n(i,e) global node number c end forc end for
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FEM Analysis – Sample Program
c c n1 number of nodes with prescribed valuesc For i = 1 to n1, input:c p(i) prescribed value of phi c nd(i) global node number c end forc c.....Initialize the matrix [K]
do 1 i = 1, nndo 1 j = 1, nn
1 k(i,j) = 0.Cc.....Start to assemble all area elements in Omega
do 4 e = 1, ne
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FEM Analysis – Sample Program
c..... Calculate b^e_i and c^e_i (i=1,2,3) i = n(1,e)j = n(2,e)m = n(3,e)be(1) = y(j) - y(m) be(2) = y(m) - y(i) be(3) = y(i) - y(j) ce(1) = x(m) - x(j) ce(2) = x(i) - x(m) ce(3) = x(j) - x(i)
Cc..... Calculate Delta^e
deltae = 0.5*(be(1)*ce(2)-be(2)*ce(1))
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FEM Analysis – Sample Program
c..... Generate the elemental matrix [K^e]do 2 i = 1, 3 do 2 j = 1, 3
if (i.eq.j) then del_ij = 1.0
else del_ij = 0.0
endif2 ke(i,j) = alphax(e)*(be(i)*be(j)
& + ce(i)*ce(j))/(4.0*deltae) & + beta(e)*(1.+del_ij)*deltae/12.
cc..... Add [K^e] to [K]
do 3 i = 1, 3 do 3 j = 1, 3
3 k(n(i,e),n(j,e)) = k(n(i,e),n(j,e))+ke(i,j) c
4 continue
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FEM Analysis – Sample Program
c.....Impose the Dirichlet boundary condition do 8 i = 1, n1
b(nd(i)) = p(i) k(nd(i),nd(i)) = 1. do 7 j = 1, nn
if(j.eq.nd(i)) go to 7 b(j) = b(j) - k(j,nd(i))*p(i) k(nd(i),j) = 0.k(j,nd(i)) = 0.
7 continue 8 continue
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Electrostatic Problems
Partial differential equation:
Boundary conditions:
Continuity conditions:
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Electrostatic Problems
Example:
Problem: To compute the characteristic impedance
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Electrostatic Problems
Mesh: Equi-potential:
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Electrostatic Problems
Axisymmetric (body of revolution):
Along the z-axis:
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Electrostatic Problems
Example:
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Electrostatic Problems
Mesh:
Equi-potential:
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Magnetostatic Problems
2-D:
Continuity conditions:
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Magnetostatic Problems
Axisymmetric (body of revolution):
Continuity conditions:
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Electrodynamic Problems
2 mm resolution14 kinds of tissue
Currents:RF shield
Head:
Currentelements
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64 MHz 128 MHz
171 MHz 256 MHz
Example – Electric field (no load)
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64 MHz 128 MHz
171 MHz 256 MHz
Example – Magnetic field (no load)
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64 MHz 128 MHz
171 MHz 256 MHz
Example – Magnetic field (with load)
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Electrodynamic ProblemsWaveguide device analysis:
Question: How to find the boundary condition at AB and CD?
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Electrodynamic Problems
At AB:
At CD:
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Electrodynamic ProblemsExample:
Equi-potential:
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Electrodynamic Problems
Scattering analysis:
Question: How to find the boundary condition on theartificial boundary?
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Electrodynamic Problems
Asymptotic expansion:
First-order ABC:
Take the derivative:
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Electrodynamic Problems
From a circular to an non-circular boundary:
First-order ABC:
For total field:
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Electrodynamic Problems
Example:
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Electrodynamic Problems
Example:
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Electrodynamic Problems
Example:
Amplitude of the scattered field