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finite element method
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V = V1 V = V2 symmetry Boundary conditions
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Assume that
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Locations & voltages are known!
?? Ve satisfies Laplace’s equation
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V1 V2 V3
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V1 V2 V3
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calculate V1 V3 V2 Internal voltage distribution
Known voltages and locations calculate
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definition Area of triangle?
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Show that the determinant of the matrix is equal to twice the area of a triangle.
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x y 1 2 3 Area of triangle?
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The determinant of this matrix is equal to twice the area of a triangle.
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definition
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Recall that this is twice the area Of the triangle
subscripts = 1 Recall that this is twice the area Of the triangle
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subscripts = 0
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Voltage distribution Within the triangle Is determined by the Voltages at the Corners and the locations of the corners are known!
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V3= 1 V1= 0 V2= 1
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MATLAB V1= 0 V2= 1 V3= 1 V1= 0 V2= 1 V3= 1
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V1= 0 V2= 1 V3= 1
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V1= 0 V2= 1 V3= 1
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constants is a Dirichlet matrix
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V1= 0 V2= 1 V3= 1 is a Dirichlet matrix
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V1= 0 V2= 1 V3= 1
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V6= 3 V4= 1 V5= 1
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V6= 3 V4= 1 V5= 1
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V1= 0 V2= 1 V3= 1 V6= 3 V4= 1 V5= 1
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V1= 0 V2= 1 V3= 1 V6= 3 V4= 1 V5= 1
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V1= 0 V2= 1 V3= 1 V6= ??? V4= 1 V5= 1
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In order to minimize this energy, V must be a solution of Laplace’s equation.
Let there be another solution U that satisfies the boundary conditions. Linear media implies superposition!
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Either 1) V is specified (U = 0) or 2) the normal derivative of V = 0
Either 1) V is specified (U = 0) or 2) the normal derivative of V = 0.<-- symmetry
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Minimum with the Actual solution
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is a Dirichlet matrix Known & unknown
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symmetry symmetry
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V = 10 V = 0 Locate nodes # of nodes = 11 2) Draw triangles
3) Identify nodes of triangles # of elements = 12 V = 0 4) Specify the known potentials at the nodes = 8 5) Specify the initial conditions at the 3 internal nodes 6) Calculate the potentials at the 3 internal nodes
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Finer mesh
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