FEA of Electrostatics

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FEA of Electrostatics
Contents

• Governing Equation for Electrostatic field
• Variational Formulation
• Domain Discretization
• Element Interpolation
• Formulation via the Ritz Method
• Assembly to Form the System of Equations
• ANSYS Elements for Electrostatics
• Example

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Governing Equation(1)

• To explain physical phenomena near charge(s)

• Essential terms

Dirichlet B.C
Neumann B.C
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Governing Equation(2)
Maxwell Equations for Electrostatic Field
Introduction of Scalar Potential
D Electric flux density E Electric field
intensity
Possions Equation
permittivity matrix
4
Variational Formulation(1)
Define an inner product as
And an operator
Divergence theorem
Then the possions eq. will be
From the minimum functional theorem () the
solution of Eq(2) is equivalent to the minimizer
of following functional,
() J. N. Reddy, Applied functional analysis
and variational methods in engineering
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Domain Discretization
Discretization
bilinear
Approximation
triangular
t
s
Iso-parametric
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Element Interpolation (1)
Triangular element will be discussed for the
simplicity
Linear approximation
thus
where
Contd
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Element Interpolation (2)
in which
and
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Formulation via the Ritz Method
Differentiating w.r.t
or
where
and the stiffness matrix is
and the load vector is
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Assembly to Form the System of Equations
To obtain the system of equations, it is
necessary to find
With the Eq(4), we can assemble all M elements
where all vectors and matrices folllowing the
summation signs have been expanded or augmented
by zero filling. That is,
Finally, the minimizer of the functional is
obtained by,
or
where
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ANSYS Elements for Electrostatics
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Example
Comb drive actuator
Si
air
V
FE Model PLANE121
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Example
Nodal Plot Scalar potential
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Example
Vector Plot Electric field intensity
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Example
Vector Plot Electrostatic force