Chapter 7, page 1

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Recap Where are we?

• Element properties, interpolation functions
  (Chapters 1-6)
• Evaluating element matrices for 2-D field
  equations (Chapter 7)
• Specific applications (Chapter 8 ?)

?
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Chapter 7 Two-Dimensional Field Equation

• General Form of the governing differential
  equation

This equation covers several physical problems
which will be briefly mentioned.
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1. Torsion (twisting of an object due to an
applied torque) of non-circular sections
Where
g shear modulus
Shear stresses w/in the shaft are related to the
derivative of ø with respect to x and y.
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• 2. Fluid mechanics
• Streamline (tangent to the flow) and potential
  (perpendicular to flow) formulations for an ideal
  irrotational flow

streamline function
potential function
Velocity derivatives of ø or ? with respect to
x and y.
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• 3. Groundwater Flow
• a) seepage under a dam or a retaining wall

F Piezometric head
Dx, Dy Permeability of the soil
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b) seepage w/in a confined aquifer (near a well
during pumping)
Impermeable layer
Piezometric head
Permeabilities of the soil
Ø constant
Pump Q
Q point source term
Ø constant
3,000 m
Impermeable layer
5,000 m
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4. Heat Transfer
2-D fin heat lost to a surrounding fluid by
convection
Where Dx, Dy thermal coefficients h
convection coefficient t fin thickness (assumed
thin so heat loss from edges is neglected) Tf
ambient temperature of surrounding fluid
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Heat Transfer

• (c) Long Two-Dimensional Bodies (heat loss in x
  and y directions only) - conduction

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• Helmholtz equation (Glt0, Q0)
• Seiche motion (standing waves on a bounded
  shallow body of water) and
• acoustic vibrations (fluid vibrating within a
  closed volume)

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• Goal to derive the integral equations for the
  element matrices associated with the 2-D field
  equation

Since ø does not have continuous derivatives
between the elements, the second order terms must
be replaced by first-derivative terms. We will
use Greens Theorem and the Chain Rule. Then, we
can get a solution, using Galerkins formulation.
Recall that element contributions are given by
R, written for each node.
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• From the discussion on continuity
  only is continuous
• and ? does not have continuous derivative between
  elements

To assure continuity for ?
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Can be replaced by an integral around the
boundary by using Greens Theorem.
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angle of outward normal to the surface
boundary (gamma) boundary of the elem.
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So,
Similarly, the y-term
can be evaluated to replace the
second-derivative term.
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• Putting everything together

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Substitute
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• Or, in the general, condensed form

Where
I(e) will occur in derivative b.c., so we do
not remove it. Derivative boundary conditions
will be discussed in Chapter 9.
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• and

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• Notation (to simplify )

Define 2 terms
Gradient Vector
so
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Now,
so

• Element Stiffness matrix for field boundary value
  problem

or
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• Element Matrices Triangular Elements

?
?
?
a, b, and c were defined in Chapter 5
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• The gradient vector for this element is

or
Note that B and D consist of constants.
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so
sym.
sym.
Note that the main diagonal must be positive.
Can also be expanded, assuming that G is constant
in the element.
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Assume that G is constant, then
sym.
Since Ni L1, Nj L2, and Nk L3 for the
linear triangle.
(symmetric)
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• Use the factorial equation to evaluate each
  integral. There are a total of 9 integrals using
  Nß, in .

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Where Q constant

• Similarly, for

or
So, to calculate k(e) and f(e) you need Dx,
Dy, A, G, Q bi, bj, bk, ci,
cj, ck
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Example 2D heat transfer from a fin (defined by
eq. 7.7) Calculate elem. matrices for the given
2D ? elem.
?
?
?
?
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Example is given in the text.
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Substitution gives
Element force vector
or
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2 Rectangular Elements - not as easy as ? (no
area coordinates and simple factorials) - the
shape functions were developed relative to the
st-coordinate system, but all the integrals are
defined relative to the xy coordinate system
(need to change variables).
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?
?
?
?
So,
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chain rule
or
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Note st xy are // and a unit length in both
systems are the same. So,
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Similarly, for other Ns, so
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or
For
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So,
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so
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Recap - Chapter 7 (field problems)

• Governing diff. eq.
• The derivations are fairly complex, but the
  results, shown on the following slides, should
  look familiar.

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and
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For the 2-D triangular element,
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For the 2-D rectangular element,
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Assignment
Page 97 -- 7.2c, 7.3c, 7.9c Due Monday, Oct. 2,
2006