Modeling Heat Transfer Problems with Generalized Boundary Conditions

1
Modeling Heat Transfer Problems with Generalized
Boundary Conditions

• Heidi Chen
• 26 October, 2009

2
Overview

• 2D Convection heat transfer homework problem
• Problem definition
• Derivation of weak form, element stiffness matrix
  and element load vector
• Modeling of geometry
• MATLAB program modifications
• Solution verification with ANSYS

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Problem Definition
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Derivation of the Weak Form

• Given the governing equation

• With essential boundary condition of specified
  temperature T on GT
• And natural boundary condition of specified heat
  flux qn on Gq such that

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Derivation of the Weak Form

• Multiplication by w and integrating by parts
  yields

• Application of the boundary conditions gives the
  final weak form

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Ke and fe

• Converting to matrix notation and summing over
  finite elements

Ke
fe
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Geometry and Boundary Conditions
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2
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MATLAB Program

• 2dBVP User Input Functions
• InputGrid Input the geometry and set the element
  type and number of spatial dimensions.
• InputData Choose plots, and specify the sides
  with essential and natural boundary conditions.
• Conductivity
• specifyEssBCValues
• specifyNaturalBCValues
• ff Heat generation (body force)

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MATLAB Program

• 2dBVP Solving
• Preprocess
• Assemble K and f matrices using Gauss quadrature
  method for numerical integration
• Apply boundary conditions, modify K and f
  matrices according to the weak form
• Partition the matrix and solve for u
  (temperature) and r (reaction fluxes)
• Postprocess Plot temperature contour and heat
  flux vectors

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MATLAB Program Modifications

• InputGrid
• Create mesh of geometry

if strcmpi(load_grid,'no')1 Use default box
grid xl 0.0 left location
of the range in the x direction xr 2.0
right location of the range in the x
direction yb 0.0 bottom
location of the range in the y direction yt
8.0 top location of the range in the
y direction nx 8 number
of elements in x direction ny 8
number of elements in y direction
ElementType 1 1 Q4 2 T3 nsd
2 number of spatial dimensions (2
(default) or 3) BoxGrid_2D(xl,xr,yb,yt,nx
,ny) Use the provided box grid generator end
Input geometry
8x8 mesh
Quadrilateral 4 Nodes
2D
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MATLAB Program Modifications

• InputData

Top surface has non-zero natural BC
IsBoundaryCondition 0 0 1 0

• specifyEssBCValue.m

No specified temperature on boundaries
dof vals

• specifyNaturalBCValue.m

a
ß
Convection BC on top surface
if ( sideInd 3 ) Add the NC value below
alpha -5 beta 5(-5) end
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MATLAB Program Modifications

• Conductivity
• Input the conductivity matrix

D 10 0 0 15
kx 10 and ky15

• ff
• Modify heat generation term

No internal heat generation
value 0

• NodalSoln
• Input heat source

125 on node 63
R(63) R(63) 125
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MATLAB Program Modifications

• Integrands4side
• Modified K and f to account for the boundary
  conditions

alpha,beta specifyNaturalBCValue(sideInd,felm.
x) if ( alpha 0) Ke Ke - felm.N'
felm.N alpha felm.detJxW end if ( beta
0) fe fe felm.N' beta
felm.detJxW end
Modify K
Modify f
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Results

• MATLAB Output

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Results

• Comparing the Temperature contour plot with ANSYS

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Results

• Check convergence

8x8 mesh
16x16 mesh
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Results

• Adaptive mesh using ANSYS
• Refine mesh around top surface and point source

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Thank you