1. HEAT TRANSFER FINITE ELEMENT FORMULATION
VIJAYAVITHAL BONGALE
DEPARTMENT OF MECHANICAL ENGINEERING
MALNAD COLLEGE OF ENGINEERING
HASSAN
Mobile :
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3. What is Finite Element Method?
FEM is a numerical analysis technique for obtaining approximate solutions to a wide variety of engineering problems.
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4. Applications of FEM:
Equilibrium problems or time independent problems. e.g. i) To find displacement distribution and stress distribution for a mechanical or thermal loading in solid mechanics. ii) To find pressure, velocity, temperature, and density distributions of equilibrium problems in fluid mechanics.
Eigenvalue problems of solid and fluid mechanics.
e.g. i) Determination of natural frequencies and modes of vibration of solids and fluids. ii) Stability of structures and the stability of laminar flows.
3.Time-dependent or propagation problems of continuum mechanics.
e.g. This category is composed of the problems that results when the time dimension is added to the problems of the first two categories.
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5. Similarities that exists between various types of engineering problems:
1. Solid Bar under Axial Load
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6. 2. One – dimensional Heat Transfer
3. One dimensional fluid flow
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7. How the Finite Element Method Works
Discretize the continuum: Divide the continuum or solution region into elements.
Select interpolation functions: Assign nodes to each element and then choose the interpolation function to represent the variation of the field variable over the element.
Find the Element Properties: Determine the matrix equations expressing the properties of the individual elements. For this one of the three approaches can be used. i) The direct approach ii) The variational approach or iii) the weighted residuals approach.
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8. Assemble the Element Properties to Obtain the System equations: Combine the matrix equations expressing the behavior of the elements and form the matrix equations expressing the behavior of the entire system.
Impose the Boundary Conditions: Before the system equations are ready for solution they must be modified to account for the boundary conditions of the problem.
Solve the System Equations: Solve the System Equations to obtain the unknown nodal values like displacement, temperature etc.
Make Additional Computations If Desired: From displacements calculate element strains and stresses, from temperatures calculate heat fluxes if required.
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10. General Steps to be followed while solving a problem on heat transfer by using FEM:
Discretize and select the element type
Choose a temperature function
Define the temperature gradient / temperature and heat flux/ temperature gradient relationships
Derive the element conduction matrix and equations by using either variational approach or by using Galerkin’s approach
Assemble the element equations to obtain the global equations and introduce boundary conditions
Solve for the nodal temperatures
Solve for the element temperature gradients and heat fluxes
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11. ONE DIMENSIONAL FINITE ELEMENT FORMULATION USING VARIATIONAL APPROACH
Step 1.Select element type.
Step 2.Choose a temperature function. t1 t2 1 x
One –D element L 2 t1 t2
T = N1t1+ N2 t2 1 2
Temperature Variation along the length of element
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12. We choose, T (x) = N1 t1 + N2 t2 --------------------------- (1)
N1 & N2 are shape functions given by,
In matrix form
and
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13. Temperature gradient matrix is given by,
Step 3. Define the temperature gradient / temperature and heat flux/ temperature gradient relationships
Temperature gradient matrix is given by,
Where matrix B is,
Where matrix B is,
The heat flux /temperature gradient relationship is,
The heat flux /temperature gradient relationship is,
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14. The material property matrix is,
Step 4. Derive the element conduction matrix and equations
Consider the following equations
q x = 0 L
T=TB S1
Insulated L
T=TB S1
q x = + q*x S2
With T =TB on surface S1,
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15. is the heat flow / unit area
Where, Q is heat generated / unit volume
is the heat flow / unit area
is positive when heat is flowing into body
is negative when heat is flowing out of the body
is Zero on an insulated boundary
Consider
With the first boundary condition of above equation and /or second boundary condition and /or loss of heat by convection from the ends of 1-D body, we have
Insulated h T 2
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16. Minimize the following functional :
(Analogous to the potential energy functional Π)
Where,
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17. Important:
q* and h on the same surface cannot be specified simultaneously because they cannot occur on the same surface
We now have the functional given by
Consider the first term,
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18. Similarily, fourth term gives,
Second term gives,
Third term gives,
Similarily, fourth term gives,
Substituting equations (11),(12),(13) and (14) in equation (10) we obtain
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19. Equation (15) has to be minimized with respect to and equated to zero
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20. The above equation is of the form,
On simplifying,
The above equation is of the form,
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21. Element Conduction matrix is
Where,
Element Conduction matrix is
The first term is conduction part of K and
second term represent convection part of K
And the force matrices have been defined by,
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22. Consider the conduction part,
Substituting for B, D and dV in the above equation,
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23. Therefore, element conduction matrix is,
The convection part is,
Therefore, element conduction matrix is,
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24. The force matrix terms will be,
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25. Convection force from the end of the element
By adding we get,
Convection force from the end of the element 1 2 h
T00
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26. N1 = 0 and N2 = 1 at right end and
We have an additional convection term contribution to the stiffness matrix and is,
N1 = 0 and N2 = 1 at right end and
The convection force from the free end
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27. The global structure conduction matrix is
Step 5. Assemble the element equations to obtain the global equations and introduce boundary conditions.
The global structure conduction matrix is
The global force matrix is
and global equations are
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28. Step 6. Solve for the nodal temperatures
Step 7. Solve for the element temperature gradients and heat fluxes.
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29. ONE DIMENSIONAL FINITE ELEMENT FORMULATION GALERKIN’S APPROACH
Equation representing one dimensional formulation of conduction with convection is given by,
Element – 1-D linear 2 noded element with temperature function
T (x) = N1 t1 + N2 t (2)
Where,
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30. The residual equations for the equation (I) are
Where i = 1,2
Integrating the first term of equation (4) by parts and rearranging,
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31. Integration by parts:
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32. Substituting for T , we get
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33. Let x1= 0 and x2 = L, then, The above equations are of the form, And
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34. If one substitute for N1 & N2 in equation (9) and solve, then,
The forcing function vectors on the right hand side of the equation (7) are given by
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36. Two Dimensional Finite Element Formulation:
1-d elements are lines
2-d elements are either triangles, quadrilaterals, or a mixture as shown
Label the nodes so that the difference between two nodes on any element is minimized.
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37. Three noded triangular element:
Assume (Choose) a Temperature Function:
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38. Define Temperature Gradient Relationships
Analogous to strain matrix: {g}=[B]{t}
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39. [B] is derivative of [N]:
Heat flux/ temperature gradient relationship is:
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40. Derive the element conduction matrix and equations:
Where,
If thickness is assumed constant and all terms of integrand constant, then the conduction portion of the total stiffness matrix is,
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41. Now the convection portion of the total stiffness matrix is,
Consider the side between nodes i and j of the element subjected to convection, then, Nm = 0 along side i-j
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42. Where Li-j is the length of side i-j
We Obtain
Where Li-j is the length of side i-j
Force Matrices:
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43. Can be found in a same manner by simply replacing
The integral
Can be found in a same manner by simply replacing
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45. Thank You
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