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APPLICATIONS IN HEAT TRANSFER &FLUID MECHANICS
ONE DIMENSIONAL HEAT TRANSFER ELEMENT
In structural problem displacement at each nodel point is obtained. By using these displacement solutions, stresses and strains are calculated for each element. In structural problems, the unknowns are represented by the components of vector field. For example, in a two dimensional plate, the unknown quantity is the vector field u(x,y),where u is a (2x1)displacement vector.
Heat transfer can be defined as the transmission of energy from one region another region due to temperature difference. A knowledge of the temperature distribution within a body is important in many engineering problems. There are three modes of heat transfer.
They are: (i) Conduction
1Strong Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions
Following the same procedure as in Section, the portion of the boundary where the temperature is prescribed, i.e. the essential boundary is denoted by T and the boundary where the flux is prescribed is recommended for Science and Engineering Track. Denoted by q ; these are the boundaries with natural boundary conditions. These boundaries are complementary, so
With the unit normal used in , we can express the natural boundary condition as qn ¼ q. For example, positive flux q causes heat inflow (negative q ) on the left boundary point where qn ¼ q ¼ q and heat outflow (positive q ) on the right boundary point where qn ¼ q ¼ q.
Strong form for 1D heat conduction problems
2Weak Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions
We again multiply the first two equations in the strong form by the weight function and integrate over the domains over which they hold, the domain for the differential equation and the domain q for the flux boundary condition, which
yields ws dx with w ¼
T and combining with gives
Recalling that w ¼ 0 on
Weak form for 1D heat conduction problems
Find T ðxÞ 2 U such that
Notice the similarity between
APPLICATION TO HEAT TRANSFER TWO-DIMENTIONAL
1Strong Form for Two-Point Boundary Value Problems
The equations developed in this chapter for heat conduction, diffusion and elasticity problems are all of the following form:
Such one-dimensional problems are called two-point boundary value problems. gives the particular meanings of the above variables and parameters for several applications. The natural boundary conditions can also be generalized as (based on Becker et al. (1981))
Equation is a natural boundary condition because the derivative of the solution appears in it. reduces to the standard natural boundary conditions considered in the previous sections when bðx Þ ¼ 0. Notice that the essential boundary condition can be recovered as a limiting case of when bðxÞ is a penalty parameter, i.e. a large number In this case, and Equation is called a generalized boundary condition.
An example of the above generalized boundary condition is an elastic bar with a spring attached as shown in In this case, bðlÞ ¼ k and reduces to
where ¼ k is the spring constant. If the spring stiffness is set to a very large value, the above boundary condition enforces ¼ u; if we let k ¼ 0, the above boundary condition corresponds to a prescribed traction boundary. In practice, such generalized boundary conditions are often used to model the influence of the surroundings. For example, if the bar is a simplified model of a building and its foundation, the spring can represent the stiffness of the soil.
2 Two-Point Boundary Value Problem With Generalized Boundary Conditions
An example of the generalized boundary for elasticity problem.
Another example of the application of this boundary condition
is convective heat transfer, where energy is tran
sferred between the
surface of the wall and the surrounding medium. Suppose convective heat
transfer occurs at x ¼ l. Let T ðlÞ be the wall temperature at x ¼ l and T be the temperature in the medium. Then the flux at the boundary x
¼ l is given by qð lÞ ¼ hðT ðlÞ T Þ, so bðlÞ ¼
h and the boundary condition is
where h is convection coefficient, which has dimensions of W m 2 o C 1 . Note that when the convection
coefficient is very large, the temperature T is immediately felt at x ¼ l and
thus the essential boundary condition is again enforced as a limiting case of
the natural boundary condition.
There are two approaches to deal with the boundary condition . We will call them the penalty and partition methods. In the penalty method, the essential boundary condition is enforced as a limiting case of the natural boundary condition by equating bðxÞ to a penalty parameter. The resulting strong form for the penalty method is given in.
General strong form for 1D problems-penalty method
þ f ¼ 0 on ;
In the partition approach, the total boundary is partitioned into the natural boundary, and the complementary essential boundary, The natural boundary condition has the generalized form defined by The resulting strong form for the partition method is summarized in.
3 Weak Form for Two-Point Boundary Value Problems
In this section, we will derive the general weak form for two-point boundary value problems. Both the penalty and partition methods described in will be considered. To obtain the general weak form for the penalty method, we multiply the two equations in the strong by the weight function and integrate over the domains over which they hold: the domain for the differential equation and the domain for the generalized boundary condition.
SCALE VARIABLE PROBLEM IN 2 DIMENSIONS
Ø Equation of Stiffness Matrix for 4 noded isoparametric quadrilateral element
N –Shape function,
Fx –load or force along x direction,
Fy –load or force along y direction.
Ø Numerical Integration (Gaussian Quadrature)
The Gauss quadrature is one of the numerical integration methods to calculate the definite integrals. In FEA, this Gauss quadrature method is mostly preferred. In this method the numerical integration is achieved by the following expression,
Table gives gauss points for integration from -1 to 1.
Ø Problem (I set)
1. Evaluate by applying 3 point Gaussian quadrature and compare with exact solution.
2. 2. Evaluate using one point and two point Gaussian quadrature. Compare with exact solution.
3. 3. For the isoparametric quadrilateral element shown in figure, determine the local co –ordinates of the point P which has Cartesian co-ordinates (7, 4).
4. A four noded rectangular element is in figure. Determine (i) Jacobian
matrix, (ii) Strain – Displacement matrix and (iii) Element Stresses. Take
E=2x105N/mm2,υ= 0.25,u=[0,0,0.003,0.004,0.006,.004,0,0]T, Ɛ= 0, ɳ=0.
Assume plane stress condition.
2 DIMENTIONAL FLUID MECHANICS
The problem of linear elastostatics described in detail in can be extended to include the effects of inertia. The resulting equations of motion take the form
where u = u(x1 , x2, x3 , t) is the unknown displacement field, ρ is the mass density, and I = (0, T ) with T being a given ti me. Also, u0 and v0 are the prescribed initi al displacement and velocity fields. Clearly, two sets of boundary conditions are set on Γu and Γq , respectively, and are assumed to hold throughout the time interval I . Likew ise, two sets of initial conditions are set for the whole domain Ω at time t = 0. The stron g form of the resulting initial/boundary- value problem is stated as follows: given functions f , t, u¯ , u0 and v0, as well as a constitutive equation for σ, find u in Ω × I , such that the equations are satisfied.
A Galerkin-based weak form of the linear elastostatics problem has been derived in Sec-tion In the elastodynamics case, the only substantial difference involves the inclusion of the term RΩ w • ρu¨ dΩ, as long as one adopts the semi-discrete approach. As a result, the weak form at a fixed time can be expressed as
Following a standard proced ure, the contribution of the forcing vector Fi nt,e due to interele- ment tractions is neglected upon assembly of the global equations . As a result, the equations is give rise to their assembled counterparts in the form
Mu + Kuˆ = F ,
where uˆ is the global unknown displacement vector1 . The preceding equat ions are, of course, subject to initial conditions t hat can be written in vectorial form as uˆ(0) = uˆ0 and vˆ(0) = vˆ0
The most commonly emp loyed method for the numerical solution of t he system of coupled linear second-order ordi nary differential equations is the Newmark m ethod. This
method is based on a time series expansion of ˆu and ˆ u˙ := v.ˆ Concentrating on the time interval (tn ,tn+1], the New mark method is defined by the equations
It is clear that the Newmark equations define a whole family of time inte grators.
It is important to distinguish this family into two categories, namely implicit and explicit integrators, corresponding to β > 0 and β = 0, respectively.
The overhead “hat” symbol is used to distinguish between the vector field u and the solution vector uˆemanating fr om the finite element approximation of the vector field u.
The general implicit Newmar k integration method may be implemented as follows: first, solve (9.18)1 for aˆn+1 , namely write
Then, substitute (9.19) into the semi-discrete form (9.17) evaluated at tn+1 to find that
After solving for uˆn+1, one ma y compute the acceleration aˆn+1 from and the velocity vˆn+1 from.
Finally, the general explicit N ewmark integration method may be implemented as follows: starting from the semi-discrete e quations evaluated at tn+1, one may substitute uˆn+1from to find that
If M is rendered diagonal (see discussion in our pages ), then aˆn+1 can be determined without solving any coupled linear algebraic equations. Then, ˆ are the velocities bvn+1 immediately computed from (9. 18)2. Also, the displacements uˆ n+1 are computed from indepen-dently of the acceleratio ns aˆn+1 .
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