1 Variational and Weighted Residual Methods

2 Introduction The Finite Element method can be used to solve various problems, including: Steady-state field problems Time-dependent field problems (Transient)

3 Steady-state Heat Flow Problem Consider a problem that involves a two dimensional steady-state temperature distribution where the temperature along one dimension z does not vary. The temperature distribution is on the boundary Γ of a certain region Ω. The rate of the supplied heat at all positions of Ω is known Other types of boundary condition may be specified

4 Steady-state Heat Flow Problem If {Φ} is the vector of nodal temperature and Q is a function defining the heat input, then the matrix expression is of the form [K] is called the stiffness matrix This produces a system of simultaneous equations that can be solved to find {Φ}. The column matrix {f} depends on Q

5 Governing Equations for Physical Problem For the conduction of heat in one dimension, q is the rate of heat flow across a plane normal to the x-axis k is the thermal conductivity is the temperature gradient If the material is isotropic, similar equations with the same constant k apply to heat flow in the y and z directions

6 Governing Equations for Physical Problem Consider a small rectangular block:

7 Governing Equations for Physical Problem Under steady-state conditions: The heat flowing out across the faces of the block equals the rate at which heat is being fed into the block; Qδxδyδz where Q is the volumetric rate of the heat input

8 Governing Equations for Physical Problem In the x-direction, the heat flowing into the block across face 1 is; That flowing out of the block across face 2 is; The net outflow of heat in the x-direction is thus;

9 Governing Equations for Physical Problem The heat balance for the entire block is; or The equation is the governing partial differential equation for the steady-state conduction of heat in an isotropic body.

10 Governing Equations for Physical Problem Governing equations that are different in form apply to other physical phenomena: Gravitational field Electrostatic fields, Fluid flow Torsion of prismatic bars

11 Variational Methods In a steady state heat conduction problem in two dimensions The solution using variational methods is a temperature distribution  (x,y) This distribution is one that makes a certain integral over the region to which it applies take on an extreme value The integral is called a functional The mathematics of variational methods is called the calculus of variation

12 Variational Methods The functional often embodies some physical principles It is the potential energy of the deformed body and the forces acting on the body in elastic problems The functional takes on a minimum value at equilibrium in this case The use of variational methods is equivalent to the use of the Principle of Virtual Work

13 Variational Methods For any functional there is a corresponding equation (the Euler equation) The solution of which, subject to the appropriate boundary conditions, gives rise to a stationary value of the functional The reverse is not true (a functional cannot always be found to correspond to a particular governing equation).

14 Variational Methods Consider the functional which is to be made stationary, subject to the following conditions  =  0 at x=0 and  =  L at x=L k is a constant Q is a given function of position, x  is to be determined in the range 0 < x <L

15 Variational Methods If Π(  ) is stationary when  (x)=u(x) Any small permissible change in , will cause a change in Π (  ) that will tend to zero as the magnitude in the change in  decreases This is called the first variation in Π (  ). d Π (  ) and thus d Π (  )=0 is merely an alternative way of expressing the requirement of stationarity

16 Variational Methods The variation away from u(x), that is away from the value of  giving rise to the extremum of the functional, is expressed as; where ε is a number η(x) is a function of x satisfying the condition η(0) = η(L) = 0

17 Variational Methods The functional can be written as The stationary condition to be satisfied is that

18 Variational Methods Differentiating inside the integral sign, we obtain Hence

19 Variational Methods Evaluating the first term by integration by parts and noting that η(0) = η(L) = 0, we have Hence

20 Variational Methods Since η is arbitrary, the equation; is satisfied only if Hence the function u(x) causes Π(  ) to be stationary Also satisfies the differential equation

21 Variational Methods The differential equation is the governing equation for 1-D heat conduction and is the variational form of this equation

22 Variational Methods The boundary condition that specify the value of temperature  are called Dirichlet boundary conditions The derivative  in the direction of the outward normal to part of the boundary, δ  / δn, this is called a Neumann boundary condition

23 Variational Methods In heat conduction δ  / δn specifies the rate at which heat flows into or out of the region enclosed by the boundary If the total heat flux into the region is fixed, the Neumann condition can be prescribed on only part of the boundary

24 Variational Methods The form of the functional depends on the type of boundary conditions that apply. To show this: let the boundary conditions be  =  0 at x = 0, but at x=L let the condition to be satisfied be In this case, the functional to be made stationary is where  L is the value  of at x = L

25 Variational Methods If we proceed as before, the condition to be satisfied by η is η(0) but at x = L If  = u is the function causing Π(  ) to be stationary, we have and

26 Variational Methods By integration by parts and hence we arrive at the governing equation Note if the conditions were at x = 0, the additional term would be

27 Variational Methods In this particular case The boundary condition δ  / δn at x = L is called a natural boundary condition The supplementary conditions,  (0) =  0 is called an essential boundary condition.

28 Numerical Solution of Variational Problem In the Rayleigh-Ritz methods the function is represented in a series form by a trial function, , The a i are coefficient to be determined The f i are members of a family of functions such that  satisfies the essential boundary conditions for all n. The f i must be capable of adequately representing the exact solution

29 Numerical Solution of Variational Problem The coefficients a i (i=1....n) are selected so that the trial function best approximates to u. If the trial function is substituted into the functional and the integration is performed, Π is obtained as a quantity that depends on the coefficients a a n.

30 Numerical Solution of Variational Problem Since Π is required to be stationary, the derivatives of Π with respect to any of the coefficients a a n will be zero A series of n simultaneous equations is obtained from which the n coefficients can be calculated

31 Example 1 Find the temperature distribution,  (x) in a rod of constant cross section where Heat is generated in the rod at the rate Q per unit volume The temperature at the rod ends, x = ±L, is constant at  0 The coefficient of thermal conductivity is k.

32 Example 1 The solution proceeds as follows We choose trial functions that satisfy the essential boundary condition The temperature distribution must be symmetrical about the y- axis  must be an even function of x Hence suitable trial functions are of the form:

33 Example 1 Taking the approximation with a single constant a 1 to be determined, the functional whose extreme value is required becomes Substituting for  and for This expression gives

34 Example 1 We now differentiate with respect to a 1 under the integral sign and perform the integration with respect to x Equating to zero and rearranging gives In this case, the exact solution is obtained - as the trial function is capable of representing the exact solution a 1 = -Q/2k

35 Example 2 Find the temperature distribution,  (x) in a rod of constant cross section, where: Heat is generated in the rod at the rate Q per unit volume The temperature at the rod end; x = -L is constant,  0 at x = +L then Coefficient of thermal conductivity, k.

36 Example 2 The functional whose extreme value to be found is now given by Possible trial functions are which satisfy the essential boundary condition at x = -L

37 Example 2 We shall take Proceeding as before, but noting the extra term in the functional Substituting the above equations into the functional equation gives

38 Example 2 Differentiating with respect to a 1 gives Neglecting terms,since the integral is zero

39 Example 2 The condition thus Gives the equation

40 Example 2 Similarly This can be written as

41 Example 2 Evaluating the integral gives The condition Gives the equation

42 Example 2 Solving for a 1 and a 2 gives The exact solution is obtained again. The boundary condition at x=+L is automatically satisfied by the solution in accordance with it being a natural boundary condition