ABSTRACT
In this chapter we present the Þnite element analysis of 3D conduction heat transfer and related diffusion problems. Though conduction is the least complex mode of thermal energy transport, it plays a major role in industrial design, manufacturing and engineering. The prediction of temperature distributions, thermal performance and thermal stresses in complex geometries has become a routine part of engineering analysis, due in large part to the Þnite element method and availability of general purpose computer programs. The conduction problem also provides a convenient framework to discuss various advanced aspects of the Þnite element method and numerical algorithms. The introduction to the Þnite element method presented in Chapter 2 aids the development presented here for the general 3D heat conduction problem. As described in the previous chapters, the appropriate mathematical description
of thermal conduction in a 3D solid (or stationary ßuid) region, Ω, is given by Eq. (1.5.4). In the interest of brevity, the subscript “s,” which refers to a solid region, is omitted throughout the chapter. The Cartesian component form of the equation is [see Eq.(1.5.10)]
ρC ∂T
∂t =
∂
à kij
∂T
! +Q (3.1.1)
where T is the temperature, ρ is the density, C is the speciÞc heat, kij are the Cartesian components of the conductivity tensor (symmetric), and Q is the internal heat generation per unit volume. Summation on repeated subscripts is assumed (i.e., i and j are summed over the range of i, j = 1, 2, 3). All of the variables may be functions of position x = (x1, x2, x3) and time t. We wish to solve Eq. (3.1.1) under appropriate boundary and initial conditions. The boundary conditions for this equation are given by the equations [see Eq. (1.10.10a,b)]:
T = fT (sk, t) on ΓT , for t > 0 (3.1.2a)
− Ã kij
∂T
! ni = qa + qc + qr = f
q(sk, t) on Γq, for t > 0 (3.1.2b)
Here sk denotes the coordinates of a point on the boundary surface, qa is an applied ßux and the convective and radiative ßux are
qc = hc(sk, T, t)(T − Tc), qr = hr(sk, T, t)(T − Tr) (3.1.2c) The initial condition on the temperature is given by
T (xj , 0) = T0(xj) (3.1.3)
differs from discussed in Chapter 2 in several respects. First, the problem considered here is for general three-dimensional geometries. Second, the boundary conditions are general enough to include conduction, convection, and radiation heat transfer between the solid regions and the surrounding environment. Third, the material coefficients, ρ, C, kij , hc, and hr can be functions of temperature, making the problem a nonlinear one. As described in Section 2.8, the Þnite element model of the initial-boundary
value problem described by Eqs. (3.1.1)—(3.1.3) is developed in two steps: (1) spatial discretization, in which the weak form of Eq. (3.1.1) over a typical element is developed and the spatial approximation of the dependent variable (i.e., temperature) T of the problem is assumed to obtain a set of ordinary differential equations in time among the nodal values T ej of the dependent variable; (2) temporal approximation of the ordinary differential equations obtained in the Þrst step is carried out using some appropriate method such as a Þnite difference approximation. This step leads to a set of algebraic equations involving the nodal values T ej at time tn+1 [= (n + 1)∆t, where n is an integer and ∆t is the time increment] in terms of known values from the previous time step. Note that time-independent (steady) boundary value problems replace the second or temporal approximation step with the invocation of some type of iterative algorithm depending on the nonlinearity of the application. See Section 2.8 for an account of these two steps when applied to a two-dimensional problem. Here we develop the Þnite element models of three-dimensional, transient heat conduction problems using the two-step procedure described above. We begin with the development of the semidiscrete Þnite element model of Eqs. (3.1.1) and (3.1.2).
