Antecedentes

The finite element method is a numerical analysis technique for obtaining approximate solutions to engineering problems. It offers a way to solve a complex continuum problem by allowing it to be subdivided into a series of simpler interrelated problems and gives a consistent technique for modelling the whole as an assemblage of discrete parts. The ‘whole’

may be a body of matter or a region of space in which some phenomenon of interest is occurring.

In the heat transfer problem, temperature field is the field variable which is the function of each generic point in the body or solution region.

Consequently, the problem is one with an infinite number of unknowns.

Finite element analysis reduces the problem to one of a finite number of unknowns by dividing the solution region into elements and by express-ing the temperature field in terms of assumed interpolation functions within each element. The interpolation functions are defined in terms of the values of the temperature field at specified points called nodes. The nodal values of the temperature field and the interpolation functions for the elements completely define the behaviour of the temperature field within the elements. For the finite element representation of the heat transfer problem the nodal values of the temperature field become the unknowns. The matrix equations expressing the properties of the indi-vidual elements are determined from the governing equation by using the weighted residual approach. The individual element matrix equations are then combined to form the global matrix equations for the complete system. Once the boundary conditions have been imposed, the global matrix equations can be solved numerically. Once the nodal values of the temperature field are found, the interpolation functions define the temperature field throughout the assemblage of elements.

Assume that the solution domain  is divided into M elements and each element has n nodes. Thus, the temperature within each element can be expressed as follows

θ(x, y, z, t)= n i=1

Ni(x, y, z)θi(t)= N(x, y, z)θe(t) (6.6)

where N_i(x, y, z) is the interpolation function defined at node i, θ_i is the value of the temperature at node i, N(x, y, z) is the interpolation matrix andθe(t) is the vector of element nodal temperatures. The element matrix equation is obtained from the governing Eq. (6.3) by using the method of weighted residuals in which the weighting function is assumed to be the same as the interpolation function, that is,

where eis the domain of element e. Using Gauss’s theorem, for element domain e of boundary ethe following expression may be derived

whereˆn is the normal of the element boundary e. Using Eq. (6.8), Eq. (6.7) can be simplified as follows:

After some manipulation the resulting element matrix equation becomes

Ce

N^TρcNd= element capacitance matrix

Kce =

e

∇N^Tλ∇Nd = element conductance matrix

Rqe=

d= element nodal vector of heat flow

RQe =

d= element nodal vector of internal heat source

Equation (6.10) is the general formulation of element matrix equa-tion for transient heat conducequa-tion in an isotropic medium. Note that the element nodal temperatures cannot be solved from the element matrix Eq. (6.10). This is because the nodal vector of heat flow in the right-hand side of Eq. (6.10) is also an unknown. However, this unknown will be eliminated during the assembly of the element matrix equations or can be identified when applying boundary conditions. Therefore, the integration in calculating Rqe can apply only to the boundaries with

the prescribed heat flux. The global finite element matrix equation is obtained by the assembly of element matrix equations, which can be expressed as

Ce = global capacitance matrix

Kc= M e=1

Kce= global conductance matrix

Rq = M e=1

Rqe= global nodal vector of heat flow

RQ= M e=1

RQe = global nodal vector of internal heat source

where the summation implies correct addition of the matrix elements in the global coordinates and degrees of freedom. Note that, Rqis other than zero only when it is in the position corresponding to the node that is on a boundary. For a boundary that has prescribed temperatures, Rq is unknown but θ is known; whereas for a boundary that has prescribed heat fluxes θ is unknown but Rq is known. Thus, the total number of unknowns in the global finite element equation is always equal to the total number of nodes. It should be noted that, for the prescribed heat flux boundary condition the expression of Rqe may involve the unknown surface temperatures which need to be decomposed out from Rqe. According to the definition of Rqe in Eq. (6.10) and noticing that,

−(q · ˆn) = ˙hnet= ˙hnet,c+ ˙hnet,r, Rqemay be rearranged into

in which,

α_eff = ˙h_net θ_g− θm

= αc+ εmε_fσ

(θg+ 273)²+ (θm+ 273)²

(θg+ 273) + (θm+ 273) where α_eff is the combined convection and radiation coefficient which is temperature dependent. Let,

Rqe= Rqθe− Kqθeθe (6.13) in which,

Rqθe=

_e

N^Tα_effθ_gd

Kqθeθe =

_e

N^Tα_effθ_md=

_e

N^Tα_effNθed

Similarly, the global nodal vector of heat flow can be rewritten into

Rq= Rqθ − Kqθθ (6.14)

in which,

Rqθ = M e=1

Rqθe

Kqθ = M e=1

Kqθe

Thus, Eq. (6.11) becomes

C∂θ(t)

∂t +

Kc+ Kqθ

θ(t) = Rqθ+ RQ (6.15)

Equation (6.15) is the finite element formulation of non-linear tran-sient heat transfer problems, in which C, Kc, Kqθ and Rqθ are all

temperature dependent. The temperature dependence of C is due to the specific heat that is the function of temperature; the temperature depen-dence of Kcis due to the conductivity that is the function of temperature;

while temperature dependence of Kqθ and Rqθ is due to the bound-ary conditions involving radiation. To solve Eq. (6.15), time integration techniques must be employed.

Integration techniques for transient non-linear solutions are typically a combination of the methods for linear transient solutions and steady-state non-linear solutions (Huebner, Thornton and Byrom, 1995; Zienkiewicz and Taylor, 2000). The transient solution of the non-linear ordinary differential equations is computed by a numerical integration method with iterations at each time step to correct for non-linearities. Explicit or implicit one-parameter β schemes are often used as the time integration method, and Newton–Raphson or modified Newton–Raphson methods are used for the iteration. Let t_k denote a typical time in the response so that t_k+1= tk+ t, where k = 0, 1, 2, . . ., N. A general family of algo-rithms results by introducing a parameter β such that tβ = tk+βt where 0≤ β ≤ 1. Equation (6.15) at time tβ can be written as

C(θβ)∂θβ

∂t + K(θβ, tβ)θβ = R(θβ, tβ) (6.16) where K = Kc+ Kqθ, R = Rqθ + RQ are defined at temperatureθβ and time tβ, and the subscript β indicates the temperature vectorθβ at time tβ. By using the following approximations

θβ = (1 − β)θk+ βθk+1

∂θβ

∂t = θ_k+1− θk

t (6.17)

R(θβ, tβ)= (1 − β)R(θk, tk)+ βR(θk+1, tk+1)

Equation (6.16) can be rewritten into

βK(θβ, tβ)+ 1

tC(θβ)



θk+1 (6.18)

=

(β− 1)K(θβ, tβ)+ 1

tC(θβ)



θk+ (1 − β)R(θk, tk)+ βR(θk+1, tk+1)

where θk+1 and θβ are unknowns and θk is known from the previous time step.

Equation (6.18) represents a general family of recurrence relations;

a particular algorithm depends on the value of β selected. If β = 0, the algorithm is the forward difference method in which if the capacitance matrices are further lumped, it becomes explicit and reduces to a set of uncoupled algebraic equations; if β = 1/2, the algorithm is the Crank–

Nicolson method; if β= 2/3, the algorithm is the Galerkin method; and if β= 1, the algorithm is the backward difference method.

For a given β, Eq. (6.18) is a recurrence relation for calculating the vector of nodal temperaturesθk+1 at the end of time step from known values ofθkat the beginning of the time step. For β > 0, the algorithm is implicit and requires solution of a set of coupled algebraic equations using iterations because the coefficient matrices K, C and nodal heat vector R are functions ofθ. The Newton–Raphson iteration method is often used to solve the non-linear equations at each time step.

Hughes (1977) shows the algorithm to be unconditionally stable for β≥ 1/2 as in the corresponding linear algorithm. For β < 1/2 the algo-rithm is only conditionally stable, and the time step must be chosen smaller than a critical time step given by

tcr = 2 1− 2β

1

λ_m (6.19)

where λm is the largest eigenvalue of the current eigenvalue prob-lem. The explicit and implicit algorithms have the same trade-offs as occur for linear transient solutions. The explicit algorithm requires less computational effort, but it is conditionally stable; the implicit algo-rithm is computationally expensive, but it is unconditionally stable.

The non-linear implicit algorithm requires even greater computational effort than in linear implicit solutions because of the need for iterations at each time step. Thus the selection of a transient solution algorithm for a non-linear thermal problem is even more difficult than in linear solutions.

There are a number of finite element computer codes that can be used to solve the non-linear heat transfer equation with the fire boundary condition. Three of the most commonly used are FIRES-T3 from National Institute of Standards and Technology, USA (Iding, Bresler and Nizamuddin, 1977a), SAFIR from the University of Liège, Belgium (Franssen, 2003) and TASEF from Lund Institute of Technology, Sweden (Sterner and Wickström, 1990). In addition to these special codes developed for structures exposed to fire, there are some gen-eral finite element programs such as ABAQUS, ANSYS, DIANA and Comsol Multiphysics which can also be used to conduct the heat transfer analysis.