Finite Elements for Thermal Analysis

Transcription

1 Chapter 4 Finite Elements for Thermal Analysis 4.1 Governing Equations for the Thermal Problem CONSTITUTIVE EQUATIONS Introducing the temperature field T(x i,t ), the heat flux vector q(x i,t) q n (x i,t) and the thermal conductivity in the n direction λ n, th following holds: q n T = λ n n and in the (O,x i ) space q q q x y z T = λx x T = λy y T = λz z For an isotropic media qi = λt, i i = 1, 2,3. (4.1) (4.2) (4.3) THERMAL BALANCE The balance of thermal energy in an infinitesimal volume dv = dxdydz of the body in the x direction for an infinitesimal time increment dt reads dq in = q dydzdt x q x x dqout = qx + dx dydzdt q = x x dqin dqout dvdt (4.4) Taking into account the heat fluxes in the other directions the thermal balance is dq () dq +,- = q 0 x dvdt = 0 q (,(dvdt = 0 Considering the case of heat generation within the body (4.5)

2 dq = q dvdt gen gen the equation becomes (4.6) dq () dq +,- + dq 78) = q 0 x + q 78) dvdt = 0 q (,( + q 78) dvdt = 0 (4.7) FOURIER EQUATION Introducing the body density ρ and its c specific heat for unit mass c, the Fourier Equation reads: q (,( + q 78) = cρ T t q <= = q ( n ( on S B (4.8) T = T on S C The previous equation can be recast (using the 4.3) in the form λ ( T,(,( + q 78) = cρ T t (4.9) and, in the homogeneous an isotropic case λt,(( + q 78) = cρ T t Finally λ K T + q 78) = cρ T t or q (,( + q 78) = 0, with q 78) = q 78) cρ T t (4.10) (4.11) 4.2 Virtual Temperature Principle It is possible to draw an analogy between the thermal problem and 3D solid mechanics STATE VARIABLES Displacement u i T Temperature Strains ε ij T,i Stresses Ϭ ij q i Temperature gradient Heat Flux LOADS Internal Loads (per unit volume) X i q gen Generated Heat (per unit volume per unit time) Surface Loads (per unit area) f i q i Thermal flux on S q

3 Prescribed Displacement u i T Prescribed Temperatures on S T GOVERNING EQUATIONS Equilibrium σ (O,O + X ( = 0 q (,( + q 78) = 0 Compatibility ε (O,O = R K u (,O + u O,( grad T = T,( Constitutive σ (O = C (OWX q ( = λt,( Virtual work (displacement) Principle (Theorem) σ (O ε (O dv = X ( u ( dv + f ( u ( ds < [ (4.12) Virtual temperature Principle (Theorem) q ( T,O dv = q 78) T ( dv + q <= TdS (4.13) The latter equation can be obtained considering the thermal energy balance equation q + = i, i qgen 0 and imposing the following expression to be zero (4.14) q (,( + q 78) TdV = 0 (4.15) where T is the virtual temperature field (null on S T ). The following holds: q ( T,( = q (,( T + q ( T,( q (,( T = q ( T,( + q ( T,( (4.16) and, considering the Gauss theorem we have (q ( T),( dv = q ( Tn ( ds < q (,( + q 78) TdV end so using the 4.16 = 0 q (,( TdV = q 78) TdV (4.17) (4.18) q ( T,( dv + q ( T,( dv = q 78) TdV (4.19) using the 4.17

4 q ( T,( dv + q ( Tn ( < ds = q 78) TdV (4.20) Considering that q s =-q i n i on S q and that T = 0 on S t we can write q ( T,( dv = q 78) TdV + q <= T ds (4.21) The matrix form of this equation is T C q dv = q 78) TdV + q <= T ds (4.22) where q C = q 0, q^, q _, T C = T x, T y, T z (4.23) 4.3 Shape functions In the finite element approach, the aim is to discretize, that is to condense the distributed properties of each element on some boundary nodes that "communicate" with the rest of the body and with external actions. The solution is determined in a weak form because the equilibrium is only satisfied in a mediated sense on the domain of the element. In order to distribute properties on nodes, we wish to approximate a function T(x,y,z) defined in an interval [a,b] by some set of basis functions ) T x, y, z = α ( φ ( x, y, z (4.24) (ar where i is the number of grid points (the edges of our elements) defined at locations x i, y i, z i. The functions are usually polynomials, but in some case (e.g. in acoustic analysis) can be useful to use trigonometric functions. These functions are called interpolating functions or shape functions. Unless in particular cases, the linear combination of these functions will not provide an exact but only approximate solution. The set of interpolating functions can in theory be chosen ad libitum, but in order for the convergence of the result to be monotonic with the condensation, four fundamental requirements must be respected: 1. ability to have same variation of temperature of the element without the occurrence of heat flux within (equivalent to rigid motions) 2. allowing states of constant gradients of temperature throughout the element (patch test) 3. compatibility along the contour lines or faces of contiguous elements, no breaks or interpenetrations should occur 4. continuity or absence of singular points in the description of the temperature field.

5 One time the interpolating functions φ ( x, y, z for each internal point of the element are identified, it is possible to write T x, y, z = α R φ R x, y, z + α K φ K x, y, z + + α ) φ ) x, y, z or in matrix form (4.25) T x, y, z = φ R x, y, z φ K x, y, z φ ) x, y, z T x, y, z = φ x, y, z α α R α K α ) (4.26) We apply the previous interpolation on particular points, the nodes, considering all the degreeof-freedom defined for each node Θ O = α R φ R x O, y O, z O + α K φ K x O, y O, z O + + α ) φ ) x O, y O, z O (4.27) Θ = A α where j = 1,2,,k with k degree-of-freedom of the element and A = φ x O, y O, z O Note that, given the positions of the nodes in the local reference, A is a known matrix, composed of defined numerical values. If we make the number of coefficients of the series of α coincident with in number of degree-offreedom of the element (n = k), the matrix A is square, a necessary condition for its inversion. This allows to solve for α α = A gr Θ that replaced in the expression of the internal temperature field (4.27) bring to T x, y, z = φ x, y, z A gr Θ (4.28) (4.29) N x, y, z = φ x, y, z A gr is the shape function matrix (in reality is a vector). T x, y, z = N x, y, z Θ Θ is the vector of the nodal temperatures of the element. (4.30)

6 4.4 Finite element formulation Introducing D C = x, y, z (4.31) we can write T = DT T C = TD C Considering a finite element approximation T x = N x Θ where N = N R, N K,, N ), Θ C = Θ R, Θ K,, Θ ) where Θ is the nodal temperature vector and N is the shape functions matrix: (4.32) (4.33) T = D N Θ T C = N Θ C D C = Θ C N C D C (4.34) The (4.22) become: Θ C N C D C qdv = q 78) N ΘdV + q <= N Θds = (4.35) = Θ C N C q 78) dv + Θ C N C q <= ds Considering the expression q = Γ T and the constitutive equations qi = λit, i we can write Θ C N C D C Γ D N C Θ dv = Θ C N C q 78) dv + Θ C N C q <= ds (4.36) Denoting B = D N, B C = N C D C Θ C B C Γ B ΘdV = Θ C N C q 78) dv + Θ C N C q <= ds (4.37) B C Γ B ΘdV = N C q 78) dv + N C q <= ds (4.38) and denoting

7 K n = B C Γ BdV and Q = N C q 78) dv + N C q <= ds (4.39) we can write K n Θ = Q where K n is the thermal coefficient matrix. (4.40)

8

9

10

11