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Chapter 11SCALAR FIELDS11.1 IntroductionThe physical behavior governing a variety of problems in engineering can bedescribed as scalar field problems. That is, where a scalar quantity varies over acontinuum. We usually need to compute the value of the scalar quantity, its gradient, andsometimes its integral over the solution domain. Typical applications of scalar fieldsincludes: electrical conduction, heat transfer, irrotational fluid flow, magnetostatics,seepage in porous media, torsion stress analysis, etc. Often these problems are governedby the well known Laplace and Poisson differential equations. The analytic solution ofthese equations in two- and three-dimensional field problems can present a formidabletask, especially in the case where there are complex boundary conditions and irregularlyshaped regions. The finite element formulation of this class of problems by usingGalerkin or variational methods has proven to be a very effective and versatile approachto the solution. Previous difficulties associated with irregular geometry and complexboundary conditions are virtually eliminated. The following development will beintroduced through the details of formulating the solution to the steady-state heatconduction problem. The approach is general, however, and by redefining the physicalquantities involved the formulation is equally applicable to other problems involving thePoisson equation.11.2 Variational formulationWe can obtain from any book on heat transfer the governing differential equationfor steady and un-steady (transient) state heat conduction. The most general form of theheat conduction equation, in the material principal coordinate directions is the transientthree-dimensional equation:(11.1)∂∂x(kx∂θ∂x) +∂∂y(ky∂θ∂y) +∂∂z(kz∂θ∂z) + Q =∂∂t(ρcpθ)where, kx,ky,kz= thermal conductivity coefficients,θ= temperature, Q = heatgeneration per unit volume,ρ= density, and cp= specific heat at constant pressure. If wefocus our attention to the two-dimensional (∂ /∂z = 0) steady-state (∂ /∂t = 0) problem,such as Fig. 11.2.1, the governing equation becomes4.3 Draft − 5/27/04© 2004 J.E. Akin 280Finite Elements, Scalar Fields 281∂∂x(kx∂θ∂x) +∂∂y(ky∂θ∂y) + Q = 0in which kx,ky, and Q are known. Equations 11.1 or 11.2, together with the appropriateboundary conditions specify the problem completely. The most commonly encounteredboundary conditions are those in which the temperature,θ, is specified on the boundary,(11.2)θ=θ(s)onΓD,or the normal heat flux into the boundary, qs, is specified:kx∂θ∂xnx+ ky∂θ∂yny+ qs= kn∂θ∂n+ qs= 0onΓqor a normal heat flux due to convection:(11.3)kn∂θ∂n+ h(θ−θ∞) = 0, on Γhwhere nxand nyare the direction cosines of the outward normal to the boundary surface,qsrepresents the known heat flux per unit of surface, and h(θ−θ∞) is the convectionheat loss per unit area due to a convection coefficient h and a surrounding fluid at atemperature ofθ∞. Only one of these two last two items is non-zero on a particularsurface. Note that the last two surface integrals could be written in a more general formif we combine them into a mixed or Robin condition written as:(11.4)kn∂θ∂n+ hθ+ g = 0,where g is either a known influx (when h = 0), or hθ∞on a convection surface.In Section 2.13.2 we illustrated how to apply the Galerkin method to this equation.As stated previously, an alternative formulation to the above heat conduction problem ispossible using the calculus of variations. It has been shown that if a variational formexists for a differential equation then both the Galerkin form and the Euler variationalform will yield exactly the same element matrix definitions. Euler’s theorem of thecalculus of variations states that if the integral(11.5)I(u) =Ω∫f (x, y,z,u,∂u∂x,∂u∂y,∂u∂z)dΩ+Γ∫(gu + hu2/2) dΓis to be minimized, the necessary and sufficient condition for this minimum to be reachedis that the unknown function u(x, y,z) satisfy the following differential equation(11.6)∂∂x∂ f∂(∂u /∂x)+∂∂y∂ f∂(∂u /∂y)+∂∂z∂ f∂(∂u /∂z)−∂ f∂u= 0within the region Ω, provided u satisfies the essential boundary conditions on ΓDandnxkx∂θ∂x+ nyky∂θ∂y+ nzkz∂θ∂z+ g + hθ= 0 = kn∂θ∂n+ g + hθon the remainder of Γ. We can verify that the minimization of the volume integral4.3 Draft − 5/27/04 © 2004 J.E. Akin. All rights reserved.282 J. E. AkinThickness, tVolumetricsource, QUnit normal, nNormalflux, q nConductivities, K x, K yGiven temperature, T0Convection,qh = h (T - Tref)h, TrefT (X, Y) *Figure 11.2.1 An anisotropic heat transfer region(11.7)I =Ω∫12kx(∂θ∂x)2+ ky(∂θ∂y)2+ kz(∂θ∂z)2− QθdΩ+Γ∫gθ+ hθ2/2dΓleads directly to the formulation equivalent to Eq. 11.2 for the steady-state case. It shouldalso be noted that the surface Γ will be split into different regions for each distinct set ofsurface input. One of those segments will usually be a Dirichlet region, ΓD, and thatsurface integral represents the unknown resultant reaction fluxes at the nodes that getlumped into the RHS of the algebraic system. The functional volume contribution isf =12kx(∂θ∂x)2+ ky(∂θ∂y)2+ kz(∂θ∂z)2− Qθ.Thus, if f is to be minimized it must satisfy Eq. 11.6. Here∂ f∂(∂θ/∂x)= kx∂θ∂x,∂ f∂(∂θ/∂y)= ky∂θ∂y,∂ f∂(∂θ/∂z)= kz∂θ∂z,∂ f∂θ=−Qso Eq. 11.6 results in4.3 Draft − 5/27/04 © 2004 J.E. Akin. All rights reserved.Finite Elements, Scalar Fields 283∂∂x(kx∂θ∂x) +∂∂y(ky∂θ∂y) +∂∂z(kz∂θ∂z) + Q = 0verifying that the function f does lead to correct steady state formulation, if Eq. 11.3 isalso satisfied. Euler also stated that the natural boundary condition associated withEq. 11.5 on a surface with a unit normal vector→n isnx∂ f∂(∂u /∂x)+ ny∂ f∂(∂u /∂y)+ nz∂ f∂(∂u /∂z)+ g + hu = 0on the boundary where the value of u is not prescribed. If both g and h are non-zero thistype of boundary condition is a Robin, or mixed, condition since it imposes a linearcombination on the solution and the normal gradient on part of the boundary.The element and boundary matrices arising from Eq. 11.7 and the Galerkin methodwill be identical. Therefore we have the tools to build non-homogeneous,


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Rice MECH 517 - Scalar Fields

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