Part I. Discrete Models. Part I: Discrete Models. Scientific Computing I. Motivation: Heat Transfer. A Wiremesh Model (2) A Wiremesh Model

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1 Part I: iscrete Models Scientific Computing I Module 5: Heat Transfer iscrete and Continuous Models Tobias Neckel Winter 04/05 Motivation: Heat Transfer Wiremesh Model A Finite Volume Model Time ependent Model Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 Motivation: Heat Transfer Part I iscrete Models objective: compute the temperature distribution of some object under certain prerequisites: temperature at object boundaries given heat sources material parameters observation from psical experiments: q k δt heat flux proportional to temperature differences Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 3 Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 4 A Wiremesh Model A Wiremesh Model consider rectangular plate as fine mesh of wires compute temperature x at nodes of the mesh x + x i,j x x i+,j x model assumption: temperatures in equilibrium at every mesh node equilibrium: steady state of temperature, energy balance inflow = outflow in each node of the mesh incoming temperature fluxes at point via the four wires: from the left: k x i,j x from the right: k x i+,j x from below: k x x from above: k x + x equation for steady state: sum over all fluxes = zero: for all temperatures x. x = xi,j + x i+,j + x + x + 4 Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 5 Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 6 A Wiremesh Model 3 temperature known at part of the boundary; for example x 0,j = T j A Finite Volume Model object: a rectangular metal plate again model as a collection of small connected rectangular cells models a heated/cooled wall with constant temperature T j at the left boundary. temperature flux known at part of the boundary; for example x i,0 = x i, x i, x i,0 = 0 models an isolated wall at the lower boundary. heat sources: temperature given at a certain position i, j: x = T s. task: solve Linear System of Equations Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 7 examine the heat flux across the cell edges Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 8

2 A Finite Volume Model A Steady-State Model model assumption: temperatures in equilibrium in every grid cell heat flux across a given edge is proportional to temperature difference T T 0 between the adjacent cells length h of the edge e.g.: heat flux across the left edge: q left T T i,j note: heat flux out of the cell and k x > 0 heat flux across all edges determines change of heat energy: q ij Tij T i,j + k x Tij T i+,j + k y Tij T + k y Tij T + heat sources: consider additional source term F due to external heating radiation F = f f heat flux per area equilibrium with source term requires q + F = 0: f = k x T T i,j T i+,j again, Linear System of Equations k y T T T + Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 9 Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 0 Towards a Time ependent Model idea: set up an OE for each cell simplification: no external heat sources or sinks, i.e. f = 0 change of temperature per time is proportional to heat flux q t into the cell no longer 0: d dt T t = c q t solve a system of OEs = κ x Tij t + T i,j t + T i+,j t + κ y Tij t + T t + T + t Boundary Conditions Finite Volume Models temperature known in boundary layer cells; for example q left,j T,j T 0,j T,j T x 0,j with T 0,j = T x 0,j not an unknown! models a heated/cooled wall with constant temperature T x 0,j at the left boundary temperature flux known in boundary layer cells; e.g. q left,j = 0: f,j = k x T,j T,j k y T,j T,j T,j+ models an isolated wall at the left boundary. Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 Part II: A Continuous Model The Heat Equation From iscrete to Continuous erivation of the Heat Equation Variants of the Heat Equation Part II A Continuous Model The Heat Equation Boundary and Initial Conditions Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 3 Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 4 From iscrete to Continuous remember the discrete model: f = k x T T i,j T i+,j k y T T T + assumption: heat flux across edges is proportional to temperature difference q left T T i,j in reality: heat flux proportional to temperature gradient q left k T T i,j From iscrete to Continuous replace k x by k/, k y by k/, and get: f = k T T i,j T i+,j k T T T + consider arbitrarily small cells:, 0: T T f = k k leads to a partial differential equation PE: T x, y k + T x, y = f x, y Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 5 Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 6

3 erivation of the Heat Equation finite volume model, but with arbitrary control volume change of heat energy per time is a result of transfer of heat energy across s surface, heat sources and sinks in external influences resulting integral equation: ρct dv = t q dv + k T n ds density ρ, specific heat c, and heat conductivity k are material parameters heat sources and sinks are modelled in term q erivation of the Heat Equation according to theorem of Gauß: k T n ds = k T dv leads to integral equation for any domain : ρct t q k T dv = 0 hence, the integrand has to be identically 0: T t = κ T + q ρc, κ := k ρc κ > 0 is called the thermal diffusion coefficient since the Laplace operator models a heat diffusion process Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 7 Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 8 Variants of the Heat Equation Boundary Conditions ifferent scenarios: vanishing external influence, q = 0: alternative notation t equilibrium solution, T t = 0: T t = κ T T = κ + T + T z 0 = κ T + q ρc Poisson s Equation T = q κρc irichlet boundary conditions: fix T on part of the boundary Neumann boundary conditions: T x, y, z = ϕx, y, z fix T s normal derivative on part of the boundary: special case: insulation x, y, z = ϕx, y, z n x, y, z = 0 n Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 9 Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 0 Part III iscretization: Finite ifference and Finite Volume Methods Part III: iscretization: Finite ifference and Finite Volume Methods From Continuous Back To iscrete Models Finite ifference Methods Meshes for Finite ifference iscretisation Finite ifference iscretisation Resulting Linear System of Equations iscretisation Stencils Finite Volume Methods Finite Volume Meshes Finite Volume iscretisation Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 From Continuous Back To iscrete Models The Model Problem Continuous Models: result from a limit process h 0 from discrete model wire mesh, finite volume opposite route discretisation iscretisation methods: Finite ifferences: replace derivative by difference quotients Finite Volumes: compute fluxes on boundary of control volumes and examine conservation laws Poisson Equation: ux, y + ux, y = f x, y on the unit square Ω = 0, with irichlet boundary conditions: ux, y = gx, y on Ω Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 3 Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 4

4 Meshes for Finite ifference iscretisation regular, Cartesian mesh; analogous to the wire-mesh model: Finite ifference iscretisation replace partial derivative at each mesh point by difference quotient: x + x i,j x x i+,j x u x ux i+,j ux + ux i,j u x ux + ux + ux h z leads to Linear System of Equations h := = : compute approximate value of u for each mesh point: u ij ux ij u ijk ux ijk h ui+,j + u + 4u +u + u i,j = f x x 0, ux = gx x Ω Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 5 Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 6 Resulting Linear System of Equations Resulting Linear System of Equations matrix-vector notation of the system: A h = f h x h a vector of all unknowns u ij requires numbering of the unknowns using row-wise numbering, e.g.: x h = u,,..., u,n, u,,..., u,n,..., u n,,..., u n,n A h is a sparse matrix only 5 diagonals are non-zero A h is block-tridiagonal: B h I A h =. I..... h I I B h = tridiag, 4,, where I is the unit matrix B h Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 7 Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 8 Notation: iscretisation Stencils Finite Volume Methods Meshes domain Ω subdivided into grid cells/elements Ω ij : idea: illustrate the matrix structure via a so-called discretisation stencil represents one row of the matrix matrix elements ordered according to their geometrical orientation discretisation stencil for Poisson equations: : [ ] h : 4 h u constant in Ω ij, i.e., ux, y = u ij Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 9 Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 30 Finite Volume iscretisation integrate over grid cells Ω ij : integration by parts: u x, y + u x, y dx dy = u x, y dx dy = u x, y dx dy = y j+ y j x i+ x i u u f x, y dx dy } {{ } := f ij x, y x, y x i+ x i y j+ y j dy dx Finite Volume iscretisation remember: u constant in Ω ij, i.e., ux, y = u ij thus approximation of derivatives on edges: u = u i+,j u ij u xi h x = u ij u i,j xi+ h x u = u + u ij u = u ij u yj yj+ again leads to Linear System of Equations: ui+,j u + u i,j + u+ u + u = f ij Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 3 Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 3

5 Finite Volume iscretisation More General... typical formulation for first-order PEs: u F ux, y Gux, y + + dx dy t =... and analogously: Fux, y dx dy = y j+ y j Fux, y x i+ x i dy Gux, y dx dy = x i+ x i Gux, y y j+ y j dx for Poisson Equation: F u = u, etc. Module 5: Heat Transfer iscrete and Continuous Models, Winter 04/05 33