ECE 695 Numerical Simulations Lecture 12: Thermomechanical FEM. Prof. Peter Bermel February 6, 2017

Transcription

1 ECE 695 Numerical Simulations Lecture 12: Thermomechanical FEM Prof. Peter Bermel February 6, 2017

2 Outline Static Equilibrium Dynamic Equilibrium Thermal transport mechanisms Thermal transport modeling in MATLAB 2/6/2017 ECE 695, Prof. Bermel 2

3 Static Equilibrium Newton s Law for a 1D wire : P 2 w + q = μ x2 w In static equilibrium, forces balance exactly: P 2 w x 2 + q = 0 Define residual error in solution such that: r B = P 2 w x 2 + q 2/6/2017 ECE 695, Prof. Bermel 3

4 Galerkin Method In general, we want to apply Galerkin method with trial functions η j : 0 L dx η j x r B x, t = 0 Analogy: stuff balloon into box, with each trial function a single finger. Substituting: L 0 dx ηj x P 2 w x 2 + q = 0 2/6/2017 ECE 695, Prof. Bermel 4

5 Galerkin Method Integrating by parts: L L 0 = η j P w + dx η x j q η j w P 0 0 x x Letting boundary terms vanish and substituting linear basis (w = i N i w i ) yields: 0 = 0 L dx η j q N i=1 0 L 0 = b Kw dx η j x P N i x w i 2/6/2017 ECE 695, Prof. Bermel 5

6 Static Equilibrium Numerical solution Analytical solution Numerical solution matches analytical solution closely at key 2/6/2017 ECE 695, Prof. Bermel 6

7 Dynamic Equilibrium Restoring time dependence will require tracking another second derivative term. In analogy with previous procedure, we can create another matrix, which yields: 0 = b Kw M w In absence of restoring force (q = 0), we have harmonic solutions (w = φe iωt ), so that: Kφ ω 2 Mφ = 0 2/6/2017 ECE 695, Prof. Bermel 7

8 Dynamic Equilibrium Convergence for first 5 frequencies as a function of the number of elements 2/6/2017 ECE 695, Prof. Bermel 8

9 Thermal Transport Mechanisms Convection: heat transfer by surface contact with gas or fluid molecules Conduction: volumetric heat transfer by propagation of phonons Radiative thermal transfer: emission of thermal photons from source to receiver 2/6/2017 ECE 695, Prof. Bermel 9

10 Thermal Transport: Convection Heat transfer by gas or fluid molecules Transfer rate per unit area given by: Q = h T 1 T 2 Heat transfer constant h determined by many factors, including material choice, microstructures, fluid flow environment, etc. 2/6/2017 ECE 695, Prof. Bermel 10

11 Thermal Transport: Conduction Volumetric heat transfer through phonon transfer Change in heat energy per unit time is: du dt = c t V T dv = q ds + Q dv V V Applying the divergence theorem thus yields: T c V + q Q dv = 0 t V 2/6/2017 ECE 695, Prof. Bermel 11

12 Thermal Transport: Conduction Local conservation of energy yields: u = Q q t Heat transfer rate quantified by Fourier s law: q = k T Combining with previous result yields thermal diffusion equation: u t Q = k 2 T = k 2 u α 2 u ρc v 2/6/2017 ECE 695, Prof. Bermel 12

13 Thermal Transport: Conduction 2/6/2017 ECE 695, Prof. Bermel 13

14 Thermal Transport: Conduction Can formulate residual as: T r B = c V k T Q t Writing down Galerkin method: V ηr B dv = 0 Integrating by parts and dropping boundary yields: T ηc V + η k T ηq dv = 0 t V 2/6/2017 ECE 695, Prof. Bermel 14

15 Thermal Transport: Conduction Now assume η = N i and T = i N i T i : i N j c V N i ds T i t + N j k N i ds T i N j Q ds = 0 This can be simplified as: i C ji T i t + K jit i L j = 0 2/6/2017 ECE 695, Prof. Bermel 15

16 Thermal Transport: Radiative Transfer Heat transfer via photon emission For a blackbody, total emission follows Stefan- Boltmann law: P = σt 4 Net thermal transfer between two infinite surfaces becomes: Q = σ T 1 4 T 2 4 2/6/2017 ECE 695, Prof. Bermel 16

17 Thermal Transport: Radiative Transfer Emission for real materials depends on emissivity In thermal equilibrium, Kirchoff s law states emissivity=absorptivity at each wavelength Emission spectrum is given by: dq dλ = 2πhc2 ε(λ) λ 5 e hc/λkt 1 Blackbody result recovered by setting ε λ = 1 and integrating 2/6/2017 ECE 695, Prof. Bermel 17

18 Thermal Transport: Modeling Convection amounts to a boundary condition in most problems Will thus be first combined with conduction Strategy: Create FEM grid for thermal conduction Impose BC s from convection (Optionally) include radiative transfer from disconnected bodies 2/6/2017 ECE 695, Prof. Bermel 18

19 Thermal Transport FEM Employ Galerkin method to reduce to linear algebra problem as before (see Petr Krysl s step-by-step introduction, Chapter 6): Where: C T + (K + H)T = L i i 2/6/2017 ECE 695, Prof. Bermel 19

20 FAESOR: MATLAB FEM Toolbox Full rationale / background provided here: MATLAB code download here: _publish.html Links also on course webpage 2/6/2017 ECE 695, Prof. Bermel 20

21 FAESOR: MATLAB FEM Toolbox 1D Meshing routine: Construct finite element block: 2/6/2017 ECE 695, Prof. Bermel 21

22 FAESOR: MATLAB FEM Toolbox Apply boundary conditions: Assemble and solve equations: 2/6/2017 ECE 695, Prof. Bermel 22

23 Results Steady-state solution for a thermally insulating medium, with a variable temperature placed along one surface 2/6/2017 ECE 695, Prof. Bermel 23

24 Results Steady-state solution for heat diffusion in an inhomogeneous medium 2/6/2017 ECE 695, Prof. Bermel 24

25 Thermal Conduction: Results Steady-state thermal gradient between two adjoining walls with different temperatures 2/6/2017 ECE 695, Prof. Bermel 25

26 Thermal Conduction: Results Example of adaptive mesh on T4 NAFEMS benchmark 2/6/2017 ECE 695, Prof. Bermel 26

27 Thermal Conduction: Results Coarse model High temp, adaptive mesh Low temp, adaptive mesh Coarse model Transient cooling of a shrink-fitted assembly 2/6/2017 ECE 695, Prof. Bermel 27

28 Error Evaluation Error for linear elements T3 higher overall than quadratic elements T6 Both decrease almost quadratically with mesh size T3 elements faster to compute 2/6/2017 ECE 695, Prof. Bermel 28

29 Next Class We will cover electronic transport 2/6/2017 ECE 695, Prof. Bermel 29