Code_Aster Study Mirror

Daniel Weschke

October 13, 2018

List of Symbols

\(c\ti{p}\)	J/(t °C)	Specific heat capacity at constant pressure
\(E\)	MPa	Young’s modulus
\(H\)	MPa	Elasticity matrix
\(h\)	W/(mm² °C)	Heat transfer coefficient
\(L\)	mm	Length
\(m\)	kg	Mass
\(\dot{m}\)	kg/s	Mass flow rate
\(\dot{Q}\)	W	Heat
\(\dot{q}\)	W/mm²	Heat flux density
\(r\)	mm	Radius
\(T\)	°C	Temperature
\(t\)	s	Time
\(u\)	mm	Displacement
\(v\)	mm/s	Velocity
\(x\)	mm	Beam direction, flow direction
\(\alpha\)	1/°C	Isotropic secant coefficient of thermal expansion
\(\lambda\)	W/(mm °C)	Thermal conductivity
\(\nu\)		Poisson’s ratio
\(\rho\)	t/mm³	Density

1 Objective

Deflection of a silicon mirror subjected to an incident beam with a maximum

bending radius of \(\units[100]{km}\) parallel to the beam.

2 Geometry & Mesh

2.1 Mirror

• Width \(\units[60]{mm}\)

• Height \(\units[60]{mm}\)

• Length \(\units[550]{mm}\)

• Material: Silicon

• Groove for stabilization

  • Position: At both sides, \(\units[9]{mm}\) below the top surface, that is below the cooling line

  • Width or depth \(\units[3.5]{mm}\)

  • Height \(\units[8]{mm}\)

Geometry of the mirror.

2.2 Cooling line

• Width \(\units[9]{mm}\)

• Height \(\units[9]{mm}\)

• Length \(\units[549]{mm}\)

• Material: Copper

• Position: At both sides at the same level as the top surface of the mirror and one millimetre behind the front face

• Drill for water line

  • Material: Water

  • Cross-section: Circle with radius \(r\ti{w} = \units[2]{mm}\)

  • Length \(L\ti{w} = \units[549]{mm}\)

  • Position: At the centre

2.3 Mesh

• Global element size \(\units[3]{mm}\)

• Element size at the top surface along the beam \(\units[1]{mm}\)

3 Material

3.1 Water

Material properties of water

\(T\)	\(\rho\ti{w}\)	\(c\ti{p,Si}\)	\(\lambda\ti{w}\)	\(\alpha\ti{w}\)
°C	t/mm³	J/(t °C)	W/(mm °C)	1/°C
0	1.0e-9	4219e3	5.611e-4	-6.77e-5
4				6.0e-7
10				8.81e-5
20				2.066e-4
25	9.9724e-10	4182e3	6.072e-4
30				3.029e-4
40				3.849e-4
50	9.882e-10	4180e3	6.436e-4	4.574e-4
60				5.231e-4
70				5.841e-4
75	9.7504e-10	4192e3	6.668e-4
80				6.417e-4
90				6.97e-4
99.61				7.489e-4
100	9.5858e-10	2074e3	2.508e-5
150	9.1707e-10	1986e3	2.886e-5

Zero-thermal-strain reference temperature for isotropic secant coefficient of thermal expansion is 24 °C.

Density \(\rho\ti{w}(T)\)

Specific heat capacity at constant pressure \(c\ti{p,w}(T)\)

Thermal conductivity \(\lambda\ti{w}(T)\)

Isotropic secant coefficient of thermal expansion \(\alpha\ti{w}(T)\) with zero-thermal-strain reference temperature at 24 °C

3.2 Silicon

Physical properties

• Density \(\rho\ti{Si} = \units[2.3296\times10^{-9}]{t/mm^3}\)

Material properties of silicon

\(T\)	\(c\ti{p,Si}\)	\(\lambda\ti{Si}\)	\(\alpha\ti{Si}\)
°C	J/(t °C)	W/(mm °C)	1/°C
-253.15			-5.0e-9
-243.15			-5.3e-8
-233.15			-1.6e-7
-223.15	78.5e3	2.600	-2.8e-7
-213.15	115.0e3	2.100	-3.65e-7
-203.15	152.0e3	1.700	-4.15e-7
-193.15	188.0e3	1.390	-4.65e-7
-183.15	224.0e3	1.140
-173.15	259.0e3	0.950	-3.4e-7
-153.15	328.0e3		-4.0e-8
-148.15		0.600
-123.15		0.420	5.25e-7
-98.15		0.325
-73.15	556.0e3	0.266	1.5e-6
-23.15		0.195
0.0	680.0e3	0.168
20			2.6e-6
26.85	714.0e3	0.156
100.0	770.0e3	0.108
126.85		0.105
226.85		0.08
227			3.5e-6
300.0	850.0e3
326.85		0.064
426.85		0.052
500.0	880.0e3
526.85		0.043
626.85		0.036
726.85		0.031
826.85		0.028
926.85		0.026
1026.85		0.025
1126.85		0.024
1226.85		0.023
1326.85		0.022
1407.85		0.022

Zero-thermal-strain reference temperature for isotropic secant coefficient of thermal expansion is 24 °C

3.2.1 Thermal

Specific heat capacity at constant pressure \(c\ti{p,Si}(T)\)

Thermal conductivity \(\lambda\ti{Si}(T)\) in W/(mm K)

Isotropic secant coefficient of thermal expansion \(\alpha\ti{Si}(T)\) with zero-thermal-strain reference temperature at 24 °C

• Volumetric heat capacity \(\rho\ti{Si}\,c\ti{p,Si}\)

3.2.2 Mechanical - Linear elastic

• Anisotropic Elasticity [1]

\[\begin{split}H\ti{Si} = \begin{bmatrix} 166 & 64 & 64 & 0 & 0 & 0 \\ 64 & 166 & 64 & 0 & 0 & 0 \\ 64 & 64 & 166 & 0 & 0 & 0 \\ 0 & 0 & 0 & 80 & 0 & 0 \\ 0 & 0 & 0 & 0 & 80 & 0 \\ 0 & 0 & 0 & 0 & 0 & 80 \end{bmatrix}\units[10^3]{MPa}\end{split}\]

• Derivative properties in principle directions:

  • Young’s modulus \(E\ti{Si} = (H\ti{Si}^{-1})_{11}^{-1} = (H\ti{Si}^{-1})_{22}^{-1} = (H\ti{Si}^{-1})_{33}^{-1} \approx \units[130.3826]{MPa}\)

  • Shear modulus \(G\ti{Si} = (H\ti{Si}^{-1})_{44}^{-1} = (H\ti{Si}^{-1})_{55}^{-1} = (H\ti{Si}^{-1})_{66}^{-1} = \units[80]{MPa}\)

  • Poisson’s ratio \(\nu\ti{Si} = -(H\ti{Si}^{-1})_{12}/(H\ti{Si}^{-1})_{11} = -(H\ti{Si}^{-1})_{23}/(H\ti{Si}^{-1})_{22} = -(H\ti{Si}^{-1})_{31}/(H\ti{Si}^{-1})_{33} \approx {0.2782609}\)

3.3 Copper

Physical properties

• Density \(\rho\ti{Cu} = \units[8.960\times10^{-9}]{t/mm^3}\)

• Isotropic secant coefficient of thermal expansion \(\alpha\ti{Cu} = \units[1.7\times10^{-5}]{1/\degC}\)

3.3.1 Thermal

Material properties of copper

\(T\)	\(c\ti{p,Cu}\)	\(\lambda\ti{Cu}\)
°C	J/(t °C)	W/(mm °C)
-100.0		0.420
-73.15	356.1e3
-23.15	374.1e3
-23.0		0.406
0.0		0.403
25.0	385.0e3
27.0		0.4005
76.85	392.6e3
77.0		0.396
100.0		0.3945
126.85	398.6e3
127.0		0.393
300.0		0.381

Specific heat capacity at constant pressure \(c\ti{p,Cu}(T)\)

Isotropic thermal conductivity \(\lambda\ti{Cu}(T)\)

• Volumetric heat capacity \(\rho\ti{Cu}\,c\ti{p,Cu}\)

4 Boundary Conditions and Loads

4.1 Thermal

• Heat flux density \(\dot{q}(x, y)\) on top surface: beamxy_f

Heat flux density \(\dot{q}(x, y)\)

import numpy as np
from scipy.interpolate import RectBivariateSpline

file = "beamprofile.out" # all x values for one y and so forth
nx = 936                 # number of points in x direction
ny = 1000                # number of points in y direction
                         # = total # of lines / # of points in x direction
with open(file, "r") as f:
  lines = f.readlines()
  x, y = np.zeros((nx)), np.zeros((ny))
  z = np.zeros((nx, ny))
  j, k = 0, -1
  for i, line in enumerate(lines):
    line_split = line.split(' ')
    j = i%nx
    if i < nx:
      x[j] = float(line_split[0])
    if j == 0:
      k += 1
      y[k] = float(line_split[1])
    z[j][k] = float(line_split[2][:-1])

interp_spline = RectBivariateSpline(x, y, z)
x1, x2, y1, y2 = 0.0, 550.0, 26.0, 34.0
beamxy_f = lambda X, Y: float(interp_spline.ev(X, Y)) if Y >= y1 and Y <= y2 else 0.0
heat_total = interp_spline.integral(x1, x2, y1, y2)

Heat flux density of the beam \(\dot{q}(x,y)\) at the top face.

Heat flux density of the beam \(\dot{q}(30,y)\) along the top face.

Heat flux density of the beam \(\dot{q}(x,275)\) along the top face.

• Heat transfer between water (external) and copper

  • External temperature \(T\ti{w}(x) = T\ti{0,w} + \Delta T\ti{w} (1 - x/L\ti{w})\)

    • Mass flow rate \(\dot{m}\ti{w} = \units[8.3333\times10^{-2}]{kg/s}\)

    • Total heat \(\dot{Q} = \iint \dot{q} dxdy = \units[3014.63736347]{W}\): heat_total

    • Temperature change in °C \(\Delta T\ti{w} = \dot{Q} / (2 \, \dot{m}\ti{w} \, c\ti{p,w})\)

    • Temperature at the beginning of the line \(T\ti{0,w} = \units[20.0]{\degC}\)

  • Heat transfer coefficient \(h\ti{w} = \units[0.0297]{W/(mm^2.\degC)}\)

Steady-state one-dimensional heat transfer equation

\[\begin{split}v\frac{dT}{dx} = \frac{1}{c} \biggl( \underbrace{\dot{q}}_{\substack{\text{through}\\\text{wall}}} + \underbrace{\frac{\lambda}{d} \frac{v^2 |v|}{2}}_{\substack{\text{Dissipation}\\\approx\,0}} \biggr)\end{split}\]

• \(x\) flow direction

• \(v(r)\) velocity

\[\begin{split}\begin{split} v\frac{dT}{dx} &= \frac{1}{c} \dot{q} \\ & \quad\text{with}\quad v = \frac{L}{t} ,~ \dot{m} = \frac{m}{t} ,~ \dot{q} = \frac{\dot{Q}}{m_t} = \frac{\dot{Q}}{2m} \\ L\frac{\dot{m}}{m}\frac{dT}{dx} &= \frac{1}{c} \frac{\dot{Q}}{2m} \\ \frac{dT}{dx} &= \frac{1}{c} \frac{\dot{Q}}{2\dot{m}}\frac{1}{L} = \text{const.} \\ \Delta T = T - T_0 &= \frac{1}{c} \frac{\dot{Q}}{2\dot{m}}\frac{x}{L} \\ \Delta T\ti{L} &= \frac{1}{c} \frac{\dot{Q}}{2\dot{m}} \\ \Delta T\ti{L} &= \frac{\units[3014.63736347]{W}}{2\cdot\units[4182.0]{J/(kg.\degC)}\cdot\units[8.333\times10^{-2}]{kg/s}} \\ \Delta T\ti{L} &= \units[4.325334224]{\degC} \end{split}\end{split}\]

Introduce a correction function.

\[\begin{split}\begin{split} \delta T(x) &= \Delta T(x) \, k(x) \\ & \qquad \text{with} \quad k(x) = ax + b \\ &= \frac{\Delta T\ti{L}}{L}(ax^2 + bx) \\ \end{split}\end{split}\]

The condition of same integral with the correction function.

\[\begin{split}\begin{split} \int\limits_0^L \delta T(x) \dif x &= \int\limits_0^L \Delta T \dif x \\ \frac{\Delta T\ti{L}}{L}\left(\frac{a}{3}L^3 + \frac{b}{2}L^2\right) &= \frac{\Delta T\ti{L}}{2L}L^2 \\ \frac{a}{3}L + \frac{b}{2} &= \frac{1}{2} \\ b &= 1 - \frac{2}{3}aL \\ \end{split}\end{split}\]

\[\begin{split}\begin{split} k(x) &= ax + 1 - \frac{2}{3}aL \\ &= aL\left(\frac{x}{L} - \frac{2}{3}\right) + 1 \\ \delta T(x) &= \Delta T\ti{L} \left(aL\left(\frac{x^2}{L^2} - \frac{2}{3}\frac{x}{L}\right) + \frac{x}{L}\right) \\ & \qquad \text{with} \quad a = \frac{3}{9L} = \frac{1}{3L} \\ \delta T(x) &= \Delta T\ti{L} \left(\frac{1}{3}\frac{x^2}{L^2} + \frac{7}{9}\frac{x}{L}\right) \\ \end{split}\end{split}\]

Temperature change profile of the water pipe.

5 Results

Temperature.

Mesh	Nodes	Temperature range	Temperature range
		\(\Delta{}T\) in °C	\(\delta{}T\) in °C
normal	592341	[26.0746, 102.843]	[26.0375, 36.0593], [32.1906, 103.096]
fine	883616	[]
finer	1774863	[26.0344, 102.311]
finest	2877747	[26.0347, 102.339]
reference		[26.126, 103.48]

Displacement in z direction.

Displacement \(u\ti{z}\) and fit of the displacement in z direction along the top face.

Compared displacement with finer meshes

Curvature via polynomial fit of the top face in z direction.

The peak-to-peak value is around 403.46 nm.

The maximum curvature is around 18.4 per pm. Therefore a minimum radius of around 54.4 km.

References

Notes

[1]

Mason, W. P. Physical acoustics and the properties of solids. The Journal of the Acoustical Society of America 28, 6 (1956), 1197–1206.