Commit 4778ee69 authored by Benoit Urruty's avatar Benoit Urruty

optim_mesh

parent 198c0856
......@@ -4,7 +4,7 @@
## Initialization
We need to interpolate the data on the mesh. These data contains :
We need to interpolate the data on the mesh. These data contain :
- The bed topography, the thickness (both from BedMachine)
- The surface mass balance (SMB from a the MAR model)
......@@ -24,34 +24,71 @@ The models is now able to compute others variables.
The goal of the inversion is to find the viscosity and the friction parameter which will permit
to approach as best as possible the observed surface velocity.
### Body Force
### SSA
The SSA is compute with the first guess on the value to optimize. We
solve the equation :
### Material
Viscosity exponent :
$$
\frac{\partial}{\partial x}(2\eta h (2 \frac{\partial u}{\partial x} + \eta \frac{\partial v}{\partial y})) + \frac{\partial}{\partial y}(\eta h (\frac{\partial u}{\partial y} + \eta \frac{\partial v}{\partial x})) - \beta^2u =\rho_igh\frac{\partial s}{\partial x}
Viscosity\ exponent = \frac{1}{n}\ avec\ n=
$$
Critical shear stress :
$$
\frac{\partial}{\partial y}(2\eta h (2 \frac{\partial v}{\partial y} + \eta \frac{\partial u}{\partial x})) + \frac{\partial}{\partial x}(\eta h (\frac{\partial v}{\partial x} + \eta \frac{\partial u}{\partial y})) - \beta^2v =\rho_igh\frac{\partial s}{\partial y}
\tau_c = 1.0\times10^{-12}
$$
Mean density :
$$
\rho_i = 917 kg/m^{-3}
$$
Mean Viscosity :
$$
\eta=prefactor ^2 \times \mu
$$
Mean Viscosity Derivative :
$$
\partial\eta=2\times prefactor \times\mu
$$
Friction Parameter :
$$
\beta = 10^\alpha
$$
Friction parameter derivative :
$$
\frac{\partial J}{\partial\alpha}=\frac{\partial J}{\partial\beta}\times 10 ^ {\alpha}\times ln 10
$$
Grounded area coefficient : give the value 1 if it's grounded or 0 if it the grounding line
Flux:
$$
$$
### SSA
The SSA is computed with the first guess on the value $\eta$ and $\beta$. We solve the equation :
$$
\frac{\partial}{\partial x}(2\eta h (2 \frac{\partial u}{\partial x} + \eta \frac{\partial v}{\partial y})) + \frac{\partial}{\partial y}(\eta h (\frac{\partial u}{\partial y} + \eta \frac{\partial v}{\partial x})) - \beta^2u =\rho_igh\frac{\partial s}{\partial x}
$$
The effective viscosity :
$$
\eta = \frac{1}{2}A^{-1/n_{\epsilon_e}(1-n)/n}
\frac{\partial}{\partial y}(2\eta h (2 \frac{\partial v}{\partial y} + \eta \frac{\partial u}{\partial x})) + \frac{\partial}{\partial x}(\eta h (\frac{\partial v}{\partial x} + \eta \frac{\partial u}{\partial y})) - \beta^2v =\rho_igh\frac{\partial s}{\partial y}
$$
The value of the velocity u and v are exported to be used in the cost function.
The value of the velocity $u$ and $v$ are exported to computed in the cost function.
### The cost function
The cost function is the comparison of the compute surface velocity
and the observed one.
and the observed one. We compute here the integer of all the difference
$$
J = \int{\frac{1}{2}((u_{SSA} - u_{Obs})^2+(v_{SSA} - v_{Obs})^2)} d\Omega
$$
......@@ -80,8 +117,8 @@ $$
We need to compute the gradient for each parameters we have to optimize. Theoretically, we have the right solution when the gradient is equal to zero but we never obtains this value. So we define a lower limit which is define a the gradient is enough low to have a good result.
$$
\nabla J= \left\{ \begin{array}{ll}
\frac{\partial J}{\partial Beta}=\\
\frac{\partial J}{\partial Eta}=
\frac{\partial J}{\partial Beta}=\frac{J(i)-J(i-1)}{Beta(i)-Beta(i-1)}\\
\frac{\partial J}{\partial Eta}=\frac{J(i)-J(i-1)}{Eta(i)-Eta(i-1)}
\end{array}
\right\}
$$
......@@ -90,20 +127,13 @@ $$
### The regularization terms
The regularization can be done one the first derivative. We gave a weight $\lambda$ to this the value computed. This weight have to be choice according the
The regularization can be done one the first derivative. We gave a weight $\lambda$ to this the value computed. This weight have to be choice according to the order of the optimized parameter.
$$
J_{reg} = \int_{\Omega} 0.5 (|dV/dx|)^2 d\Omega
\\with\ V \ the\ nodal\ variable
$$
$$
J_{reg} = \int_{\Omega} 0.5 ((V-V^{prior})/s^2)^2 d\Omega
\\with\ V \ the\ nodal\ variable, V^{prior}\ is \ the\ estimate\ and\ s^2\ the\ variance.
$$
### The optimization M1QN3
The optimization will take the gradient and try to found the best choice by doing simulations to find the best direction and will pass to the next iteration.
\ No newline at end of file
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202 2535
303 335454
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303 275809
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1.000000000000E+000 168995 6.375949530874E+015 2.697290591472E+014
2.000000000000E+000 122072 6.379676541478E+015 2.696764958384E+014
3.000000000000E+000 123462 6.379706477498E+015 2.695996586492E+014
4.000000000000E+000 126500 6.380117957237E+015 2.694969702347E+014
5.000000000000E+000 129856 6.380395535351E+015 2.696278229463E+014
6.000000000000E+000 133359 6.379883829224E+015 2.694896330371E+014
7.000000000000E+000 136451 6.379772514343E+015 2.696067397066E+014
8.000000000000E+000 139344 6.379127361973E+015 2.695786798181E+014
9.000000000000E+000 142148 6.378743986633E+015 2.694373545293E+014
1.000000000000E+001 144529 6.380209511903E+015 2.696153846251E+014
Elmer version: 8.4
Elmer revision: 6cb7d614
Elmer compilation date: 2020-04-07
Solver input file: mesh_optim.sif
File started at: 2020/05/19 15:20:31
Variables in columns of matrix: ./f_1.dat
1: value: time scalar variable
2: nodes: time
3: int: h
4: int: vobs
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