geometrical optimization of a reactor's core of gen iv
TRANSCRIPT
Geometrical optimization of a reactor's core of gen IV using
FreeFem++– Geometrical optimization of a reactor’s core of gen IV
using FreeFem++–
joint-work 1 between SINETICS department at EDF R&D, Clamart and CMAP, Ecole Polytechnique
1. PEPS agreement
III. Numerical results Conclusion
Aims and context
Apply a shape optimization method to the conception of a nuclear reactor of generation IV having a reduced sodium void effect
Implement the method with the finite element software FreeFem++
Show to what extent the method could be applied in this context
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 2 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
2 II. Shape optimization equations
3 III. Numerical results
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 3 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
Modeling of the reactor
Description of the core
We assume the reactor can be simplified to a cylinder and that we have an axisymetric geometry = [0,R]× [0, h]
= ∪4 i=1i is defined as the reunion of 4 regions having some averaged
physical properties, each of them being characterized by a set of homogenized coefficients (D,Σa,Σf )
The active core is composed by 3 different materials : we denote by f the fissile material
The last region is the reflector surrounding the active core : absorbing region made of steel material
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 4 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
State equation
Using the notations : ∂D = {(r , z) ∈ ∂; r 6= 0}, ∂N = {(r , z) ∈ ∂; r = 0} and u the neutrons flux
We consider the simplified 1 group energy diffusion equation formulated on the space X = {φ ∈ H1();φ|∂D
= 0} on a bounded domain = ∪i i :
− D 4 u + Σau = λνΣf u in i
transmissions conditions on ∂i ∩ ∂j
u = 0 on ∂D ∩ ∂i
∂u
(1)
Remark : With this convention for λ, we have λ = 1 keff
keff is a key parameter linked to the neutronic equilibrium of the core (generaly close to 1)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 5 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
2 II. Shape optimization equations
3 III. Numerical results
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 6 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Variational form
The state equation represents an equilibrium between the reactions describing the behaviour of neutrons at steady states :
−D 4 u leakage rate
+ Σau absorption rate
Variational form
We are looking for weak solutions of the variational form : Find u ∈ X such that :∫
∫
a(u, v) =
b(u, v) =
(3)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 7 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Cost function
Estimation of the sodium void effect
We want to determine the first eigenvalue of this problem, which is simple, since this is the only to have a physical meaning (eigenvector keep the same sign over )
By considering two set of homogenized coefficients for this equation in a nominal state (with index ”n”) and a peturbated one (index ”v”), we calculate the sodium void effect as δρNa = λn − λv
Our cost function is :
+
(4)
The last term is not differentiable so we approximate it by the following quantity :
V (f )1− 1 s
(∫ f
1/s
(5)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 8 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Cost function
Estimation of the sodium void effect
We want to determine the first eigenvalue of this problem, which is simple, since this is the only to have a physical meaning (eigenvector keep the same sign over )
By considering two set of homogenized coefficients for this equation in a nominal state (with index ”n”) and a peturbated one (index ”v”), we calculate the sodium void effect as δρNa = λn − λv
Our cost function is :
+
(4)
The last term is not differentiable so we approximate it by the following quantity :
V (f )1− 1 s
(∫ f
1/s
(5)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 8 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Cost function
Estimation of the sodium void effect
We want to determine the first eigenvalue of this problem, which is simple, since this is the only to have a physical meaning (eigenvector keep the same sign over )
By considering two set of homogenized coefficients for this equation in a nominal state (with index ”n”) and a peturbated one (index ”v”), we calculate the sodium void effect as δρNa = λn − λv
Our cost function is :
+ (λn − λ0)2 target reactivity
(4)
The last term is not differentiable so we approximate it by the following quantity :
V (f )1− 1 s
(∫ f
1/s
(5)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 8 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Cost function
Estimation of the sodium void effect
We want to determine the first eigenvalue of this problem, which is simple, since this is the only to have a physical meaning (eigenvector keep the same sign over )
By considering two set of homogenized coefficients for this equation in a nominal state (with index ”n”) and a peturbated one (index ”v”), we calculate the sodium void effect as δρNa = λn − λv
Our cost function is :
+ (λn − λ0)2 target reactivity
(4)
The last term is not differentiable so we approximate it by the following quantity :
V (f )1− 1 s
(∫ f
1/s
(5)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 8 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Cost function
Estimation of the sodium void effect
We want to determine the first eigenvalue of this problem, which is simple, since this is the only to have a physical meaning (eigenvector keep the same sign over )
By considering two set of homogenized coefficients for this equation in a nominal state (with index ”n”) and a peturbated one (index ”v”), we calculate the sodium void effect as δρNa = λn − λv
Our cost function is :
+ (λn − λ0)2 target reactivity
(4)
The last term is not differentiable so we approximate it by the following quantity :
V (f )1− 1 s
(∫ f
1/s
(5)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 8 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
What the algorithm does
Figure: Initial shape Figure: Shape after 500 iterations
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 9 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Admissible shape transformation of the bounded set
We move the shape during a time t with a lipschitzian velocity field satisfying tθn∞ ≤ 1 moving the points with the following equation :
x → x + t θn(x)
Thus, we have n+1 = (Id + t θn)(n)
We impose θ · n = 0 on ∂ and on the external boundary of the active core
The volume constraint expresses as vol(T (f )) = vol(f )
T = Id + t θ
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 10 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Admissible shape transformation of the bounded set
We move the shape during a time t with a lipschitzian velocity field satisfying tθn∞ ≤ 1 moving the points with the following equation :
x → x + t θn(x)
Thus, we have n+1 = (Id + t θn)(n)
We impose θ · n = 0 on ∂ and on the external boundary of the active core
The volume constraint expresses as vol(T (f )) = vol(f )
T = Id + t θ
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 10 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Gradient method
We characterize the optimal shape by first order optimality conditions
We define the shape derivative in of the function J() as the derivative at the point 0 of the application :
{ W 1,∞(f ,R2)→ R
θ → J((Id + t θ)())
That is also the linear form on W 1,∞(f ,R2) such that :
J((Id + t θ)(f )) = J(f ) + J ′(f ), θ+ (θ) (6)
Once we know the shape derivative J ′(f )(θ), we apply a gradient descent algorithm by solving the following problem :
∀θ ∈ H1(f ,R2) : J ′(f )(θ) + θn, θ = 0 (7)
for the scalar product on X × X , it gives J ′(f )(θn) = −θn2
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 11 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Gradient method
We characterize the optimal shape by first order optimality conditions
We define the shape derivative in of the function J() as the derivative at the point 0 of the application :
{ W 1,∞(f ,R2)→ R
θ → J((Id + t θ)())
That is also the linear form on W 1,∞(f ,R2) such that :
J((Id + t θ)(f )) = J(f ) + J ′(f ), θ+ (θ) (6)
Once we know the shape derivative J ′(f )(θ), we apply a gradient descent algorithm by solving the following problem :
∀θ ∈ H1(f ,R2) : J ′(f )(θ) + θn, θ = 0 (7)
for the scalar product on X × X , it gives J ′(f )(θn) = −θn2
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 11 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Gradient method
We characterize the optimal shape by first order optimality conditions
We define the shape derivative in of the function J() as the derivative at the point 0 of the application :
{ W 1,∞(f ,R2)→ R
θ → J((Id + t θ)())
That is also the linear form on W 1,∞(f ,R2) such that :
J((Id + t θ)(f )) = J(f ) + J ′(f ), θ+ (θ) (6)
Once we know the shape derivative J ′(f )(θ), we apply a gradient descent algorithm by solving the following problem :
∀θ ∈ H1(f ,R2) : J ′(f )(θ) + θn, θ = 0 (7)
for the scalar product on X × X , it gives J ′(f )(θn) = −θn2
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 11 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Gradient method
We characterize the optimal shape by first order optimality conditions
We define the shape derivative in of the function J() as the derivative at the point 0 of the application :
{ W 1,∞(f ,R2)→ R
θ → J((Id + t θ)())
That is also the linear form on W 1,∞(f ,R2) such that :
J((Id + t θ)(f )) = J(f ) + J ′(f ), θ+ (θ) (6)
Once we know the shape derivative J ′(f )(θ), we apply a gradient descent algorithm by solving the following problem :
∀θ ∈ H1(f ,R2) : J ′(f )(θ) + θn, θ = 0 (7)
for the scalar product on X × X , it gives J ′(f )(θn) = −θn2
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 11 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Gradient method
We characterize the optimal shape by first order optimality conditions
We define the shape derivative in of the function J() as the derivative at the point 0 of the application :
{ W 1,∞(f ,R2)→ R
θ → J((Id + t θ)())
That is also the linear form on W 1,∞(f ,R2) such that :
J((Id + t θ)(f )) = J(f ) + J ′(f ), θ+ (θ) (6)
Once we know the shape derivative J ′(f )(θ), we apply a gradient descent algorithm by solving the following problem :
∀θ ∈ H1(f ,R2) : J ′(f )(θ) + θn, θ = 0 (7)
for the scalar product on X × X , it gives J ′(f )(θn) = −θn2
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 11 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Lagrangian associated to the minimization problem
J does not depend explicitly on the shape
Use of the adjoint method
We introduce the Lagrangian L that transforms the initial problem into a new one without constraints on the variables :
L(, un, uv , pn, pv , λn, λv ) = λn − λv sodium void effect
+ (λn − λ0) 2
states equations
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 12 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Adjoint method
Let (uk , λk ) be the solution of the state equation in the state k
We have the relation :
J() = L(, un, uv , λn, λv , p, q) ∀ (p, q) ∈ X 2 (8)
By composed derivation, we get :
∂J ∂
, θ
The adjoint state is defined in the following manner : ∂L ∂λk
(uk , pk , λk ) = 0
⇐⇒ pk := adjoint state of uk (9)
Finally, we obtain :
, θ (10)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 13 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
1 I. Setting of the problem
2 II. Shape optimization equations
3 III. Numerical results
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 14 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
The two studied configurations
J(, e) = α1 J1 + α2 J2 + α3 J3 (11)
We consider two cases : one with the same weights for the parametric and geometric optimization, another one with distinct weights taking into account the physical dependency of each criterium Ji to e and
With an admissible error on the volume of 10% and an initial value of ePu = 20% on f
Thanks to a program given by O. PANTZ we apply a change of topology of the mesh when needed.
We look at the following quantities to asses the convergence : For e : en − en−1 =
∫ f
For : J′(n)(θn) J′(0)(θ0)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 15 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
The behaviour of our algorithm in images
(Loading video.mpg)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 16 / 24
video.mpg
III. Numerical results Conclusion
Figure: Initial condition
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 17 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Parameters 2. Some results
After a first step (2)
Figure: Case 1 : δρNa = −29, 06pcm Figure: Case 2 : δρNa = 5, 88pcm
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 18 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
Cost function
Figure: J = αi Ji Figure: J′(n)(θn) J′(0)(θ0)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 19 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
Figure: Case 1 Figure: Case 2
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 20 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
Figure: Case 1 : δρNa = −35pcm Figure: Case 2 : δρNa = 4pcm
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 21 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
Convergence indexes
Figure: J = αi Ji Figure: J′(n)(θn) J′(0)(θ0)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 22 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
Conclusion and perspectives
Simulation must still be done with the 2 groups energy diffusion model
Initialize with different shape to check the robustness of the algorithm
Generalize the remeshing with Level-Set to much more complex situations
Apply a penalization step to obtain feasible design
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 23 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
Thank you for your attention
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 24 / 24
I. Setting of the problem
State Equation
III. Numerical results
joint-work 1 between SINETICS department at EDF R&D, Clamart and CMAP, Ecole Polytechnique
1. PEPS agreement
III. Numerical results Conclusion
Aims and context
Apply a shape optimization method to the conception of a nuclear reactor of generation IV having a reduced sodium void effect
Implement the method with the finite element software FreeFem++
Show to what extent the method could be applied in this context
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 2 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
2 II. Shape optimization equations
3 III. Numerical results
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 3 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
Modeling of the reactor
Description of the core
We assume the reactor can be simplified to a cylinder and that we have an axisymetric geometry = [0,R]× [0, h]
= ∪4 i=1i is defined as the reunion of 4 regions having some averaged
physical properties, each of them being characterized by a set of homogenized coefficients (D,Σa,Σf )
The active core is composed by 3 different materials : we denote by f the fissile material
The last region is the reflector surrounding the active core : absorbing region made of steel material
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 4 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
State equation
Using the notations : ∂D = {(r , z) ∈ ∂; r 6= 0}, ∂N = {(r , z) ∈ ∂; r = 0} and u the neutrons flux
We consider the simplified 1 group energy diffusion equation formulated on the space X = {φ ∈ H1();φ|∂D
= 0} on a bounded domain = ∪i i :
− D 4 u + Σau = λνΣf u in i
transmissions conditions on ∂i ∩ ∂j
u = 0 on ∂D ∩ ∂i
∂u
(1)
Remark : With this convention for λ, we have λ = 1 keff
keff is a key parameter linked to the neutronic equilibrium of the core (generaly close to 1)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 5 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
2 II. Shape optimization equations
3 III. Numerical results
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 6 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Variational form
The state equation represents an equilibrium between the reactions describing the behaviour of neutrons at steady states :
−D 4 u leakage rate
+ Σau absorption rate
Variational form
We are looking for weak solutions of the variational form : Find u ∈ X such that :∫
∫
a(u, v) =
b(u, v) =
(3)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 7 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Cost function
Estimation of the sodium void effect
We want to determine the first eigenvalue of this problem, which is simple, since this is the only to have a physical meaning (eigenvector keep the same sign over )
By considering two set of homogenized coefficients for this equation in a nominal state (with index ”n”) and a peturbated one (index ”v”), we calculate the sodium void effect as δρNa = λn − λv
Our cost function is :
+
(4)
The last term is not differentiable so we approximate it by the following quantity :
V (f )1− 1 s
(∫ f
1/s
(5)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 8 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Cost function
Estimation of the sodium void effect
We want to determine the first eigenvalue of this problem, which is simple, since this is the only to have a physical meaning (eigenvector keep the same sign over )
By considering two set of homogenized coefficients for this equation in a nominal state (with index ”n”) and a peturbated one (index ”v”), we calculate the sodium void effect as δρNa = λn − λv
Our cost function is :
+
(4)
The last term is not differentiable so we approximate it by the following quantity :
V (f )1− 1 s
(∫ f
1/s
(5)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 8 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Cost function
Estimation of the sodium void effect
We want to determine the first eigenvalue of this problem, which is simple, since this is the only to have a physical meaning (eigenvector keep the same sign over )
By considering two set of homogenized coefficients for this equation in a nominal state (with index ”n”) and a peturbated one (index ”v”), we calculate the sodium void effect as δρNa = λn − λv
Our cost function is :
+ (λn − λ0)2 target reactivity
(4)
The last term is not differentiable so we approximate it by the following quantity :
V (f )1− 1 s
(∫ f
1/s
(5)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 8 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Cost function
Estimation of the sodium void effect
We want to determine the first eigenvalue of this problem, which is simple, since this is the only to have a physical meaning (eigenvector keep the same sign over )
By considering two set of homogenized coefficients for this equation in a nominal state (with index ”n”) and a peturbated one (index ”v”), we calculate the sodium void effect as δρNa = λn − λv
Our cost function is :
+ (λn − λ0)2 target reactivity
(4)
The last term is not differentiable so we approximate it by the following quantity :
V (f )1− 1 s
(∫ f
1/s
(5)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 8 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Cost function
Estimation of the sodium void effect
We want to determine the first eigenvalue of this problem, which is simple, since this is the only to have a physical meaning (eigenvector keep the same sign over )
By considering two set of homogenized coefficients for this equation in a nominal state (with index ”n”) and a peturbated one (index ”v”), we calculate the sodium void effect as δρNa = λn − λv
Our cost function is :
+ (λn − λ0)2 target reactivity
(4)
The last term is not differentiable so we approximate it by the following quantity :
V (f )1− 1 s
(∫ f
1/s
(5)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 8 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
What the algorithm does
Figure: Initial shape Figure: Shape after 500 iterations
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 9 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Admissible shape transformation of the bounded set
We move the shape during a time t with a lipschitzian velocity field satisfying tθn∞ ≤ 1 moving the points with the following equation :
x → x + t θn(x)
Thus, we have n+1 = (Id + t θn)(n)
We impose θ · n = 0 on ∂ and on the external boundary of the active core
The volume constraint expresses as vol(T (f )) = vol(f )
T = Id + t θ
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 10 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Admissible shape transformation of the bounded set
We move the shape during a time t with a lipschitzian velocity field satisfying tθn∞ ≤ 1 moving the points with the following equation :
x → x + t θn(x)
Thus, we have n+1 = (Id + t θn)(n)
We impose θ · n = 0 on ∂ and on the external boundary of the active core
The volume constraint expresses as vol(T (f )) = vol(f )
T = Id + t θ
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 10 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Gradient method
We characterize the optimal shape by first order optimality conditions
We define the shape derivative in of the function J() as the derivative at the point 0 of the application :
{ W 1,∞(f ,R2)→ R
θ → J((Id + t θ)())
That is also the linear form on W 1,∞(f ,R2) such that :
J((Id + t θ)(f )) = J(f ) + J ′(f ), θ+ (θ) (6)
Once we know the shape derivative J ′(f )(θ), we apply a gradient descent algorithm by solving the following problem :
∀θ ∈ H1(f ,R2) : J ′(f )(θ) + θn, θ = 0 (7)
for the scalar product on X × X , it gives J ′(f )(θn) = −θn2
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 11 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Gradient method
We characterize the optimal shape by first order optimality conditions
We define the shape derivative in of the function J() as the derivative at the point 0 of the application :
{ W 1,∞(f ,R2)→ R
θ → J((Id + t θ)())
That is also the linear form on W 1,∞(f ,R2) such that :
J((Id + t θ)(f )) = J(f ) + J ′(f ), θ+ (θ) (6)
Once we know the shape derivative J ′(f )(θ), we apply a gradient descent algorithm by solving the following problem :
∀θ ∈ H1(f ,R2) : J ′(f )(θ) + θn, θ = 0 (7)
for the scalar product on X × X , it gives J ′(f )(θn) = −θn2
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 11 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Gradient method
We characterize the optimal shape by first order optimality conditions
We define the shape derivative in of the function J() as the derivative at the point 0 of the application :
{ W 1,∞(f ,R2)→ R
θ → J((Id + t θ)())
That is also the linear form on W 1,∞(f ,R2) such that :
J((Id + t θ)(f )) = J(f ) + J ′(f ), θ+ (θ) (6)
Once we know the shape derivative J ′(f )(θ), we apply a gradient descent algorithm by solving the following problem :
∀θ ∈ H1(f ,R2) : J ′(f )(θ) + θn, θ = 0 (7)
for the scalar product on X × X , it gives J ′(f )(θn) = −θn2
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 11 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Gradient method
We characterize the optimal shape by first order optimality conditions
We define the shape derivative in of the function J() as the derivative at the point 0 of the application :
{ W 1,∞(f ,R2)→ R
θ → J((Id + t θ)())
That is also the linear form on W 1,∞(f ,R2) such that :
J((Id + t θ)(f )) = J(f ) + J ′(f ), θ+ (θ) (6)
Once we know the shape derivative J ′(f )(θ), we apply a gradient descent algorithm by solving the following problem :
∀θ ∈ H1(f ,R2) : J ′(f )(θ) + θn, θ = 0 (7)
for the scalar product on X × X , it gives J ′(f )(θn) = −θn2
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 11 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Gradient method
We characterize the optimal shape by first order optimality conditions
We define the shape derivative in of the function J() as the derivative at the point 0 of the application :
{ W 1,∞(f ,R2)→ R
θ → J((Id + t θ)())
That is also the linear form on W 1,∞(f ,R2) such that :
J((Id + t θ)(f )) = J(f ) + J ′(f ), θ+ (θ) (6)
Once we know the shape derivative J ′(f )(θ), we apply a gradient descent algorithm by solving the following problem :
∀θ ∈ H1(f ,R2) : J ′(f )(θ) + θn, θ = 0 (7)
for the scalar product on X × X , it gives J ′(f )(θn) = −θn2
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 11 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Lagrangian associated to the minimization problem
J does not depend explicitly on the shape
Use of the adjoint method
We introduce the Lagrangian L that transforms the initial problem into a new one without constraints on the variables :
L(, un, uv , pn, pv , λn, λv ) = λn − λv sodium void effect
+ (λn − λ0) 2
states equations
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 12 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
Adjoint method
Let (uk , λk ) be the solution of the state equation in the state k
We have the relation :
J() = L(, un, uv , λn, λv , p, q) ∀ (p, q) ∈ X 2 (8)
By composed derivation, we get :
∂J ∂
, θ
The adjoint state is defined in the following manner : ∂L ∂λk
(uk , pk , λk ) = 0
⇐⇒ pk := adjoint state of uk (9)
Finally, we obtain :
, θ (10)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 13 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Cost function J 2. Shape optimization variable 3. Shape derivative 4. How we calculate the shape derivative
1 I. Setting of the problem
2 II. Shape optimization equations
3 III. Numerical results
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 14 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
The two studied configurations
J(, e) = α1 J1 + α2 J2 + α3 J3 (11)
We consider two cases : one with the same weights for the parametric and geometric optimization, another one with distinct weights taking into account the physical dependency of each criterium Ji to e and
With an admissible error on the volume of 10% and an initial value of ePu = 20% on f
Thanks to a program given by O. PANTZ we apply a change of topology of the mesh when needed.
We look at the following quantities to asses the convergence : For e : en − en−1 =
∫ f
For : J′(n)(θn) J′(0)(θ0)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 15 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
The behaviour of our algorithm in images
(Loading video.mpg)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 16 / 24
video.mpg
III. Numerical results Conclusion
Figure: Initial condition
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 17 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
1. Parameters 2. Some results
After a first step (2)
Figure: Case 1 : δρNa = −29, 06pcm Figure: Case 2 : δρNa = 5, 88pcm
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 18 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
Cost function
Figure: J = αi Ji Figure: J′(n)(θn) J′(0)(θ0)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 19 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
Figure: Case 1 Figure: Case 2
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 20 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
Figure: Case 1 : δρNa = −35pcm Figure: Case 2 : δρNa = 4pcm
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 21 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
Convergence indexes
Figure: J = αi Ji Figure: J′(n)(θn) J′(0)(θ0)
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 22 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
Conclusion and perspectives
Simulation must still be done with the 2 groups energy diffusion model
Initialize with different shape to check the robustness of the algorithm
Generalize the remeshing with Level-Set to much more complex situations
Apply a penalization step to obtain feasible design
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 23 / 24
I. Setting of the problem II. Shape optimization equations
III. Numerical results Conclusion
Thank you for your attention
Emmanuel DOMBRE Geometrical optimization of a reactor’s core of gen IV using FreeFem++ 6 decembre 2011 24 / 24
I. Setting of the problem
State Equation
III. Numerical results