Mixing
Source for learning Functional derivative , Gateaux derivative
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I have started doing research in optimal control theory. My area of research is PDE constrained optimization.
I am facing significant difficulty in deriving the Gateaux Derivative of functionals required for optimization.
For example I have to find out $U(sigma,u,w;C) = 1/2 int (sigma - C:epsilon[u]):{C}^{-1}:(sigma - C:epsilon[u]))$, where $C$ is a 4th order constitutive tensor and $sigma$ is 2nd order stress tensor. The operator $:$ is the double dot product operator. All quantities are function of space ($R^3$).
How can I calculate derivative of $U$ with respect to $sigma$ ?? I have tried to expand this using index notation. But it made the task very difficult.
Could you provide a source where I can have exposure to such kind of mathematics?
Thank you.
optimization optimal-control
asked Jan 30 at 15:41
archangel89archangel89
438422
$endgroup$
add a comment |
$begingroup$
I have started doing research in optimal control theory. My area of research is PDE constrained optimization.
I am facing significant difficulty in deriving the Gateaux Derivative of functionals required for optimization.
For example I have to find out $U(sigma,u,w;C) = 1/2 int (sigma - C:epsilon[u]):{C}^{-1}:(sigma - C:epsilon[u]))$, where $C$ is a 4th order constitutive tensor and $sigma$ is 2nd order stress tensor. The operator $:$ is the double dot product operator. All quantities are function of space ($R^3$).
How can I calculate derivative of $U$ with respect to $sigma$ ?? I have tried to expand this using index notation. But it made the task very difficult.
Could you provide a source where I can have exposure to such kind of mathematics?
Thank you.
optimization optimal-control
asked Jan 30 at 15:41
archangel89archangel89
438422
$endgroup$
1
$begingroup$
Another term for Gâteaux derivative is shape derivative, often used in connection with shape optimization. Are these the sort of problems you are interested in?
$endgroup$
– hardmath
Jan 30 at 19:36
$begingroup$
I am not directly doing shape optimization right now. But the work does involve topological derivative applied to structural mechanics. In short, I have a cost functional that includes U mentioned above. Finding the first order optimality is the first step of my task. Any insights??
$endgroup$
– archangel89
Jan 31 at 3:43
add a comment |
$begingroup$
I have started doing research in optimal control theory. My area of research is PDE constrained optimization.
I am facing significant difficulty in deriving the Gateaux Derivative of functionals required for optimization.
For example I have to find out $U(sigma,u,w;C) = 1/2 int (sigma - C:epsilon[u]):{C}^{-1}:(sigma - C:epsilon[u]))$, where $C$ is a 4th order constitutive tensor and $sigma$ is 2nd order stress tensor. The operator $:$ is the double dot product operator. All quantities are function of space ($R^3$).
How can I calculate derivative of $U$ with respect to $sigma$ ?? I have tried to expand this using index notation. But it made the task very difficult.
Could you provide a source where I can have exposure to such kind of mathematics?
Thank you.
optimization optimal-control
asked Jan 30 at 15:41
archangel89archangel89
438422
$endgroup$
I have started doing research in optimal control theory. My area of research is PDE constrained optimization.
I am facing significant difficulty in deriving the Gateaux Derivative of functionals required for optimization.
For example I have to find out $U(sigma,u,w;C) = 1/2 int (sigma - C:epsilon[u]):{C}^{-1}:(sigma - C:epsilon[u]))$, where $C$ is a 4th order constitutive tensor and $sigma$ is 2nd order stress tensor. The operator $:$ is the double dot product operator. All quantities are function of space ($R^3$).
How can I calculate derivative of $U$ with respect to $sigma$ ?? I have tried to expand this using index notation. But it made the task very difficult.
Could you provide a source where I can have exposure to such kind of mathematics?
Thank you.
optimization optimal-control
optimization optimal-control
asked Jan 30 at 15:41
archangel89archangel89
438422
asked Jan 30 at 15:41
archangel89archangel89
438422
edited Jan 30 at 19:29
archangel89
asked Jan 30 at 15:41
archangel89archangel89
438422
asked Jan 30 at 15:41
archangel89archangel89
438422
asked Jan 30 at 15:41
archangel89archangel89
438422
438422
1
$begingroup$
Another term for Gâteaux derivative is shape derivative, often used in connection with shape optimization. Are these the sort of problems you are interested in?
$endgroup$
– hardmath
Jan 30 at 19:36
$begingroup$
I am not directly doing shape optimization right now. But the work does involve topological derivative applied to structural mechanics. In short, I have a cost functional that includes U mentioned above. Finding the first order optimality is the first step of my task. Any insights??
$endgroup$
– archangel89
Jan 31 at 3:43
add a comment |
1
$begingroup$
Another term for Gâteaux derivative is shape derivative, often used in connection with shape optimization. Are these the sort of problems you are interested in?
$endgroup$
– hardmath
Jan 30 at 19:36
$begingroup$
I am not directly doing shape optimization right now. But the work does involve topological derivative applied to structural mechanics. In short, I have a cost functional that includes U mentioned above. Finding the first order optimality is the first step of my task. Any insights??
$endgroup$
– archangel89
Jan 31 at 3:43
1
1
$begingroup$
Another term for Gâteaux derivative is shape derivative, often used in connection with shape optimization. Are these the sort of problems you are interested in?
$endgroup$
– hardmath
Jan 30 at 19:36
$begingroup$
Another term for Gâteaux derivative is shape derivative, often used in connection with shape optimization. Are these the sort of problems you are interested in?
$endgroup$
– hardmath
Jan 30 at 19:36
$begingroup$
I am not directly doing shape optimization right now. But the work does involve topological derivative applied to structural mechanics. In short, I have a cost functional that includes U mentioned above. Finding the first order optimality is the first step of my task. Any insights??
$endgroup$
– archangel89
Jan 31 at 3:43
$begingroup$
I am not directly doing shape optimization right now. But the work does involve topological derivative applied to structural mechanics. In short, I have a cost functional that includes U mentioned above. Finding the first order optimality is the first step of my task. Any insights??
$endgroup$
– archangel89
Jan 31 at 3:43
add a comment |
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$begingroup$
Another term for Gâteaux derivative is shape derivative, often used in connection with shape optimization. Are these the sort of problems you are interested in?
$endgroup$
– hardmath
Jan 30 at 19:36
$begingroup$
I am not directly doing shape optimization right now. But the work does involve topological derivative applied to structural mechanics. In short, I have a cost functional that includes U mentioned above. Finding the first order optimality is the first step of my task. Any insights??
$endgroup$
– archangel89
Jan 31 at 3:43