1. Geometric Optimization

This application generates reduced basis for a electric and biot-savart problem and does geometric optimization.

1.1. Problem

In the context of high field magnets, we want to optimize the cuts of a magnet in order to reach a better homogeneity H:

\[H(\mathbf{B}) = \frac{\max_{\mathbf{x}\in\Omega_M} \mathbf{B}_z(\mathbf{x})}{\min_{\mathbf{x}\in\Omega_M} \mathbf{B}_z(\mathbf{x})} - 1\]

For this we want to parametrize the geometry with the following geometrical transformation:

\[\phi_\mu\begin{pmatrix} x\\y\\z\end{pmatrix} = \begin{pmatrix} x\cos(\alpha_{\mu}(z))+y\sin(\alpha_{\mu}(z))\\ -x\sin(\alpha_{\mu}(z))+y\cos(\alpha_{\mu}(z))\\ z \end{pmatrix}\]

where \(\alpha\) is a Bezier curve which control points are the parameters \(\mu\):

\[\alpha_\mu(t) = \sum_{k=0}^n B_k^n(t) \mu_k = \sum_{k=0}^n C_n^k t^k(1-t)^{n-k} \mu_k\]

with \(\mu_0=\mu_n=0\) to impose continuity.

Then the magnetic field \(\mathbf{B}\) and the electric potential \(V\) depends on the parameters \(\mu\). Using a reference domain, where all control points are zero, we can write the weak formulation for the electric problem as:

\[\begin{aligned} \int_{\Omega} \sigma\left(\nabla{V}\cdot J_{\phi_{\mu}}^{-1}\right)&\cdot\left(\nabla{\varphi_V}\cdot J_{\phi_{\mu}}^{-1}\right)\left|J_{\phi_{\mu}}\right| - \int_{\Gamma_D} \sigma \left(\left(\nabla{V}\cdot J_{\phi_{\mu}}^{-1}\cdot\mathbf{n}\right)\varphi_V + \left(\nabla{\varphi_V}\cdot J_{\phi_{\mu}}^{-1}\cdot\mathbf{n}\right)V\right)\left|J_{\phi_{\mu}}\right| - \int_{\Gamma_D} \sigma\frac{\gamma}{h_F}V\varphi_V\left|J_{\phi_{\mu}}\right| \nonumber\\ =& - \int_{\Gamma_O} \sigma V_D \left(\nabla{\varphi_V}\cdot J_{\phi_{\mu}}^{-1}\cdot\mathbf{n}\right)\left|J_{\phi_{\mu}}\right| + \int_{\Gamma_O} \sigma V_D \frac{\gamma}{h_F} \varphi_V\left|J_{\phi_{\mu}}\right| \qquad\qquad (1) \end{aligned}\]

and the following formulation for the Biot-Savart law:

\[ \mathbf{B}(\mathbf{x},\mu) = \frac{\mu_0}{4\pi}\int_{\Omega_C} \frac{-\sigma\nabla{V}\cdot J_{\phi_{\mu}}^{-1}\times (\phi_{{\mu}}(\mathbf{x})-\mathbf{r})}{|\phi_{{\mu}}(\mathbf{x})-\mathbf{r}|^3}\left|J_{\phi_{\mu}}\right|d\mathbf{r} \qquad\qquad (2)\]

where \(V\) is solution of problem (1).

Thus the optimization problem is to find the parameters \(\bar{\mu}\) such that:

\[\bar{\mu} = \text{arg }\min_\mu H(\mathbf{B}(\mu))\]

The domain \(\Omega\) can be decomposed into three type of materials:

  • where the optimization takes place and is actually deformed (the helices)

  • where the optimization takes place but with no deformation (the rings)

  • where no optimization takes place, used for background field

1.2. Options

The applicaiton can be piloted via a set of options, the more important are:

thermoelectric.filename

path of the model file in json format, see the following section

thermoelectric.deim-size

size of the trainset for DEIM of right hand side

thermoelectric.mdeim-size

size of the trainset for DEIM of left hand side

thermoelectric.db.base

the name of the database for the electric problem

toolbox.filename

path of the model file for the background field, can be the same as thermoelectric.filename

toolbox.mesh.filename

path of the geometry

electric.crb.error-type

error type (0: residual, 1: residual with scm, 2: random, 3: empirical)

electric.crb.dimension-max

maximum dimension of the reduced basis

electric.crb.rebuild-database

boolean to rebuild the electric RB database

vec.deim.dimension-max

maximum dimension of the RHS DEIM basis

vec.deim.rebuild-database

boolean to rebuild the RHS DEIM database

mat.deim.dimension-max

maximum dimension of the LHS DEIM basis

mat.deim.rebuild-database

boolean to rebuild the LHS DEIM database

biotsavart.filename

path of the model file for biotsavart, can be the same as thermoelectric.filename

biotsavart.db.base

the name of the database for BiotSavart

biotsavart.trainset-deim-size

size of the trainset for DEIM of BiotSavart

bs.deim.dimension-max

maximum dimension of the BiotSavart DEIM basis

bs.deim.rebuild-database

boolean to rebuild the BiotSavart DEIM database

biotsavart.do-opt

boolean to do optimization or else convergence

biotsavart.use-bg-field

boolean to use or not a background field

nlopt.algo

algorithm to use for the optimization

nlopt.maxeval

maximume number of iterations for the optimization

1.3. Model file

The model to use, parameters, materials and boundary conditions, is set via a model file with the following sections.

Parameters
"Parameters": (1)
{
    "pA": (2)
    {
        "min":-3.14, (3)
        "max":3.14, (4)
        "value":0 (5)
    },

  • <1>: name of the section

  • <2>: name of the parameter

  • <3>: minimum of the parameter

  • <4>: maximum of the parameter

  • <5>: value of the parameter, might be used as default parameter

Materials
"Materials": (1)
{
    "H1_Cu": (2)
    {
        "physics":"electric-geo", (3)
        "sigma":58e6, (4)
        "zmin":-0.08651, (5)
        "zmax":0.08651, (6)
        "params":["pA","pB","pC","pD","pE","pF","pG","pH"] (7)
    }
    "H3_Cu":
    {
        "physics":"electric-geo",
        "sigma":58e6
    },
    "Copper":
    {
        "physics":"electric", (8)
        "markers": (9)
        {
            "name":["H%1%_Cu","R%2%"],
            "index1":"3:5",
            "index2":"2:4"
        },
        "sigma":"58e6"
    },

  • <1>: name of the section

  • <2>: marker of the material

  • <3>: electric-geo means the optimization takes place in this material

  • <4>: electric conductivity

  • <5>: z minimum from where the deformation begins, if not present, do not deform

  • <6>: z maximum to where the deformation ends, if not present, do not deform

  • <7>: list of parameters to use for this material, if not present, do not deform

  • <8>: electric means background field

  • <9>: markers and parameters can be used as in the toolbox

Boundary conditions
"BoundaryConditions": (1)
{
    "potential": (2)
    {
        "Dirichlet": (3)
        {
            "H1_V0": (4)
            {
                "expr":"0", (5)
                "material":"H1_Cu" (6)
            },
        }
    },
    "electric-potential": (7)
    {
        "Dirichlet":
        {
            "H2_V0":
            {
                "expr":"-16"
            },

  • <1>: name of the section

  • <2>: electric potential in materials with deformation

  • <3>: type of the condition, for now only Dirichlet

  • <4>: marker of the condition

  • <5>: expression of the condition

  • <6>: name of the material to use for weak Dirichlet conditions

  • <7>: electric-potential for boundary conditions of the background field, same as in toolbox