{
  "name": "FillingConstraint",
  "main": {
    "name": "FillingConstraint",
    "namespace": "sofa::component::constraintset",
    "module": "SofaTestPlugin",
    "include": "constraint/phases/FillingConstraint.h",
    "doc": "Surf Constraint",
    "inherits": [
      "BaseOpenSurfaceConstraint"
    ],
    "templates": [
      "sofa::defaulttype::Vec3Types"
    ],
    "data_fields": [
      {
        "name": "d_Pin",
        "type": "Real",
        "xmlname": "P_in",
        "help": "Input pressure at time t+dt"
      },
      {
        "name": "d_Kin",
        "type": "Real",
        "xmlname": "K_in",
        "help": "Hemodynamic coefficient between pressure and flow"
      }
    ],
    "links": [],
    "methods": [
      {
        "name": "init",
        "return_type": "void",
        "params": [],
        "is_virtual": false,
        "is_pure_virtual": false,
        "is_static": false,
        "access": "public"
      },
      {
        "name": "getConstraintResolution",
        "return_type": "void",
        "params": [
          {
            "name": "cParams",
            "type": "const ConstraintParams *"
          },
          {
            "name": "resTab",
            "type": "int &"
          },
          {
            "name": "offset",
            "type": "unsigned int &"
          }
        ],
        "is_virtual": true,
        "is_pure_virtual": false,
        "is_static": false,
        "access": "public"
      }
    ]
  },
  "desc": {
    "description": "The `FillingConstraint` is part of the SofaTestPlugin module and inherits from `BaseOpenSurfaceConstraint`. It serves to apply surface constraints in simulations, particularly focusing on pressure and flow dynamics related to hemodynamic coefficients. This constraint is used to model filling phases where input pressure (`P_in`) and a hemodynamic coefficient (`K_in`) between pressure and flow are essential parameters.\n\nThe component interacts with other SOFA components through its API, primarily by implementing the `getConstraintResolution` method inherited from the base class. It also relies on the context's time and delta-time to compute relevant values during initialization.\n\nUsers can configure this constraint using two data fields: `P_in`, representing input pressure at a given time step (`t+dt`), and `K_in`, denoting the hemodynamic coefficient between pressure and flow. The component is initialized through its `init` method, which sets up necessary parameters and conditions for subsequent operations."
  },
  "maths": {
    "maths": "<maths_md_description>\n\nThe `FillingConstraint` is designed to apply surface constraints in simulations, particularly focusing on pressure and flow dynamics related to hemodynamic coefficients. It models filling phases where the input pressure (`P_in`) and a hemodynamic coefficient (`K_in`) between pressure and flow are essential parameters.\n\n### Key Parameters\n- **Input Pressure ($P_{in}$)**: Represents the input pressure at time $t + \\Delta t$.\n- **Hemodynamic Coefficient ($K_{in}$)**: Denotes the hemodynamic coefficient that relates changes in pressure to flow.\n\n### Mathematical Formulation\nThe `FillingConstraint` applies a surface constraint by solving for the Lagrange multiplier ($\\lambda$) using the following equation:\n\n\begin{align*}\n  W \\cdot \\lambda &= d_{target} - d_{free} \\\\\n                    &= \\Delta t \\cdot K_{in} \\cdot (\\lambda - P_{in}) - d_{free} \\\\\n  \\Rightarrow \\quad \\lambda &= \\frac{\\Delta t \\cdot K_{in} \\cdot P_{in} + d_{free}}{\\Delta t \\cdot K_{in} - W}\n\tag{1}\\label{eq:lagrange_multiplier}\n\tag{2}\\label{eq:solve_lambda}\n\tag{3}\\label{eq:final_lambda}\n\nWhere:\n- $W$ is a weight factor.\n- $\\Delta t$ is the time step (obtained from context).\n- $d_{target}$ represents the target displacement.\n- $d_{free}$ is the free displacement component due to external forces or other constraints.\n\n### Initialization and Setup\nThe `init` method sets up necessary parameters and conditions for subsequent operations. Specifically, it initializes the following:\n- `m_hemo_container.time`: Current simulation time.\n- `m_hemo_container.dt`: Time step (`\\Delta t`).\n- `m_hemo_container.Qtarget`: Target flow rate (though this is currently set but not used in the equations).\n\n### Constraint Resolution\nThe method `getConstraintResolution` adds the corresponding `FillingConstraintResolution` instance to the constraint resolution algorithms. This involves:\n- Creating an instance of `FillingConstraintResolution`, passing it necessary data containers.\n- Adding the newly created instance to the result vector (`resTab`).\n\n### Constraint Resolution Implementation\nThe `resolution` method in the `FillingConstraintResolution` class computes $\\lambda$ using equation \\eqref{eq:final_lambda}. The key steps are:\n1. Retrieve necessary parameters from data containers.\n2. Solve for $\\lambda$ as per equation \\eqref{eq:final_lambda}.\n3. Assign the computed value of $\\lambda$ to `lambda[line]`.\n\n### Summary\nThe `FillingConstraint` effectively models the filling phase dynamics by solving a linear constraint equation involving input pressure and hemodynamic coefficients, ensuring that the simulation accurately reflects the intended behavior in this context.\n</maths_md_description>"
  },
  "summary": {
    "abstract": "The `FillingConstraint` applies surface constraints in simulations focusing on pressure and flow dynamics during filling phases by solving for the Lagrange multiplier with input pressure and hemodynamic coefficients.",
    "sheet": "# FillingConstraint\n\n## Overview\n\nThe `FillingConstraint` is part of the SofaTestPlugin module, inheriting from `BaseOpenSurfaceConstraint`. It models surface constraints in simulations related to pressure and flow dynamics during filling phases.\n\n## Mathematical Model\n\nThe `FillingConstraint` applies a surface constraint by solving for the Lagrange multiplier ($\\lambda$) using the following equation:\n\n\begin{align*}\n  W \\cdot \\lambda &= d_{target} - d_{free} \\\\\n                    &= \\Delta t \\cdot K_{in} \\cdot (\\lambda - P_{in}) - d_{free} \\\\\n  \\Rightarrow \\quad \\lambda &= \\frac{\\Delta t \\cdot K_{in} \\cdot P_{in} + d_{free}}{\\Delta t \\cdot K_{in} - W}\n\tag{1}\\label{eq:lagrange_multiplier}\n\tag{2}\\label{eq:solve_lambda}\n\tag{3}\\label{eq:final_lambda}\n\nWhere:\n- $W$ is a weight factor.\n- $\\Delta t$ is the time step (obtained from context).\n- $d_{target}$ represents the target displacement.\n- $d_{free}$ is the free displacement component due to external forces or other constraints.\n\n## Parameters and Data\n\nThe significant data fields exposed by the `FillingConstraint` are:\n\n- **P_in**: Input pressure at time $t + \\Delta t$ (type: Real).\n- **K_in**: Hemodynamic coefficient between pressure and flow (type: Real)."
  }
}