{
  "name": "CentralDifferenceSolver",
  "main": {
    "name": "CentralDifferenceSolver",
    "namespace": "sofa::component::odesolver::forward",
    "module": "Sofa.Component.ODESolver.Forward",
    "include": "sofa/component/odesolver/forward/CentralDifferenceSolver.h",
    "doc": "Explicit time integrator using central difference (also known as Verlet of Leap-frog).\n\nExplicit time integrator using central difference (also known as Verlet of Leap-frop).\n@see http://www.dynasupport.com/support/tutorial/users.guide/time.integration\n@see http://en.wikipedia.org/wiki/Leapfrog_method",
    "inherits": [
      "OdeSolver"
    ],
    "templates": [],
    "data_fields": [
      {
        "name": "d_rayleighMass",
        "type": "SReal",
        "xmlname": "rayleighMass",
        "help": "Rayleigh damping coefficient related to mass"
      },
      {
        "name": "d_threadSafeVisitor",
        "type": "bool",
        "xmlname": "threadSafeVisitor",
        "help": "If true, do not use realloc and free visitors in fwdInteractionForceField."
      }
    ],
    "links": [],
    "methods": [
      {
        "name": "solve",
        "return_type": "void",
        "params": [
          {
            "name": "params",
            "type": "const core::ExecParams *"
          },
          {
            "name": "dt",
            "type": "SReal"
          },
          {
            "name": "xResult",
            "type": "sofa::core::MultiVecCoordId"
          },
          {
            "name": "vResult",
            "type": "sofa::core::MultiVecDerivId"
          }
        ],
        "is_virtual": true,
        "is_pure_virtual": false,
        "is_static": false,
        "access": "public"
      },
      {
        "name": "getIntegrationFactor",
        "return_type": "SReal",
        "params": [
          {
            "name": "inputDerivative",
            "type": "int"
          },
          {
            "name": "outputDerivative",
            "type": "int"
          }
        ],
        "is_virtual": true,
        "is_pure_virtual": false,
        "is_static": false,
        "access": "public"
      },
      {
        "name": "getSolutionIntegrationFactor",
        "return_type": "SReal",
        "params": [
          {
            "name": "outputDerivative",
            "type": "int"
          }
        ],
        "is_virtual": true,
        "is_pure_virtual": false,
        "is_static": false,
        "access": "public"
      }
    ]
  },
  "desc": {
    "description": "The `CentralDifferenceSolver` is an explicit time integrator in the SOFA framework, specifically designed for forward integration methods such as central difference (also known as Verlet or Leap-frog). This solver computes the next position and velocity based on current and previous positions using explicit integration steps. It inherits from the base `OdeSolver` class and provides a `solve` method that updates the mechanical state by integrating forces over time with optional Rayleigh damping.\n\nThe component interacts with other SOFA components through its inherited methods, such as `solve`, which takes parameters like the current execution context (`core::ExecParams*`), time step (`SReal dt`), and result identifiers for positions (`sofa::core::MultiVecCoordId`) and velocities (`sofa::core::MultiVecDerivId`). It also provides methods to calculate integration factors between different derivative orders.\n\nThe `CentralDifferenceSolver` has two data fields: `d_rayleighMass`, which is a Rayleigh damping coefficient related to mass, and `d_threadSafeVisitor`, a boolean flag that controls whether realloc and free operations are used in the interaction force field. The solver supports both optimized and unoptimized versions for different computational requirements."
  },
  "maths": {
    "maths": "<h3>Description</h3>\n<p>The <code>CentralDifferenceSolver</code>, also known as the Verlet or Leap-frog integrator, is an explicit time integration method used to advance a system's state in time. It uses both the current and previous positions of particles to calculate their future positions.</p>\n\n<h4>Mathematical Formulation</h4>\n<p>The central difference method can be described by the following steps:</p>\n<ul>\n<li><strong>Initialization</strong>: Given initial position \\(\\mathbf{x}_0\\) and velocity \\(\\mathbf{v}_0\\).</li>\n<li><strong>Force Calculation**: At time step \\(t_k = k \\Delta t\\), the forces acting on particles are computed based on their positions. The force vector is denoted as \\(\\mathbf{F}(\\mathbf{x}_k)\\).\n    <ul>\n        <li>For Rayleigh damping, an additional term is added to the force:</li>\n    </ul>\n</li>\n<li><strong>Rayleigh Damping**: The effective acceleration \\(\\ddot{\\mathbf{x}}_k^{eff}\\) takes into account the Rayleigh damping coefficient \\(r\\):\n    <p>\\[\\ddot{\\mathbf{x}}_k^{eff} = M^{-1}(\\mathbf{F}(\\mathbf{x}_k) + r (M \\dot{\\mathbf{x}}_{k-1}))\\]\\</li>\n<li><strong>Velocity Update**: The velocity at the next time step is given by:</li>\n    <ul>\n        <li>If there's no Rayleigh damping, we have a standard Verlet integration:\n            \\[\\mathbf{v}_{k+1} = \\mathbf{v}_k + \\Delta t \\ddot{\\mathbf{x}}_k^{eff}\\]\\</li>\n    </ul>\n</li>\n<li><strong>Position Update**: The position at the next time step is computed as:</li>\n    <p>\\[\\mathbf{x}_{k+1} = \\mathbf{x}_k + \\Delta t \\mathbf{v}_{k+1}\\]\\</p>\n<p>The method is symplectic, meaning it conserves energy in the long term for conservative systems. For Rayleigh damping, the force updates are modified to account for the additional dissipative forces:</p>\n\\[\\ddot{\\mathbf{x}}_k^{eff} = M^{-1}(\\mathbf{F}(\\mathbf{x}_k) + r (M \\dot{\\mathbf{x}}_{k-1}))\\]\n\n<p>The Rayleigh damping term modifies the effective acceleration by introducing a linear function of velocity, which models viscous damping.</p>\n<h4>Integration Factors</h4>\n<p>To determine how much an input derivative affects an output derivative, integration factors are defined. For instance, for position (0th order), velocity (1st order), and acceleration (2nd order):</p>\n\\[\\text{matrix}[3][3] = \\begin{bmatrix} 1 & dt & dt^2 \\\\ 0 & 1 & dt \\\\ 0 & 0 & 0 \\end{bmatrix}\\]\n<p>where the integration factors for each derivative order are:</p>\n\\[\\text{vect}[3] = [dt^2, dt, 1]\\]</p>"
  },
  "summary": {
    "abstract": "The CentralDifferenceSolver is an explicit time integrator using central difference (Verlet or Leap-frog) for forward integration, handling Rayleigh damping and providing methods to solve ODEs explicitly.",
    "sheet": "<h1>CentralDifferenceSolver</h1>\n\n<h2>Overview</h2>\n<p>The CentralDifferenceSolver is an explicit time integrator in the SOFA framework that uses central difference (also known as Verlet or Leap-frog) for forward integration. It inherits from the base <code>OdeSolver</code> class and provides methods to solve ordinary differential equations (ODEs) explicitly using current and previous positions.</p>\n\n<h2>Mathematical Model</h2>\n<p>The central difference method can be described by the following steps:</p>\n<ul>\n<li><strong>Initialization</strong>: Given initial position \\(\\mathbf{x}_0\\) and velocity \\(\\mathbf{v}_0\\).</li>\n<li><strong>Force Calculation**: At time step \\(t_k = k \\Delta t\\), the forces acting on particles are computed based on their positions. The force vector is denoted as \\(\\mathbf{F}(\\mathbf{x}_k)\\).\n    <ul>\n        <li>For Rayleigh damping, an additional term is added to the force:</li>\n    </ul>\n</li>\n<li><strong>Rayleigh Damping**: The effective acceleration \\(\\ddot{\\mathbf{x}}_k^{eff}\\) takes into account the Rayleigh damping coefficient \\(r\\):\n    <p>\\[\\ddot{\\mathbf{x}}_k^{eff} = M^{-1}(\\mathbf{F}(\\mathbf{x}_k) + r (M \\dot{\\mathbf{x}}_{k-1}))\\]</p>\n</li>\n<li><strong>Velocity Update**: The velocity at the next time step is given by:</li>\n    <ul>\n        <li>If there's no Rayleigh damping, we have a standard Verlet integration:\n            \\[\\mathbf{v}_{k+1} = \\mathbf{v}_k + \\Delta t \\ddot{\\mathbf{x}}_k^{eff}\\]</p>\n</li>\n<li><strong>Position Update**: The position at the next time step is computed as:</li>\n    <p>\\[\\mathbf{x}_{k+1} = \\mathbf{x}_k + \\Delta t \\mathbf{v}_{k+1}\\]</p>\n<p>The method is symplectic, meaning it conserves energy in the long term for conservative systems. For Rayleigh damping, the force updates are modified to account for the additional dissipative forces:</p>\n\\[\\ddot{\\mathbf{x}}_k^{eff} = M^{-1}(\\mathbf{F}(\\mathbf{x}_k) + r (M \\dot{\\mathbf{x}}_{k-1}))\\]\n\n<p>The Rayleigh damping term modifies the effective acceleration by introducing a linear function of velocity, which models viscous damping.</p>\n<h2>Parameters and Data</h2>\n<ul>\n<li><strong>d_rayleighMass (SReal)</strong>: Rayleigh damping coefficient related to mass. Default value is not specified but can be set via the XML configuration.</li>\n<li><strong>d_threadSafeVisitor (bool)</strong>: If true, do not use realloc and free visitors in fwdInteractionForceField. This flag ensures thread safety during operations involving interaction force fields.</li>\n</ul>"
  }
}