Data Fields

Methods

{
  "name": "SphereQuadTopology",
  "namespace": "sofa::component::topology::container::constant",
  "module": "Sofa.Component.Topology.Container.Constant",
  "include": "sofa/component/topology/container/constant/SphereQuadTopology.h",
  "doc": "Sphere topology constructed with deformed quads.",
  "inherits": [
    "CubeTopology"
  ],
  "templates": [],
  "data_fields": [
    {
      "name": "d_center",
      "type": "Vec3",
      "xmlname": "center",
      "help": "Center of the sphere"
    },
    {
      "name": "d_radius",
      "type": "SReal",
      "xmlname": "radius",
      "help": "Radius of the sphere"
    }
  ],
  "links": [],
  "methods": [
    {
      "name": "getPoint",
      "return_type": "Vec3",
      "params": [
        {
          "name": "x",
          "type": "int"
        },
        {
          "name": "y",
          "type": "int"
        },
        {
          "name": "z",
          "type": "int"
        }
      ],
      "is_virtual": true,
      "is_pure_virtual": false,
      "is_static": false,
      "access": "public"
    }
  ],
  "description": "The `SphereQuadTopology` is a topology container in SOFA that generates a spherical mesh using deformed quads. It inherits from the `CubeTopology`, which provides basic grid-based topological functionalities. The primary purpose of this component is to define the geometry and structure of a sphere for use in simulations, particularly where quad-based discretization is required.\n\nThe `SphereQuadTopology` includes two data fields: `center` (of type `Vec3`) representing the center of the sphere, and `radius` (of type `SReal`) defining its radius. It provides a method `getPoint(x, y, z)` to retrieve the coordinates of points on the spherical surface based on grid indices.\n\nThis component interacts with other SOFA components that require topological information for simulation purposes, such as mechanical models, collision detection, and visualization pipelines. Practical usage involves setting the sphere's center and radius through XML configuration or programmatically.",
  "maths": "The `SphereQuadTopology` component in the SOFA framework is responsible for generating a spherical mesh using deformed quads. This topology container inherits from the `CubeTopology`, which provides the basic grid-based topological functionalities. The primary purpose of this component is to define the geometry and structure of a sphere, suitable for use in simulations where quad-based discretization is required.\n\n### Mathematical Description\n\n#### Parameters\n- **Center of the Sphere**: \\(\\mathbf{c} = (c_x, c_y, c_z) \\in \\mathbb{R}^3\\)\n  - Represented by the data field `d_center` in the component.\n- **Radius of the Sphere**: \\(r \\in \\mathbb{R}\\)\n  - Represented by the data field `d_radius` in the component.\n\n#### Discretization and Geometry\nThe sphere is discretized using a grid with indices \\((x, y, z)\\). The coordinates of points on the spherical surface are determined using these grid indices. Specifically, for each point \\(p_i\\) indexed by \\((x, y, z)\\), its coordinates are given by:\n\n\begin{align*}\n  p_i = \\mathbf{c} + r \\cdot \\frac{(2x - (d_nx-1)), (2y - (d_ny-1)), (2z - (d_nz-1))}{\\sqrt{(2x - (d_nx-1))^2 + (2y - (d_ny-1))^2 + (2z - (d_nz-1))^2}}\n\\end{align*}\n\nwhere \\(d_nx, d_ny, d_nz\\) are the number of grid points along each axis.\n\n#### Role in FEM Pipeline\nThe `SphereQuadTopology` component plays a crucial role in defining the mesh and connectivity information required for subsequent stages of the finite element simulation pipeline. This includes:\n\n- **Mesh Generation**: The component generates the nodal coordinates and topological relationships (e.g., which nodes form quads) necessary to discretize the sphere.\n- **Integration with Mechanical Models**: Other SOFA components, such as mechanical models, collision detection modules, and visualization pipelines, rely on this topology information for their respective operations. For instance, the mechanical model will use these points and connectivity to assemble mass matrices, stiffness matrices, and other operators required in the weak formulation of the governing equations.\n\n#### Numerical Methods\nThe `SphereQuadTopology` does not directly involve numerical methods such as time integration or nonlinear solution techniques. Instead, it provides a geometric representation that is used by downstream components for these purposes.\n\n### Physical Description\nFrom a physical perspective, the `SphereQuadTopology` component defines the geometry of a deformable sphere that can be used in simulations involving soft tissues or other materials requiring quad-based discretization. The generated mesh serves as the basis for defining material properties and boundary conditions within the simulation framework.",
  "abstract": "The `SphereQuadTopology` generates a spherical mesh using deformed quads, defining the geometry and structure of a sphere for simulations. It inherits from `CubeTopology`, providing basic grid-based topological functionalities.",
  "sheet": "<h1>SphereQuadTopology</h1>\n\n<h2>Overview</h2>\n<p>The <code>SphereQuadTopology</code> is responsible for generating the topology of a spherical mesh using deformed quads. It inherits from <code>CubeTopology</code>, which provides basic grid-based topological functionalities. This component defines the geometry and structure of a sphere, suitable for simulations where quad-based discretization is required.</p>\n\n<h2>Mathematical Model</h2>\n<p>The sphere is discretized using a grid with indices <span class=\"math-inline\">\\((x, y, z)\\)</span>. The coordinates of points on the spherical surface are determined based on these grid indices. Specifically, for each point <span class=\"math-inline\">\\(p_i\\)</span> indexed by <span class=\"math-inline\">\\((x, y, z)\\)</span>, its coordinates are given by:</p>\n<p><span class=\"math-display\">\\[ p_i = \\mathbf{c} + r \\cdot \\frac{(2x - (d_nx-1)), (2y - (d_ny-1)), (2z - (d_nz-1))}{\\sqrt{(2x - (d_nx-1))^2 + (2y - (d_ny-1))^2 + (2z - (d_nz-1))^2}} \\]</span></p>\n<p>where <span class=\"math-inline\">\\(\\mathbf{c} = (c_x, c_y, c_z)\\)</span> is the center of the sphere and <span class=\"math-inline\">\\(r\\)</span> is its radius.</p>\n\n<h2>Parameters and Data</h2>\n<p>The significant data fields exposed by this component are:</p>\n<ul>\n<li><strong>Center of the Sphere (d_center):</strong> <code>Vec3</code>, representing the center of the sphere. Default value is not specified.</li>\n<li><strong>Radius of the Sphere (d_radius):</strong> <code>SReal</code>, defining its radius. Default value is not specified.</li>\n</ul>\n\n<h2>Dependencies and Connections</h2>\n<p>The `SphereQuadTopology` component interacts with other SOFA components that require topological information for simulation purposes, such as mechanical models, collision detection, and visualization pipelines.</p>"
}