{
  "name": "IdentityTopologicalMapping",
  "main": {
    "name": "IdentityTopologicalMapping",
    "namespace": "sofa::component::topology::mapping",
    "module": "Sofa.Component.Topology.Mapping",
    "include": "sofa/component/topology/mapping/IdentityTopologicalMapping.h",
    "doc": "TopologicalMapping where the destination topology should be kept identical to the source topology. The implementation currently assumes that both topology have been initialized identically.\n\nThis class is a specific implementation of TopologicalMapping where the destination topology should be kept identical to the source topology.\nThe implementation currently assumes that both topology have been initialized identically.",
    "inherits": [
      "TopologicalMapping"
    ],
    "templates": [],
    "data_fields": [],
    "links": [],
    "methods": [
      {
        "name": "init",
        "return_type": "void",
        "params": [],
        "is_virtual": true,
        "is_pure_virtual": false,
        "is_static": false,
        "access": "public"
      },
      {
        "name": "updateTopologicalMappingTopDown",
        "return_type": "void",
        "params": [],
        "is_virtual": true,
        "is_pure_virtual": false,
        "is_static": false,
        "access": "public"
      },
      {
        "name": "getFromIndex",
        "return_type": "int",
        "params": [
          {
            "name": "ind",
            "type": "int"
          }
        ],
        "is_virtual": false,
        "is_pure_virtual": false,
        "is_static": false,
        "access": "public"
      }
    ]
  },
  "desc": {
    "description": "The `IdentityTopologicalMapping` is a specific implementation within the SOFA framework, used for maintaining identical topology between a source and destination model. It inherits from `sofa::core::topology::TopologicalMapping`. This mapping ensures that any topological changes in the source topology are directly replicated in the destination topology without modification.\n\nThe component interacts with other SOFA components through its methods:\n- `init`: Initializes the mapping by updating links between models and ensuring both the source (`fromModel`) and target (`toModel`) are set up correctly.\n- `updateTopologicalMappingTopDown`: Translates topological changes from the source model to the destination, maintaining consistency in topology across the two models. It handles various types of changes such as adding or removing points, edges, and triangles.\n- `getFromIndex`: Returns the corresponding index from the source topology for a given index in the destination topology (currently implemented as a direct return).\n\nPractical usage involves configuring this component to ensure consistent topological representations between models. It is particularly useful when requiring identical topologies across different simulation elements or models."
  },
  "maths": {
    "maths": "The `IdentityTopologicalMapping` in the SOFA framework ensures that the destination topology remains identical to the source topology. This mapping is essential for maintaining consistency across different models or simulation elements where topological changes need to be propagated without any modifications.\n\n### Governing Equations and Operators\n\nThis component does not directly contribute to governing equations such as mass matrix \\(M\\), stiffness matrix \\(K\\), internal force \\(f_{int}\\), or residual \\(R\\). Instead, it focuses on ensuring that the topological structure remains consistent between two models. Therefore, there are no direct constitutive laws or kinematic relationships to describe in terms of strains and stresses.\n\n### Topological Mapping Role\n\nThe `IdentityTopologicalMapping` plays a crucial role in the assembly phase by maintaining the consistency of topology across different components or sub-components within a simulation framework. It ensures that any changes (such as adding, removing, or renumbering points, edges, or triangles) made to the source model are directly replicated in the destination model.\n\n### Numerical Methods and Discretization Choices\n\nThe `IdentityTopologicalMapping` does not involve traditional numerical methods such as time integration or linear solvers. Instead, it focuses on propagating topological changes by maintaining a one-to-one correspondence between indices of elements (points, edges, triangles) in the source and destination models.\n\n### Role in Global FEM Pipeline\n\nIn the broader context of the finite element method (FEM), this component ensures that the topology remains consistent during various stages such as:\n\n- **Assembly**: Ensuring that the global system matrices (mass matrix \\(M\\), stiffness matrix \\(K\\)) are assembled correctly with respect to the same topological structure.\n- **Time Integration**: Propagating changes in topology consistently over time steps, ensuring that any updates to the mesh or elements are synchronized across models.\n\n### Variational / Lagrangian Mechanics Framework\n\nWhile `IdentityTopologicalMapping` does not directly contribute to variational formulations or Lagrangian mechanics, it supports the broader framework by maintaining topological consistency. This ensures that any physical simulations based on variational principles and Lagrangian mechanics are performed on consistent topologies across different simulation elements.\n\n### Methods Overview\n\n- **Initialization (`init`)**: Ensures both source (`fromModel`) and destination (`toModel`) models are properly initialized and linked.\n- **Topological Change Propagation (`updateTopologicalMappingTopDown`)**: Translates topological changes (such as adding or removing points, edges, triangles) from the source to the destination model.\n- **Index Mapping (`getFromIndex`)**: Provides a direct mapping of indices between the source and destination models, ensuring that each index in the destination topology corresponds directly to an index in the source topology.\n\n### Conclusion\n\nThe `IdentityTopologicalMapping` component is crucial for maintaining consistent topologies across different simulation elements or models within the SOFA framework. It does not contribute to the traditional numerical aspects of FEM but ensures that any changes made to one model are accurately and consistently replicated in another, thereby supporting the overall variational and Lagrangian mechanics framework."
  },
  "summary": {
    "abstract": "The `IdentityTopologicalMapping` ensures that the destination topology remains identical to the source topology, maintaining consistency across different simulation elements or models in the SOFA framework.",
    "sheet": "# IdentityTopologicalMapping\n\n## Overview\n\nThe `IdentityTopologicalMapping` is a specific implementation within the SOFA framework that maintains identical topology between a source and destination model. It inherits from `sofa::core::topology::TopologicalMapping`. This mapping ensures that any topological changes in the source topology are directly replicated in the destination topology without modification.\n\n## Dependencies and Connections\n\nThis component typically requires both a source (`fromModel`) and a destination (`toModel`). The implementation assumes that both topologies have been initialized identically. It fits into the SOFA scene graph by ensuring consistent topological representations between models, which is particularly useful when requiring identical topologies across different simulation elements.\n\n## Practical Notes\n\nThe `IdentityTopologicalMapping` assumes that both source and destination topologies are initialized identically. Any deviation from this assumption can lead to inconsistencies in the mapping process."
  }
}