Methods

{
  "name": "ParallelMeshSpringForceField",
  "namespace": "multithreading::component::solidmechanics::spring",
  "module": "MultiThreading",
  "include": "MultiThreading/component/solidmechanics/spring/ParallelMeshSpringForceField.h",
  "doc": "Parallel stiff springs acting along the edges of a mesh.",
  "inherits": [
    "ParallelSpringForceField"
  ],
  "templates": [
    "sofa::sofa::defaulttype::Vec3Types"
  ],
  "data_fields": [],
  "links": [],
  "methods": [
    {
      "name": "init",
      "return_type": "void",
      "params": [],
      "is_virtual": false,
      "is_pure_virtual": false,
      "is_static": false,
      "access": "public"
    }
  ],
  "description": "The `ParallelMeshSpringForceField` is a force field component in the SOFA framework designed for parallel computation, specifically for applying stiff spring forces along the edges of a mesh. It resides within the `MultiThreading` module and inherits from both `MeshSpringForceField` and `ParallelSpringForceField`, leveraging multi-threading capabilities to enhance computational efficiency.\n\n### Role and Purpose:\n- This component is used to simulate stiff springs acting between nodes in a mesh, which can be beneficial for modeling various physical scenarios, such as the elastic properties of materials or the deformation behavior under external forces. The parallel implementation allows for faster computation times on multi-core processors.\n\n### Interactions with Other Components:\n- `ParallelMeshSpringForceField` interacts with other SOFA components through its API and inherits functionalities from `MeshSpringForceField` and `ParallelSpringForceField`. It integrates into the overall mechanical simulation pipeline by applying forces to the mesh nodes based on their connectivity and spring properties.\n\n### Practical Usage Guidance:\n- This component should be used when simulating scenarios where stiff springs along the edges of a mesh are required. The multi-threading capabilities ensure that these simulations can run efficiently on modern hardware with multiple cores. It is important to configure the topology and mechanical state of the simulation correctly for optimal performance.\n\n### Data Fields:\n- No additional data fields are explicitly defined; it inherits its properties from `MeshSpringForceField` and `ParallelSpringForceField`. The component initializes itself by calling the base class's `init()` method and setting up a task scheduler for parallel execution.",
  "maths": "The `ParallelMeshSpringForceField` component in the SOFA framework is designed to apply stiff spring forces along the edges of a mesh. This force field is particularly useful for simulating elastic properties and deformation behavior under external forces. The component leverages parallel computation capabilities, enhancing computational efficiency on multi-core processors.\n\n### Governing Equations:\nThe springs are modeled using Hooke's law, where each edge between two nodes in the mesh represents a spring with stiffness $k$. For an edge connecting node $i$ and node $j$, the force along this edge is given by:\n\begin{equation}\n\\mathbf{f}_{ij} = - k (\\mathbf{x}_j - \\mathbf{x}_i)\\cdot(\\hat{\\mathbf{n}}_{ij})\n\\end{equation}\nwhere $\\mathbf{x}_i$ and $\\mathbf{x}_j$ are the positions of nodes $i$ and $j$, respectively, and $\\hat{\\mathbf{n}}_{ij}$ is the unit vector along the edge. The negative sign indicates that the force acts in the opposite direction to the displacement.\n\n### Constitutive Law:\nThe constitutive law for each spring follows Hooke's law, where the spring stiffness $k$ determines the magnitude of the restoring force. This linear elastic behavior can be summarized as:\n\begin{equation}\n\\mathbf{f}_{ij} = k (\\mathbf{x}_i^0 - \\mathbf{x}_j^0)\n\\end{equation}\nwhere $\\mathbf{x}_i^0$ and $\\mathbf{x}_j^0$ are the initial positions of nodes $i$ and $j$.\n\n### Role in FEM Pipeline:\nThe `ParallelMeshSpringForceField` component contributes to the global internal force vector $\\mathbf{f}_{int}$ by computing the forces along each edge. The internal forces can be expressed as a sum over all edges:\n\begin{equation}\n\\mathbf{f}_{int} = \\sum_{ij} \\mathbf{f}_{ij}\n\\end{equation}\nThe component integrates into the mechanical simulation pipeline by applying these forces to the nodes based on their connectivity and spring properties.\n\n### Numerical Methods:\nThe parallel implementation of this force field leverages multi-threading capabilities, distributing the computation of forces across multiple cores. This is achieved through inheritance from `ParallelSpringForceField`, which manages task scheduling for efficient parallel execution.\n\n### Variational Mechanics Framework:\nThe stiffness matrix $K$ associated with these springs can be derived using a variational approach. The potential energy stored in each spring is given by:\n\begin{equation}\nV_{ij} = \\frac{1}{2} k (\\mathbf{x}_j - \\mathbf{x}_i)^T (\\hat{\\mathbf{n}}_{ij})^T (\\hat{\\mathbf{n}}_{ij})(\\mathbf{x}_j - \\mathbf{x}_i)\n\\end{equation}\nThe stiffness matrix $K$ is then derived from the second derivative of this potential energy with respect to nodal displacements.\n\n### Summary:\nThe `ParallelMeshSpringForceField` component implements stiff springs along mesh edges using Hooke's law and leverages parallel computation for efficient simulation. It contributes to the internal forces in the FEM pipeline, ensuring realistic elastic behavior while optimizing performance on multi-core hardware.",
  "abstract": "The `ParallelMeshSpringForceField` applies stiff spring forces along mesh edges in parallel, enhancing computational efficiency through multi-threading.",
  "sheet": "\n# ParallelMeshSpringForceField\n\n## Overview\n\nThe `ParallelMeshSpringForceField` is a force field component designed to apply stiff spring forces along the edges of a mesh. It leverages multi-threaded computation for enhanced performance and inherits from `ParallelSpringForceField`, integrating into the SOFA simulation pipeline by contributing to the internal forces acting on nodes.\n\n## Mathematical Model\n\nThe springs are modeled using Hooke's law, where each edge between two nodes in the mesh represents a spring with stiffness $k$. For an edge connecting node $i$ and node $j$, the force along this edge is given by:\n\\[ \\mathbf{f}_{ij} = - k (\\mathbf{x}_j - \\mathbf{x}_i) \\cdot (\\hat{\\mathbf{n}}_{ij}) \\]\nwhere $\\mathbf{x}_i$ and $\\mathbf{x}_j$ are the positions of nodes $i$ and $j$, respectively, and $\\hat{\\mathbf{n}}_{ij}$ is the unit vector along the edge. The negative sign indicates that the force acts in the opposite direction to the displacement.\n\nThe constitutive law for each spring follows Hooke's law, where the spring stiffness $k$ determines the magnitude of the restoring force:\n\\[ \\mathbf{f}_{ij} = k (\\mathbf{x}_i^0 - \\mathbf{x}_j^0) \\]\nwhere $\\mathbf{x}_i^0$ and $\\mathbf{x}_j^0$ are the initial positions of nodes $i$ and $j$. The internal forces can be expressed as a sum over all edges:\n\\[ \\mathbf{f}_{int} = \\sum_{ij} \\mathbf{f}_{ij} \\]\n\nThe stiffness matrix $K$ associated with these springs is derived from the potential energy stored in each spring, given by:\n\\[ V_{ij} = \\frac{1}{2} k (\\mathbf{x}_j - \\mathbf{x}_i)^T (\\hat{\\mathbf{n}}_{ij})^T (\\hat{\\mathbf{n}}_{ij})(\\mathbf{x}_j - \\mathbf{x}_i) \\]\n"
}