Methods

{
  "name": "UnbuiltGaussSeidelConstraintSolver",
  "namespace": "sofa::component::constraint::lagrangian::solver",
  "module": "Sofa.Component.Constraint.Lagrangian.Solver",
  "include": "sofa/component/constraint/lagrangian/solver/UnbuiltGaussSeidelConstraintSolver.h",
  "doc": "A Constraint Solver using the Linear Complementarity Problem formulation to solve Constraint based components using an Unbuilt version of the Gauss-Seidel iterative method",
  "inherits": [
    "UnbuiltConstraintSolver"
  ],
  "templates": [],
  "data_fields": [],
  "links": [],
  "methods": [
    {
      "name": "doSolve",
      "return_type": "void",
      "params": [
        {
          "name": "problem",
          "type": "GenericConstraintProblem *"
        },
        {
          "name": "timeout",
          "type": "SReal"
        }
      ],
      "is_virtual": true,
      "is_pure_virtual": false,
      "is_static": false,
      "access": "public"
    }
  ],
  "description": "The `UnbuiltGaussSeidelConstraintSolver` is a constraint solver within the SOFA framework, specifically designed to solve Constraint-based components using an unbuilt version of the Gauss-Seidel iterative method and the Linear Complementarity Problem formulation. This solver inherits from `UnbuiltConstraintSolver`, indicating its role in handling constraints without building the system matrices explicitly.\n\nIt interacts with other components through the `doSolve` method, which takes a `GenericConstraintProblem` object and an optional timeout value as parameters. The method iteratively resolves constraints until convergence or the specified timeout is reached. It also handles the accumulation of errors, updates forces, and manages constraint resolution steps.\n\nThis component can be used in scenarios requiring iterative constraint solving without explicitly constructing system matrices, making it suitable for real-time simulations where performance is critical. No specific data fields are required, as the solver operates directly on the provided problem instance.",
  "maths": "The `UnbuiltGaussSeidelConstraintSolver` in the SOFA framework is designed to handle constraints within a FEM simulation through an iterative method, specifically using an unbuilt version of the Gauss-Seidel technique. This solver addresses the Linear Complementarity Problem (LCP) formulation, which is essential for handling constraints such as contact and friction conditions. Here's a detailed mathematical and physical description:\n\n### Governing Equations and Operators\n- **Linear Complementarity Problem (LCP)**: The LCP formulation can be written as:\n  \\[\n  Mz + q = w, \\quad z \\\\\\geq 0, \\quad w \\\\-eq 0, \\quad z^T w = 0,\n  \\]\n  where \\(M\\) is a matrix derived from the constraints, \\(q\\) represents external forces or displacements, \\(z\\) are the constraint forces (Lagrange multipliers), and \\(w\\) are the primal variables representing the system states.\n- **Gauss-Seidel Iteration**: The solver iteratively updates the Lagrange multipliers \\(z_i^{k+1}\\) at iteration step \\(k+1\\):\n  \\[\n  z_i^{k+1} = \\\\- (M_{ii})^{-1} \\left( q_i + \\sum_{j=1}^{i-1} M_{ij} z_j^{k+1} + \\sum_{j=i+1}^N M_{ij} z_j^k \\right),\n  \\]\n  ensuring that the updates respect the complementarity conditions.\n\n### Constitutive or Kinematic Laws Involved\n- **Constraint Jacobians**: The matrix \\(M\\) often represents the constraint Jacobian, which captures how constraints change with respect to system states. For example, for a contact problem, it might include the normal vector of contacting surfaces and their derivatives.\n- **Damping Models**: In some cases, damping terms may be included in \\(q\\), representing dissipative forces that help stabilize the iterative solution process.\n\n### Role in the Global FEM Pipeline\n- **Assembly Phase**: The solver does not explicitly build global matrices (unbuilt version), making it suitable for large-scale problems where matrix assembly can be computationally expensive. Instead, it operates directly on the constraint problem instance provided by other components.\n- **Nonlinear Resolution**: The solver performs iterative updates to resolve constraints until convergence is achieved or a timeout occurs. This aligns with the broader nonlinear resolution step in FEM simulations.\n\n### Numerical Methods and Discretization Choices\n- **Iterative Method**: Gauss-Seidel iteration is chosen for its simplicity and potentially faster convergence compared to Jacobi methods, especially when variables are ordered strategically (e.g., by dependency).\n- **Unbuilt Version**: The solver avoids constructing the entire system matrix \\(M\\), which reduces memory usage and computational overhead.\n\n### Variational / Lagrangian Mechanics Framework\n- **Lagrangian Multipliers**: The LCP formulation naturally fits into the Lagrangian mechanics framework, where constraints are enforced using Lagrange multipliers. This solver ensures that these multipliers converge to values that satisfy both the complementarity conditions and the system's equilibrium.\n\n### Summary\nThe `UnbuiltGaussSeidelConstraintSolver` is a crucial component for enforcing constraints in FEM simulations within SOFA. It leverages an iterative Gauss-Seidel method to solve LCPs without explicitly constructing large matrices, making it efficient for real-time simulations.",
  "abstract": "The `UnbuiltGaussSeidelConstraintSolver` solves Constraint-based components iteratively using an unbuilt version of the Gauss-Seidel method and Linear Complementarity Problem (LCP) formulation without constructing system matrices explicitly.",
  "sheet": "# UnbuiltGaussSeidelConstraintSolver\n\n## Overview\n\nThe `UnbuiltGaussSeidelConstraintSolver` is a constraint solver within SOFA that uses an unbuilt version of the Gauss-Seidel iterative method to solve Constraint-based components. It operates on the Linear Complementarity Problem (LCP) formulation and does not explicitly build system matrices, making it efficient for real-time simulations.\n\n## Mathematical Model\n\n### Linear Complementarity Problem (LCP)\n\nThe LCP formulation is given by:\n\n\begin{align*}\nMz + q &= w, \\\nz &\\geq 0, \\\nw &\\geq 0, \\\nz^T w &= 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Gauss-Seidel Iteration\n\nThe solver iteratively updates the Lagrange multipliers \\(z_i^{k+1}\\) at iteration step \\(k+1\\):\n\n\begin{align*}\n  z_i^{k+1} &= - (M_{ii})^{-1} \\\\left( q_i + \\sum_{j=1}^{i-1} M_{ij} z_j^{k+1} + \\sum_{j=i+1}^N M_{ij} z_j^k \\\right),\n  \\\nz &\\geq 0,\n  \\\nw &\\geq 0,\n  \\\nz^T w &= 0.\n\tag{25}\n\tag{26}\n\tag{27}\n\tag{28}\n\tag{29}\n\tag{30}\n\tag{31}\n\tag{32}\n\tag{33}\n\tag{34}\n\tag{35}\n\tag{36}\n\tag{37}\n\tag{38}\n\tag{39}\n\tag{40}\n\tag{41}\n\tag{42}\n\tag{43}\n\tag{44}\n\tag{45}\n\tag{46}\n\tag{47}\n\tag{48}\n\tag{49}\n\tag{50}\n\tag{51}\n\tag{52}\n\tag{53}\n\tag{54}\n\tag{55}\n\tag{56}\n\tag{57}\n\tag{58}\n\tag{59}\n\tag{60}\n\tag{61}\n\tag{62}\n\tag{63}\n\tag{64}\n\tag{65}\n\tag{66}\n\tag{67}\n\tag{68}\n\tag{69}\n\tag{70}\n\tag{71}\n\tag{72}\n\tag{73}\n\tag{74}\n\tag{75}\n\tag{76}\n\tag{77}\n\tag{78}\n\tag{79}\n\tag{80}\n\tag{81}\n\tag{82}\n\tag{83}\n\tag{84}\n\tag{85}\n\tag{86}\n\tag{87}\n\tag{88}\n\tag{89}\n\tag{90}\n\tag{91}\n\tag{92}\n\tag{93}\n\tag{94}\n\tag{95}\n\tag{96}\n\tag{97}\n\tag{98}\n\tag{99}\n\tag{100}\n\n## Practical Notes\n\nThe `UnbuiltGaussSeidelConstraintSolver` is efficient for real-time simulations due to its iterative nature and avoidance of explicit matrix construction. However, convergence can be sensitive to the ordering of variables; strategic variable ordering may improve performance."
}