{
  "name": "PrecomputedMatrixSystem",
  "namespace": "sofa::component::linearsolver::preconditioner",
  "module": "Sofa.Component.LinearSolver.Preconditioner",
  "include": "sofa/component/linearsolver/preconditioner/PrecomputedMatrixSystem.h",
  "doc": "Precomputed matrix system.",
  "inherits": [
    "TypedMatrixLinearSystem"
  ],
  "templates": [
    "linearalgebra::CompressedRowSparseMatrix<SReal>, FullVector<SReal>"
  ],
  "data_fields": [],
  "links": [],
  "methods": [],
  "description": "The `PrecomputedMatrixSystem` is a linear solver component in the SOFA framework designed to handle precomputed matrix systems for preconditioning purposes within the `Sofa.Component.LinearSolver.Preconditioner` module. This component inherits from `TypedMatrixLinearSystem`, indicating that it deals with specific types of matrices and vectors, namely `CompressedRowSparseMatrix<SReal>` and `FullVector<SReal>`. The role of this component is to facilitate efficient matrix operations by leveraging precomputed systems, which can improve the performance of linear solvers in simulations involving complex mechanical behaviors. It interacts with other components within the SOFA ecosystem through its inheritance from `TypedMatrixLinearSystem`, enabling it to work seamlessly with various matrix and vector types used in linear algebra operations.",
  "maths": "The `PrecomputedMatrixSystem` is a linear solver component within the SOFA framework that facilitates preconditioning in the context of solving large sparse systems of equations. This component inherits from `TypedMatrixLinearSystem`, which indicates its role in handling specific matrix and vector types, namely `CompressedRowSparseMatrix<SReal>` and `FullVector<SReal>`. The purpose is to improve the efficiency of linear solvers by precomputing matrix systems that can be reused during iterative solution processes.\n\n### Mathematical Role\nThe primary mathematical objects involved are matrices and vectors. Specifically:\n- **Mass Matrix** (\\(M/**): Represents the mass distribution in the mechanical system, often contributing to the global stiffness or tangent stiffness matrix in FEM simulations.\n- **Stiffness Matrix** (\\(K/**): Encodes the elastic properties of the material under consideration, derived from the constitutive law and spatial discretization (e.g., via shape functions).\n\n### Governing Equations and Operators\nIn the context of linear systems, the component deals with solving equations of the form:\n\\[\nA x = b,\n\\]\nwhere \\(A\\) is a precomputed matrix system, typically derived from the mass or stiffness matrices. The vector \\(b\\) represents external forces or constraints, and \\(x\\) is the solution vector containing nodal displacements.\n\n### Constitutive Laws and Kinematic Models\nWhile this component does not directly implement constitutive laws (e.g., stress-strain relationships), it supports efficient handling of precomputed systems derived from such models. The matrices involved are typically assembled based on variational formulations and FEM discretizations, where strain measures (\\(\\varepsilon\\)) and stress tensors (\\(\\sigma\\)) play critical roles.\n\n### Role in the Global FEM Pipeline\n- **Assembly**: The component is used during assembly to store precomputed systems that can be reused for iterative solves.\n- **Time Integration**: Although it does not directly handle time integration, its efficiency improvements support faster convergence of implicit schemes like Backward Euler or Newmark-type integrators.\n- **Nonlinear Solve**: It assists in the nonlinear solution process by providing efficient preconditioning matrices to accelerate Krylov subspace methods (e.g., Conjugate Gradient).\n\n### Numerical Methods and Discretization Choices\nThe `PrecomputedMatrixSystem` employs a compressed row sparse matrix format (`CompressedRowSparseMatrix<SReal>`) for storage efficiency. This is particularly important in large-scale simulations where memory management and computational efficiency are critical.\n\n### Variational Mechanics Framework\nIn the broader variational mechanics framework, this component ensures that precomputed systems maintain consistency with the underlying weak form of the equations of motion. By providing efficient matrix operations, it supports the iterative solution process while preserving the mechanical invariants and constraints defined by Lagrangian mechanics.",
  "abstract": "The `PrecomputedMatrixSystem` handles precomputed matrix systems for preconditioning in linear solvers within SOFA simulations, improving efficiency through specific matrix types.",
  "sheet": "\n# PrecomputedMatrixSystem\n\n## Overview\n\nThe `PrecomputedMatrixSystem` is a component designed to handle precomputed matrix systems for preconditioning purposes. It inherits from `TypedMatrixLinearSystem`, indicating its role in managing specific matrix and vector types, such as `CompressedRowSparseMatrix<SReal>` and `FullVector<SReal>`. This component facilitates efficient linear solver operations by leveraging precomputed systems.\n\n## Mathematical Model\n\nThe primary mathematical objects involved are matrices and vectors. Specifically:\n- **Mass Matrix** (\\(M/**): Represents the mass distribution in the mechanical system, often contributing to the global stiffness or tangent stiffness matrix in FEM simulations.\n- **Stiffness Matrix** (\\(K/**): Encodes the elastic properties of the material under consideration, derived from the constitutive law and spatial discretization (e.g., via shape functions).\n\nThe component deals with solving equations of the form:\n\\[\nA x = b,\n\\]\nwhere \\(A\\) is a precomputed matrix system, typically derived from the mass or stiffness matrices. The vector \\(b\\) represents external forces or constraints, and \\(x\\) is the solution vector containing nodal displacements.\n\n## Dependencies and Connections\n\nThe `PrecomputedMatrixSystem` interacts with other components within the SOFA ecosystem through its inheritance from `TypedMatrixLinearSystem`, enabling it to work seamlessly with various matrix and vector types used in linear algebra operations. It typically requires or exchanges data with components that manage matrix assembly, iterative solvers, and preconditioning strategies."
}