VolumeAlgorithms
Compute open surface volume and volume derivative with respect to DOFs
The `VolumeAlgorithms` component computes the volume and its derivative with respect to degrees of freedom (DOFs) for open surfaces, such as ventricles, using mesh data from a topology container.
- module
- SofaTestPlugin
- namespace
- sofa::component::engine
- include
- utils/VolumeAlgorithms.h
- inherits
-
- BaseObject
- templates
-
- sofa::defaulttype::Vec3Types
- description
The VolumeAlgorithms component in the SOFA framework is designed to compute the volume and its derivative with respect to degrees of freedom (DOFs) for an open surface, typically representing anatomical structures like ventricles. This component operates within the SofaTestPlugin module and interacts closely with a topology container (TriangleSetTopologyContainer) to access mesh data necessary for computations.
Governing Equations and Operators:
Volume Calculation:
The volume of an open surface (e.g., ventricle) is computed as follows:
egin{equation}
V = \frac{1}{6} \left| \sum_{i=0}^{N-1} \text{tripleProduct}(x[t_i[0]], x[t_i[1]], x[t_i[2]]) + \sum_j \text{holeContribution}j(x) \right|
\end{equation}
where:
- $t_i$ are triangles from the mesh.
- $N$ is the number of triangles.
- The triple product $\text{tripleProduct}(a, b, c)$ computes the volume contribution of a triangle formed by vertices $a$, $b$, and $c$ as $(a - c) \cdot ((b - c) \times (a - c))$.
- Holes contribute to the volume using a centroid method where each hole's triangles are summed around a central point:
egin{equation}
\text{holeContribution}_j(x) = \sum{k=0}^{M_j-1} \text{tripleProduct}(x[e_k[0]], C, x[e_k[1]])
\end{equation}
where $e_k$ are edges of the hole and $C$ is the centroid of all vertices on the hole.
Volume Derivative:
The derivative of the volume with respect to each vertex position is computed as follows:
egin{equation}
dV_i = \frac{1}{6} \sum_{j=0}^{T_i-1} (x[t_j[0]] - x[t_j[2]]) \times (x[t_j[1]] - x[t_j[2]]) + \text{holeDerivative}i(x)
\end{equation}
where:
- $t_j$ are triangles around vertex $i$.
- $T_i$ is the number of triangles adjacent to vertex $i$.
- The hole derivative contribution involves a centroid method where each hole's edges contribute to the volume change at vertex $i$:
egin{equation}
\text{holeDerivative}_i(x) = \sum{k=0}^{H_i-1} \frac{dv_c}{24} + \frac{dv_c}{24}
\end{equation}
where $dv_c$ is the sum of cross products around hole edges.
Constitutive and Kinematic Laws:
The component does not directly involve constitutive laws or stress tensors, as it primarily focuses on geometric volume computations. However, the derivatives computed can be used in various nonlinear mechanics formulations where the geometry's changes affect mechanical behavior.
Role in FEM Pipeline:
The VolumeAlgorithms component fits into the broader variational and Lagrangian mechanics framework by providing essential geometric properties that can influence the overall simulation dynamics. Specifically, it contributes to:
- Assembly Phase: Computes volume and its derivatives with respect to vertex positions, which are crucial for nonlinear solve steps.
- Time Integration: Provides flow calculations based on velocity data at each time step, aiding in implicit integration schemes.
- Nonlinear Solve: The computed derivative $dV$ can be used as part of the Jacobian matrix in Newton-Raphson iterations to handle constraints and ensure physical consistency.
Numerical Methods:
The component uses standard mesh-based geometry operations like triple product calculations for volume contributions. It leverages SOFA's topology container capabilities to efficiently manage triangle and edge data, ensuring accurate computation of volumes even with holes present in the surface.
Summary:
The VolumeAlgorithms component is a geometric engine that provides critical volume and derivative computations for open surfaces within the SOFA framework. By interfacing with the mesh topology, it enables detailed analysis and simulation of anatomical structures' volumetric changes over time.
Data Fields
| Name | Type | Default | Help |
|---|---|---|---|
d_volume |
Real | |
Volume of the ventricle |
d_dV |
VecDeriv | |
Derivative of the volume with respect to endocardium vertices |
d_holes |
Holes | |
list of list of edges |
d_flow |
Real | |
flow computed at the beginning of the time step |
d_previous_flow |
Real | |
flow computed at the previous time step |
d_show_boundary_edges |
bool | |
Links
| Name | Type | Help |
|---|---|---|
l_topology |
Link to topology |
Methods
void
init
()
virtual
void
get_boundary_edges
()
void
handleEvent
(sofa::core::objectmodel::Event * event)
virtual
void
update_volume
()
Real
compute_flow
(const VecCoord & x, const VecDeriv & v)
Real
compute_flow_dV
(const VecDeriv & dV, const VecDeriv & v)
void
draw
(const core::visual::VisualParams * vparams)
virtual
SReal
compute_volume
(const VecCoord & x)
void
compute_volume_derivative
(const VecCoord & x, VecDeriv & dvol)
bool
all_used
(const VecBool & x)
int
get_first
(const VecBool & x)
{
"name": "VolumeAlgorithms",
"namespace": "sofa::component::engine",
"module": "SofaTestPlugin",
"include": "utils/VolumeAlgorithms.h",
"doc": "Compute open surface volume and volume derivative with respect to DOFs",
"inherits": [
"BaseObject"
],
"templates": [
"sofa::defaulttype::Vec3Types"
],
"data_fields": [
{
"name": "d_volume",
"type": "Real",
"xmlname": "volume",
"help": "Volume of the ventricle"
},
{
"name": "d_dV",
"type": "VecDeriv",
"xmlname": "dV",
"help": "Derivative of the volume with respect to endocardium vertices"
},
{
"name": "d_holes",
"type": "Holes",
"xmlname": "edges_on_border",
"help": "list of list of edges"
},
{
"name": "d_flow",
"type": "Real",
"xmlname": "flow",
"help": "flow computed at the beginning of the time step"
},
{
"name": "d_previous_flow",
"type": "Real",
"xmlname": "previous_flow",
"help": "flow computed at the previous time step"
},
{
"name": "d_show_boundary_edges",
"type": "bool",
"xmlname": "show_boundary_edges",
"help": ""
}
],
"links": [
{
"name": "l_topology",
"target": "TriangleSetTopologyContainer",
"kind": "single",
"xmlname": "topology",
"help": "Link to topology"
}
],
"methods": [
{
"name": "init",
"return_type": "void",
"params": [],
"is_virtual": true,
"is_pure_virtual": false,
"is_static": false,
"access": "public"
},
{
"name": "get_boundary_edges",
"return_type": "void",
"params": [],
"is_virtual": false,
"is_pure_virtual": false,
"is_static": false,
"access": "public"
},
{
"name": "handleEvent",
"return_type": "void",
"params": [
{
"name": "event",
"type": "sofa::core::objectmodel::Event *"
}
],
"is_virtual": true,
"is_pure_virtual": false,
"is_static": false,
"access": "public"
},
{
"name": "update_volume",
"return_type": "void",
"params": [],
"is_virtual": false,
"is_pure_virtual": false,
"is_static": false,
"access": "public"
},
{
"name": "compute_flow",
"return_type": "Real",
"params": [
{
"name": "x",
"type": "const VecCoord &"
},
{
"name": "v",
"type": "const VecDeriv &"
}
],
"is_virtual": false,
"is_pure_virtual": false,
"is_static": false,
"access": "public"
},
{
"name": "compute_flow_dV",
"return_type": "Real",
"params": [
{
"name": "dV",
"type": "const VecDeriv &"
},
{
"name": "v",
"type": "const VecDeriv &"
}
],
"is_virtual": false,
"is_pure_virtual": false,
"is_static": false,
"access": "public"
},
{
"name": "draw",
"return_type": "void",
"params": [
{
"name": "vparams",
"type": "const core::visual::VisualParams *"
}
],
"is_virtual": true,
"is_pure_virtual": false,
"is_static": false,
"access": "public"
},
{
"name": "compute_volume",
"return_type": "SReal",
"params": [
{
"name": "x",
"type": "const VecCoord &"
}
],
"is_virtual": false,
"is_pure_virtual": false,
"is_static": false,
"access": "public"
},
{
"name": "compute_volume_derivative",
"return_type": "void",
"params": [
{
"name": "x",
"type": "const VecCoord &"
},
{
"name": "dvol",
"type": "VecDeriv &"
}
],
"is_virtual": false,
"is_pure_virtual": false,
"is_static": false,
"access": "public"
},
{
"name": "all_used",
"return_type": "bool",
"params": [
{
"name": "x",
"type": "const VecBool &"
}
],
"is_virtual": false,
"is_pure_virtual": false,
"is_static": false,
"access": "public"
},
{
"name": "get_first",
"return_type": "int",
"params": [
{
"name": "x",
"type": "const VecBool &"
}
],
"is_virtual": false,
"is_pure_virtual": false,
"is_static": false,
"access": "public"
}
],
"description": "The `VolumeAlgorithms` component in SOFA is designed to compute the volume and its derivative with respect to degrees of freedom (DOFs) for an open surface, typically representing anatomical structures like ventricles. It inherits from `BaseObject` and operates within the `SofaTestPlugin`. This component interacts with a topology container (`TriangleSetTopologyContainer`) to access mesh data and compute relevant properties such as volume and flow rates.\n\n### Interactions with Other SOFA Components:\n- **Link to Topology**: The component uses a link named `topology` to connect with the triangle set topology, essential for retrieving vertex and edge information required for computations.\n- **Mechanical State Access**: It dynamically casts the mechanical state from the simulation context to access position (`x`) and velocity (`v`) data necessary for volume and flow calculations.\n\n### Data Fields:\n- `volume`: Stores the computed volume of the surface (e.g., ventricle).\n- `dV`: Holds the derivative of the volume with respect to vertex positions, critical for understanding how changes in geometry affect volume.\n- `edges_on_border`: List of edges on the boundary used for detecting holes and ensuring proper volume computation.\n- `flow` and `previous_flow`: Track fluid flow through the surface at different time steps, useful for dynamic simulations involving fluid dynamics or blood flow.\n- `show_boundary_edges`: Boolean flag to control visualization of boundary edges during rendering.\n\n### Practical Usage Guidance:\nThe component is typically used in scenarios where tracking changes in volume and associated flows are essential, such as in biomechanical models of heart function. It provides functionalities like `compute_volume`, `compute_flow`, and visualization tools through the `draw` method to integrate seamlessly into larger simulation workflows.",
"maths": "The `VolumeAlgorithms` component in the SOFA framework is designed to compute the volume and its derivative with respect to degrees of freedom (DOFs) for an open surface, typically representing anatomical structures like ventricles. This component operates within the `SofaTestPlugin` module and interacts closely with a topology container (`TriangleSetTopologyContainer`) to access mesh data necessary for computations.\n\n### Governing Equations and Operators:\n\n#### Volume Calculation:\nThe volume of an open surface (e.g., ventricle) is computed as follows:\n\begin{equation}\nV = \\frac{1}{6} \\left| \\sum_{i=0}^{N-1} \\text{tripleProduct}(x[t_i[0]], x[t_i[1]], x[t_i[2]]) + \\sum_j \\text{holeContribution}_j(x) \\right|\n\\end{equation}\nwhere:\n- $t_i$ are triangles from the mesh.\n- $N$ is the number of triangles.\n- The triple product $\\text{tripleProduct}(a, b, c)$ computes the volume contribution of a triangle formed by vertices $a$, $b$, and $c$ as $(a - c) \\cdot ((b - c) \\times (a - c))$.\n- Holes contribute to the volume using a centroid method where each hole's triangles are summed around a central point:\n\begin{equation}\n\\text{holeContribution}_j(x) = \\sum_{k=0}^{M_j-1} \\text{tripleProduct}(x[e_k[0]], C, x[e_k[1]])\n\\end{equation}\nwhere $e_k$ are edges of the hole and $C$ is the centroid of all vertices on the hole.\n\n#### Volume Derivative:\nThe derivative of the volume with respect to each vertex position is computed as follows:\n\begin{equation}\ndV_i = \\frac{1}{6} \\sum_{j=0}^{T_i-1} (x[t_j[0]] - x[t_j[2]]) \\times (x[t_j[1]] - x[t_j[2]]) + \\text{holeDerivative}_i(x)\n\\end{equation}\nwhere:\n- $t_j$ are triangles around vertex $i$.\n- $T_i$ is the number of triangles adjacent to vertex $i$.\n- The hole derivative contribution involves a centroid method where each hole's edges contribute to the volume change at vertex $i$:\n\begin{equation}\n\\text{holeDerivative}_i(x) = \\sum_{k=0}^{H_i-1} \\frac{dv_c}{24} + \\frac{dv_c}{24}\n\\end{equation}\nwhere $dv_c$ is the sum of cross products around hole edges.\n\n### Constitutive and Kinematic Laws:\nThe component does not directly involve constitutive laws or stress tensors, as it primarily focuses on geometric volume computations. However, the derivatives computed can be used in various nonlinear mechanics formulations where the geometry's changes affect mechanical behavior.\n\n### Role in FEM Pipeline:\nThe `VolumeAlgorithms` component fits into the broader variational and Lagrangian mechanics framework by providing essential geometric properties that can influence the overall simulation dynamics. Specifically, it contributes to:\n- **Assembly Phase**: Computes volume and its derivatives with respect to vertex positions, which are crucial for nonlinear solve steps.\n- **Time Integration**: Provides flow calculations based on velocity data at each time step, aiding in implicit integration schemes.\n- **Nonlinear Solve**: The computed derivative $dV$ can be used as part of the Jacobian matrix in Newton-Raphson iterations to handle constraints and ensure physical consistency.\n\n### Numerical Methods:\nThe component uses standard mesh-based geometry operations like triple product calculations for volume contributions. It leverages SOFA's topology container capabilities to efficiently manage triangle and edge data, ensuring accurate computation of volumes even with holes present in the surface.\n\n### Summary:\nThe `VolumeAlgorithms` component is a geometric engine that provides critical volume and derivative computations for open surfaces within the SOFA framework. By interfacing with the mesh topology, it enables detailed analysis and simulation of anatomical structures' volumetric changes over time.",
"abstract": "The `VolumeAlgorithms` component computes the volume and its derivative with respect to degrees of freedom (DOFs) for open surfaces, such as ventricles, using mesh data from a topology container.",
"sheet": "# VolumeAlgorithms\n\n## Overview\nThe `VolumeAlgorithms` component in SOFA is designed to compute the volume and its derivative with respect to degrees of freedom (DOFs) for an open surface. It interacts closely with a topology container (`TriangleSetTopologyContainer`) to access mesh data necessary for computations.\n\n## Mathematical Model\n### Volume Calculation\nThe volume of an open surface (e.g., ventricle) is computed as follows:\n\\begin{equation}\nV = \\frac{1}{6} \\left| \\sum_{i=0}^{N-1} \\text{tripleProduct}(x[t_i[0]], x[t_i[1]], x[t_i[2]]) + \\sum_j \\text{holeContribution}_j(x) \\right|\n\\end{equation}\nwhere:\n- $t_i$ are triangles from the mesh.\n- $N$ is the number of triangles.\n- The triple product $\\text{tripleProduct}(a, b, c)$ computes the volume contribution of a triangle formed by vertices $a$, $b$, and $c$ as $(a - c) \\cdot ((b - c) \\times (a - c))$.\n- Holes contribute to the volume using a centroid method where each hole's triangles are summed around a central point:\n\\begin{equation}\n\\text{holeContribution}_j(x) = \\sum_{k=0}^{M_j-1} \\text{tripleProduct}(x[e_k[0]], C, x[e_k[1]])\n\\end{equation}\nwhere $e_k$ are edges of the hole and $C$ is the centroid of all vertices on the hole.\n\n### Volume Derivative\nThe derivative of the volume with respect to each vertex position is computed as follows:\n\\begin{equation}\ndV_i = \\frac{1}{6} \\sum_{j=0}^{T_i-1} (x[t_j[0]] - x[t_j[2]]) \\times (x[t_j[1]] - x[t_j[2]]) + \\text{holeDerivative}_i(x)\n\\end{equation}\nwhere:\n- $t_j$ are triangles around vertex $i$.\n- $T_i$ is the number of triangles adjacent to vertex $i$.\n- The hole derivative contribution involves a centroid method where each hole's edges contribute to the volume change at vertex $i$:\n\\begin{equation}\n\\text{holeDerivative}_i(x) = \\sum_{k=0}^{H_i-1} \\frac{dv_c}{24} + \\frac{dv_c}{24}\n\\end{equation}\nwhere $dv_c$ is the sum of cross products around hole edges.\n\n## Parameters and Data\nThe significant data fields exposed by the component are:\n- `volume`: Stores the computed volume of the surface (e.g., ventricle).\n- `dV`: Holds the derivative of the volume with respect to vertex positions, critical for understanding how changes in geometry affect volume.\n- `edges_on_border`: List of edges on the boundary used for detecting holes and ensuring proper volume computation.\n- `flow` and `previous_flow`: Track fluid flow through the surface at different time steps, useful for dynamic simulations involving fluid dynamics or blood flow.\n- `show_boundary_edges`: Boolean flag to control visualization of boundary edges during rendering."
}