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fespace

Module: fespace.py

This module provides abstract base functionality for finite element space implementations, defining core interfaces for fe computations and analysis.

Classes:

Name Description
Fespace

Abstract base class for finite element space implementations
Dependencies
  • numpy: For numerical computations
  • abc: For abstract base class functionality
Key Features
  • Mesh handling and cell management
  • Shape function evaluation and manipulation
  • Boundary condition implementation
    • Dirichlet boundary handling
    • Boundary function management
  • Quadrature point management
  • Forcing function integration
  • Sensor data handling for inverse problems
  • Reference and physical space transformations
  • Gradient computation in both spaces
Authors

Thivin Anandh D (https://thivinanandh.github.io)

Version Info

27/Dec/2024: Initial version - Thivin Anandh D

Notes

This module serves as the foundation for specific finite element space implementations. It defines the minimum interface required for fe computations while allowing flexibility in concrete implementations.

Fespace

Abstract base class defining the interface for finite element spaces.

This class provides the foundation for implementing finite element spaces, including mesh handling, element operations, and solution computations.

Attributes:

Name Type Description
mesh

Mesh object containing geometric information
cells ndarray

Array of cell indices
boundary_points dict

Dictionary of boundary point information
cell_type str

Type of finite element cell
fe_order int

Order of finite element approximation
fe_type str

Type of finite element basis
quad_order int

Order of quadrature rule
quad_type str

Type of quadrature formula
fe_transformation_type str

Type of geometric transformation
bound_function_dict dict

Dictionary of boundary condition functions
bound_condition_dict dict

Dictionary of boundary condition types
forcing_function callable

Source term function
output_path str

Path for output files
Example

class MyFespace(Fespace): ... def set_finite_elements(self): ... # Implementation ... pass ... def generate_dirichlet_boundary_data(self): ... # Implementation ... pass ... # Implement other abstract methods

Notes
  • All coordinate transformations must be implemented
  • Shape function values and gradients are available in both reference and physical spaces
  • Supports both internal and external sensor data for inverse problems
  • Boundary conditions must be properly specified through the boundary dictionaries
Source code in scirex/core/sciml/fe/fespace.py 
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class Fespace:
    """
    Abstract base class defining the interface for finite element spaces.

    This class provides the foundation for implementing finite element spaces,
    including mesh handling, element operations, and solution computations.

    Attributes:
        mesh: Mesh object containing geometric information
        cells (ndarray): Array of cell indices
        boundary_points (dict): Dictionary of boundary point information
        cell_type (str): Type of finite element cell
        fe_order (int): Order of finite element approximation
        fe_type (str): Type of finite element basis
        quad_order (int): Order of quadrature rule
        quad_type (str): Type of quadrature formula
        fe_transformation_type (str): Type of geometric transformation
        bound_function_dict (dict): Dictionary of boundary condition functions
        bound_condition_dict (dict): Dictionary of boundary condition types
        forcing_function (callable): Source term function
        output_path (str): Path for output files

    Example:
        >>> class MyFespace(Fespace):
        ...     def set_finite_elements(self):
        ...         # Implementation
        ...         pass
        ...     def generate_dirichlet_boundary_data(self):
        ...         # Implementation
        ...         pass
        ...     # Implement other abstract methods

    Notes:
        - All coordinate transformations must be implemented
        - Shape function values and gradients are available in both
        reference and physical spaces
        - Supports both internal and external sensor data for
        inverse problems
        - Boundary conditions must be properly specified through
        the boundary dictionaries
    """

    def __init__(
        self,
        mesh,
        cells,
        boundary_points,
        cell_type: str,
        fe_order: int,
        fe_type: str,
        quad_order: int,
        quad_type: str,
        fe_transformation_type: str,
        bound_function_dict: dict,
        bound_condition_dict: dict,
        forcing_function,
        output_path: str,
    ) -> None:
        """
        The constructor of the Fespace2D class.

        Args:
            mesh: The mesh object.
            cells: The cells of the mesh.
            boundary_points: The boundary points of the mesh.
            cell_type: The type of the cell.
            fe_order: The order of the finite element.
            fe_type: The type of the finite element.
            quad_order: The order of the quadrature.
            quad_type: The type of the quadrature.
            fe_transformation_type: The type of the finite element transformation.
            bound_function_dict: The dictionary of the boundary functions.
            bound_condition_dict: The dictionary of the boundary conditions.
            forcing_function: The forcing function.
            output_path: The path to the output directory.

        Returns:
            None
        """
        self.mesh = mesh
        self.boundary_points = boundary_points
        self.cells = cells
        self.cell_type = cell_type
        self.fe_order = fe_order
        self.fe_type = fe_type
        self.quad_order = quad_order
        self.quad_type = quad_type

        self.fe_transformation_type = fe_transformation_type
        self.output_path = output_path
        self.bound_function_dict = bound_function_dict
        self.bound_condition_dict = bound_condition_dict
        self.forcing_function = forcing_function

    @abstractmethod
    def set_finite_elements(self) -> None:
        """
        Assigns the finite elements to each cell.

        This method initializes the finite element objects for each cell in the mesh.
        It creates an instance of the `FE2D_Cell` class for each cell, passing the necessary parameters.
        The finite element objects store information about the basis functions, gradients, Jacobians,
        quadrature points, weights, actual coordinates, and forcing functions associated with each cell.

        After initializing the finite element objects, this method prints the shape details of various matrices
        and updates the total number of degrees of freedom (dofs) for the entire mesh.

        Args:
            None

        Returns:
            None
        """

    @abstractmethod
    def generate_dirichlet_boundary_data(self) -> np.ndarray:
        """
        Generate Dirichlet boundary data.

        This function returns the boundary points and their corresponding values.

        Args:
            None

        Returns:
            np.ndarray: The boundary points and their values.

        Notes:
            The boundary points and values are stored in the `boundary_points` attribute of the `Fespace` object.
        """

    @abstractmethod
    def get_shape_function_val(self, cell_index: int) -> np.ndarray:
        """
        Get the actual values of the shape functions on a given cell.

        Args:
            cell_index (int): The index of the cell.

        Returns:
            np.ndarray: An array containing the actual values of the shape functions.

        Raises:
            ValueError: If the cell_index is greater than the number of cells.
        """

    @abstractmethod
    def get_shape_function_grad_x(self, cell_index: int) -> np.ndarray:
        """
        Get the gradient of the shape function with respect to the x-coordinate.

        Args:
            cell_index (int): The index of the cell.

        Returns:
            np.ndarray: An array containing the gradient of the shape function with respect to the x-coordinate.

        Raises:
            ValueError: If the cell_index is greater than the number of cells.
        """

    @abstractmethod
    def get_shape_function_grad_x_ref(self, cell_index: int) -> np.ndarray:
        """
        Get the gradient of the shape function with respect to the x-coordinate on the reference element.

        Args:
            cell_index (int): The index of the cell.

        Returns:
            np.ndarray: An array containing the gradient of the shape function with respect to the x-coordinate on the reference element.

        Raises:
            ValueError: If the cell_index is greater than the number of cells.
        """

    @abstractmethod
    def get_shape_function_grad_y(self, cell_index: int) -> np.ndarray:
        """
        Get the gradient of the shape function with respect to y at the given cell index.

        Args:
            cell_index (int): The index of the cell.

        Returns:
            np.ndarray: An array containing the gradient of the shape function with respect to y.

        Raises:
            ValueError: If the cell_index is greater than the number of cells.
        """

    @abstractmethod
    def get_shape_function_grad_y_ref(self, cell_index: int):
        """
        Get the gradient of the shape function with respect to y at the reference element.

        Args:
            cell_index (int): The index of the cell.

        Returns:
            np.ndarray: An array containing the gradient of the shape function with respect to y at the reference element.

        Raises:
            ValueError: If the cell_index is greater than the number of cells.

        Notes:
            This function returns the gradient of the shape function with respect to y at the reference element
            for a given cell. The shape function gradient values are stored in the `basis_grady_at_quad_ref` array
            of the corresponding finite element cell. The `cell_index` parameter specifies the index of the cell
            for which the shape function gradient is required. If the `cell_index` is greater than the total number
            of cells, a `ValueError` is raised. The returned gradient values are copied from the `basis_grady_at_quad_ref` array to ensure immutability.
        """

    @abstractmethod
    def get_quadrature_actual_coordinates(self, cell_index: int) -> np.ndarray:
        """
        Get the actual coordinates of the quadrature points for a given cell.

        Args:
            cell_index (int): The index of the cell.

        Returns:
            np.ndarray: An array containing the actual coordinates of the quadrature points.

        Raises:
            ValueError: If the cell_index is greater than the number of cells.
        """

    @abstractmethod
    def get_forcing_function_values(self, cell_index: int) -> np.ndarray:
        """
        Get the forcing function values at the quadrature points.

        Args:
            cell_index (int): The index of the cell.

        Returns:
            np.ndarray: An array containing the forcing function values at the quadrature points.

        Raises:
            ValueError: If the cell_index is greater than the number of cells.

        Notes:
            This function computes the forcing function values at the quadrature points for a given cell.
            It loops over all the basis functions and computes the integral using the actual coordinates
            and the basis functions at the quadrature points. The resulting values are stored in the
            `forcing_at_quad` attribute of the corresponding `fe_cell` object. The forcing function is evaluated using the `forcing_function` method of the `fe_cell`
            object.
        """

    @abstractmethod
    def get_sensor_data(self, exact_solution, num_points: int) -> np.ndarray:
        """
        Obtain sensor data (actual solution) at random points.

        Args:
            exact_solution (ndarray): The exact solution values.
            num_points (int): The number of points to sample from the domain.

        Returns:
            np.ndarray: The sensor data at the given points.

        Notes:
            This method is used in the inverse problem to obtain the sensor data at random points within the domain. Currently, it only works for problems with an analytical solution.
            Methodologies to obtain sensor data for problems from a file are not implemented yet.
            It is also not implemented for external or complex meshes.
        """

    @abstractmethod
    def get_sensor_data_external(
        self, exact_sol, num_points: int, file_name: str
    ) -> np.ndarray:
        """
        This method is used to obtain the sensor data from an external file when there is no analytical solution available.

        Args:
            exact_sol: The exact solution values.
            num_points: The number of points to sample from the domain.
            file_name: The name of the file containing the sensor data.

        Returns:
            np.ndarray: The sensor data at the given points based on the external file.

        """

__init__(mesh, cells, boundary_points, cell_type, fe_order, fe_type, quad_order, quad_type, fe_transformation_type, bound_function_dict, bound_condition_dict, forcing_function, output_path)

The constructor of the Fespace2D class.

Parameters:

Name Type Description Default
mesh

The mesh object.

 required
cells

The cells of the mesh.

 required
boundary_points

The boundary points of the mesh.

 required
cell_type str

The type of the cell.

 required
fe_order int

The order of the finite element.

 required
fe_type str

The type of the finite element.

 required
quad_order int

The order of the quadrature.

 required
quad_type str

The type of the quadrature.

 required
fe_transformation_type str

The type of the finite element transformation.

 required
bound_function_dict dict

The dictionary of the boundary functions.

 required
bound_condition_dict dict

The dictionary of the boundary conditions.

 required
forcing_function

The forcing function.

 required
output_path str

The path to the output directory.

 required

Returns:

Type Description
None

None
Source code in scirex/core/sciml/fe/fespace.py 
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def __init__(
    self,
    mesh,
    cells,
    boundary_points,
    cell_type: str,
    fe_order: int,
    fe_type: str,
    quad_order: int,
    quad_type: str,
    fe_transformation_type: str,
    bound_function_dict: dict,
    bound_condition_dict: dict,
    forcing_function,
    output_path: str,
) -> None:
    """
    The constructor of the Fespace2D class.

    Args:
        mesh: The mesh object.
        cells: The cells of the mesh.
        boundary_points: The boundary points of the mesh.
        cell_type: The type of the cell.
        fe_order: The order of the finite element.
        fe_type: The type of the finite element.
        quad_order: The order of the quadrature.
        quad_type: The type of the quadrature.
        fe_transformation_type: The type of the finite element transformation.
        bound_function_dict: The dictionary of the boundary functions.
        bound_condition_dict: The dictionary of the boundary conditions.
        forcing_function: The forcing function.
        output_path: The path to the output directory.

    Returns:
        None
    """
    self.mesh = mesh
    self.boundary_points = boundary_points
    self.cells = cells
    self.cell_type = cell_type
    self.fe_order = fe_order
    self.fe_type = fe_type
    self.quad_order = quad_order
    self.quad_type = quad_type

    self.fe_transformation_type = fe_transformation_type
    self.output_path = output_path
    self.bound_function_dict = bound_function_dict
    self.bound_condition_dict = bound_condition_dict
    self.forcing_function = forcing_function

generate_dirichlet_boundary_data() abstractmethod

Generate Dirichlet boundary data.

This function returns the boundary points and their corresponding values.

Returns:

Type Description
ndarray

np.ndarray: The boundary points and their values.
Notes

The boundary points and values are stored in the boundary_points attribute of the Fespace object.

Source code in scirex/core/sciml/fe/fespace.py 
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@abstractmethod
def generate_dirichlet_boundary_data(self) -> np.ndarray:
    """
    Generate Dirichlet boundary data.

    This function returns the boundary points and their corresponding values.

    Args:
        None

    Returns:
        np.ndarray: The boundary points and their values.

    Notes:
        The boundary points and values are stored in the `boundary_points` attribute of the `Fespace` object.
    """

get_forcing_function_values(cell_index) abstractmethod

Get the forcing function values at the quadrature points.

Parameters:

Name Type Description Default
cell_index int

The index of the cell.

 required

Returns:

Type Description
ndarray

np.ndarray: An array containing the forcing function values at the quadrature points.

Raises:

Type Description
ValueError

If the cell_index is greater than the number of cells.
Notes

This function computes the forcing function values at the quadrature points for a given cell. It loops over all the basis functions and computes the integral using the actual coordinates and the basis functions at the quadrature points. The resulting values are stored in the forcing_at_quad attribute of the corresponding fe_cell object. The forcing function is evaluated using the forcing_function method of the fe_cell object.

Source code in scirex/core/sciml/fe/fespace.py 
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@abstractmethod
def get_forcing_function_values(self, cell_index: int) -> np.ndarray:
    """
    Get the forcing function values at the quadrature points.

    Args:
        cell_index (int): The index of the cell.

    Returns:
        np.ndarray: An array containing the forcing function values at the quadrature points.

    Raises:
        ValueError: If the cell_index is greater than the number of cells.

    Notes:
        This function computes the forcing function values at the quadrature points for a given cell.
        It loops over all the basis functions and computes the integral using the actual coordinates
        and the basis functions at the quadrature points. The resulting values are stored in the
        `forcing_at_quad` attribute of the corresponding `fe_cell` object. The forcing function is evaluated using the `forcing_function` method of the `fe_cell`
        object.
    """

get_quadrature_actual_coordinates(cell_index) abstractmethod

Get the actual coordinates of the quadrature points for a given cell.

Parameters:

Name Type Description Default
cell_index int

The index of the cell.

 required

Returns:

Type Description
ndarray

np.ndarray: An array containing the actual coordinates of the quadrature points.

Raises:

Type Description
ValueError

If the cell_index is greater than the number of cells.
Source code in scirex/core/sciml/fe/fespace.py 
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@abstractmethod
def get_quadrature_actual_coordinates(self, cell_index: int) -> np.ndarray:
    """
    Get the actual coordinates of the quadrature points for a given cell.

    Args:
        cell_index (int): The index of the cell.

    Returns:
        np.ndarray: An array containing the actual coordinates of the quadrature points.

    Raises:
        ValueError: If the cell_index is greater than the number of cells.
    """

get_sensor_data(exact_solution, num_points) abstractmethod

Obtain sensor data (actual solution) at random points.

Parameters:

Name Type Description Default
exact_solution ndarray

The exact solution values.

 required
num_points int

The number of points to sample from the domain.

 required

Returns:

Type Description
ndarray

np.ndarray: The sensor data at the given points.
Notes

This method is used in the inverse problem to obtain the sensor data at random points within the domain. Currently, it only works for problems with an analytical solution. Methodologies to obtain sensor data for problems from a file are not implemented yet. It is also not implemented for external or complex meshes.

Source code in scirex/core/sciml/fe/fespace.py 
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@abstractmethod
def get_sensor_data(self, exact_solution, num_points: int) -> np.ndarray:
    """
    Obtain sensor data (actual solution) at random points.

    Args:
        exact_solution (ndarray): The exact solution values.
        num_points (int): The number of points to sample from the domain.

    Returns:
        np.ndarray: The sensor data at the given points.

    Notes:
        This method is used in the inverse problem to obtain the sensor data at random points within the domain. Currently, it only works for problems with an analytical solution.
        Methodologies to obtain sensor data for problems from a file are not implemented yet.
        It is also not implemented for external or complex meshes.
    """

get_sensor_data_external(exact_sol, num_points, file_name) abstractmethod

This method is used to obtain the sensor data from an external file when there is no analytical solution available.

Parameters:

Name Type Description Default
exact_sol

The exact solution values.

 required
num_points int

The number of points to sample from the domain.

 required
file_name str

The name of the file containing the sensor data.

 required

Returns:

Type Description
ndarray

np.ndarray: The sensor data at the given points based on the external file.
Source code in scirex/core/sciml/fe/fespace.py 
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@abstractmethod
def get_sensor_data_external(
    self, exact_sol, num_points: int, file_name: str
) -> np.ndarray:
    """
    This method is used to obtain the sensor data from an external file when there is no analytical solution available.

    Args:
        exact_sol: The exact solution values.
        num_points: The number of points to sample from the domain.
        file_name: The name of the file containing the sensor data.

    Returns:
        np.ndarray: The sensor data at the given points based on the external file.

    """

get_shape_function_grad_x(cell_index) abstractmethod

Get the gradient of the shape function with respect to the x-coordinate.

Parameters:

Name Type Description Default
cell_index int

The index of the cell.

 required

Returns:

Type Description
ndarray

np.ndarray: An array containing the gradient of the shape function with respect to the x-coordinate.

Raises:

Type Description
ValueError

If the cell_index is greater than the number of cells.
Source code in scirex/core/sciml/fe/fespace.py 
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@abstractmethod
def get_shape_function_grad_x(self, cell_index: int) -> np.ndarray:
    """
    Get the gradient of the shape function with respect to the x-coordinate.

    Args:
        cell_index (int): The index of the cell.

    Returns:
        np.ndarray: An array containing the gradient of the shape function with respect to the x-coordinate.

    Raises:
        ValueError: If the cell_index is greater than the number of cells.
    """

get_shape_function_grad_x_ref(cell_index) abstractmethod

Get the gradient of the shape function with respect to the x-coordinate on the reference element.

Parameters:

Name Type Description Default
cell_index int

The index of the cell.

 required

Returns:

Type Description
ndarray

np.ndarray: An array containing the gradient of the shape function with respect to the x-coordinate on the reference element.

Raises:

Type Description
ValueError

If the cell_index is greater than the number of cells.
Source code in scirex/core/sciml/fe/fespace.py 
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@abstractmethod
def get_shape_function_grad_x_ref(self, cell_index: int) -> np.ndarray:
    """
    Get the gradient of the shape function with respect to the x-coordinate on the reference element.

    Args:
        cell_index (int): The index of the cell.

    Returns:
        np.ndarray: An array containing the gradient of the shape function with respect to the x-coordinate on the reference element.

    Raises:
        ValueError: If the cell_index is greater than the number of cells.
    """

get_shape_function_grad_y(cell_index) abstractmethod

Get the gradient of the shape function with respect to y at the given cell index.

Parameters:

Name Type Description Default
cell_index int

The index of the cell.

 required

Returns:

Type Description
ndarray

np.ndarray: An array containing the gradient of the shape function with respect to y.

Raises:

Type Description
ValueError

If the cell_index is greater than the number of cells.
Source code in scirex/core/sciml/fe/fespace.py 
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@abstractmethod
def get_shape_function_grad_y(self, cell_index: int) -> np.ndarray:
    """
    Get the gradient of the shape function with respect to y at the given cell index.

    Args:
        cell_index (int): The index of the cell.

    Returns:
        np.ndarray: An array containing the gradient of the shape function with respect to y.

    Raises:
        ValueError: If the cell_index is greater than the number of cells.
    """

get_shape_function_grad_y_ref(cell_index) abstractmethod

Get the gradient of the shape function with respect to y at the reference element.

Parameters:

Name Type Description Default
cell_index int

The index of the cell.

 required

Returns:

Type Description

np.ndarray: An array containing the gradient of the shape function with respect to y at the reference element.

Raises:

Type Description
ValueError

If the cell_index is greater than the number of cells.
Notes

This function returns the gradient of the shape function with respect to y at the reference element for a given cell. The shape function gradient values are stored in the basis_grady_at_quad_ref array of the corresponding finite element cell. The cell_index parameter specifies the index of the cell for which the shape function gradient is required. If the cell_index is greater than the total number of cells, a ValueError is raised. The returned gradient values are copied from the basis_grady_at_quad_ref array to ensure immutability.

Source code in scirex/core/sciml/fe/fespace.py 
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@abstractmethod
def get_shape_function_grad_y_ref(self, cell_index: int):
    """
    Get the gradient of the shape function with respect to y at the reference element.

    Args:
        cell_index (int): The index of the cell.

    Returns:
        np.ndarray: An array containing the gradient of the shape function with respect to y at the reference element.

    Raises:
        ValueError: If the cell_index is greater than the number of cells.

    Notes:
        This function returns the gradient of the shape function with respect to y at the reference element
        for a given cell. The shape function gradient values are stored in the `basis_grady_at_quad_ref` array
        of the corresponding finite element cell. The `cell_index` parameter specifies the index of the cell
        for which the shape function gradient is required. If the `cell_index` is greater than the total number
        of cells, a `ValueError` is raised. The returned gradient values are copied from the `basis_grady_at_quad_ref` array to ensure immutability.
    """

get_shape_function_val(cell_index) abstractmethod

Get the actual values of the shape functions on a given cell.

Parameters:

Name Type Description Default
cell_index int

The index of the cell.

 required

Returns:

Type Description
ndarray

np.ndarray: An array containing the actual values of the shape functions.

Raises:

Type Description
ValueError

If the cell_index is greater than the number of cells.
Source code in scirex/core/sciml/fe/fespace.py 
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@abstractmethod
def get_shape_function_val(self, cell_index: int) -> np.ndarray:
    """
    Get the actual values of the shape functions on a given cell.

    Args:
        cell_index (int): The index of the cell.

    Returns:
        np.ndarray: An array containing the actual values of the shape functions.

    Raises:
        ValueError: If the cell_index is greater than the number of cells.
    """

set_finite_elements() abstractmethod

Assigns the finite elements to each cell.

This method initializes the finite element objects for each cell in the mesh. It creates an instance of the FE2D_Cell class for each cell, passing the necessary parameters. The finite element objects store information about the basis functions, gradients, Jacobians, quadrature points, weights, actual coordinates, and forcing functions associated with each cell.

After initializing the finite element objects, this method prints the shape details of various matrices and updates the total number of degrees of freedom (dofs) for the entire mesh.

Returns:

Type Description
None

None
Source code in scirex/core/sciml/fe/fespace.py 
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@abstractmethod
def set_finite_elements(self) -> None:
    """
    Assigns the finite elements to each cell.

    This method initializes the finite element objects for each cell in the mesh.
    It creates an instance of the `FE2D_Cell` class for each cell, passing the necessary parameters.
    The finite element objects store information about the basis functions, gradients, Jacobians,
    quadrature points, weights, actual coordinates, and forcing functions associated with each cell.

    After initializing the finite element objects, this method prints the shape details of various matrices
    and updates the total number of degrees of freedom (dofs) for the entire mesh.

    Args:
        None

    Returns:
        None
    """