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Function 3D

Module: basis_function_3d.py

This module provides the abstract base class for all 3D finite element basis functions. It defines the interface for computing basis functions and their derivatives in three-dimensional reference coordinates.

Classes:

Name Description
BasisFunction3D

Abstract base class for 3D finite element basis functions
Dependencies
  • abc: For abstract base class functionality
Key Features
  • Abstract interface for 3D basis function evaluation
  • Support for first and second order derivatives
  • Reference coordinate system (xi, eta, zeta) implementation
  • Unified interface for different polynomial bases
  • Systematic derivative computation in three dimensions
Authors

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

Version Info

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

References

None

BasisFunction3D

An abstract base class defining the interface for three-dimensional finite element basis functions.

This class serves as a template for implementing various types of 3D basis functions used in finite element computations. It defines the required methods for function evaluation and derivatives in three dimensions.

Attributes:

Name Type Description
num_shape_functions int

The number of shape functions in the 3D element. Must be specified during initialization.

Methods:

Name Description
value

Evaluates the basis function at given coordinates Args: xi (np.ndarray): First reference coordinate eta (np.ndarray): Second reference coordinate zeta (np.ndarray): Third reference coordinate Returns: float: Value of basis function at (xi, eta, zeta)
gradx

Computes derivative w.r.t. xi Args: xi (np.ndarray): First reference coordinate eta (np.ndarray): Second reference coordinate zeta (np.ndarray): Third reference coordinate Returns: float: Partial derivative w.r.t. xi
grady

Computes derivative w.r.t. eta Args: xi (np.ndarray): First reference coordinate eta (np.ndarray): Second reference coordinate zeta (np.ndarray): Third reference coordinate Returns: float: Partial derivative w.r.t. eta
gradxx

Computes second derivative w.r.t. xi Args: xi (np.ndarray): First reference coordinate eta (np.ndarray): Second reference coordinate zeta (np.ndarray): Third reference coordinate Returns: float: Second partial derivative w.r.t. xi
gradxy

Computes mixed derivative w.r.t. xi and eta Args: xi (np.ndarray): First reference coordinate eta (np.ndarray): Second reference coordinate zeta (np.ndarray): Third reference coordinate Returns: float: Mixed partial derivative w.r.t. xi and eta
gradyy

Computes second derivative w.r.t. eta Args: xi (np.ndarray): First reference coordinate eta (np.ndarray): Second reference coordinate zeta (np.ndarray): Third reference coordinate Returns: float: Second partial derivative w.r.t. eta
Notes
  • All coordinate inputs (xi, eta, zeta) should be in the reference element range
  • All methods are abstract and must be implemented by derived classes
  • Implementation should ensure proper handling of 3D tensor-product bases
Source code in scirex/core/sciml/fe/basis_function_3d.py 
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class BasisFunction3D:  # pragma: no cover
    """
    An abstract base class defining the interface for three-dimensional finite element basis functions.

    This class serves as a template for implementing various types of 3D basis functions
    used in finite element computations. It defines the required methods for function
    evaluation and derivatives in three dimensions.

    Attributes:
        num_shape_functions (int): The number of shape functions in the 3D element.
            Must be specified during initialization.

    Methods:
        value(xi, eta, zeta): Evaluates the basis function at given coordinates
            Args:
                xi (np.ndarray): First reference coordinate
                eta (np.ndarray): Second reference coordinate
                zeta (np.ndarray): Third reference coordinate
            Returns:
                float: Value of basis function at (xi, eta, zeta)

        gradx(xi, eta, zeta): Computes derivative w.r.t. xi
            Args:
                xi (np.ndarray): First reference coordinate
                eta (np.ndarray): Second reference coordinate
                zeta (np.ndarray): Third reference coordinate
            Returns:
                float: Partial derivative w.r.t. xi

        grady(xi, eta, zeta): Computes derivative w.r.t. eta
            Args:
                xi (np.ndarray): First reference coordinate
                eta (np.ndarray): Second reference coordinate
                zeta (np.ndarray): Third reference coordinate
            Returns:
                float: Partial derivative w.r.t. eta

        gradxx(xi, eta, zeta): Computes second derivative w.r.t. xi
            Args:
                xi (np.ndarray): First reference coordinate
                eta (np.ndarray): Second reference coordinate
                zeta (np.ndarray): Third reference coordinate
            Returns:
                float: Second partial derivative w.r.t. xi

        gradxy(xi, eta, zeta): Computes mixed derivative w.r.t. xi and eta
            Args:
                xi (np.ndarray): First reference coordinate
                eta (np.ndarray): Second reference coordinate
                zeta (np.ndarray): Third reference coordinate
            Returns:
                float: Mixed partial derivative w.r.t. xi and eta

        gradyy(xi, eta, zeta): Computes second derivative w.r.t. eta
            Args:
                xi (np.ndarray): First reference coordinate
                eta (np.ndarray): Second reference coordinate
                zeta (np.ndarray): Third reference coordinate
            Returns:
                float: Second partial derivative w.r.t. eta

    Notes:
        - All coordinate inputs (xi, eta, zeta) should be in the reference element range
        - All methods are abstract and must be implemented by derived classes
        - Implementation should ensure proper handling of 3D tensor-product bases
    """

    def __init__(self, num_shape_functions):
        self.num_shape_functions = num_shape_functions

    @abstractmethod
    def value(self, xi, eta, zeta):
        """
        Evaluates the basis function at the given xi and eta coordinates.

        Args:
            xi (np.ndarray): The xi coordinate.
            eta (np.ndarray): The eta coordinate.
            zeta (np.ndarray): The zeta coordinate.

        Returns:
            np.ndarray: The value of the basis function at (xi, eta, zeta).
        """

    @abstractmethod
    def gradx(self, xi, eta, zeta):
        """
        Computes the partial derivative of the basis function with respect to xi.

        Args:
            xi (np.ndarray): The xi coordinate.
            eta (np.ndarray): The eta coordinate.
            zeta (np.ndarray): The zeta coordinate.

        Returns:
            np.ndarray: The partial derivative of the basis function with respect to xi.
        """

    @abstractmethod
    def grady(self, xi, eta, zeta):
        """
        Computes the partial derivative of the basis function with respect to eta.

        Args:
            xi (np.ndarray): The xi coordinate.
            eta (np.ndarray): The eta coordinate.
            zeta (np.ndarray): The zeta coordinate.

        Returns:
            np.ndarray: The partial derivative of the basis function with respect to eta.
        """

    @abstractmethod
    def gradxx(self, xi, eta, zeta):
        """
        Computes the second partial derivative of the basis function with respect to xi.

        Args:
            xi (np.ndarray): The xi coordinate.
            eta (np.ndarray): The eta coordinate.
            zeta (np.ndarray): The zeta coordinate.

        Returns:
            np.ndarray: The second partial derivative of the basis function with respect to xi.
        """

    @abstractmethod
    def gradxy(self, xi, eta, zeta):
        """
        Computes the mixed partial derivative of the basis function with respect to xi and eta.

        Args:
            xi (np.ndarray): The xi coordinate.
            eta (np.ndarray): The eta coordinate.
            zeta (np.ndarray): The zeta coordinate.

        Returns:
            np.ndarray: The mixed partial derivative of the basis function with respect to xi and eta.
        """

    @abstractmethod
    def gradyy(self, xi, eta, zeta):
        """
        Computes the second partial derivative of the basis function with respect to eta.

        Args:
            xi (np.ndarray): The xi coordinate.
            eta (np.ndarray): The eta coordinate.
            zeta (np.ndarray): The zeta coordinate.

        Returns:
            np.ndarray: The second partial derivative of the basis function with respect to eta.
        """

gradx(xi, eta, zeta) abstractmethod

Computes the partial derivative of the basis function with respect to xi.

Parameters:

Name Type Description Default
xi ndarray

The xi coordinate.

 required
eta ndarray

The eta coordinate.

 required
zeta ndarray

The zeta coordinate.

 required

Returns:

Type Description

np.ndarray: The partial derivative of the basis function with respect to xi.
Source code in scirex/core/sciml/fe/basis_function_3d.py 
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@abstractmethod
def gradx(self, xi, eta, zeta):
    """
    Computes the partial derivative of the basis function with respect to xi.

    Args:
        xi (np.ndarray): The xi coordinate.
        eta (np.ndarray): The eta coordinate.
        zeta (np.ndarray): The zeta coordinate.

    Returns:
        np.ndarray: The partial derivative of the basis function with respect to xi.
    """

gradxx(xi, eta, zeta) abstractmethod

Computes the second partial derivative of the basis function with respect to xi.

Parameters:

Name Type Description Default
xi ndarray

The xi coordinate.

 required
eta ndarray

The eta coordinate.

 required
zeta ndarray

The zeta coordinate.

 required

Returns:

Type Description

np.ndarray: The second partial derivative of the basis function with respect to xi.
Source code in scirex/core/sciml/fe/basis_function_3d.py 
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@abstractmethod
def gradxx(self, xi, eta, zeta):
    """
    Computes the second partial derivative of the basis function with respect to xi.

    Args:
        xi (np.ndarray): The xi coordinate.
        eta (np.ndarray): The eta coordinate.
        zeta (np.ndarray): The zeta coordinate.

    Returns:
        np.ndarray: The second partial derivative of the basis function with respect to xi.
    """

gradxy(xi, eta, zeta) abstractmethod

Computes the mixed partial derivative of the basis function with respect to xi and eta.

Parameters:

Name Type Description Default
xi ndarray

The xi coordinate.

 required
eta ndarray

The eta coordinate.

 required
zeta ndarray

The zeta coordinate.

 required

Returns:

Type Description

np.ndarray: The mixed partial derivative of the basis function with respect to xi and eta.
Source code in scirex/core/sciml/fe/basis_function_3d.py 
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@abstractmethod
def gradxy(self, xi, eta, zeta):
    """
    Computes the mixed partial derivative of the basis function with respect to xi and eta.

    Args:
        xi (np.ndarray): The xi coordinate.
        eta (np.ndarray): The eta coordinate.
        zeta (np.ndarray): The zeta coordinate.

    Returns:
        np.ndarray: The mixed partial derivative of the basis function with respect to xi and eta.
    """

grady(xi, eta, zeta) abstractmethod

Computes the partial derivative of the basis function with respect to eta.

Parameters:

Name Type Description Default
xi ndarray

The xi coordinate.

 required
eta ndarray

The eta coordinate.

 required
zeta ndarray

The zeta coordinate.

 required

Returns:

Type Description

np.ndarray: The partial derivative of the basis function with respect to eta.
Source code in scirex/core/sciml/fe/basis_function_3d.py 
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@abstractmethod
def grady(self, xi, eta, zeta):
    """
    Computes the partial derivative of the basis function with respect to eta.

    Args:
        xi (np.ndarray): The xi coordinate.
        eta (np.ndarray): The eta coordinate.
        zeta (np.ndarray): The zeta coordinate.

    Returns:
        np.ndarray: The partial derivative of the basis function with respect to eta.
    """

gradyy(xi, eta, zeta) abstractmethod

Computes the second partial derivative of the basis function with respect to eta.

Parameters:

Name Type Description Default
xi ndarray

The xi coordinate.

 required
eta ndarray

The eta coordinate.

 required
zeta ndarray

The zeta coordinate.

 required

Returns:

Type Description

np.ndarray: The second partial derivative of the basis function with respect to eta.
Source code in scirex/core/sciml/fe/basis_function_3d.py 
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@abstractmethod
def gradyy(self, xi, eta, zeta):
    """
    Computes the second partial derivative of the basis function with respect to eta.

    Args:
        xi (np.ndarray): The xi coordinate.
        eta (np.ndarray): The eta coordinate.
        zeta (np.ndarray): The zeta coordinate.

    Returns:
        np.ndarray: The second partial derivative of the basis function with respect to eta.
    """

value(xi, eta, zeta) abstractmethod

Evaluates the basis function at the given xi and eta coordinates.

Parameters:

Name Type Description Default
xi ndarray

The xi coordinate.

 required
eta ndarray

The eta coordinate.

 required
zeta ndarray

The zeta coordinate.

 required

Returns:

Type Description

np.ndarray: The value of the basis function at (xi, eta, zeta).
Source code in scirex/core/sciml/fe/basis_function_3d.py 
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@abstractmethod
def value(self, xi, eta, zeta):
    """
    Evaluates the basis function at the given xi and eta coordinates.

    Args:
        xi (np.ndarray): The xi coordinate.
        eta (np.ndarray): The eta coordinate.
        zeta (np.ndarray): The zeta coordinate.

    Returns:
        np.ndarray: The value of the basis function at (xi, eta, zeta).
    """