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Legendre_Special

Module: basis_2d_QN_Legendre_Special.py

This module implements a specialized basis function class for 2D Quad elements using Legendre polynomials. It provides functionality for computing basis functions and their derivatives in two dimensions, with a special formulation based on differences of consecutive Legendre polynomials.

Classes:

Name Description
Basis2DQNLegendreSpecial

Main class implementing 2D basis functions using special Legendre polynomials
Dependencies
  • numpy: For numerical computations
  • scipy.special: For Legendre polynomial calculations
  • matplotlib.pyplot: For visualization support
  • .basis_function_2d: For base class implementation
Key Features
  • Implementation of 2D Quad element basis functions using Legendre polynomials
  • Special formulation using differences of consecutive polynomials
  • Computation of function values and derivatives up to second order
  • Tensor product construction of 2D basis functions from 1D components
  • Support for variable number of shape functions
Authors

Thivin Anandh (http://thivinanandh.github.io/)

Version Info: 27/Dec/2024: Initial version: Thivin Anandh D References: None

Basis2DQNLegendreSpecial

Bases: BasisFunction2D

A specialized implementation of two-dimensional basis functions using Legendre polynomials for Q1 elements.

This class provides a complete implementation for computing basis functions and their derivatives in two dimensions. The basis functions are constructed using a special formulation based on differences of consecutive Legendre polynomials.

The class inherits from BasisFunction2D and implements all required methods for computing function values and derivatives up to second order.

Attributes:

Name Type Description
num_shape_functions int

Total number of shape functions in the 2D element. Must be a perfect square as it represents tensor product of 1D functions.

Methods:

Name Description
test_fcn

Computes test functions using Legendre polynomial differences
test_grad_fcn

Computes first derivatives of test functions
test_grad_grad_fcn

Computes second derivatives of test functions
value

Computes values of all basis functions
gradx

Computes x-derivatives of all basis functions
grady

Computes y-derivatives of all basis functions
gradxx

Computes second x-derivatives of all basis functions
gradyy

Computes second y-derivatives of all basis functions
gradxy

Computes mixed xy-derivatives of all basis functions
Implementation Details
  • Basis functions are constructed using differences of consecutive Legendre polynomials
  • Test functions are created using Pn+1(x) - Pn-1(x) where Pn is the nth Legendre polynomial
  • All computations maintain numerical precision using numpy arrays
  • Efficient vectorized operations for multiple point evaluations
  • Tensor product construction for 2D basis functions
Example 
basis = Basis2DQNLegendreSpecial(num_shape_functions=16)  # Creates 4x4 basis functions
xi = np.linspace(-1, 1, 100)
eta = np.linspace(-1, 1, 100)
values = basis.value(xi, eta)
x_derivatives = basis.gradx(xi, eta)
Source code in scirex/core/sciml/fe/basis_2d_qn_legendre_special.py 
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class Basis2DQNLegendreSpecial(BasisFunction2D):
    """
    A specialized implementation of two-dimensional basis functions using Legendre polynomials for Q1 elements.

    This class provides a complete implementation for computing basis functions and their derivatives
    in two dimensions. The basis functions are constructed using a special formulation based on
    differences of consecutive Legendre polynomials.

    The class inherits from BasisFunction2D and implements all required methods for computing
    function values and derivatives up to second order.

    Attributes:
        num_shape_functions (int): Total number of shape functions in the 2D element.
            Must be a perfect square as it represents tensor product of 1D functions.

    Methods:
        test_fcn(n_test, x): Computes test functions using Legendre polynomial differences
        test_grad_fcn(n_test, x): Computes first derivatives of test functions
        test_grad_grad_fcn(n_test, x): Computes second derivatives of test functions
        value(xi, eta): Computes values of all basis functions
        gradx(xi, eta): Computes x-derivatives of all basis functions
        grady(xi, eta): Computes y-derivatives of all basis functions
        gradxx(xi, eta): Computes second x-derivatives of all basis functions
        gradyy(xi, eta): Computes second y-derivatives of all basis functions
        gradxy(xi, eta): Computes mixed xy-derivatives of all basis functions

    Implementation Details:
        - Basis functions are constructed using differences of consecutive Legendre polynomials
        - Test functions are created using Pn+1(x) - Pn-1(x) where Pn is the nth Legendre polynomial
        - All computations maintain numerical precision using numpy arrays
        - Efficient vectorized operations for multiple point evaluations
        - Tensor product construction for 2D basis functions

    Example:
        ```python
        basis = Basis2DQNLegendreSpecial(num_shape_functions=16)  # Creates 4x4 basis functions
        xi = np.linspace(-1, 1, 100)
        eta = np.linspace(-1, 1, 100)
        values = basis.value(xi, eta)
        x_derivatives = basis.gradx(xi, eta)
        ```
    """

    def __init__(self, num_shape_functions: int):
        super().__init__(num_shape_functions)

    def test_fcn(self, n_test: int, x: np.ndarray) -> np.ndarray:
        """
        Calculate the test function values for a given number of tests and input values.

        Args:
            n_test (int): The number of test functions to calculate.
            x (np.ndarray): The input values at which to evaluate the test functions.

        Returns:
            np.ndarray: An array containing the results of the test functions at the given input values.
        """
        test_total = []
        for n in range(1, n_test + 1):
            obj1 = legendre(n + 1)
            obj2 = legendre(n - 1)
            test = obj1(x) - obj2(x)
            test_total.append(test)
        return np.asarray(test_total)

    def test_grad_fcn(self, n_test: int, x: np.ndarray) -> np.ndarray:
        """
        Calculate the gradient of the test function at a given point.

        Args:
            n_test (int): The number of test cases to evaluate.
            x (np.ndarray): The input value at which to evaluate the function.

        Returns:
            np.ndarray: An array containing the results of the test cases.
        """
        test_total = []
        for n in range(1, n_test + 1):
            obj1 = legendre(n + 1).deriv()
            obj2 = legendre(n - 1).deriv()
            test = obj1(x) - obj2(x)
            test_total.append(test)
        return np.asarray(test_total)

    def test_grad_grad_fcn(self, n_test: int, x: np.ndarray) -> np.ndarray:
        """
        Calculate the gradient of the second derivative of a function using Legendre polynomials.

        Args:
            n_test (int): The number of test cases to evaluate.
            x (np.ndarray): The input value at which to evaluate the function.

        Returns:
            np.ndarray: An array containing the results of the test cases.
        """
        test_total = []
        for n in range(1, n_test + 1):
            obj1 = legendre(n + 1).deriv(2)
            obj2 = legendre(n - 1).deriv(2)
            test = obj1(x) - obj2(x)

            test_total.append(test)
        return np.asarray(test_total)

    def value(self, xi: np.ndarray, eta: np.ndarray) -> np.ndarray:
        """
        This method returns the values of the basis functions at the given (xi, eta) coordinates.

        Args:
            xi (np.ndarray): The xi coordinates.
            eta (np.ndarray): The eta coordinates.

        Returns:
            np.ndarray: The values of the basis functions.
        """
        values = np.zeros((self.num_shape_functions, len(xi)))

        num_shape_func_in_1d = int(np.sqrt(self.num_shape_functions))

        test_function_x = self.test_fcn(num_shape_func_in_1d, xi)
        test_function_y = self.test_fcn(num_shape_func_in_1d, eta)

        # Generate an outer product of the test functions to generate the basis functions
        for i in range(num_shape_func_in_1d):
            values[i * num_shape_func_in_1d : (i + 1) * num_shape_func_in_1d, :] = (
                test_function_x[i, :] * test_function_y
            )

        return values

    def gradx(self, xi: np.ndarray, eta: np.ndarray) -> np.ndarray:
        """
        This method returns the x-derivatives of the basis functions at the given (xi, eta) coordinates.

        Args:
            xi (np.ndarray): The xi coordinates.
            eta (np.ndarray): The eta coordinates.

        Returns:
            np.ndarray: The x-derivatives of the basis functions.
        """
        values = np.zeros((self.num_shape_functions, len(xi)))

        num_shape_func_in_1d = int(np.sqrt(self.num_shape_functions))

        test_function_grad_x = self.test_grad_fcn(num_shape_func_in_1d, xi)
        test_function_y = self.test_fcn(num_shape_func_in_1d, eta)

        # Generate an outer product of the test functions to generate the basis functions
        for i in range(num_shape_func_in_1d):
            values[i * num_shape_func_in_1d : (i + 1) * num_shape_func_in_1d, :] = (
                test_function_grad_x[i, :] * test_function_y
            )

        return values

    def grady(self, xi: np.ndarray, eta: np.ndarray) -> np.ndarray:
        """
        This method returns the y-derivatives of the basis functions at the given (xi, eta) coordinates.

        Args:
            xi (np.ndarray): The xi coordinates.
            eta (np.ndarray): The eta coordinates.

        Returns:
            np.ndarray: The y-derivatives of the basis functions.
        """
        values = np.zeros((self.num_shape_functions, len(xi)))

        num_shape_func_in_1d = int(np.sqrt(self.num_shape_functions))

        test_function_x = self.test_fcn(num_shape_func_in_1d, xi)
        test_function_grad_y = self.test_grad_fcn(num_shape_func_in_1d, eta)

        # Generate an outer product of the test functions to generate the basis functions
        for i in range(num_shape_func_in_1d):
            values[i * num_shape_func_in_1d : (i + 1) * num_shape_func_in_1d, :] = (
                test_function_x[i, :] * test_function_grad_y
            )

        return values

    def gradxx(self, xi: np.ndarray, eta: np.ndarray) -> np.ndarray:
        """
        This method returns the xx-derivatives of the basis functions at the given (xi, eta) coordinates.

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

        Returns:
            np.ndarray: The xx-derivatives of the basis functions.
        """
        values = np.zeros((self.num_shape_functions, len(xi)))

        num_shape_func_in_1d = int(np.sqrt(self.num_shape_functions))

        test_function_grad_grad_x = self.test_grad_grad_fcn(num_shape_func_in_1d, xi)
        test_function_y = self.test_fcn(num_shape_func_in_1d, eta)

        # Generate an outer product of the test functions to generate the basis functions
        for i in range(num_shape_func_in_1d):
            values[i * num_shape_func_in_1d : (i + 1) * num_shape_func_in_1d, :] = (
                test_function_grad_grad_x[i, :] * test_function_y
            )

        return values

    def gradxy(self, xi: np.ndarray, eta: np.ndarray) -> np.ndarray:
        """
        This method returns the xy-derivatives of the basis functions at the given (xi, eta) coordinates.

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

        Returns:
            np.ndarray: The xy-derivatives of the basis functions.
        """
        values = np.zeros((self.num_shape_functions, len(xi)))

        num_shape_func_in_1d = int(np.sqrt(self.num_shape_functions))

        test_function_grad_x = self.test_grad_fcn(num_shape_func_in_1d, xi)
        test_function_grad_y = self.test_grad_fcn(num_shape_func_in_1d, eta)

        # Generate an outer product of the test functions to generate the basis functions
        for i in range(num_shape_func_in_1d):
            values[i * num_shape_func_in_1d : (i + 1) * num_shape_func_in_1d, :] = (
                test_function_grad_x[i, :] * test_function_grad_y
            )

        return values

    def gradyy(self, xi: np.ndarray, eta: np.ndarray) -> np.ndarray:
        """
        This method returns the yy-derivatives of the basis functions at the given (xi, eta) coordinates.

        Args:
            xi (np.ndarray): The xi coordinates.
            eta (np.ndarray): The eta coordinates.

        Returns:
            np.ndarray: The yy-derivatives of the basis functions.
        """
        values = np.zeros((self.num_shape_functions, len(xi)))

        num_shape_func_in_1d = int(np.sqrt(self.num_shape_functions))

        test_function_x = self.test_fcn(num_shape_func_in_1d, xi)
        test_function_grad_grad_y = self.test_grad_grad_fcn(num_shape_func_in_1d, eta)

        # Generate an outer product of the test functions to generate the basis functions
        for i in range(num_shape_func_in_1d):
            values[i * num_shape_func_in_1d : (i + 1) * num_shape_func_in_1d, :] = (
                test_function_x[i, :] * test_function_grad_grad_y
            )

        return values

gradx(xi, eta)

This method returns the x-derivatives of the basis functions at the given (xi, eta) coordinates.

Parameters:

Name Type Description Default
xi ndarray

The xi coordinates.

 required
eta ndarray

The eta coordinates.

 required

Returns:

Type Description
ndarray

np.ndarray: The x-derivatives of the basis functions.
Source code in scirex/core/sciml/fe/basis_2d_qn_legendre_special.py 
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def gradx(self, xi: np.ndarray, eta: np.ndarray) -> np.ndarray:
    """
    This method returns the x-derivatives of the basis functions at the given (xi, eta) coordinates.

    Args:
        xi (np.ndarray): The xi coordinates.
        eta (np.ndarray): The eta coordinates.

    Returns:
        np.ndarray: The x-derivatives of the basis functions.
    """
    values = np.zeros((self.num_shape_functions, len(xi)))

    num_shape_func_in_1d = int(np.sqrt(self.num_shape_functions))

    test_function_grad_x = self.test_grad_fcn(num_shape_func_in_1d, xi)
    test_function_y = self.test_fcn(num_shape_func_in_1d, eta)

    # Generate an outer product of the test functions to generate the basis functions
    for i in range(num_shape_func_in_1d):
        values[i * num_shape_func_in_1d : (i + 1) * num_shape_func_in_1d, :] = (
            test_function_grad_x[i, :] * test_function_y
        )

    return values

gradxx(xi, eta)

This method returns the xx-derivatives of the basis functions at the given (xi, eta) coordinates.

Parameters:

Name Type Description Default
xi ndarray

The xi coordinate.

 required
eta ndarray

The eta coordinate.

 required

Returns:

Type Description
ndarray

np.ndarray: The xx-derivatives of the basis functions.
Source code in scirex/core/sciml/fe/basis_2d_qn_legendre_special.py 
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def gradxx(self, xi: np.ndarray, eta: np.ndarray) -> np.ndarray:
    """
    This method returns the xx-derivatives of the basis functions at the given (xi, eta) coordinates.

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

    Returns:
        np.ndarray: The xx-derivatives of the basis functions.
    """
    values = np.zeros((self.num_shape_functions, len(xi)))

    num_shape_func_in_1d = int(np.sqrt(self.num_shape_functions))

    test_function_grad_grad_x = self.test_grad_grad_fcn(num_shape_func_in_1d, xi)
    test_function_y = self.test_fcn(num_shape_func_in_1d, eta)

    # Generate an outer product of the test functions to generate the basis functions
    for i in range(num_shape_func_in_1d):
        values[i * num_shape_func_in_1d : (i + 1) * num_shape_func_in_1d, :] = (
            test_function_grad_grad_x[i, :] * test_function_y
        )

    return values

gradxy(xi, eta)

This method returns the xy-derivatives of the basis functions at the given (xi, eta) coordinates.

Parameters:

Name Type Description Default
xi ndarray

The xi coordinate.

 required
eta ndarray

The eta coordinate.

 required

Returns:

Type Description
ndarray

np.ndarray: The xy-derivatives of the basis functions.
Source code in scirex/core/sciml/fe/basis_2d_qn_legendre_special.py 
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def gradxy(self, xi: np.ndarray, eta: np.ndarray) -> np.ndarray:
    """
    This method returns the xy-derivatives of the basis functions at the given (xi, eta) coordinates.

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

    Returns:
        np.ndarray: The xy-derivatives of the basis functions.
    """
    values = np.zeros((self.num_shape_functions, len(xi)))

    num_shape_func_in_1d = int(np.sqrt(self.num_shape_functions))

    test_function_grad_x = self.test_grad_fcn(num_shape_func_in_1d, xi)
    test_function_grad_y = self.test_grad_fcn(num_shape_func_in_1d, eta)

    # Generate an outer product of the test functions to generate the basis functions
    for i in range(num_shape_func_in_1d):
        values[i * num_shape_func_in_1d : (i + 1) * num_shape_func_in_1d, :] = (
            test_function_grad_x[i, :] * test_function_grad_y
        )

    return values

grady(xi, eta)

This method returns the y-derivatives of the basis functions at the given (xi, eta) coordinates.

Parameters:

Name Type Description Default
xi ndarray

The xi coordinates.

 required
eta ndarray

The eta coordinates.

 required

Returns:

Type Description
ndarray

np.ndarray: The y-derivatives of the basis functions.
Source code in scirex/core/sciml/fe/basis_2d_qn_legendre_special.py 
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def grady(self, xi: np.ndarray, eta: np.ndarray) -> np.ndarray:
    """
    This method returns the y-derivatives of the basis functions at the given (xi, eta) coordinates.

    Args:
        xi (np.ndarray): The xi coordinates.
        eta (np.ndarray): The eta coordinates.

    Returns:
        np.ndarray: The y-derivatives of the basis functions.
    """
    values = np.zeros((self.num_shape_functions, len(xi)))

    num_shape_func_in_1d = int(np.sqrt(self.num_shape_functions))

    test_function_x = self.test_fcn(num_shape_func_in_1d, xi)
    test_function_grad_y = self.test_grad_fcn(num_shape_func_in_1d, eta)

    # Generate an outer product of the test functions to generate the basis functions
    for i in range(num_shape_func_in_1d):
        values[i * num_shape_func_in_1d : (i + 1) * num_shape_func_in_1d, :] = (
            test_function_x[i, :] * test_function_grad_y
        )

    return values

gradyy(xi, eta)

This method returns the yy-derivatives of the basis functions at the given (xi, eta) coordinates.

Parameters:

Name Type Description Default
xi ndarray

The xi coordinates.

 required
eta ndarray

The eta coordinates.

 required

Returns:

Type Description
ndarray

np.ndarray: The yy-derivatives of the basis functions.
Source code in scirex/core/sciml/fe/basis_2d_qn_legendre_special.py 
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def gradyy(self, xi: np.ndarray, eta: np.ndarray) -> np.ndarray:
    """
    This method returns the yy-derivatives of the basis functions at the given (xi, eta) coordinates.

    Args:
        xi (np.ndarray): The xi coordinates.
        eta (np.ndarray): The eta coordinates.

    Returns:
        np.ndarray: The yy-derivatives of the basis functions.
    """
    values = np.zeros((self.num_shape_functions, len(xi)))

    num_shape_func_in_1d = int(np.sqrt(self.num_shape_functions))

    test_function_x = self.test_fcn(num_shape_func_in_1d, xi)
    test_function_grad_grad_y = self.test_grad_grad_fcn(num_shape_func_in_1d, eta)

    # Generate an outer product of the test functions to generate the basis functions
    for i in range(num_shape_func_in_1d):
        values[i * num_shape_func_in_1d : (i + 1) * num_shape_func_in_1d, :] = (
            test_function_x[i, :] * test_function_grad_grad_y
        )

    return values

test_fcn(n_test, x)

Calculate the test function values for a given number of tests and input values.

Parameters:

Name Type Description Default
n_test int

The number of test functions to calculate.

 required
x ndarray

The input values at which to evaluate the test functions.

 required

Returns:

Type Description
ndarray

np.ndarray: An array containing the results of the test functions at the given input values.
Source code in scirex/core/sciml/fe/basis_2d_qn_legendre_special.py 
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def test_fcn(self, n_test: int, x: np.ndarray) -> np.ndarray:
    """
    Calculate the test function values for a given number of tests and input values.

    Args:
        n_test (int): The number of test functions to calculate.
        x (np.ndarray): The input values at which to evaluate the test functions.

    Returns:
        np.ndarray: An array containing the results of the test functions at the given input values.
    """
    test_total = []
    for n in range(1, n_test + 1):
        obj1 = legendre(n + 1)
        obj2 = legendre(n - 1)
        test = obj1(x) - obj2(x)
        test_total.append(test)
    return np.asarray(test_total)

test_grad_fcn(n_test, x)

Calculate the gradient of the test function at a given point.

Parameters:

Name Type Description Default
n_test int

The number of test cases to evaluate.

 required
x ndarray

The input value at which to evaluate the function.

 required

Returns:

Type Description
ndarray

np.ndarray: An array containing the results of the test cases.
Source code in scirex/core/sciml/fe/basis_2d_qn_legendre_special.py 
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def test_grad_fcn(self, n_test: int, x: np.ndarray) -> np.ndarray:
    """
    Calculate the gradient of the test function at a given point.

    Args:
        n_test (int): The number of test cases to evaluate.
        x (np.ndarray): The input value at which to evaluate the function.

    Returns:
        np.ndarray: An array containing the results of the test cases.
    """
    test_total = []
    for n in range(1, n_test + 1):
        obj1 = legendre(n + 1).deriv()
        obj2 = legendre(n - 1).deriv()
        test = obj1(x) - obj2(x)
        test_total.append(test)
    return np.asarray(test_total)

test_grad_grad_fcn(n_test, x)

Calculate the gradient of the second derivative of a function using Legendre polynomials.

Parameters:

Name Type Description Default
n_test int

The number of test cases to evaluate.

 required
x ndarray

The input value at which to evaluate the function.

 required

Returns:

Type Description
ndarray

np.ndarray: An array containing the results of the test cases.
Source code in scirex/core/sciml/fe/basis_2d_qn_legendre_special.py 
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def test_grad_grad_fcn(self, n_test: int, x: np.ndarray) -> np.ndarray:
    """
    Calculate the gradient of the second derivative of a function using Legendre polynomials.

    Args:
        n_test (int): The number of test cases to evaluate.
        x (np.ndarray): The input value at which to evaluate the function.

    Returns:
        np.ndarray: An array containing the results of the test cases.
    """
    test_total = []
    for n in range(1, n_test + 1):
        obj1 = legendre(n + 1).deriv(2)
        obj2 = legendre(n - 1).deriv(2)
        test = obj1(x) - obj2(x)

        test_total.append(test)
    return np.asarray(test_total)

value(xi, eta)

This method returns the values of the basis functions at the given (xi, eta) coordinates.

Parameters:

Name Type Description Default
xi ndarray

The xi coordinates.

 required
eta ndarray

The eta coordinates.

 required

Returns:

Type Description
ndarray

np.ndarray: The values of the basis functions.
Source code in scirex/core/sciml/fe/basis_2d_qn_legendre_special.py 
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def value(self, xi: np.ndarray, eta: np.ndarray) -> np.ndarray:
    """
    This method returns the values of the basis functions at the given (xi, eta) coordinates.

    Args:
        xi (np.ndarray): The xi coordinates.
        eta (np.ndarray): The eta coordinates.

    Returns:
        np.ndarray: The values of the basis functions.
    """
    values = np.zeros((self.num_shape_functions, len(xi)))

    num_shape_func_in_1d = int(np.sqrt(self.num_shape_functions))

    test_function_x = self.test_fcn(num_shape_func_in_1d, xi)
    test_function_y = self.test_fcn(num_shape_func_in_1d, eta)

    # Generate an outer product of the test functions to generate the basis functions
    for i in range(num_shape_func_in_1d):
        values[i * num_shape_func_in_1d : (i + 1) * num_shape_func_in_1d, :] = (
            test_function_x[i, :] * test_function_y
        )

    return values