4.1.1.9.1.2. pyfem.util.BezierShapeFunctions module

Bezier shape functions for PyFEM.

This module implements Bezier-based shape functions for finite element analysis, providing higher-order geometric representations beyond standard polynomial shape functions. Bezier elements are useful for modeling curved geometries and achieving geometric exactness in isogeometric analysis.

Module contents:

  • getBezierLine4: Compute Bezier shape functions for 4-node cubic Bezier curve

  • calcWeight: Compute integration weight from Jacobian

  • getElemBezierData: Generate shape data for Bezier elements with integration

References

Isogeometric analysis and related computational mechanics applications.

getBezierLine4(xi: float | floating, C: ndarray) → shapeData[source]

Compute shape functions and derivatives for a cubic Bezier line element.

Evaluates the shape functions and their derivatives with respect to the parametric coordinate xi for a 4-node cubic Bezier curve. The physical coordinates are computed using the control points in matrix C.

Parameters:

  • xi (float) – Parametric coordinate in the range [-1, 1].

  • C (np.ndarray) – Control points matrix of shape (nDim, 4), where each column is a control point in physical space. nDim is typically 2 or 3.

Returns:

Object containing:

  • h: Physical coordinates at parameter xi (shape: (nDim,))

  • dhdxi: Derivative of physical coordinates w.r.t. xi (shape: (nDim, 1))

  • xi: The input parametric coordinate

Return type:

shapeData

Raises:

NotImplementedError – If xi is a 1D array or C does not have exactly 4 columns.

Notes

The cubic Bezier basis functions are: - B[0] = -0.125*(xi-1)^3 - B[1] = 0.375*(xi-1)^2*(xi+1) - B[2] = -0.375*(xi-1)*(xi+1)^2 - B[3] = 0.125*(xi+1)^3

Examples

>>> import numpy as np
>>> xi = 0.0  # Evaluate at parametric center
>>> C = np.array([[0, 1, 2, 3], [0, 1, 1, 0]])  # 2D control points
>>> sData = getBezierLine4(xi, C)
>>> sData.h.shape
(2,)
calcWeight(jac: ndarray) → floating | float[source]

Calculate integration weight from Jacobian matrix.

Computes the determinant of the Jacobian for volume integration (square matrix) or the norm for line integration (rectangular matrix), which are used to weight integration points in numerical quadrature.

Parameters:

jac (np.ndarray) – Jacobian matrix. Can be: - Square matrix (nDim, nDim) for volume integration: returns determinant - Rectangular matrix (1, 2) for 1D curve in 2D space: returns curve length

Returns:

The integration weight. For square Jacobians, returns

the determinant. For rectangular matrices, returns the norm (magnitude) of the Jacobian.

Return type:

np.floating | float

Notes

The weight is typically multiplied by integration point weights from quadrature rules to compute integrals over curved elements.

Examples

>>> import numpy as np
>>> # 2D Jacobian (square)
>>> jac_2d = np.array([[1.0, 0.0], [0.0, 1.0]])
>>> w = calcWeight(jac_2d)
>>> w
1.0

>>> # 1D curve in 2D space
>>> jac_1d = np.array([[1.0, 1.0]])
>>> w = calcWeight(jac_1d)
>>> np.isclose(w, np.sqrt(2.0))
True
getElemBezierData(elemCoords: ndarray, C: ndarray, order: int = 4, method: str = 'Gauss', elemType: str = 'default') → elemShapeData[source]

Generate shape data and integration information for Bezier elements.

Evaluates shape functions and their spatial derivatives at integration points for a Bezier element. This function is used to assemble element stiffness matrices and load vectors in isogeometric analysis.

Parameters:

  • elemCoords (np.ndarray) – Physical coordinates of element nodes, shape (nDim, nNodes). These are the physical nodes in the background mesh.

  • C (np.ndarray) – Control points matrix, shape (nDim, nControlPts). Defines the geometric mapping via Bezier basis functions.

  • order (int, optional) – Integration order (number of integration points). Defaults to 4. Higher orders give more accurate integration.

  • method (str, optional) – Quadrature rule. Defaults to “Gauss” for Gauss-Legendre quadrature.

  • elemType (str, optional) – Element type identifier (e.g., ‘Line4’, ‘Line3’). If ‘default’, automatically detected from elemCoords shape. Defaults to ‘default’.

Returns:

Container with integration point data. Attributes:

  • sData: List of shapeData objects, one per integration point, each containing: - h: Physical coordinates at integration point (shape: (nDim,)) - dhdxi: Derivatives w.r.t. parametric coord (shape: (nDim, 1)) - dhdx: Derivatives w.r.t. physical coords (shape: (nDim, nDim)) - weight: Integration weight including Jacobian determinant - xi: Parametric coordinate of integration point

Return type:

elemShapeData

Raises:

NotImplementedError – If elemType is unknown or not supported.

Notes

Integration points are obtained from Line3 element quadrature rules. The Jacobian is computed from the physical derivatives of shape functions. The spatial derivatives (dhdx) are computed only for square Jacobians.

Examples

>>> import numpy as np
>>> # Control points for a curved quadratic line
>>> C = np.array([[0, 0.5, 1.0], [0, 0.1, 0.0]])
>>> elemCoords = np.array([[0, 1.0], [0, 0.0]])
>>> elemData = getElemBezierData(elemCoords, C, order=2, elemType='Line2')
>>> len(elemData.sData)  # Number of integration points
2
>>> elemData.sData[0].weight > 0  # Positive weight
True