green-anisotropic

Python wrapped C++ code for bottom-up dynamic computation of high order gradient components of 2D linear elastic Green functions in anistropic, orthotropic, R0-orthotropic, square symmetric and isotropic media.

Author: Nicolas Venkovic.

email: [email protected].

Supporting documents:

• Main supporting document (GreenAnisotropic2D.pdf),
• Description of the bottom-up dynamic implementation (Bottom-up_DP_algo.pdf).

Compiler and dependencies:

• gcc.
• Python 2.X.
• pybind11.
• NumPy, Matplotlib, SciPy.

Installation:

$ python setup.py build_ext -i

Tests:

• Global equilibrium of random bodies: Verifies that the integral of the traction field of a body bounded by a random curve is in equilibrium with a concentrated load applied at the origin. Verification done for 10 instances of each material symmetry and 10 realizations of random curves.
• Numerical differentiation of Green functions: Verifies that the computation of specific components of several gradients of Green's function matches pre-calculated values. Verification done for 5 material instances of each material symmetry with gradients up to the eighth order.

$ python tests.py

Usage:

An example script to start working with the module is given. Run the examples as follows:

$ python example.py

C++ content wrapped in Python:

• isym : Symmetry code
• P0, P1, R0, R1, T0, T1, K : Polar invariants
• kappa, mu : Isotropic elastic constants
• Th : Phase angle
• L1111, L1122, L1112 : Stiffness components
• L2211, L2222, L2212 : Stiffness components
• L1211, L1222, L1212 : Stiffness components
• L1121, L2221, L2111 : Stiffness components
• L2122, L2112, L1221 : Stiffness components
• L2121 : Stiffness component
• S : First complete Barnett-Lothe tensor integral
• H : Second complete Barnett-Lothe tensor integral
• set_sym() : Sets up material symmetry
• get_S() : Computes first complete Barnett-Lothe tensor integral
• get_H() : Computes second complete Barnett-Lothe tensor integral
• dnGi() : Computes Cartesian component with indices i of n-th gradient Green's function

C++ content not wrapped in Python:

• dkN1_anisotropic() : Computes k-th derivative of first Barnett-Lothe integrand of an anisotropic medium
• dkN2_anisotropic() : Computes k-th derivative of second Barnett-Lothe integrand of an anisotropic medium
• dkN1_orthotropic() : Computes k-th derivative of first Barnett-Lothe integrand of an orthotropic medium
• dkN2_orthotropic() : Computes k-th derivative of second Barnett-Lothe integrand of an R0-orthotropic medium
• dkN1_r0_orthotropic() : Computes k-th derivative of first Barnett-Lothe integrand of an R0-orthotropic medium
• dkN2_r0_orthotropic() : Computes k-th derivative of second Barnett-Lothe integrand of an R0-orthotropic medium
• dkN1_square_symmetric() : Computes k-th derivative of first Barnett-Lothe integrand of an square-symmetric medium
• dkN2_square_symmetric() : Computes k-th derivative of second Barnett-Lothe integrand of an R0-orthotropic medium
• dkN1_isotropic() : Computes k-th derivative of first Barnett-Lothe integrand of an isotropic medium
• dkN2_isotropic() : Computes k-th derivative of second Barnett-Lothe integrand of an isotropic medium
• dh() : Computes k-th derivative of angle dependent part of n-th gradient of Green's function
• h() : Computes angle dependent part of k-th gradient of Green's function
• dkh1i() : Computes k-th derivative of angle dependent part of 1-st gradient the Green's function
• dknvec() : Computes k-th derivative of Cartesian component i of unit vector nvec
• dkmvec() : Computes k-th derivative of Cartesian component i of unit vector mvec,
  i.e. active counter-clockwise Pi/2 rotation of nvec
• trapzd() : Performs numerical integration of Barnett-Lothe integrand by trapezoidal rule
• qtrap() : Handles the trapezoidal rule for targeted error, i.e. 10.^-10

Prior version history:

• 0.90 (May 31st, 2017),
• 0.91 (June 7th, 2017).