Dynamics module

The dynamics module contains functionality for dynamical analyses. You can use the functionality by importing the dynamics module:

from haskoning_atr_tools.dynamics import *

Wave speed

The function calculate_wave_speed() is available to calculate the speed of a wave in a solid material. The speed of a wave is fixed by the type of wave and the physical properties of the medium in which it travels. For transverse waves in a solid (earthquake S-waves) the speed, in [m/s], is determined by:

print(calculate_wave_speed(E=210e9, nu=0.3, density=7850))
>>> 5421.932703208199

The formula used for the calculation is:

\[v = \sqrt{\frac{E}{\rho (1 - \nu^2)}}\]

where:

\(v\) is the wave speed [m/s],

\(E\) is the Young’s modulus [N/m²],

\(\rho\) is the density [kg/m³],

\(\nu\) is Poisson’s ratio [-]

Wavelength

The function calculate_wavelength() is available to calculate the length of the wave. There is a relationship between the period, wavelength and speed of the wave. The period of a cork floating in the water is affected by how fast the wave passes (wave speed) and the distance between peaks (wavelength). The relationship between speed, period and wavelength of a sine wave is given by \(v = \lambda/T\) where wavelength [m] and period for a sine wave were defined previously. This can also be written as \(v = \lambda f\) since frequency [Hz] is the inverse of period and is true for all linear waves.

Hence, the wavelength, in [m], is given by:

print(calculate_wavelength(E=210E+9, nu=0.3, density=7800, frequency=50))
>>> 108.78565864408424

The formula used in the calculation is:

\[\lambda = \frac{v}{f}\]

where:

\(\lambda\) is the wavelength [m],

\(v\) is the wave speed [m/s],

\(f\) is the frequency [Hz].

Maximum mesh size

In Finite Element Analysis (FEA), the maximum mesh size (h) depends on the specific requirements of your analysis and the level of accuracy you need. Generally, the mesh size should be small enough to capture the critical details of the geometry and the expected stress or strain gradients.

The formula used in the calculation of calculate_maximum_mesh_element_size() is:

\[h = \lambda/n\]

where:

\(n\) is the minimum number of divisions.

The minimal number of divisions per wavelength is a crucial factor in determining the mesh element size for accurate simulations. This number depends on several factors:

geometry_complexity (float): A factor representing the complexity of the geometry. This can be quantified by analysing the number of features, edges, and faces in the geometry. More complex geometries with intricate details will have higher values. LINK
material_property (float): A factor representing the material properties. This factor can be derived from the material’s mechanical properties such as Young’s modulus, Poisson’s ratio, and density. These properties are often available in material property databases or engineering handbooks. LINK
physics_factor (float): A factor representing the physics of the problem. This depends on the type of physical phenomena being modeled. For example, in structural analysis, this could relate to the type of loading conditions (static, dynamic, thermal). The more complex the physical interactions, the higher the value.
accuracy_requirement (float): A factor representing the accuracy requirements. This is determined by the precision needed for the analysis. Higher accuracy requirements will result in higher values. LINK
computational_resource (float): A factor representing the available computational resources. This factor is based on the available computational power, such as the number of processors, memory, and computational time. However, a higher number of elements used will increase the demand for computational resources such as memory and processor time. LINK

Hence, the minimum number of divisions can be calculated considering all those factors in calculate_minimum_divisions():

print(calculate_minimum_divisions(
    geometry_complexity=2.0, material_property=2.0, physics_factor=2.0, accuracy_requirement=2.0,
    computational_resource=2.0))
>>> 8

But a common guideline is to have the minimum division number equal to 8. Example of a calculation of maximum mesh size for steel with Young’s modulus of 210e9 N/m², Poisson’s ratio nu of 0.3 and density of 7800 kg/m³. The frequency of the wave of 50 Hz and minimum divisions taken as default (8 divisions). The maximum mesh element size, in [m], is:

print(calculate_maximum_mesh_element_size(E=210e9, nu=0.3, density=7800, frequency=50))
>>> 13.59820733051053

Reference

This is the documentation of the functions in the dynamics module.

haskoning_atr_tools.dynamics.generic_formulas.calculate_maximum_mesh_element_size(E: float, nu: float, density: float, frequency: float, minimum_divisions: int = 8) → float[source]

Calculate the maximum mesh element size based on material properties, frequency, and minimal number of divisions.

Input:

E (float): Young’s modulus of the material, in [N/m2].
nu (float): Poisson’s ratio of the material, in [-].
density (float): Density of the material, in [kg/m3].
frequency (float): Frequency of the wave, in [Hz].
minimum_divisions (int): Minimal number of divisions per wavelength (no units). Default value is 8. For more detailed analysis use the calculate_minimum_divisions function.

Output:

Returns the maximum mesh-element size as float, in [m].

haskoning_atr_tools.dynamics.generic_formulas.calculate_minimum_divisions(geometry_complexity: float, material_property: float, physics_factor: float, accuracy_requirement: float, computational_resource: float) → int[source]

Helper function to calculate the minimum number of divisions based on various factors, which can be used for the max_mesh_element_size function. Factors should be larger than zero.

Input:

geometry_complexity (float): A factor representing the complexity of the geometry.
material_property (float): A factor representing the material properties.
physics_factor (float): A factor representing the physics of the problem.
accuracy_requirement (float): A factor representing the accuracy requirements.
computational_resource (float): A factor representing the available computational resources.

Output:

Returns the calculated minimum number of divisions as integer.

haskoning_atr_tools.dynamics.generic_formulas.calculate_wave_speed(E: float, nu: float, density: float) → float[source]

Calculate the wave speed in a material using dynamics. [1]

Inout:

E (float): Young’s modulus of the material, in [N/m2].
nu (float): Poisson’s ratio of the material, in [-].
density (float): Density of the material, in [kg/m3].

Output:

Returns the calculated wave speed as float, in [m/s].

haskoning_atr_tools.dynamics.generic_formulas.calculate_wavelength(E: float, nu: float, density: float, frequency: float) → float[source]

Calculate the length speed based on Young’s modulus (E), Poisson’s ratio (nu), density, and frequency.

Input:

E (float): Young’s modulus of the material, in [N/m2].
nu (float): Poisson’s ratio of the material, in [-].
density (float): Density of the material, in [kg/m3].
frequency (float): Frequency of the wave, in [Hz].

Output:

Returns the calculated wavelength as float, in [m].