Nondimensionalization#

Nondimensionalization permits solving a system equations across a vast range of scales. We are primarily interested in elasticity-related boundary value problems, so we define displacement, length, rigidity, time, and temperature scales.

Scales#

The Python General object is the generic interfaces to the underlying C++ Scales object that manages the scales used to nondimensionalize values.

Tip

The default nondimensionalization is reasonable for many problems; however, it may be necessary to change the default values in some cases.

QuasistaticElasticity#

QuasistaticElasticity provides convenient scales for nondimensionalizing quasistatic elasticity and incompressible elsticity boundary values problems. The scales are specified in terms of the displacement scale, length scale, shear modulus, and time scale.

displacement_scale: Nominal size of the displacment
length_scale: Nominal size of the geometry controlling the deformation (for example, fault dimension or size of the domain)
shear_modulus: Nominal shear modulus
time_scale: Time scale of deformation (for example, viscoelastic relaxation time)

Table 18 Nondimensional scales for quasistatic elasticity boundary value problems.#
Scale	Name	Default Value
length	displacement_scale	1 m
length	length_scale	100 km
pressure	shear_modulus	10 GPa
time	time_scale	100 years

Pyre User Interface

See QuasistaticElasticity component.

DynamicElasticity#

DynamicElasticity provides convenient scales for nondimensionalizing dynamic elasticity boundary values problems. The scales are specified in terms of the displacement scale, length scale, shear modulus, and shear wave speed. The time scale is computed based on the time it takes a shear wave (\(v_s\))to propagate the length scale (\(l_o\)).

\[t_o = l_o / v_s\]

displacement_scale: Nominal size of the displacment
length_scale: Nominal size of the geometry controlling the deformation (for example, fault dimension or size of the domain)
shear_modulus: Nominal shear modulus
shear_wave_speed: Nominal shear wave speed

Table 19 Nondimensional scales for dynamic elasticity boundary value problems.#
Scale	Name	Default Value
length	displacement_scale	1 m
length	length_scale	100 km
pressure	shear_modulus	10 GPa
–	shear_wave_speed	3 km/s

Pyre User Interface

See DynamicElasticity component.

QuasistaticPoroelasticity#

QuasistaticPoroelasticity provides convenient scales for nondimensionalizing quasistatic poroelasticity boundary values problems. The scales are specified in terms of the displacement scale, length scale, shear modulus, viscosity, and permeability. The time scale is computed based on the length scale (\(l_o\)), rigidity (\(\mu_o\)), viscosity (\(\mu_{f_o}\)), and permability (\(k_o\)).

\[t_o = \frac{\mu_{f_o}}{k_o} \frac{l_o^2}{\mu_o}\]

displacement_scale: Nominal size of the displacment
length_scale: Nominal size of the geometry controlling the deformation (for example, fault dimension or size of the domain)
shear_modulus: Nominal shear modulus
viscosity: Nominal viscosity
permeability: Nominal permeability

Table 20 Nondimensional scales for quasistatic poroelasticity boundary value problems.#
Scale	Name	Default Value
length	displacement_scale	1 m
length	length_scale	100 km
pressure	shear_modulus	10 GPa
–	viscosity	0.001 Pa-s
–	permeability	1.0e-13 m\(^2\)

Pyre User Interface