Geometry

This module contains geometry related functions that can be used for seeding nodes and integration points, impose boundary conditions and to select nodes and elements. The main functionality relies on distance functions. For the construction of user-specific distance functions, there are the Rvachev operations, but there are also some pre-made signed distance functions and positive smooth distance functions. The distance functions are suitable, for example, for verifying whether nodes lie within certain regions or on specified surfaces. Additionally, the smooth distance functions can be used for constructing solution structures.

Rvachev function operations

r_equivalence(o1, o2)	Computes the logical NOT XOR operation by combining two positively evaluated smooth distance functions.
r_conjunction(o1, o2)	Computes the logical AND operation using smooth distance functions.
r_disjunction(o1, o2)	Computes the logical OR operation using smooth distance functions.
r_trimming(o, t)	Trimming operations.
signed_to_positive(o)	Absolute value using the formula jnp.sqrt(o**2).
only_positive(o)	Set negative regions to zero by manipulating values using the formula (o + jnp.sqrt(o**2)) / 2.
first_order_normalization(o, grad)	Normalizes an unnormalized distance function o with its gradient grad.
normals_from_normalized_sdf(x, sdf)	Returns a smooth vector field that equals the surface normals where the normalized smooth distance function sdf equals 0

Signed distance functions

sdf_infinite_line(x, x_p1, x_p2)	Normalized signed smooth distance function of an infinite line going through x_p1 and x_p2.
sdf_nd_sphere(x, xc, r)	Normalized signed distance function to n-dimensional sphere, positive in the interior.
sdf_nd_planes(x, x_p1, x_p2)	Normalized signed smooth distance function, positive between points (1D), lines (2D), and n-dimensional planes (nD).
sdf_normal_to_direction(x, x_p1, normal)	Normalized signed smooth distance function to a plane defined by one point and a normal vector, positive in the direction of the normal vector.
sdf_infinite_cylinder(x, x_p1, normal, radius)	Normalized signed smooth distance function, positive within an infinite cylinder in 3D.
sdf_cylinder(x, x_0, normal, radius, length)	Normalized signed smooth distance function, positive within a cylinder in 3D.
sdf_cylinder_extruded(x, x_0, normal, ...)	Normalized signed smooth distance function of a cylinder that is extruded in time (fourth dimension).
sdf_triangle_2d(x, x_p1, x_p2, x_p3)	Normalized signed smooth distance function to a triangle (2D) or infinite triprism (3D), positive in the interior.
sdf_convex_polygon_2d(x, x_p_list)	Normalized signed smooth distance function to a convex polygon (2D), positive in the interior.
sdf_infinite_triprism_3d(x, x_p1, x_p2, x_p3)	Normalized signed smooth distance function to a triangle (2D) or infinite triprism (3D), positive in the interior.
sdf_parallelogram_2d(x, x_p1, x_p2, x_p3)	Normalized signed smooth distance function to a parallelogram, positive in the interior.
sdf_cuboid(x, x_p1, x_p2)	Normalized signed smooth distance function to a cuboid, positive in the interior.

Positive smooth distance functions

psdf_infinite_line(x, x_p1, x_p2)	Positive normalized smooth distance function of an infinite line going through x_p1 and x_p2.
psdf_trimmed_line(x, x_p1, x_p2)	Positive normalized smooth distance function of a line segment going from x_p1 to x_p2.
psdf_nd_sphere(x, xc, r)	Interior positive normalized smooth distance function of an n-dimensional sphere.
psdf_nd_sphere_cutout(x, xc, r)	Exterior positive normalized smooth distance function of an n-dimensional sphere.
psdf_infinite_cylinder(x, x_p1, normal, radius)	Positive normalized smooth distance function to an infinite cylinder in 3D.
psdf_cylinder(x, x_0, normal, radius, length)	Positive normalized smooth distance function to a cylinder in 3D.
psdf_cylinder_extruded(x, x_0, normal, ...)	Positive normalized smooth distance function of a cylinder that is extruded in time.
psdf_triangle_2d(x, x_p1, x_p2, x_p3)	Interior positive normalized smooth distance function of a triangle in 2D and a tri-prism in 3D.
psdf_infinite_triprism_3d(x, x_p1, x_p2, x_p3)	Interior positive normalized smooth distance function of a triangle in 2D and a tri-prism in 3D.
psdf_parallelogram(x, x_p1, x_p2, x_p3)	Positive normalized smooth distance function to a parallelogram, positive in the interior.
psdf_arc_in_2d(x, xc, x_p1, x_p2)	First order normalized positive smooth distance function of a circular arc in 2D.
psdf_polygon(x, x_p_list)	Positive normalized smooth distance function for a polygon in 2D.

Helper functions

centroid(x_p1, x_p2)	Mean of two points in n-dimensions:
normal(x_p1, x_p2)	Normalized vector showing from x_p1 to x_p2.
project(x, normal)	Projects vector x in the direction of vector normal.
project_on_line(x_p1, x_p2, x_p3)	Projects x_p1 on the line going through x_p2 and x_p3.

Transfinite interpolation

transfinite_interpolation(x, boundary_psdfs, ...)	Transfinite interpolation using positive smooth distance functions (PSDFs) and boundary conditions.
psdf_unification(x, boundary_psdfs)	Unification of multiple positive smooth distance functions (PSDFs).

Mesh related functions

triangle_area(x_edges)	Compute the area of a triangle given its vertices (jitted).
triangle_areas(x_nodes, elements)	Compute the areas of multiple triangles given their vertices and connectivity (jitted).
in_sdf(x, sdf_fun[, tol])	Check if a point lies on the surface defined by a positive or signed distance function.
in_sdfs(x, sdf_fun[, tol])	Check if multiple points lie on the surface defined by a positive or signed distance function.
select_in_sdfs(x, sdf_fun)	Select the indices of points that lie on the surface defined by a positive or signed distance function.
in_plane(x, x_p1, normal)	Check if a point lies in a plane defined by a point and a normal vector (jitted).
in_planes(x_q, x_p1, normal)	Check if multiple points lie in a plane defined by a point and a normal vector (jitted).
select_in_plane(x_q, x_p1, normal)	Select the indices of points that lie in a plane (non-jittable).
on_point(x_q, x_p1)	Check if points are close to a given point (jitted).
select_point(x_q, x_p1)	Select the index of a point that is close to a given point (non-jittable).
on_line(x, x_p1, x_p2)	Check if a point lies on a line segment defined by two points (jitted).
on_lines(x_q, x_p1, x_p2)	Check if multiple points lie on a line segment defined by two points (jitted).
select_on_line(x_q, x_p1, x_p2)	Select the indices of points that lie on a line segment defined by two points (non-jittable).
elem_in_plane(x_nodes, x_p1, normal)	Check if an element lies in a plane defined by a point and a normal vector.
select_elements_in_plane(x_nodes, ...)	Select the indices of elements that lie in a plane.
elem_on_line(x_nodes, x_p1, x_p2)	Check if an element lies on a line segment defined by two points.
select_elements_on_line(x_nodes, ...)	Select the indices of elements that lie on a line segment defined by two points.
subelem_in_elem(subelem_to_find, in_elem)	Check if a sub-element (e.g.
subelem_in_elems(subelem_to_find, in_elems)	Find the element that contains a given sub-element.
subelems_in_elems(subelems_to_find, in_elems)	Find the elements that contain given sub-elements (one elem per sub_elem).