Quickstart Guide

This guide provides a quick introduction to using TTFEMesh. For detailed information about the meaning of the tensorized Jacobians, Dirichlet masks, and concatenation maps, we refer to the original paper arXiv:1802.02839.

Installation

pip install ttfemesh

Creating a Domain

We create a simple domain with two rectangles and an edge connecting them. We set the Dirichlet boundary for the left side and the right side.

from ttfemesh.domain import RectangleFactory, CurveConnection2D, VertexConnection2D 
from ttfemesh.domain import DirichletBoundary2D, Domain2D

# Create first rectangle
lower_left = (0, 0)
upper_right = (2, 1)
rectangle1 = RectangleFactory.create(lower_left, upper_right)

# Create second rectangle
lower_left = (2, 0)
upper_right = (3, 1)
rectangle2 = RectangleFactory.create(lower_left, upper_right)

# Connect the rectangles
domain_idxs = [0, 1]
curve_idxs = [1, 3]
edge = CurveConnection2D(domain_idxs, curve_idxs)

# Set boundary conditions
bc = DirichletBoundary2D([(0, 3), (1, 1)])

# Create the domain
domain = Domain2D([rectangle1, rectangle2], [edge], bc)
domain.plot()

Meshing a Domain

Generating a mesh for a domain is straightforward. It requires the domain, a quadrature rule, and a mesh size exponent. The mesh size exponent is used to control the size of the mesh, i.e., the number of discretization points in each direction will be \(2^{n}\), where \(n\) is the mesh size exponent.

from ttfemesh.quadrature import GaussLegendre2D
from ttfemesh.mesh import DomainBilinearMesh2D

order = 1
qrule = GaussLegendre2D(order)
mesh_size_exponent = 3

mesh = DomainBilinearMesh2D(domain, qrule, mesh_size_exponent)
print(mesh)

Working with Tensor Trains

Jacobians and Determinants

For each of the subdomains, you can retrieve the tensorized Jacobians:

subdmesh = mesh.get_subdomain_mesh(0)
jac_tts = subdmesh.get_jacobian_tensor_trains()
print(jac_tts.shape)
print(jac_tts)

jac_dets = subdmesh.get_jacobian_det_tensor_trains()
print(jac_dets.shape)
print(jac_dets)

jac_invdets = subdmesh.get_jacobian_invdet_tensor_trains()
print(jac_invdets.shape)
print(jac_invdets)

Element to Global Index Map

You can retrieve the tensorized element to global index map:

element2global_map = mesh.get_element2global_index_map()
print(element2global_map.shape)
print(element2global_map)

Dirichlet Masks

For each of the subdomains, you can retrieve the tensorized Dirichlet masks:

masks = mesh.get_dirichlet_masks()
print(masks)

Concatenation Maps

You can retrieve the concatenation maps for all pairs of connected subdomains:

concat_maps = mesh.get_concatenation_maps()
print(concat_maps)

Next Steps

• Check out the API Reference for detailed documentation of all available classes and functions.

• Visit our GitHub repository to contribute or report issues.