incidence_matrices.py¶
Building incidence matrices, , and in the 3d reference domain . We build them through the local numberings which are also implemented here in this script.
⭕ To access the source code, click on the [source] button at the right
side or click on
[incidence_matrices.py]
.
Dependence may exist. In case of error, check import and install required
packages or download required scripts. © mathischeap.com
- incidence_matrices.E_curl(N_xi, N_et, N_sg)[source]¶
The incidence matrix for mimetic polynomials constructed on three sets of nodes, , and .
- Parameters:
N_xi (positive integer) –
N_xi + 1
nodes in set .N_et (positive integer) –
N_et + 1
nodes in set .N_sg (positive integer) –
N_sg + 1
nodes in set .
- Returns:
A csc_matrix .
- Example:
>>> E = E_curl(1,1,1) >>> E.toarray() array([[ 0, 0, 0, 0, 1, 0, -1, 0, -1, 0, 1, 0], [ 0, 0, 0, 0, 0, 1, 0, -1, 0, -1, 0, 1], [-1, 0, 1, 0, 0, 0, 0, 0, 1, -1, 0, 0], [ 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 1, -1], [ 1, -1, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0], [ 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 0, 0]], dtype=int32)
- incidence_matrices.E_div(N_xi, N_et, N_sg)[source]¶
The incidence matrix for mimetic polynomials constructed on three sets of nodes, , and .
- Parameters:
N_xi (positive integer) –
N_xi + 1
nodes in set .N_et (positive integer) –
N_et + 1
nodes in set .N_sg (positive integer) –
N_sg + 1
nodes in set .
- Returns:
A csc_matrix .
- Example:
>>> E = E_div(1,1,1) >>> E.toarray() array([[-1, 1, -1, 1, -1, 1]], dtype=int32)
- incidence_matrices.E_grad(N_xi, N_et, N_sg)[source]¶
The incidence matrix for mimetic polynomials constructed on three sets of nodes, , and .
- Parameters:
N_xi (positive integer) –
N_xi + 1
nodes in set .N_et (positive integer) –
N_et + 1
nodes in set .N_sg (positive integer) –
N_sg + 1
nodes in set .
- Returns:
A csc_matrix .
- Example:
>>> E = E_grad(1,1,1) >>> E.toarray() array([[-1, 1, 0, 0, 0, 0, 0, 0], [ 0, 0, -1, 1, 0, 0, 0, 0], [ 0, 0, 0, 0, -1, 1, 0, 0], [ 0, 0, 0, 0, 0, 0, -1, 1], [-1, 0, 1, 0, 0, 0, 0, 0], [ 0, -1, 0, 1, 0, 0, 0, 0], [ 0, 0, 0, 0, -1, 0, 1, 0], [ 0, 0, 0, 0, 0, -1, 0, 1], [-1, 0, 0, 0, 1, 0, 0, 0], [ 0, -1, 0, 0, 0, 1, 0, 0], [ 0, 0, -1, 0, 0, 0, 1, 0], [ 0, 0, 0, -1, 0, 0, 0, 1]], dtype=int32)
- incidence_matrices.local_numbering_EP(N_xi, N_et, N_sg)[source]¶
Generating the local numbering for a vector of 3D edge polynomials in constructed on three sets of nodes, , and .
- Parameters:
N_xi (positive integer) –
N_xi + 1
nodes in set .N_et (positive integer) –
N_et + 1
nodes in set .N_sg (positive integer) –
N_sg + 1
nodes in set .
- Returns:
A tuple of three outputs:
- A 3d np.array contain the local numbering for the first
componement of the vector. Its three dimensions refer to three axes .
- A 3d np.array contain the local numbering for the second
componement of the vector. Its three dimensions refer to three axes .
- A 3d np.array contain the local numbering for the third
componement of the vector. Its three dimensions refer to three axes .
- Example:
>>> ln0, ln1, ln2 = local_numbering_EP(2,2,2) >>> ln0[:,:,0] array([[0, 2, 4], [1, 3, 5]]) >>> ln0[:,:,1] array([[ 6, 8, 10], [ 7, 9, 11]]) >>> ln0[:,:,2] array([[12, 14, 16], [13, 15, 17]]) >>> ln1[:,:,0] array([[18, 21], [19, 22], [20, 23]]) >>> ln1[:,:,1] array([[24, 27], [25, 28], [26, 29]]) >>> ln1[:,:,2] array([[30, 33], [31, 34], [32, 35]]) >>> ln2[:,:,0] array([[36, 39, 42], [37, 40, 43], [38, 41, 44]]) >>> ln2[:,:,1] array([[45, 48, 51], [46, 49, 52], [47, 50, 53]])
- incidence_matrices.local_numbering_FP(N_xi, N_et, N_sg)[source]¶
Generating the local numbering for a vector of 3D face polynomials in constructed on three sets of nodes, , and .
- Parameters:
N_xi (positive integer) –
N_xi + 1
nodes in set .N_et (positive integer) –
N_et + 1
nodes in set .N_sg (positive integer) –
N_sg + 1
nodes in set .
- Returns:
A tuple of three outputs:
- A 3d np.array contain the local numbering for the first
componement of the vector. Its three dimensions refer to three axes .
- A 3d np.array contain the local numbering for the second
componement of the vector. Its three dimensions refer to three axes .
- A 3d np.array contain the local numbering for the third
componement of the vector. Its three dimensions refer to three axes .
- Example:
>>> ln0, ln1, ln2 = local_numbering_FP(2,2,2) >>> ln0[:,:,0] array([[0, 3], [1, 4], [2, 5]]) >>> ln0[:,:,1] array([[ 6, 9], [ 7, 10], [ 8, 11]]) >>> ln1[:,:,0] array([[12, 14, 16], [13, 15, 17]]) >>> ln1[:,:,1] array([[18, 20, 22], [19, 21, 23]]) >>> ln2[:,:,0] array([[24, 26], [25, 27]]) >>> ln2[:,:,1] array([[28, 30], [29, 31]]) >>> ln2[:,:,2] array([[32, 34], [33, 35]])
- incidence_matrices.local_numbering_NP(N_xi, N_et, N_sg)[source]¶
Generating the local numbering for a node polynomials in constructed on three sets of nodes, , and .
- Parameters:
N_xi (positive integer) –
N_xi + 1
nodes in set .N_et (positive integer) –
N_et + 1
nodes in set .N_sg (positive integer) –
N_sg + 1
nodes in set .
- Returns:
A 3d np.array contain the local numbering. Its three dimensions refer to three axes .
- Example:
>>> ln = local_numbering_NP(3,3,3) >>> ln[:,:,0] array([[ 0, 4, 8, 12], [ 1, 5, 9, 13], [ 2, 6, 10, 14], [ 3, 7, 11, 15]]) >>> ln[:,:,1] array([[16, 20, 24, 28], [17, 21, 25, 29], [18, 22, 26, 30], [19, 23, 27, 31]]) >>> ln[:,:,2] array([[32, 36, 40, 44], [33, 37, 41, 45], [34, 38, 42, 46], [35, 39, 43, 47]]) >>> ln[:,:,3] array([[48, 52, 56, 60], [49, 53, 57, 61], [50, 54, 58, 62], [51, 55, 59, 63]])
- incidence_matrices.local_numbering_VP(N_xi, N_et, N_sg)[source]¶
Generating the local numbering for a volume polynomials in constructed on three sets of nodes, , and .
- Parameters:
N_xi (positive integer) –
N_xi + 1
nodes in set .N_et (positive integer) –
N_et + 1
nodes in set .N_sg (positive integer) –
N_sg + 1
nodes in set .
- Returns:
A 3d np.array contain the local numbering. Its three dimensions refer to three axes .
- Example:
>>> ln = local_numbering_VP(3,3,3) >>> ln[:,:,0] array([[0, 3, 6], [1, 4, 7], [2, 5, 8]]) >>> ln[:,:,1] array([[ 9, 12, 15], [10, 13, 16], [11, 14, 17]]) >>> ln[:,:,2] array([[18, 21, 24], [19, 22, 25], [20, 23, 26]])
↩️ Back to Ph.D. thesis complements (ptc).