amfe.boundary module

Module for handling the Dirichlet boundary.

class amfe.boundary.DirichletBoundary(no_of_unconstrained_dofs=nan)[source]

Bases: object

Class responsible for the Dirichlet Boundary conditions

B

sparse matrix –

Matrix for mapping the constrained dofs to the unconstrained dofs:
u_unconstr = B @ u_constr

The matrix is a sparse CSR matrix

slave_dofs

ndarray – Array of unique slave dof indices which are eliminated in the constrained dofs

row

ndarray, dtype: int – row indices of triplet sparse matrix description for C (constraints)

col

ndarray, dtype: int – col indices of triplet sparse matrix description for C (constraints)

val

ndarray, dtype: float – vals of triplet sparse matrix description for C (constraints)

no_of_unconstrained_dofs

int – number of unconstrained dofs

no_of_constrained_dofs

int – number of constrained dofs

add_constraints(slave_dofs, row, col, val)[source]

Add constraints to the system.

The slave_dofs are eliminated. The triple row, col and val expresses the triplet entries in the B-matrix mapping the constrained dofs to the unconstrained dofs:

u_unconstr = B @ u_constr
Parameters:
  • slave_dofs (array-like, dtype int) – array or list of slave dofs which should be eliminated
  • row (ndarray, dtype int, shape(n)) – row index list of master-slave mapping
  • col (ndarray, dtype int, shape(n)) – col index list of master-slave mapping
  • val (ndarray, shape(n)) – values of master-slave mapping
Returns:

B – sparse matrix performing the mapping from the constrained dofs to the unconstrained dofs.
Return type:

sparse_matrix, shape(ndim_unconstr, ndim_constr)
apply_master_slave_list(master_slave_list)[source]

Apply a master-slave list of the form

Parameters:master_slave_list (list) –

list containing the dirichlet-boundary triples (DBT)

>>> [DBT_1, DBT_2, DBT_3, ]
Returns:
Return type:None

Notes

each dirchilet_boundary_triple is itself a list containing

>>> DBT = [master_dof=None, [list_of_slave_dofs], weighting_matrix=None]
master_dof
: int / None
the dof onto which the slave dofs are projected. The master_dof will be overwritten at the end, i.e. if the master dof should participate at the end, it has to be a member in the list of slave_dofs. If the master_dof is set to None, the slave_dofs will be fixed
list_of_slave_dofs
: list containing ints
The list of the dofs which will be projected onto the master dof; the weights of the projection are stored in the B_matrix
weighting_matrix
: ndarray / None
The weighting-matrix which enables to apply complicated boundary conditions showing up in symmetry-conditions or rotational dofs. The default-value for weighting_matrix is None, which weighs all members of the list_of_slave_dofs equally with 1.

Examples

The dofs 0, 2 and 4 are fixed:

>>> DBT = [None, [0, 2, 4], None]
>>> my_boundary = DirichletBoundary([DBT, ])

The dofs 0, 1, 2, 3, 4, 5, 6 are fixed and the dofs 100, 101, 102, 103 have all the same displacements:

>>> DBT_fix = [None, np.arange(7), None]
>>> DBT_disp = [100, [101, 102, 103], None]
>>> my_boundary = DirichletBoundary([DBT_fix, DBT_disp])

Symmetry: The displacement of dof 21 is negativ equal to the displacement of dof 20, i.e. u_20 + u_21 = 0

>>> DBT_symm = [20, [21], np.array([-1])]
>>> my_boundary = DirichletBoundary([DBT_symm, ])
b_matrix()[source]
Parameters:None –
Returns:B – Matrix (is usually Boolean, when no weightingfactors are assigned), which links the dofs of the constrained system to the dofs of the unconstrained system:
>>> u = B @ u_constr

If no Dirichlet Boundaries are chosen, B is identity.
Return type:scipy.sparse.csr_matrix

Examples

Apply the constraints to a random stiffness matrix:

>>> ndim = 100
>>> K = sp.sparse.random(100,100, format='csr')
# TODO: CHECK: The next line can be deleted because deprecated, right?
>>> my_dirichlet_boundary = DirichletBoundary(ndim, [[None, [0,1,2,3,4], None]])
>>> B = my_dirichlet_boundary.b_matrix()
>>> B.T.dot(K.dot(B)) # B.T @ K @ B
... <95x95 sparse matrix of type '<class 'numpy.float64'>'
        with 93 stored elements in Compressed Sparse Column format>

Notes

Wichtig: Für die Dirichlet-Randbedingungen werden Freiheitsgrade des globalen Systems und nicht die Knotenfreiheitsgrade berücksichtigt. Die Indexwerte der Knoten müssen stets in DOFs des globalen Sytems umgerechnet werden

constrain_dofs(dofs)[source]

Take the dofs to the constrained dofs.

Parameters:dofs (ndarray) – Array containing the dofs to be constrained
Returns:
Return type:None
constrain_matrix(M_unconstr)[source]

Constrain a matrix with the given Dirichlet boundary conditions.

Parameters:M_unconstr (sp.sparse.sparse_matrix) – Sparse unconstrained matrix
Returns:M – Sparse constrained matrix
Return type:sp.sparse.sparse_matrix
constrain_vec(vec_unconstr)[source]

Constrain a vector with the given Dirichlet boundary conditions.

Parameters:vec_unconstr (ndarray) – vector of unconstrained system
Returns:vec – vector of constrained system
Return type:ndarray

Notes

The dimension of the returned vec is smaller than of vec_unconstr, as the fixed dofs are removed from the vector.

unconstrain_vec(vec)[source]

Remove the constraints of a vector. Returns B @ u

Parameters:vec (ndarray) – vector of the finite element system where constraints are imposed on.
Returns:vec_unconstr – Vector of the finite element system where no constraints are imposed on. All dofs correspond to the dofs of the mesh.
Return type:ndarray

Notes

The dimension of vec becomes larger, as the constrained dofs are added to the vector vec.

update()[source]
update internal variables according to internally saved boundary list:
  • calls self.b_matrix()
  • update self.no_of_constrained_dofs, self.no_of_unconstrained_dofs
Parameters:None –
Returns:
Return type:None