General assembly function. Many of the functions below use more than one mesh_fem: the main mesh_fem (mf_u) used for the main unknown, and data mesh_fem (mf_d) used for the data. It is always assumed that the Qdim of mf_d is equal to 1: if mf_d is used to describe vector or tensor data, you just have to "stack" (in fortran ordering) as many scalar fields as necessary.
M = gf_asm('mass matrix', mesh_im mim, mesh_fem mf1[, mesh_fem mf2[, boundary_num]])
M = gf_asm('lsneuman matrix', mesh_im mim, mesh_fem mf1, mesh_fem mf2, levelset ls)
M = gf_asm('nlsgrad matrix', mesh_im mim, mesh_fem mf1, mesh_fem mf2, levelset ls)
M = gf_asm('stabilization patch matrix', @tm mesh, mesh_fem mf, mesh_im mim, real ratio, real h)
L = gf_asm('laplacian', mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, vec a)
Le = gf_asm('linear elasticity', mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, vec lambda_d, vec mu_d)
TRHS = gf_asm('nonlinear elasticity', mesh_im mim, mesh_fem mf_u, vec U, string law, mesh_fem mf_d, mat params, {'tangent matrix'|'rhs'|'incompressible tangent matrix', mesh_fem mf_p, vec P|'incompressible rhs', mesh_fem mf_p, vec P})
{K, B} = gf_asm('stokes', mesh_im mim, mesh_fem mf_u, mesh_fem mf_p, mesh_fem mf_d, vec nu)
A = gf_asm('helmholtz', mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, vec k)
A = gf_asm('bilaplacian', mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, vec a)
A = gf_asm('bilaplacian KL', mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, vec a, vec nu)
V = gf_asm('volumic source', mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, vec fd)
B = gf_asm('boundary source', int bnum, mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, vec G)
{HH, RR} = gf_asm('dirichlet', int bnum, mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, mat H, vec R [, threshold])
Q = gf_asm('boundary qu term',int boundary_num, mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, mat q)
gf_asm('define function', string name, int nb_args, string expression[, string expression_derivative_t[, string expression_derivative_u]])
gf_asm('undefine function', string name)
{...} = gf_asm('generic', mesh_im mim, int order, string expression, int region, [string varname, int is_variable[, {mesh_fem mf, mesh_imd mimd}], value], ...)
{...} = gf_asm('volumic' [,CVLST], expr [, mesh_ims, mesh_fems, data...])
{...} = gf_asm('boundary', int bnum, string expr [, mesh_im mim, mesh_fem mf, data...])
Mi = gf_asm('interpolation matrix', mesh_fem mf, {mesh_fem mfi | vec pts})
Me = gf_asm('extrapolation matrix',mesh_fem mf, {mesh_fem mfe | vec pts})
B = gf_asm('integral contact Uzawa projection', int bnum, mesh_im mim, mesh_fem mf_u, vec U, mesh_fem mf_lambda, vec vec_lambda, mesh_fem mf_obstacle, vec obstacle, scalar r [, {scalar coeff | mesh_fem mf_coeff, vec coeff} [, int option[, scalar alpha, vec W]]])
B = gf_asm('level set normal source term', int bnum, mesh_im mim, mesh_fem mf_u, mesh_fem mf_lambda, vec vec_lambda, mesh_fem mf_levelset, vec levelset)
General assembly function. Many of the functions below use more than one mesh_fem: the main mesh_fem (mf_u) used for the main unknown, and data mesh_fem (mf_d) used for the data. It is always assumed that the Qdim of mf_d is equal to 1: if mf_d is used to describe vector or tensor data, you just have to "stack" (in fortran ordering) as many scalar fields as necessary.
M = gf_asm('mass matrix', mesh_im mim, mesh_fem mf1[, mesh_fem mf2[, boundary_num]])
Assembly of a mass matrix. Return a spmat object.
M = gf_asm('lsneuman matrix', mesh_im mim, mesh_fem mf1, mesh_fem mf2, levelset ls)
Assembly of a level set Neuman matrix. Return a spmat object.
M = gf_asm('nlsgrad matrix', mesh_im mim, mesh_fem mf1, mesh_fem mf2, levelset ls)
Assembly of a nlsgrad matrix. Return a spmat object.
M = gf_asm('stabilization patch matrix', @tm mesh, mesh_fem mf, mesh_im mim, real ratio, real h)
Assembly of stabilization patch matrix . Return a spmat object.
L = gf_asm('laplacian', mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, vec a)
Assembly of the matrix for the Laplacian problem.
with
a
a scalar.
Return a spmat object.
Le = gf_asm('linear elasticity', mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, vec lambda_d, vec mu_d)
Assembles of the matrix for the linear (isotropic) elasticity problem.
with
defined via
lambda_d
and mu_d
.
Return a spmat object.
TRHS = gf_asm('nonlinear elasticity', mesh_im mim, mesh_fem mf_u, vec U, string law, mesh_fem mf_d, mat params, {'tangent matrix'|'rhs'|'incompressible tangent matrix', mesh_fem mf_p, vec P|'incompressible rhs', mesh_fem mf_p, vec P})
Assembles terms (tangent matrix and right hand side) for nonlinear elasticity.
The solution U
is required at the current time-step. The law
may be choosen among:
- 'SaintVenant Kirchhoff':
Linearized law, should be avoided). This law has the two usual
Lame coefficients as parameters, called lambda and mu.
- 'Mooney Rivlin':
This law has three parameters, called C1, C2 and D1.
Can be preceded with the words 'compressible' or 'incompressible' to force
a specific version. By default, the incompressible version is considered
which requires only the first two material coefficients.
- 'neo Hookean':
A special case of the 'Mooney Rivlin' law that requires one material
coefficient less (C2 = 0). By default, its compressible version is used.
- 'Ciarlet Geymonat':
This law has 3 parameters, called lambda, mu and gamma, with
gamma chosen such that gamma is in ]-lambda/2-mu, -mu[.
The parameters of the material law are described on the mesh_fem mf_d
.
The matrix params
should have nbdof(mf_d)
columns, each row
correspounds to a parameter.
The last argument selects what is to be built: either the tangent
matrix, or the right hand side. If the incompressibility is
considered, it should be followed by a mesh_fem mf_p
, for the
pression.
Return a spmat object (tangent matrix), vec object (right hand
side), tuple of spmat objects (incompressible tangent matrix), or
tuple of vec objects (incompressible right hand side).
{K, B} = gf_asm('stokes', mesh_im mim, mesh_fem mf_u, mesh_fem mf_p, mesh_fem mf_d, vec nu)
Assembly of matrices for the Stokes problem.
with
(
nu
), the fluid's dynamic viscosity.
On output, K
is the usual linear elasticity stiffness matrix with
and
.
B
is a matrix
corresponding to .
K
and B
are spmat object's.
A = gf_asm('helmholtz', mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, vec k)
Assembly of the matrix for the Helmholtz problem.
= 0, with
k
complex scalar.
Return a spmat object.
A = gf_asm('bilaplacian', mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, vec a)
Assembly of the matrix for the Bilaplacian problem.
with
a
scalar.
Return a spmat object.
A = gf_asm('bilaplacian KL', mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, vec a, vec nu)
Assembly of the matrix for the Bilaplacian problem with Kirchoff-Love formulation.
with
a
scalar.
Return a spmat object.
V = gf_asm('volumic source', mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, vec fd)
Assembly of a volumic source term.
Output a vector V
, assembled on the mesh_fem mf_u
, using the data
vector fd
defined on the data mesh_fem mf_d
. fd
may be real or
complex-valued.
Return a vec object.
B = gf_asm('boundary source', int bnum, mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, vec G)
Assembly of a boundary source term.
G
should be a [Qdim x N] matrix, where N is the number of dof
of mf_d
, and Qdim is the dimension of the unkown u (that is set
when creating the mesh_fem).
Return a vec object.
{HH, RR} = gf_asm('dirichlet', int bnum, mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, mat H, vec R [, threshold])
Assembly of Dirichlet conditions of type h.u = r
.
Handle h.u = r
where h is a square matrix (of any rank) whose
size is equal to the dimension of the unkown u. This matrix is
stored in H
, one column per dof in mf_d
, each column containing
the values of the matrix h stored in fortran order:
.. math::
H(:,j) = [h11(x_j) h21(x_j) h12(x_j) h22(x_j)]
if u is a 2D vector field.
Of course, if the unknown is a scalar field, you just have to set
H = ones(1, N)
, where N is the number of dof of mf_d
.
This is basically the same than calling gf_asm('boundary qu term')
for H
and calling gf_asm('neumann') for R
, except that this
function tries to produce a 'better' (more diagonal) constraints
matrix (when possible).
See also gf_spmat_get(spmat S, 'Dirichlet_nullspace').
Q = gf_asm('boundary qu term',int boundary_num, mesh_im mim, mesh_fem mf_u, mesh_fem mf_d, mat q)
Assembly of a boundary qu term.
q
should be be a [Qdim x Qdim x N] array, where N is the number
of dof of mf_d
, and Qdim is the dimension of the unkown u (that
is set when creating the mesh_fem).
Return a spmat object.
gf_asm('define function', string name, int nb_args, string expression[, string expression_derivative_t[, string expression_derivative_u]])
Define a new function name
which can be used in high level
generic assembly. The function can have one or two parameters.
In expression
all available predefined function or operation
of the generic assembly can be used. However, no reference to
some variables or data can be specified. The argument of the
function is t
for a one parameter function and t
and u
for a two parameter function. For instance 'sin(pi*t)+2*t*t'
is a valid expression for a one parameter function and
'sin(max(t,u)*pi)' is a valid expression for a two parameters
function. expression_derivative_t
and expression_derivative_u
are optional expressions for the derivatives with respect
to t
and u
. If they are not furnished, a symbolic derivation
is used.
gf_asm('undefine function', string name)
Cancel the definition of a previously defined function name
for the high level generic assembly.
{...} = gf_asm('generic', mesh_im mim, int order, string expression, int region, [string varname, int is_variable[, {mesh_fem mf, mesh_imd mimd}], value], ...)
High-level generic assembly procedure for volumic assembly.
Performs the generic assembly of expression
with the integration
method mim
on the mesh region of index region
(-1 means all
the element of the mesh). The same mesh should be shared by
the integration method and all the finite element methods or
mesh_im_data corresponding to the variables.
order
indicates either that the (scalar) potential
(order = 0) or the (vector) residual (order = 1) or the
tangent (matrix) (order = 2) is to be computed.
The variables and constant (data) are listed after the
region number. For each variable/constant, first the variable/constant
name should be given (as it is referred in the assembly string), then
1 if it is a variable or 0 for a constant, then the finite element
method if it is a fem variable/constant or the mesh_im_data if it is
data defined on integration points, and the vector representing
the value of the variable/constant. It is possible to give an arbitrary
number of variable/constant. The difference between a variable and a
constant is that automatic differentiation is done with respect to
variables only (see GetFEM++ user documentation). Test functions are
only available for variables, not for constants.
Note that if several variables are given, the assembly of the
tangent matrix/residual vector will be done considering the order
in the call of the function (the degrees of freedom of the first
variable, then of the second, and so on).
For example, the L2 norm of a vector field "u" can be computed with::
gf_compute('L2 norm') or with the square root of:
gf_asm('generic', mim, 0, 'u.u', -1, 'u', 1, mf, U);
The nonhomogeneous Laplacian stiffness matrix of a scalar field can be evaluated with::
gf_asm('laplacian', mim, mf, mf_data, A) or equivalently with:
gf_asm('generic', mim, 2, 'A*Grad_Test2_u.Grad_Test_u', -1, 'u', 1, mf, U, 'A', 0, mf_data, A);
{...} = gf_asm('volumic' [,CVLST], expr [, mesh_ims, mesh_fems, data...])
Low-level generic assembly procedure for volumic assembly.
The expression expr
is evaluated over the mesh_fem's listed in the
arguments (with optional data) and assigned to the output arguments.
For details about the syntax of assembly expressions, please refer
to the getfem user manual (or look at the file getfem_assembling.h
in the getfem++ sources).
For example, the L2 norm of a field can be computed with::
gf_compute('L2 norm') or with the square root of:
gf_asm('volumic','u=data(#1); V()+=u(i).u(j).comp(Base(#1).Base(#1))(i,j)',mim,mf,U)
The Laplacian stiffness matrix can be evaluated with::
gf_asm('laplacian',mim, mf, mf_data, A) or equivalently with:
gf_asm('volumic','a=data(#2);M(#1,#1)+=sym(comp(Grad(#1).Grad(#1).Base(#2))(:,i,:,i,j).a(j))', mim,mf,mf_data,A);
{...} = gf_asm('boundary', int bnum, string expr [, mesh_im mim, mesh_fem mf, data...])
Low-level generic boundary assembly. See the help for gf_asm('volumic').
Mi = gf_asm('interpolation matrix', mesh_fem mf, {mesh_fem mfi | vec pts})
Build the interpolation matrix from a mesh_fem onto another mesh_fem or a set of points.
Return a matrix Mi
, such that V = Mi.U
is equal to
gf_compute('interpolate_on',mfi). Useful for repeated interpolations.
Note that this is just interpolation, no elementary integrations
are involved here, and mfi
has to be lagrangian. In the more
general case, you would have to do a L2 projection via the mass
matrix.
Mi
is a spmat object.
Me = gf_asm('extrapolation matrix',mesh_fem mf, {mesh_fem mfe | vec pts})
Build the extrapolation matrix from a mesh_fem onto another mesh_fem or a set of points.
Return a matrix Me
, such that V = Me.U
is equal to
gf_compute('extrapolate_on',mfe). Useful for repeated
extrapolations.
Me
is a spmat object.
B = gf_asm('integral contact Uzawa projection', int bnum, mesh_im mim, mesh_fem mf_u, vec U, mesh_fem mf_lambda, vec vec_lambda, mesh_fem mf_obstacle, vec obstacle, scalar r [, {scalar coeff | mesh_fem mf_coeff, vec coeff} [, int option[, scalar alpha, vec W]]])
Specific assembly procedure for the use of an Uzawa algorithm to solve contact problems. Projects the term $-(\lambda - r (u_N-g))_-$ on the finite element space of $\lambda$. Return a vec object.
B = gf_asm('level set normal source term', int bnum, mesh_im mim, mesh_fem mf_u, mesh_fem mf_lambda, vec vec_lambda, mesh_fem mf_levelset, vec levelset)
Performs an assembly of the source term represented by vec_lambda
on mf_lambda
considered to be a component in the direction of the
gradient of a levelset function (normal to the levelset) of a vector
field defined on mf_u
on the boundary bnum
.
Return a vec object.
Y. Collette