gf_asm

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.

# Calling Sequence

    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('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)

    Me = gf_asm('extrapolation matrix',mesh_fem mf, mesh_fem mfe)

    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)

    B = gf_asm('Nitsche contact rigid obstacle rhs', int bnum, mesh_im mim, mesh_fem mf_u, vec U, mesh_fem mf_obs, vec obs, scalar fcoeff, scalar r, scalar theta, scalar clambda, scalar cmu)

## Description

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.

## Command list

- `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. \nabla\cdot(a(x)\nabla u) 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. \nabla\cdot(C(x):\nabla u) with C 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. -\nu(x)\Delta u + \nabla p = 0 \nabla\cdot u = 0 with \nu (`nu`), the fluid's dynamic viscosity. On output, `K` is the usual linear elasticity stiffness matrix with \lambda = 0 and 2\mu = \nu. `B` is a matrix corresponding to \int p\nabla\cdot\phi. `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. \Delta u + k^2 u = 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. \Delta(a(x)\Delta u) = 0 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. \Delta(a(x)\Delta u) = 0 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('volumic' [,CVLST], expr [, mesh_ims, mesh_fems, data...])`

  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: 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, 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, A);

- `{...} = gf_asm('boundary', int bnum, string expr [, mesh_im mim, mesh_fem mf, data...])`

  Generic boundary assembly. See the help for gf_asm('volumic').

- `Mi = gf_asm('interpolation matrix', mesh_fem mf, mesh_fem mfi)`

  Build the interpolation matrix from a mesh_fem onto another mesh_fem. 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)`

  Build the extrapolation matrix from a mesh_fem onto another mesh_fem. 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.

- `B = gf_asm('Nitsche contact rigid obstacle rhs', int bnum, mesh_im mim, mesh_fem mf_u, vec U, mesh_fem mf_obs, vec obs, scalar fcoeff, scalar r, scalar theta, scalar clambda, scalar cmu)`

  Compute the right hand side (residual) of the Nitsche term for contact with friction on a rigid obstacle of a linearly elastic body. Experimental. Return a vec object.

## See Also

- getfem types

## Authors

Y. Collette