assembleFEMatrices

Assemble finite element matrices

Syntax

FEM = assembleFEMatrices(model)

FEM = assembleFEMatrices(model,bcmethod)

Description

FEM = assembleFEMatrices(model) returns a structure containing finite element matrices for the PDE problem in model.

FEM = assembleFEMatrices(model,bcmethod) assembles finite element matrices and imposes boundary conditions using the method specified by bcmethod.

Examples

Assemble Finite Element Matrices for a 2-D Problem

Create a PDEModel for the Poisson equation on the L-shaped membrane with zero Dirichlet boundary conditions.

model = createpde(1);
geometryFromEdges(model,@lshapeg);
specifyCoefficients(model,'m',0,'d',0,'c',1,'a',0,'f',1);
applyBoundaryCondition(model,'edge',1:model.Geometry.NumEdges,'u',0);

Generate a mesh and obtain the default finite element matrices for the problem and mesh.

generateMesh(model,'Hmax',0.2);
FEM = assembleFEMatrices(model)

FEM = struct with fields:
    K: [401x401 double]
    A: [401x401 double]
    F: [401x1 double]
    Q: [401x401 double]
    G: [401x1 double]
    H: [80x401 double]
    R: [80x1 double]
    M: [401x401 double]

Assemble Finite Element Matrices for a 2-D Problem With Boundary Conditions Imposed Using nullspace Method and Obtain PDE Solution

Create a PDEModel for the Poisson equation on the L-shaped membrane with zero Dirichlet boundary conditions.

model = createpde(1);
geometryFromEdges(model,@lshapeg);
specifyCoefficients(model,'m',0,'d',0,'c',1,'a',0,'f',1);
applyBoundaryCondition(model,'edge',1:model.Geometry.NumEdges,'u',0);

Generate a mesh and obtain the nullspace finite element matrices for the problem and mesh.

generateMesh(model,'Hmax',0.2);
FEM = assembleFEMatrices(model,'nullspace')

FEM = struct with fields:
    Kc: [321x321 double]
    Fc: [321x1 double]
     B: [401x321 double]
    ud: [401x1 double]
     M: [321x321 double]

Obtain the solution to the PDE.

u = FEM.B*(FEM.Kc\FEM.Fc) + FEM.ud;

Compare to the solution using solvepde. The two solutions are identical.

u1 = solvepde(model);
norm(u - u1.NodalSolution)

ans = 0

Input Arguments

model — PDE model

PDEModel object

PDE model, specified as a PDEModel object.

Example: model = createpde

bcmethod — Method for including boundary conditions

'none' (default) | 'nullspace' | 'stiff-spring'

Method for including boundary conditions, specified as 'none', 'nullspace', or 'stiff-spring'. For the meaning and use of bcmethod, see Algorithms.

Example: FEM = assembleFEMatrices(model,'nullspace')

Data Types: char | string

Output Arguments

FEM — Finite element matrices

structure

Finite element matrices, returned as a structure. The fields in the structure depend on bcmethod. (These fields correspond to the legacy assempde outputs with the same names, except that there are now both A and M matrices.)

• 'none' or no bcmethod given — The fields are K, A, F, Q, G, H, R, M.

• 'nullspace' — The fields are Kc, Fc, B, ud, M.

• 'stiff-spring' — The fields are Ks, Fs, M.

For the meaning and use of FEM, see Algorithms.

Tips

• The mass matrix M is nonzero when the model is time-dependent. By using this matrix, you can solve a model with Raleigh damping. See Dynamics of Damped Cantilever Beam.

Algorithms

The full finite element matrices and vectors are the following:

• K is the stiffness matrix, the integral of the c coefficient against the basis functions.

• M is the mass matrix, the integral of the m or d coefficient against the basis functions.

• A is the integral of the a coefficient against the basis functions.

• F is the integral of the f coefficient against the basis functions.

• Q is the integral of the q boundary condition against the basis functions.

• G is the integral of the g boundary condition against the basis functions.

• The H and R matrices come directly from the Dirichlet conditions and the mesh.

Given these matrices, the 'nullspace' technique generates the combined finite element matrices [Kc,Fc,B,ud] as follows. The combined stiffness matrix is for the reduced linear system, Kc = K + M + Q. The corresponding combined load vector is Fc = F + G. The B matrix spans the null space of the columns of H (the Dirichlet condition matrix representing hu = r). The R vector represents the Dirichlet conditions in Hu = R. The ud vector represents boundary condition solutions for the Dirichlet conditions.

From the 'nullspace' matrices, you can compute the solution u as

u = B*(Kc\Fc) + ud.

Note

Internally, for time-independent problems, solvepde uses the 'nullspace' technique, and calculates solutions using u = B*(Kc\Fc) + ud.

Alternatively, the 'stiff-spring' technique returns a matrix Ks and a vector Fs that together represent a different type of combined finite element matrices. The approximate solution u is u = Ks\Fs.

The 'stiff-spring' technique generates matrices more quickly than the 'nullspace' technique, but the 'stiff-spring' technique generally gives less accurate solutions.