assema

(Not recommended) Assemble area integral contributions

assema is not recommended. Use assembleFEMatrices instead.

Syntax

[K,M,F] = assema(model,c,a,f)

[K,M,F] = assema(p,t,c,a,f)

Description

[K,M,F] = assema(model,c,a,f) assembles the stiffness matrix K, the mass matrix M, and the load vector F using the mesh contained in model, and the PDE coefficients c, a, and f.

[K,M,F] = assema(p,t,c,a,f) assembles the matrices from the mesh data in p and t.

Examples

Assemble Finite Element Matrices

Assemble finite element matrices for an elliptic problem on complicated geometry.

The PDE is Poisson's equation,

$$-\nabla \cdot \nabla u=1.$$

Partial Differential Equation Toolbox™ solves equations of the form

$$-\nabla \cdot (c\nabla u)+au=f.$$

So, represent Poisson's equation in toolbox syntax by setting c = 1, a = 0, and f = 1.

c = 1;
a = 0;
f = 1;

Create a PDE model container. Import the ForearmLink.stl file into the model and examine the geometry.

model = createpde;
importGeometry(model,'ForearmLink.stl');
pdegplot(model,'FaceAlpha',0.5)

Create a mesh for the model.

generateMesh(model);

Create the finite element matrices from the mesh and the coefficients.

[K,M,F] = assema(model,c,a,f);

The returned matrix K is quite sparse. M has no nonzero entries.

disp(['Fraction of nonzero entries in K is ',num2str(nnz(K)/numel(K))])

Fraction of nonzero entries in K is 0.00046964

disp(['Number of nonzero entries in M is ',num2str(nnz(M))])

Number of nonzero entries in M is 0

Assemble Finite Element Matrices Using [p,e,t] Mesh

Assemble finite element matrices for the 2-D L-shaped region, using the [p,e,t] mesh representation.

Define the geometry using the lshapeg function included your software.

g = @lshapeg;

Use coefficients c = 1, a = 0, and f = 1.

c = 1;
a = 0;
f = 1;

Create a mesh and assemble the finite element matrices.

[p,e,t] = initmesh(g);
[K,M,F] = assema(p,t,c,a,f);

The returned matrix M has all zeros. The K matrix is quite sparse.

disp(['Fraction of nonzero entries in K is ',num2str(nnz(K)/numel(K))])

Fraction of nonzero entries in K is 0.042844

disp(['Number of nonzero entries in M is ',num2str(nnz(M))])

Number of nonzero entries in M is 0

Input Arguments

model — PDE model

PDEModel object

PDE model, specified as a PDEModel object.

Example: model = createpde

c — PDE coefficient

scalar | matrix | character vector | character array | string scalar | string vector | coefficient function

PDE coefficient, specified as a scalar, matrix, character vector, character array, string scalar, string vector, or coefficient function. c represents the c coefficient in the scalar PDE

$$-\nabla \cdot \left(c\nabla u\right)+au=f$$

or in the system of PDEs

$$-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$$

Example: 'cosh(x+y.^2)'

Data Types: double | char | string | function_handle
Complex Number Support: Yes

a — PDE coefficient

scalar | matrix | character vector | character array | string scalar | string vector | coefficient function

PDE coefficient, specified as a scalar, matrix, character vector, character array, string scalar, string vector, or coefficient function. a represents the a coefficient in the scalar PDE

$$-\nabla \cdot \left(c\nabla u\right)+au=f$$

or in the system of PDEs

$$-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$$

Example: 2*eye(3)

Data Types: double | char | string | function_handle
Complex Number Support: Yes

f — PDE coefficient

scalar | matrix | character vector | character array | string scalar | string vector | coefficient function

PDE coefficient, specified as a scalar, matrix, character vector, character array, string scalar, string vector, or coefficient function. f represents the f coefficient in the scalar PDE

$$-\nabla \cdot \left(c\nabla u\right)+au=f$$

or in the system of PDEs

$$-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$$

Example: char('sin(x)';'cos(y)';'tan(z)')

Data Types: double | char | string | function_handle
Complex Number Support: Yes

p — Mesh points

matrix

Mesh points, specified as a 2-by-Np matrix of points, where Np is the number of points in the mesh. For a description of the (p,e,t) matrices, see Mesh Data as [p,e,t] Triples.

Typically, you use the p, e, and t data exported from the PDE Modeler app, or generated by initmesh or refinemesh.

Example: [p,e,t] = initmesh(gd)

Data Types: double

t — Mesh triangles

matrix

Mesh triangles, specified as a 4-by-Nt matrix of triangles, where Nt is the number of triangles in the mesh. For a description of the (p,e,t) matrices, see Mesh Data as [p,e,t] Triples.

Typically, you use the p, e, and t data exported from the PDE Modeler app, or generated by initmesh or refinemesh.

Example: [p,e,t] = initmesh(gd)

Data Types: double

Output Arguments

K — Stiffness matrix

sparse matrix

Stiffness matrix, returned as a sparse matrix. See Elliptic Equations.

Typically, you use K in a subsequent call to assempde.

M — Mass matrix

sparse matrix

Mass matrix. returned as a sparse matrix. See Elliptic Equations.

Typically, you use M in a subsequent call to a solver such as assempde or hyperbolic.

F — Load vector

vector

Load vector, returned as a vector. See Elliptic Equations.

Typically, you use F in a subsequent call to assempde.

Version History

Introduced before R2006a

R2016a: Not recommended

assema is not recommended. Use assembleFEMatrices instead. There are no plans to remove assema.